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(§t>o Reprint Editio.n^S) 


Analytical 

Mechanics 

A Comprehensive Treatise 
on the 

Dynamics of Constrained Systems 


John G Papastavridjs 



(Jbo Reprint Edition 


Analytical 

Mechanics 


A Comprehensive Treatise 
on the 

Dynamics of Constrained Systems 


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(£bo Reprint Edition 


Analytical 

Mechanics 

A Comprehensive Treatise 
on the 

Dynamics of Constrained Systems 


John G. Papastavridis, PhD. 


v> World Scientific 


NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI 

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Library of Congress Cataloging-in-Publication Data 

Papastavridis, J. G. (John G.) 

Analytical mechanics : a comprehensive treatise on the dynamics of constrained systems / by John G Papastavridis. 
— Reprint edition, 
pages cm 

Reprint of 2002 edition. 

Includes bibliographical references and index. 

ISBN 978-981-4338-71-4 (hardcover : alk. paper) 

1. Mechanics, Analytic. I. Title. 

QA805.P355 2014 
531.01*515—dc22 

2013017316 


British Library Cataloguing-in-Publication Data 

A catalogue record for this book is available from the British Library. 


This edition is a corrected reprint of the work first published by Oxford University Press in 2002. 


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To the living and loving memory of my father, 

GEORGE S. PAP AST A VRIDIS 
(rEOPriOY £. ITAITALTAYPIAH) 

A lawyer and fearless maverick, who, throughout 
his life, fought with fortitude, conviction, and class 
to better his world; 

and to whom I owe a critical part of my Weltanschauung. 


[The author’s father is shown here in the traditional Greek foustanella, 
in Athens, Greece (Themistokleous street, near Lqfos Strefi ), ca.1924] 


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PREFACE TO THE CORRECTED REPRINT 


This is a corrected reprint of a work first published in early 2002, by Oxford University 
Press, and which went out of print shortly thereafter. 

A few sign misprints and similar errors have been corrected; some notations have been, 
hopefully, improved (especially in Chs. 1, 2); a useful addition has been made on p. 336, 
and a couple of sections have been thoroughly revised (e.g. §3.12, §8.13). 

I am grateful to (a) the many reviewers, in some of the most prestigious professional 
journals and elsewhere (e.g. Bulletin of the American Mathematical Society, IEEE Control 
Systems Magazine, Zentralhlatt fiir Mathematit, amazon.com and amazon.co.uk, private 
communications, and references in advanced works of mechanics), for their enthusiastic 
comments; (b) the American Association of Publishers for selecting, in January 2003, the 
book for their “Annual Award for Outstanding Professional and Scholarly Titles of 2002, 
in Engineering”; and (c) last but not least, my WSPC editor, Dr. S. W. Lim, and his most 
capable and courteous staff, for their continuous and effective support. All these have been 
of essential moral and practical help to me in the making of this “new” edition! 

Here, I take the opportunity to restate that, in this book, I have: 

(a) Sought to combine the best of the old and new, i.e. no age discrimination; no knee-jerk 
disdain for “dusty old stuff” nor automatic following of “progress/modernity” — even 
in the exact sciences, and especially in mechanics, new is not necessarily or uniformly 
better; and 

(b) Avoided developments of considerable but nevertheless purely mathematical interest, 
especially those of the a-historical and intuition-deadening (“epsilonic”) type. 

May this treatise, as well as my other two works on mechanics [“Tensor Calculus 
and Analytical Dynamics ” (CRC, 1999) and “ Elementary Mechanics ” (WSPC, under 
production)], keep making many and loyal readers! 


John G. Papastavridis 
Atlanta, Georgia, Spring 2014 


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PREFACE 


Many of the scientific treatises today are formulated in a half- 
mystical language, as though to impress the reader with the 
uncomfortable feeling that he is in the permanent presence of a 
superman. The present book is conceived in a humble spirit and 
is written for humble people. 

(Lanczos, 1970, pp. vii-viii) 


GENERAL DESCRIPTION 

This book is a classical and detailed introduction to advanced analytical mechanics 
(AM), with special emphasis on its basic principles and equations of motion, as they 
apply to the most general constrained mechanical systems with a finite number of 
degrees of freedom (this term is explained in Chapter 2). For the reasons detailed 
below, and in spite of the age of the subject, I think that no other single volume exists, 
in English and in print, that is comparable to the one at hand in breadth and depth of the 
material covered — and, in this nontrivial sense, this ca. 1400-page and 174-figure long 
work is unicpie. 

The book is addressed to graduate students, professors, and researchers, in the 
areas of applied mechanics, engineering science, and mechanical, aerospace, struc¬ 
tural, (even) electrical engineering, as well as physics and applied mathematics. 
Advanced undergraduates are also very welcome to browse, and thus get initiated 
into higher dynamics. The sole technical prerequisite here, a relatively modest one, is 
a solid working knowledge of “elementary/intermediate” (i.e., undergraduate) 
dynamics; roughly, equivalent to the (bulk of the) material covered in, say, 
Spiegel’s Theoretical Mechanics, part of the well-known Schaum’s outline series. 
Also, familiarity with the simplest aspects of Lagrange’s equations, that is, how to 
take the partial and total derivatives of scalar energetic functions, would be helpful; 
although, strictly, it is not necessary. [See also “Suggestions to the Reader” 
(Introduction, §3).] 


CONTENTS 

Specifically, the book covers in what I consider to be a most logical and pedagogical 
sequence, the following topics: 


Introduction: Introduction to analytical mechanics, brief summary of the history of 
theoretical mechanics; suggestions to the reader; and list of symbols/notations, abbre¬ 
viations, and basic formulae. 


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PREFACE 


Chapter 1: Background'. Algebra of vectors and Cartesian tensors, and basic concepts 
and equations of Newton-Euler (or momentum) mechanics of particles and rigid 
bodies; that is, a highly selective compendium of undergraduate dynamics, and (some 
of) its mathematics, from a mature viewpoint. 

Chapter 2: Kinematics of constrained systems (i.e., Lagrangean kinematics); including 
the general theory of up to linear velocity (i.e., Pfaffian) constraints, in both holonomic 
(or true) and nonholonomic (or quasi) coordinates', and a uniquely readable account of 
the fundamental theorem of Frobenius, for testing the nonholonomicity of such con¬ 
straints. 

Chapter 3: Kinetics of constrained systems (i.e., Lagrangean kinetics); including the 
fundamental principles of AM; that is, those of d’Alembert-Lagrange and of relaxation 
of the constraints, the central equation of Heun-Hamel; equations of motion with or 
without reactions, with or without multipliers, in true or quasi system variables; an 
introduction to servoconstraints (theories of Appell-Beghin, et al.); and rigid-body 
applications. This is the key chapter of the entire book, as far as engineering readers 
are concerned. 

Chapter 4: Impulsive motion, under ideal constraints; including the associated extremum 
theorems of Carnot, Kelvin, Bertrand, Robin, et al. 

Chapter 5: Nonlinear nonholonomic constraints; that is, kinematics and kinetics under 
nonlinear, and generally nonholonomic, velocity constraints. 

Chapter 6: Differential variational principles, of Jourdain, Gauss, Hertz, et al., and their 
derivative higher-order equations of motion of Nielsen, Tsenov, et al. 

Chapter 7: Time-integral theorems and variational principles, of Lagrange, Hamilton, 
Jacobi, O. Holder, Voss, Suslov, Voronets, Hamel, et ah; for linear and nonlinear 
velocity constraints in true and quasi variables, with or without multipliers; plus energy 
and virial theorems. 

Chapter 8: Introduction to Hamiltonian/Canonical methods; that is, equations of 
Hamilton and Routh-Helmholtz, cyclic systems, steady motion and its stability, varia¬ 
tion of constants, canonical transformations and Poisson’s brackets, Hamilton-Jacobi 
integration theory, integral invariants, Noether’s theorem, and action-angle variables 
and their applications to adiabatic invariants and perturbation theory. 

Chapters 2-8 each contain a large number of completely solved examples, and 
problems with their answers (and, occasionally, hints), to illustrate and extend the 
previous theories; short ones are integrated within each chapter section, and longer, 
more synthetic, ones are collected at each chapter’s end; and also, critical comments/ 
references for further study. The exposition ends with a relatively extensive, cumu¬ 
lative, and alphabetical list of References and Suggested Reading, including every¬ 
thing from standard textbooks all the way to epoch-making memoirs of the last 
(more than) two hundred years. This list complements those found in such well- 
known references as Neimark and Fufaev (1967/1972) and Roberson and 
Schwertassek (1988). 

Parts of the text have, unavoidably, state-of-the-art flavor. However, as far as 
fundamental ideas go, very little, if anything, of the topics covered is truly new — 
today, no one can claim much originality in classical mechanics! The newness here, a 
nontrivial one, I think, consists in restoring, clarifying, putting together, and pre¬ 
senting, in what I hope is a readable form, material most of which has appeared over 
the past one hundred fifty, or so, years; frequently in little known, and/or hard to 
find and decipher, sources. (In view of the thousands of books, lecture notes, articles, 
and so on, used in the writing of this work, failure to acknowledge an author’s 

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PREFACE 


particular contribution is not intentional, merely an oversight.) But, given the aston¬ 
ishing unfamiliarity, confusion, and intellectual provincialism so prevalent in many 
theoretical and applied mechanics circles today, even in the fundamental concepts 
and principles of analytical dynamics (like virtual displacements/work and principle 
of d’Alembert-Lagrange, which is, by far, the most misunderstood “principle” of 
physics!), I felt very strongly that this noble, beautiful, and powerful body of knowl¬ 
edge, that diamond of our cultural heritage, should be accurately preserved and 
represented, so as to benefit present and future workers in dynamics. 

No single volume can even pretend to cover satisfactorily all aspects of this vast 
and fascinating subject; in particular, both its theoretical and applied aspects, let 
alone the currently popular computational ones. Since this is not an encyclopedia of 
theoretical and applied dynamics, an inescapable and necessary selection has oper¬ 
ated, and so, the following important topics are not covered: applications of differ¬ 
ential forms/exterior calculus (of Cartan, Gallissot, et al.) and symplectic geometry 
to Lagrangean and Hamiltonian mechanics; group theoretic applications; nonlinear dynam¬ 
ics (inch regular and stochastic/chaotic motion) and stability of motion; theory of orbits; 
and computational/numerical techniques. For all these, there already exists an enormous 
and competent literature (see “Suggestions to the Reader”). However, with the help of this 
treatise, the conscientious reader will be able to move quickly and confidently into any par¬ 
ticular theoretical and/or computational area of modern dynamics. In this sense, the work 
at hand constitutes an optimal investment of the reader’s precious energies. 


RAISON D'ETRE, AND SOME PHILOSOPHY 

The customary words of explanation, or apology, for writing “another” book on 
advanced dynamics are now in order. The main theme of this work, like a Wagnerian 
leitmotiv, is deductive order, formal structure, and physical ideas, as they pertain to 
that particular energetic form of mechanics of constrained systems founded by 
Lagrange and known as analytical (= deductive) mechanics; to be differentiated 
from the also analytical but momentum, or “elementary,” form of system mechanics, 
founded by Euler. It is a book for people who place theory (theories), ideas, knowledge, 
and understanding above all else—and do not apologize for it. Here, AM is studied 
not as the “maid” of some other (allegedly) more important discipline, but as a sub¬ 
ject worth knowing in its own right; that is, as a “king or queen.” As such, it will 
attract those with a qualitative and theoretical bent of mind; while it may not be as 
agreeable to those with purely computational and/or intellectually local predilections. [In 
the words of the late Professor R.M. Rosenberg (University of California, Berkeley): 
“The held of dynamics is plowed by two classes of people: those who enjoy the 
inherent beauty, symmetry and consistency of this discipline, and those who are 
satisfied with having a machine that manufactures equations of motion of complex 
mechanisms” (private communication, 1986).] Generally, science is more than a 
collection of particular problems and special techniques, even involved ones — it is 
much more than mere information. However, practical people should be reminded 
that theory and application are mutually complementary rather than adversarial; in 
fact, contemporary important practical problems and the availability of powerful 
computational capabilities have made the thorough understanding of the fundamen¬ 
tal principles of mechanics more necessary today than before. Applications and 
computers have, among other things, helped resurrect, restore, and sharpen 
old academic curiosities (for engineers anyway), such as the differential variational 

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PREFACE 


principles of Jourdain and Gauss (which have found applications in such “un¬ 
related” areas as multibody dynamics, nonlinear oscillations, even the elasto-plastic 
buckling of shells); and Hamilton’s canonical equations in quasi variables (which 
have found applications in robotic manipulators). 

A more concrete reason for writing this book is that, outside of the truly monu¬ 
mental British treatise of Pars (1965) and the English translations of the beautiful 
(former) Soviet monographs of Neimark and Fufaev (1967/1972) and Gantmacher 
(1966/1970), there is no comprehensive exposition of advanced engineering-oriented 
dynamics in print, in the entire English language literature! True, the famous treatise 
of Whittaker (1904/1917/1927/1937), for many years out of print, has recently been 
reprinted (1988). However, even Whittaker, although undeniably a classic and in 
many respects the single most influential dynamics volume of the twentieth century 
(primarily, to celestial and quantum mechanics), nevertheless leaves a lot to be 
desired in matters of logic, fundamental principles, and their earthly applications; 
for example, there is no clear and general formulation of the principle of 
d’Alembert-Lagrange and its use, in connection with Hamel’s method of quasi 
variables, to uncouple the equations of motion and obtain constraint reactions; 
also, Whittaker would be totally unacceptable with the better of today’s educational 
philosophies. Such drawbacks have plagued most British texts of that era; for exam¬ 
ple, the otherwise excellent works of Thomson/Tait, Routh, Lamb, Ramsey, Smart, 
and many of their U.S.-made descendants. [In a way, Whittaker et al. have been 
pretty lucky in that most of the great continental European works on advanced 
dynamics — for example, those of Boltzmann, Heun, Maggi, Appell, Marcolongo, 
Suslov, Nordheim et al. (vol. 5 of Handhuch der Physik, 1927), Winkelmann (vol. 1 
of Handbuch der Physikalischen und Technischen Mechanik , 1929), Prange (vol. 4 of 
Encyclopddie der Mathematischen Wissenschaften, 1935), Rose, Hamel, Peres, Lur’e, 
et al. were never translated into English.] Next, the comprehensive three-volume 
work of MacMillan (late 1920s to early 1930s) and the encyclopedic treatise of 
Webster (early 1900s), probably the two best U.S.-made mechanics texts, are, unfor¬ 
tunately, out of print. The very lively and deservedly popular monograph of Lanczos 
(1949-1970) does not go far enough in areas of engineering importance; for example, 
nonholonomic variables and constraints; and, also, lacks in examples and problems. 
Only the excellent encyclopedic article of Synge (1960) comes close to our objectives; 
but, that, too, has Lanczos’ drawbacks for engineering students and classroom use. 

The existing contemporary expositions on advanced dynamics, in English and in 
print, fall roughly into the following three groups: 

Formalistic I Abstract, of the by-and-for-mathematicians variety, and, as such, of next to 
zero relevance and/or usefulness to most nonmathematicians. Considering the high 
mental effort and time that must be expended toward their mastery vis-a-vis their 
meager results in understanding mechanics better and/or solving new and nontrivial 
problems, these works represent a pretty poor investment of ever scarce intellectual 
resources; that is, they are not worth their “money.” The effort should be commensurate 
to the returns. And, contrary to the impression given by authors of this group, even in 
the most exact sciences, books are written by and for concrete people; not by super- 
logical, detached, and cold machines. As Winner puts it: "The accepted form of ‘objec¬ 
tivity’ in scientific and technical reports (one can also include books and articles in social 
science) requires that the prose read as if there were no person in the room when the 
writing took place” (1986, p. 71). Also, I categorically reject soothing apologies of the 
type “oh, well, that is a book for mathematicians”; that is, the book has little or no 
consideration for ordinary folk. The distinguished physicist F. J. Dyson confirms our 

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PREFACE 


suspicions that “the marriage of mathematics and physics [about which we have been 
told so many nice things since our high school days] has ended in divorce” (quoted 
in M. Kline’s Mathematics, The Loss of Certainty, Oxford University Press, 1980, 
pp. 302-303). 

Applied, which either emphasize the numerical/computational aspects of mechanics, but, 
perhaps unavoidably, are soft and/or sketchy on its fundamental principles; or are so 
theoretically/conceptually impoverished and unmotivated that the reader is soon led to 
a narrow and dead-end view of mechanics. [Notable and refreshing exceptions to this 
style are the recent compact but rich-in-ideas works by Bremer et al. [1988(a), (b), 1992] 
in dynamics/control/flexible multibody systems.] 

Mainstream or traditionalist; for example, those by (alphabetically): Arya, Baruh, Calkin, 
Crandall et al., Corben et al., Desloge, Goldstein, Greenwood, Kilmister et al., 
Konopinski, Kuypers, Lanczos, Marion, McCauley, Meirovitch, Park, Rosenberg, 
Woodhouse. The problem with this group, however, is that its representatives either 
do not go far and deep enough (somehow, the more advanced topics seem to be mono¬ 
polized by the expositions of the first group); or they could use some improvements in 
the quality and/or quantity of their engineering^ relevant examples and problems. 

The book at hand belongs squarely and unabashedly to this last group, and aims 
to remedy its above-mentioned shortcomings by bridging the space between it and 
some of the earlier-mentioned classics, such as (chronologically): Heun (1906, 1914), 
Prange (1933-1935), Hamel (1927, 1949), Peres (1953), Lur’e (1961/1968), 
Gantmacher (1966/1970), Neimark and Fufaev (1967/1972), Dobronravov (1970, 
1976), and Novoselov (1966, 1967, 1979). Hence, my earlier claim that this treatise 
is unique in the entire contemporary literature; and my strong belief that it does meet 
real and long overdue needs of students and teachers of advanced (engineering) 
dynamics of the international community. I have sought to combine the best of 
the old and new — no age discrimination here — and I hope that this work will 
help counter the very real and disturbing trend, brought about by the proponents 
of the first two groups, toward a dynamical tower of Babel. 


ON NOTATION 

To make the exposition accessible to as many willing and able readers as possible, 
and following the admirable and ever applicable example of Lanczos (1949-1970), I 
have chosen, wherever possible, an informal approach; and I have, thus, deliberately 
avoided all set-theoretic and functional-analytic formalisms, all unnecessary rigor 
(“epsilonics”) and similar ahistorical/unmotivating/intuition-deadening tools and 
methods. For the same reasons, 1 have also avoided the currently popular direct/ 
dyadic (coordinate-free) and matrix notations (except in a very small number of 
truly useful situations); and I have, instead, chosen good old-fashioned elementary/ 
geometrical (undergraduate) form, for vectors, and/or indicial Cartesian tensorial 
notation for vectors, tensors, etc. 

The ad nauseam advertised “advantages” of the coordinate-free (“direct”) nota¬ 
tion and matrices are vastly exaggerated and misguiding. To begin with, it is no 
accident that the solution of all concrete physical problems is intimately connected 
with a specific and convenient (or natural, or canonical) system of coordinates. 
Indicial tensorial notation seems to kill two birds with one stone: it combines both 
coordinate invariance (generality) and coordinate specificity; that is, one knows 
exactly what to do in a given set of coordinates/axes; see, for example, Korenev 

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PREFACE 


(1979), MaiBer (1988) for robotics applications. However, the systematic use of 
general tensors in dynamics has been kept out of this book. [That is carried out in 
my monograph. Tensor Calculus and Analytical Dynamics (CRC Press, 1999).] The 
only thing tensorial used here amounts to nothing more than the earlier-mentioned 
indicial Cartesian tensor notation; and for reasons that will become clear later, not 
even the well-known summation convention is employed. Indicial tensorial notation 
turns out to be the best tool in “unknown and rugged terrain”; and frequently it is 
the only available notation, for example, in dealing with nonvectorial/tensorial “geo¬ 
metrical objects,” such as the Christoffel symbols and the Ricci/Boltzmann/Hamel 
coefficients. Once the fundamental theory is thoroughly understood, and the numer¬ 
ical implementation of a (frequently large-scale) concrete problem is sought, then 
one can profitably use matrices, and so on. Heavy use of matrices, with their non¬ 
commutativity “straitjacket,” at an early stage [e.g., Haug, 1992(a)] is likely to 
restrict creativity and replace physical understanding with the local mechanical 
manipulation of symbols. 


FURTHER PHILOSOPHY: On Computerization, Applications, and 
Ultimate Coals of Research 

1 do not think that the author of a book on analytical mechanics (AM) should be 
constantly defending it as simply a means to some other allegedly higher ends [e.g., a 
prerequisite to quantum mechanics, as Goldstein (1980) does], or in terms of its 
current “real life” applications in space or earth (e.g., artificial satellites, rocketry, 
robotics, etc.; i.e., in terms of dollars to be made); although, clearly, such connec¬ 
tions do exist and can be helpful. What should worry us is that these days, under 
what B. Schwartz calls “economic imperialism,” or what R. Bellah calls “market 
totalitarianism” (i.e., the penetration of purely monetary values into every aspect of 
social life; or, to regard all aspects of human relations as matters of economic self- 
interest, and model them after the market) every activity is fast becoming a means 
for something else, preferably quantifiable and monetary. In the process, daily work, 
craftsmanship, and the pleasure derived from the practice of that activity, have all 
been degraded. Unless we restore some internal, or intrinsic, goals and rewards to our 
subject and disseminate them to our young students, pretty soon such an activity will 
be no different from clerical or assembly-line work; that is, just a paycheck. As stated 
earlier, we view AM as a course worth pursuing in its own terms. We study it because 
it is worth learning, and because it is a grand and glorious part of our intellectual/ 
cultural heritage—those who do not care about the past cannot possibly care about 
the present, let alone the future. 

On a more practical level, a few years from now such applied areas as multibody 
dynamics, a subject with which so many dynamicists are preoccupied today, will be 
exhausted — some say that that has already happened. What are the practical 
mechanicians going to do then? Most of their expositions (second of the earlier 
groups) are too narrow and do not prepare the reader for the long haul. But there 
is a more fundamental reason for adopting “my” particular approach to mechanics: 
I strongly believe that every generation has to rediscover ( better, reinvent) AM, and most 
other areas of knowledge for that matter, anew and on its own terms; that is, replow 
the soil and not just be handed down from their predecessors, discontinuously, 
prepackaged and predigested “information” in a diskette (the electronic equivalent 
of ashes in an urn). To squeeze the “entire” mechanics into a huge master computer 

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PREFACE 


program, which (according to common but nevertheless vulgar advertisements) 
“does everything,” and makes it available to the reader (“user”) in the form of 
data inputs, is not only dangerous for the present (e.g., accidents, screw-ups, 
which are especially consequential in today’s large-scale systems — recall the omni¬ 
present Murphy’s laws), but also, being a degradation and dehumanization of 
knowledge, it guarantees the intellectual death of our society. If the job makes the 
person (mentally, psychologically, and physically), then how are we going to answer 
the question “What are people for?” 

Typical of such contemporary one-dimensional, or “digital,” approaches to 
dynamics are sweeping statements like: “pre-computer analytical methods for 
deriving the system equations must be replaced by systematic computer oriented 
formalisms, which can be translated conveniently into efficient computer codes for 
* generating the system equations based on simple user data describing the system 
model, * solving those complex equations yielding results ready for design evalua¬ 
tion” and “Emphasis is on computer based derivation of the system equations thus 
freeing the user from the time consuming and error-prone task of developing equa¬ 
tions of motion for various problems again and again.” [From advertisement of 
Roberson and Schwertassek (1988) in Ingenieur-Archiv, 59, p.A.3, 1989.] Here, the 
advertisers hide the well-known fact of how much error prone is the formulating and 
running of any complicated program; and how the combination of this with the 
absence of any physically simple and meaningful checks for finding errors — some¬ 
thing of a certainty for the structureless/formless mechanics of Newton-Euler, on 
which so much of multibody dynamics rests — is a recipe for chaos (=> arbitrariness)! 
Our reading of this ad is that the whole process will, eventually, “free” the user from 
thinking at all — first, we replace the human functions and then we replace humans 
altogether [first industrial revolution: mechanization of muscles, second (current) 
industrial revolution: automation of both muscles and brains]; and anyone who 
dares to criticize, or inquire about choices (i.e., politics), is summarily and arrogantly 
dismissed as a technophobe or, worse, a neo-Luddite! 

As the mathematicians Davis and Hersh put it accurately: 

By turning attention away from underlying physical mechanisms and towards the pos¬ 
sibility of once-for-all algorithmization, it encourages the feeling that the purpose of 
computation is to spare mankind of the necessity of thinking deeply.... Excessive 
computerization would lead to a life of formal actions devoid of meaning, for the 
computer lives by precise languages, precise recipes, abstract and general programs 
wherein the underlying significance of what is done becomes secondary. [Inimitably 
captured in M. McLuhan’s well-known dictum: The medium is the message.] It fosters 
a spirit-sapping formalism. The computer is often described as a neutral but willing 
slave. The danger is not that the computer is a robot but that humans will become 
robotized as they adapt to its abstractions and rigidities (1986, pp. 293, 16). 

And, in a similar vein, H. R. Post adds: “You understand a subject when you have 
grasped its structure, not when you are merely informed of specific numerical 
results” (quoted in Truesdell, 1984, p. 601). 

The issue is not whether the complete computer codification of (some version of ) 
dynamics can be achieved or not; it clearly can , somehow. The issue is the desirability 
of it; that is, the could versus the should , its scale compared with the other 
approaches, and the temporal order of such a presentation to the student (“user”). 
The only safe way for using such heavily computerized schemes is for the student to 
already possess a very thorough grounding in the fundamentals of mechanics—like 

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PREFACE 


vaccination against a virus! There is no painless and short way to bypass several 
centuries of hard thinking by a handful of great fellow humans — no royal road to 
mechanics! Otherwise, we are headed for more confusion, degradation, errors, and 
accidents, and eventual disengagement from our subject. [For iconoclastic, devastating, 
and sobering critiques of the contemporary mindless and rabid computeritis, see, for 
example, Truesdell (1984, pp. 594-631), Davis and Flersh (1986), and Mander (1985).] 
As for the applications of mechanics, there is nothing wrong with them; as long as 
they do not hurt or exploit people and nature — alas, several such contemporary 
applications do just that. Those preoccupied with them rarely, if ever, ask the natural 
question: What are the (most likely) applications of the applications; namely, their 
social/environmental consequences? In this light, common statements like “the com¬ 
puter is only a tool” are utterly naive and meaningless. I should also add that the 
current relentless emphasis, even in the academia, on applied research with quick 
tangible results — that is, dollars at the expense of every other nonmonetary aspect — 
is a relatively recent phenomenon imposed on us from outside ; it is neither intrinsic 
nor accidental to science, but instead is an intensely socio-economic activity — 
technology is neither autonomous nor neutral! [And as Truesdell concurs, with 
depressing accuracy (1987, p. 91): “It is not philosophers of science who will enforce 
one kind of research or another. No, it will be the national funding agencies, the 
sources of manna, nectar, and ambrosia for the corrupted scientists. The directors of 
funds are birds of a feather, chattering mainly to each other and at any one moment 
singing more or less the same cacophonous tune. There may come a time when even 
the scholarly foundations will give preference to those who claim to promote 
national ‘defense’ by research on the basic principles governing some new, as yet 
totally secret — that is, known only to the directors of war in the U.S. and Russia — 
allegedly secret idea for a broader and more effective death by torture in a world full 
of humanitarians and their -isms.”] 

If applications, even worthwhile ones, are but one motive for studying mechanics, 
and science in general, then what else is? Here are some plausible (existential?) 
reasons offered by Einstein, which I have found particularly inspiring, since my 
high school years: 

Man tries to make for himself in the fashion that suits him best a simplified and 
intelligible picture of the world; he then tries to some extent to substitute this cosmos 
of his for the world of experience, and thus to overcome it. This is what the painter, the 
poet, the speculative philosopher, and the natural philosopher do, each in his own 
fashion. Each makes this cosmos and its construction the pivot of his emotioned life, in 
order to find in this way the peace and security which he cannot find in the narrow whirlpool 
of personal experience (emphasis added; from "Principles of Research,” an address 
delivered in 1918, on the occasion of M. Planck’s sixtieth birthday). 

From a broader perspective, I am convinced that the quality of our lives depends not 
so much on specific gadgets/artifacts, no matter how technically advanced they may 
be (e.g., from artificial hearts to space stations), but on our collective abilities to 
formulate simple, clear, and unifying ideas that will allow us to understand (and then 
change gently and gracefully — sustainably) our increasingly complicated, unstable 
and fragile societies; and, in the process, understand ourselves. The resulting psycho¬ 
logical and intellectual peace of mind from such a liberal arts (= liberating) approach 
cannot be overstated. It is this kind of activity and attitude that gives human life 
meaning — we do not do science just to make money, merely to exchange and con¬ 
sume. This book is intended as a small but tangible contribution to this lofty goal. 

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PREFACE 


SOME PERSONAL HISTORY 

My interest in AM began during my undergraduate studies (mid-to-late 1960s) upon 
reading in Hamel (1949, pp. 233-236, 367) about the differences between the calculus 
of variations (mathematics) and Hamilton’s variational principle (mechanics) for 
nonholonomic systems. The need for a deeper understanding of the underlying 
kinematical concepts led me, about twenty years later, to the study of the original 
epoch-making memoirs of such mechanics masters as Appell, Boltzmann, Heun, and 
Hamel. Then, in the spring of 1986, in related studies on variational calculus, I had 
the good fortune to stumble upon the virtually unknown but excellent papers of 
Schaefer (1951) and Stiickler (1955), which, along with my earlier acquaintance with 
tensors, showed me the way toward the correct understanding of everything virtual: 
virtual displacements and virtual work/Lagrange’s principle; that is, 1 arrived at AM 
via the calculus of variations, just like Lagrange in the 1760s! Finally, the emphasis 
on the fundamental distinction between particle and system quantities I owe to 
the writings of Heun, the founder of theoretical engineering dynamics (early 20th 
century), and especially to those of his students: Winkelmann and the great Hamel. 

In closing, 1 would like to recommend the reading of the preface(s) of Lanczos 
(1949-1970); the present work has been conceived and driven by a similar overall 
philosophy. 

May this book make many and loyal friends! 


john.papas@me.gatech.edu 

Atlanta, Georgia J. G. P. 

Autumn 2001 


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ACKNOWLEDGMENTS 


(Where the author recognizes, with gratitude and pleasure, 
the social dimension of his activity) 


Every book on analytical mechanics is better off the closer it comes to the simplicity, 
clarity, and thoroughness of Georg Hamel’s classic Theoretische Mechanik , arguably 
the best (broadest and deepest) single work on mechanics; and, secondarily, of 
Anatolii E Lur’e’s outstanding Analiticheskaya Mekhanika. I hope that this treatise 
follows closely and loyally the tradition created by these great masters. My indebt¬ 
edness to their monumental works is hereby permanently registered. 

Next, I express my deep appreciation and thanks to 

Ms. Katharine L. Calhoun, of the Georgia Tech library, for her most courteous 
and capable help in locating and obtaining for me, over the past several years (1986 
to present), hundreds of rare and critically needed references, from all over the 
country. Katharine is an oasis of humanity and graciousness in an otherwise arid 
and grim campus. 

Drs. Feng Xiang Mei, professor at the Beijing Institute of Technology, China, and 
Zhen Wu, professor at the University of Tsiao Tong, China; and Drs. Sergei A. 
Zegzhda and Mikhail P. Yushkov, professors at the Mathematics and Mechanics 
faculty of St. Petersburg University, Russia [birthplace of the first treatise on 
theoretical/analytical mechanics (Euler’s, Mechanica Sive Motus Scientia ..., 
2 vols, 1736)] for making available to me copies of their excellent textbooks and 
papers on advanced dynamics, which are virtually unavailable in the West (see the 
list of References and Suggested Reading). 

Dr. John L. Junkins, chaired (and distinguished) professor of aeronautical 
engineering at Texas A&M University, for early encouragement on a previous 
version (outline) of the manuscript (1986) — what John aptly dubbed “the zeroth 
approximation.” 

Dr. Donald T. Greenwood, professor of aerospace engineering at the University 
of Michigan (Ann Arbor) and a true Nestor among American dynamicists, also 
author of internationally popular and instructive graduate texts on dynamics, for 
his detailed and mature (and very time- and energy-consuming) comments on the 
entire technical part of the manuscript; and for sharing with me his (soon to appear 
in book form) notes on Special Advanced Topics in Dynamics. 

Dr. Wolfram Stadler, professor and scholar of mechanics at San Francisco State 
University and author of uniquely encyclopaedic work on robotics/mechatronics, 
for qualitative and constructive criticism on Chapter 1 (Background). Wolf 
remains a staunch and creative individualist, in an age of ruthless academic 
collectivization. 

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xviii 



ACKNOWLEDGMENTS 

A very special expression of indebtedness to my friend Dr. Hartmut Bremer (O. 
Univ.-Prof. Dr.-Ing. habil.), professor of mechatronics at the University of Linz, 
Austria (formerly, professor of mechanics at the Technical University of Munich, 
Germany) and author of two dense and comprehensive textbooks on dynamics/ 
control/multibody systems, for extensive, thoughtful and critical discussions/com¬ 
ments on several topics of theoretical and applied dynamics, including their histor¬ 
ical, cultural, and educational aspects; and for supplying me with rare and precious 
references on the subject. To Hartmut’s persistent efforts (1989-1993), 1 owe a very 
rare photograph of Karl Heun (to appear in a special gallery of photos of mechanics 
masters, which will be included in my forthcoming Elementary Mechanics). 

In addition, I consider myself very fortunate to have benefited from the “mechan¬ 
ical” knowledge and wisdom of my friend Dr. Leon Y. Bahar, professor of mechan¬ 
ical engineering and mechanics at Drexel University and another American Nestor. 
Leon, a veritable engineering science scholar, craftsman, and above all dedicated 
teacher (i.e., representative of an academic species that is somewhere between endan¬ 
gered and extinct, as professionalism goes up and scholarship goes down), has over 
the past several years selflessly provided me with critical and enlightening quanta of 
knowledge and insight (lecture notes, papers, and extensive letters), and much valued 
mentorship. 

These are not the best of times for writing “another” book on advanced theore¬ 
tical mechanics — to put it mildly. However, and this provides a certain consolation, 
even such all-time titans of mathematics and mechanics as Euler, Lagrange, and 
Gauss had considerable difficulties in publishing, respectively, their Theoria Motus 
... (1765), Mechanique Analitique (1788), and Disquisitiones Arithmeticae (1801). 
[According to Truesdell (1984, p. 352), all that Euler received for his masterpiece 
on rigid-body dynamics (1765) was ... twelve free copies of it!] Aspiring academic 
writers in this area are forewarned that the contemporary “research” university is not 
particularly supportive to such scholarly activities; these latter, obviously “interfere” 
with the more lucrative business (to the university bureaucrats, but not necessarily to 
students and society at large) of contracts and grants from big business and big 
government. Such an inimical “academic” environment makes it much more natural 
than usual that I reserve the strongest expression of gratitude, by far, to my family, 
here in Atlanta, Georgia: my wife Kim Ann and daughter Julia Constantina; and in 
my native Athens, Greece: my mother Konstantina and brother Stavros (in my 
nonobjective but fair view, one of the brightest mathematicians of contemporary 
Greece, and an island of moral and intellectual nobility, in an archipelago of petty 
greed and irrationality) — for their continual and critical moral and material support 
throughout the several long and solitary years of writing of the book. 

Last, this volume (as well as my other two mechanics books) could not have been 
written without (i) the institution of academic tenure (much maligned and curtailed 
recently by reactionary ideologues, demagogues, and ignoramuses) and (ii) the (alas, 
fast disappearing) policy of most university libraries, of open , direct, and free access 
to books and journals. Regrettably, and in spite of high-tech millennarian promises, 
the next generation of scholarly authors will not be as lucky as I have been, in both 
these areas! 


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Contents 


INTRODUCTION 3 

1 Introduction to Analytical Mechanics 4 

2 History of Theoretical Mechanics: A Bird’s-Eye View 9 

3 Suggestions to the Reader 13 

4 Abbreviations, Symbols, Notations, Formulae 14 


BACKGROUND: BASIC CONCEPTS AND EQUATIONS OF PARTICLE AND 
RIGID-BODY MECHANICS 71 

1.1 Vector and (Cartesian) Tensor Algebra 72 

1.2 Space-Time Axioms; Particle Kinematics 89 

1.3 Bodies and their Masses 98 

1.4 Force; Law of Newton-Euler 101 

1.5 Space-Time and the Principle of Galilean Relativity 104 

1.6 The Fundamental Principles (or Balance Laws) of General 
System Mechanics 106 

1.7 Accelerated (Noninertial) Frames of Reference (or Relative 
Motion, or Moving Axes); Angular Velocity and 
Acceleration 113 

1.8 The Rigid Body: Introduction 138 

1.9 The Rigid Body: Geometry of Motion and Kinematics 
(Summary of Basic Theorems) 140 

1.10 The Rigid Body: Geometry of Rotational Motion; 

Finite Rotation 155 

1.11 The Rigid Body: Active and Passive Interpretations of a Proper 
Orthogonal Tensor; Successive Finite Rotations 178 

1.12 The Rigid Body: Eulerian Angles 192 

1.13 The Rigid Body: Transformation Matrices (Direction Cosines) 
Between Space-Fixed and Body-Fixed Triads; and Angular 
Velocity Components along Body-Fixed Axes, for All Sequences 
of Eulerian Angles 205 

1.14 The Rigid Body: An Introduction to Quasi Coordinates 212 

1.15 The Rigid Body: Tensor of Inertia, Kinetic Energy 214 

1.16 The Rigid Body: Linear and Angular Momentum 222 

1.17 The Rigid Body: Kinetic Energy and Kinetics of Translation and 
Rotation (Eulerian “Gyro Equations”) 225 

1.18 The Rigid Body: Contact Forces, Friction 237 

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xx CONTENTS 

2 KINEMATICS OF CONSTRAINED SYSTEMS 
(i.e., LAGRANGEAN KINEMATICS) 242 

2.1 Introduction 242 

2.2 Introduction to Constraints and their Classifications 243 

2.3 Quantitative Introduction to Nonholonomicity 257 

2.4 System Positional Coordinates and System Forms of the 
Flolonomic Constraints 270 

2.5 Velocity, Acceleration, Admissible and Virtual Displacements; in 
System Variables 278 

2.6 System Forms of Linear Velocity (Pfaffian) Constraints 286 

2.7 Geometrical Interpretation of Constraints 291 

2.8 Noncommutativity versus Nonholonomicity; Introduction to 
the Theorem of Frobenius 296 

2.9 Quasi Coordinates, and their Calculus 301 

2.10 Transitivity, or Transpositional, Relations; Hamel 
Coefficients 312 

2.11 Pfaffian (Velocity) Constraints via Quasi Variables, and their 
Geometrical Interpretation 323 

2.12 Constrained Transitivity Equations, and Hamel’s Form of 
Frobenius’ Theorem 334 

2.13 General Examples and Problems 345 


3 KINETICS OF CONSTRAINED SYSTEMS 
(i.e., LAGRANGEAN KINETICS) 381 

3.1 Introduction 381 

3.2 The Principle of Lagrange (LP) 382 

3.3 Virtual Work of Inertial Forces (61), and Related 
Kinematico-Inertial Identities 399 

3.4 Virtual Work of Forces: Impressed (6'W) and Constraint 
Reactions (6 1 W R ) 405 

3.5 Equations of Motion via Lagrange’s Principle: General 
Forms 409 

3.6 The Central Equation (The Zentralgleichung of Heun and 
Hamel) 461 

3.7 The Principle of Relaxation of the Constraints (The Lagrange- 
Hamel Befreiungsprinzip ) 469 

3.8 Equations of Motion: Special Forms 486 

3.9 Kinetic and Potential Energies; Energy Rate, or Power, 
Theorems 511 

3.10 Lagrange’s Equations: Explicit Forms; and Linear Variational 
Equations (or Method of Small Oscillations) 537 

3.11 Appell’s Equations: Explicit Forms 563 

3.12 Equations of Motion: Integration and Conservation 
Theorems 566 

3.13 The Rigid Body: Lagrangean-Eulerian Kinematico-Inertial 
Identities 581 

3.14 The Rigid Body: Appellian Kinematico-Inertial Identities 594 

3.15 The Rigid Body: Virtual Work of Forces 597 

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CONTENTS 


XXI 



3.16 Relative Motion (or Moving Axes/Frames) via Lagrange’s 
Method 606 

3.17 Servo (or Control) Constraints 636 

3.18 General Examples and Problems 650 

APPENDIX 3.A1 

Remarks on the History of the Hamel-type Equations of Analytical 
Mechanics 702 

APPENDIX 3.A2 

Critical Comments on Virtual Displacements/Work; and Lagrange’s 
Principle 708 

4 IMPULSIVE MOTION 718 

4.1 Introduction 718 

4.2 Brief Overview of the Newton-Euler Impulsive Theory 718 

4.3 The Lagrangean Impulsive Theory; Namely, Constrained 
Discontinuous Motion 721 

4.4 The Appellian Classification of Impulsive Constraints, and 
Corresponding Equations of Impulsive Motion 724 

4.5 Impulsive Motion via Quasi Variables 751 

4.6 Extremum Theorems of Impulsive Motion (of Carnot, Kelvin, 
Bertrand, Robin, et al.) 784 

5 NONLINEAR NONHOLONOMIC CONSTRAINTS 817 

5.1 Introduction 818 

5.2 Kinematics; The Nonlinear Transitivity Equations 819 

5.3 Kinetics: Variational Equations/Principles; General and Special 
Equations of Motion (of Johnsen, Hamel, et al.) 831 

5.4 Second- and Higher-Order Constraints 871 

6 DIFFERENTIAL VARIATIONAL PRINCIPLES, AND ASSOCIATED GENERALIZED 
EQUATIONS OF MOTION OF NIELSEN, TSENOV, ET AL. 875 

6.1 Introduction 875 

6.2 The General Theory 876 

6.3 Principle of Jourdain, and Equations of Nielsen 879 

6.4 Introduction to the Principle of Gauss and the Equations of 
Tsenov 884 

6.5 Additional Forms of the Equations of Nielsen and Tsenov 894 

6.6 The Principle of Gauss (Extensive Treatment) 911 

6.7 The Principle of Hertz 930 

7 TIME-INTEGRAL THEOREMS AND VARIATIONAL PRINCIPLES 934 

7.1 Introduction 935 

TIME-INTEGRAL THEOREMS 936 

7.2 Time-Integral Theorems: Pfaffian Constraints, Holonomic 
Variables 936 

7.3 Time-Integral Theorems: Pfaffian Constraints, Linear 
Nonholonomic Variables 948 

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xxii 



CONTENTS 

7.4 Time-Integral Theorems: Nonlinear Velocity Constraints, 
Holonomic Variables 957 

7.5 Time-Integral Theorems: Nonlinear Velocity Constraints, 
Nonlinear Nonholonomic Variables 958 

TIME-INTEGRAL VARIATIONAL PRINCIPLES (IVP) 960 

7.6 Hamilton’s Principle versus Calculus of Variations 960 

7.7 Integral Variational Equations of Mechanics 966 

7.8 Special Integral Variational Principles (of Suslov, 

Voronets, et al.) 974 

7.9 Noncontemporaneous Variations; Additional IVP Forms 990 
APPENDIX 7.A 

Extremal Properties of the Hamiltonian Action (Is the Action 
Really a Minimum; Namely, Least?) 1055 

8 INTRODUCTION TO HAMILTONIAN/CANONICAL METHODS: 

EQUATIONS OF HAMILTON AND ROUTH; CANONICAL FORMALISM 1070 

8.1 Introduction 1070 

8.2 The Hamiltonian, or Canonical, Central Equation and 
Hamilton’s Canonical Equations of Motion 1073 

8.3 The Routhian Central Equation and Routh’s Equations of 
Motion 1087 

8.4 Cyclic Systems; Equations of Kelvin-Tait 1097 

8.5 Steady Motion (of Cyclic Systems) 1115 

8.6 Stability of Steady Motion (of Cyclic Systems) 1119 

8.7 Variation of Constants (or Parameters) 1143 

8.8 Canonical Transformations (CT) 1161 

8.9 Canonicity Conditions via Poisson’s Brackets (PB) 1176 

8.10 The Hamilton-Jacobi Theory 1192 

8.11 Hamilton’s Principal and Characteristic Functions, and 
Associated Variational Principles/Differential Equations 1218 

8.12 Integral Invariants 1230 

8.13 Noether’s Theorem 1243 

8.14 Periodic Motions; Action-Angle Variables 1250 

8.15 Adiabatic Invariants 1290 

8.16 Canonical Perturbation Theory in Action-Angle Variables 1305 
References and Suggested Reading 1323 

Index 1371 


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Words of Wisdom and Beauty 


On Rigor It is not so much important to be rigorous as to be right. 

—A. N. KOLMOGOROV 

On Theory There is nothing more practical than a good theory. 

—L. BOLTZMANN 

We have no access to a theory-independent world — that is, a world 

unconditioned by our point of view_The world we see is ... 

theory-laden: it already bears the ineliminable mark of our involve¬ 
ment in it .... Knowledge is always a representation of reality from 

within a particular perspective_We cannot assume ... “the view 

from nowhere.” 

—T. W. CLARK 

I really do not at all like the now fashionable “positivistic” 
tendency of clinging to what is observable ... I think ... that theory 
cannot be fabricated out of the results of observation, but that it can 
only be invented. 

—A. EINSTEIN 

On Method In the sciences the subject is not only set by the method; at the same 
time it is set into the method and remains subordinate to the method 
.... In the method lies all the power of knowledge. The subject belongs 
to the method, (emphasis added) 

—M. HEIDEGGER 

The core of the practice of science — the thread that keeps it going 
as a coherent and developing activity—lies in the actions of those 
whose goals are internal to the practice. And these internal goals are 
all noneconomic, (emphasis added.) 

—B. SCHWARTZ 


On Beauty My own students, few they have been, I have tried to teach how to 
ask questions humbly and to see ways to some taste in a vulgar, 
obscene epoch. Taste is acquired by those who can face questions, 
especially insoluble questions. 

C. A. TRUESDELL 

It is by the steady elimination of everything which is ugly — 
thoughts and words no less than tangible objects — and by the sub¬ 
stitution of things of true and lasting beauty that the whole progress 
of humanity proceeds. 

—A. PAVLOVA 


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(Jbo Reprint Edition 


Analytical 

Mechanics 


A Comprehensive Treatise 
on the 

Dynamics of Constrained Systems 


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Introduction 


KOIMON TONAE, TON AYTON AI1ANTQN, OYTE Til 0EQN 
OYTE ANGPQnnN EIIOIHIEN, AAA'HN AEI KAI EZTIN KAI EITAI 
IlYP AEIZOON, AllTOMENON METPA KAI AIIOIBENNYMENON 
METPA. 

[HPAKAEITOZ (Herakleitos, Creek philosopher; Ephesos, Ionia, 

late 6th century b.c.)] 

[Translation: "This world [order], which is the same for all 
[beings], no one of gods or humans have created; but it was 
ever, is now, and ever shall be an ever-living Fire, that starts and 
goes out according to certain rules [laws]." 

This magnificent statement marks the beginning of science—one 
of the countless, fundamental, and original gifts of Greece to the 
world. (See, e.g., Burnet, 1930, p. 134; also Frankfort et al., 

1946, chap. 8.)] 

Die Mechanik ist die Wissenschaft von der Bewegung; als ihre 
Aufgabe bezeichnen wir: die in der Natur vor sich gehenden 
Bewegungen vollstandig und auf die einfachste Weise zu 
beschreiben. 

(Translation: Mechanics is the science of motion; we define as its 
task the complete description and in the simplest possible manner 
of such motions as occur in nature.) 

(Kirchhoff, 1876, p. 1, author's emphasis) 

Dynamics or Mechanics is the science of motion .... The 
problem of dynamics according to Kirchhoff, is to describe all 
motions occurring in nature in an unambiguous and the simplest 
manner. In addition it is our object to classify them and to 
arrange them on the basis of the simplest possible laws. 

The success which has attended the efforts of physicists, 
mathematicians, and astronomers in achieving this object from the 
time of Calileo and Newton through that of Lagrange and Laplace 
to that of Helmhoitz and Kelvin, constitutes one of the greatest 
triumphs of the human intellect. 

(Webster, 1912, p. 3, emphasis added) 

Die Mechanik ist ein Teil der Physik. 

(Translation: Mechanics is a part of physics.) 

(Foppl, 1898, vol. 1, p. 1) 


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INTRODUCTION 


1 INTRODUCTION TO ANALYTICAL MECHANICS 

What Is Analytical Mechanics? 

Classical mechanics (CM)—that is, the exact science of nonrelativistic and non¬ 
quantum motion (effects) and forces (causes)—was founded in the 17th century 
(Galileo, 1638; Newton, 1687), and was brought to fruition and generality during 
the next century, almost single-handedly, by Euler (1752: principle of linear momen¬ 
tum; 1775: principle of angular momentum). [D’Alembert too had formulated sepa¬ 
rate laws of linear and angular momentum (1743, 1758), but his approach came 
nowhere near that of Euler in generality and power.] That was the first complete 
dynamical theory in history. We shall call it, conveniently (even though not quite 
accurately), the Newton-Euler method of mechanics (NEM). 

The second such theory was also initiated in the (late) 17th century, this time by 
Huygens and Jakob Bernoulli; it was further developed during the 18th century by 
Johann Bernoulli (Jakob’s brother) and d’Alembert (early 1740s), and was finally 
brought to relative mathematical and physical completion by the other great math¬ 
ematician of that century, Lagrange (1760: principle of “least” action; 1764: princi¬ 
ple of d’Alembert in Lagrange’s form, or Lagrange's principle', 1780: central equation 
and Lagrange’s equations', 1788: Mechanique Analitique; 1811-1812: transitivity equa¬ 
tions). This second approach, what we shall call the method of d’Alembert Lagrange, 
or, simply and more accurately, the method of Lagrange, forms the basis of what has 
come to be known as analytical mechanics (AM); or, equivalently, Lagrangean 
mechanics (LM). Although both these methods are, roughly, theoretically equivalent, 
since there is only one classical mechanics, the second approach proved much more 
influential and fertile to the subsequent development not only of mechanics, but also 
of practically all areas of physics: from generalized coordinates and configuration 
space to Riemannian geometry and tensors, and from there to general relativity; and 
similarly for quantum mechanics. 

Analytical mechanics proved particularly significant and useful to engineers, 
although it took another century after Lagrange for this to be fully realized (see 
§2). The reason for this is that AM was specifically designed by its inventors to 
handle constrained (earthly) systems— the concept of constraint is central to AM. 
Not that NEM could not handle such systems, but AM proved incomparably 
more expedient both for formulating their simplest (or minimal) equations of 
motion, and also for offering numerous theoretical and practical insights and 
tools for their solution (e.g., theorems of conservation and invariance, variational 
“principles” and associated direct methods of approximation, etc.—detailed in 
chaps. 3-8). 

In NEM, the basic principles (or axioms) are those of linear and angular momen¬ 
tum, and, secondarily, that of action-reaction, for the internal (or mutual) forces (see 
chap. 1); that is, 

NEM is a mechanics of systems based on momentum principles. 

In LM, on the other hand, the primary axiom is the kinetic principle of virtual work 
for the constraint reactions [=Lagrange’s principle (LP)] and, secondarily, the principle 
of relaxation (or liberation, or freeing) of the constraints (see chap. 3); that is, 

LM is a mechanics of systems based on energetic principles. 

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§1 INTRODUCTION TO ANALYTICAL MECHANICS 


With the help of his LP, Lagrange and many others later (see §2) formulated the 
most general equations of motion of systems subject to general positional and/or 
motional constraints. [The former are called holonomic, while the latter, if they 
cannot be brought (integrated) to positional form are called nonholonomic (see 
chap. 2).] 

Last, from the viewpoint of applications, AM constitutes the theoretical founda¬ 
tion of advanced engineering dynamics ; which, in turn, is very useful to the following: 
structural dynamics (e.g., bridges, airport runways); machine dynamics (e.g., internal 
combustion engines); vehicle dynamics (e.g., automobiles, locomotives); rotor 
dynamics (e.g., turbines); robot dynamics (e.g., robotic manipulators); aero-jastro- 
dynamics (e.g., airplanes, artihcial satellites); control theoryjsystem dynamics (e.g., 
electromechanical systems, valves); celestial dynamics (e.g., astronomy), and so on. 


Comments on the Methodology of AM 

1. From the otherwise physically complete (particle) mechanics of Newton two 
things were missing: rotation and constraints (and, of course, deformation, but we do 
not deal with continua here). The first was taken care of by Euler, Mozzi, Cauchy, 
Chasles, Rodrigues et al. (1750s to the mid 19th century), and the second by 
Lagrange (1760s 1780s) and later many others (1870-1910). Of course, special 
cases of both problems had been examined earlier: for example, Newton discussed 
motions on specified curves and the associated forces, and, as Heun points out, with 
the help of his third law of action/reaction, he could have built a constrained particle 
mechanics, had he pursued that possibility; d’Alembert worked with particles “con¬ 
strained in rigid body connections”; and even Huygens had such pendula involving 
several constrained particles. Much later (early 1810s), Lagrange brought rotation 
under his energetic plan (genesis of nonholonomic, or ywa,si-coordinates; special tran¬ 
sitivity equations—see bridge between Euler and Lagrange below). 

2. Analytical versus synthetical, Euler versus Lagrange. To begin with, CM holds 
quite satisfactorily for sizes, or lengths, from 1CT 10 m (atom) to 10 20 m (galaxy), and 
for speeds up to c/10 (c = speed of light « 300,000 km/s). Outside of these broad 
ranges, CM is replaced by relativity (high speeds) and quanta (small sizes) (see, e.g., 
French, 1971, p. 8). Now, depending on the method adopted, CM can be classified as 
follows: 


Experimental 


MECHANICS 


Synthetical 


Theoretical 


Analytical 


kal 


Newton/Euler (momentum) 
D'Alembert/Lagrange (energy) 


This classification, a logically possible one out of many (see below; e.g., Hamel, 
1917), stresses the following: 

2(a). Contrary to popular declarations, and Lagrange himself is partly to blame 
for this, AM does not mean mechanics via mathematical analysis; that is, it does not 

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INTRODUCTION 


mean an ageometrical and figureless mechanics. [Even such 20th century mechanics 
authorities as Whittaker state that “The name Analytical Dynamics is given to that 
branch of knowledge in which the motions of material bodies,..., are discussed by 
the aid of mathematical analysis” (1937, p. 1).] Instead, and in the sense used in 
philosophy/logic, AM means a deductive mechanics: everything flowing from a few 
selected initial postulates/principles/axioms by logical (mathematical) reasoning; 
that is, from the general to the particular—as contrasted with inductive, or synthetic, 
mechanics; that is, from the particular to the general. As such, AM is by no means 
ageometrical (and, similarly, synthetic mechanics does not necessarily mean geome¬ 
trical and nonmathematical mechanics). Also, in the past (mainly 19th century) the 
terms theoretical, rational, and analytical have frequently been used synonymously. 

2(b). In such a classification, the mechanics of Euler also deserves to be called 
analytical! The reason that we in this book, and most everybody else, do not have 
more to do with historical tradition and usage rather than with strict logic: today 
AM has come to mean, specifically, 

Lagrangean method = Principle of Lagrange 

(= Principle of d’Alembert + Johann Bernoulli’s principle 
of virtual work) 

+ Principle of relaxation of the constraints 
(Hamel’s Befreiungsprinzip) 

After more than 200 years, AM is defined by its practice —that is, by its methods, 
tools, and range of problems dealt with by its practitioners—and because, contrary 
to the mechanics of Newton-Euler, it is capable of extending to other areas of 
physics: for example, statistical mechanics, electrodynamics. As the distinguished 
applied mathematician Gantmacher puts it 

[A]nalytical mechanics is characterized both by a specific system of presentation and 
also by a definite range of problems investigated. The characteristic feature ... is that 
general principles (differential or integral) serve as the foundation; then the basic differ¬ 
ential equations of motion are derived from these principles analytically. The basic 
content of analytical mechanics consists in describing the general principles of 
mechanics, deriving from them the fundamental differential equations of motion, and 
investigating the equations obtained and methods of integrating them (1970, p. 7). 

2(c). Frequently, one is left with the impression that Eulerian mechanics is vec¬ 
torial, whereas Lagrangean mechanics is scalar. This, however, is only superficially 
true: LM can be quite geometrical and vectorial, but in generalized nonphysical/non- 
Euclidean (Riemannian and beyond) spaces [see, e.g., Papastavridis (1999), Synge 
(1926-1927, 1936), and references therein]. 

2(d). Euler and Lagrange should be viewed as mutually complementary, not as 
adversarial—as some historians of mechanics do. And although it is undeniably true 
that, of the two, Euler was the greater “geometer” in both quantity and quality, yet it 
was the method of Lagrange that shaped and drove the subsequent epoch-making 
developments of theoretical physics and a fair part of applied mathematics (i.e., 
differential geometry/tensors —> relativity; Hamiltonian mechanics/phase space —> 
statistical mechanics, quantum mechanics). Lagrange himself, shortly before his 
death (in 1813), succeeded in building the bridge between his method and that of 
Euler (rigid-body equations) by obtaining a special case of “transitivity equations” 

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§1 INTRODUCTION TO ANALYTICAL MECHANICS 


[so named by Heun (early 20th century) because they allow the transition from 
Lagrangean to Eulerian], which appeared in the second volume of the second edition 
of his Mecanique Analytique (1815). And that is why the great mechanician Hamel, 
in 1903-1904, dubbed his own famous equations the “Lagrange-Euler equations”; 
and in his magnum opus Theoretische Mechanik (1949) he founded the entire 
mechanics on Lagrangean principles. [With the exception of Neimark and Fufaev 
(1972), the transitivity equations are completely and conspicuously absent from the 
entire English and French literature!] 

3. Newtonian particles versus Eulerian continua. There is a certain viewpoint, 
particularly popular among celestial dynamicists/astronomers, (particle) physicists, 
and some applied mathematicians, according to which classical mechanics is the 
study of the motions of systems of particles under mutually attractive/repulsive 
forces, whose intensities depend only on the distances among these particles (mole¬ 
cules, etc.); and that, eventually, all physical phenomena are to be explained by such 
a “mechanistic” model. This Newtonian mindset dominated 19th century mechanics 
and physics almost completely, and obscured the fact that such a “central force + 
particle(s)” mechanics [launched, mainly, by P. S. de Laplace in his monumental five- 
volume Traite de Mecanique Celeste (1799-1825)] is but one possibility, even within 
the nonrelativistic and nonquantum confines of the 19th century. Under other, 
physically more realistic, possibilities the total interparticle force, generally, consists 
of a reaction to the geometrical and/or kinematical constraints imposed, and an 
impressed, or physical, part that can depend explicitly on time, position(s), and 
velocity(-ies) of some or all of the system particles. However, the introduction of 
such forces to mechanics creates effects that cannot be accounted by mechanics 
alone, such as thermal and/or electromagnetic phenomena; whereas, the conse¬ 
quences of Newtonian forces stay within mechanics. 

The “mechanistic theory of matter”—namely, to explain all nonmechanical phe¬ 
nomena via simple models of internal nonvisible (concealed) motions of the system’s 
molecules (second half of 19th century, proposed by physicists like W. Thomson, 
J. Thomson, Helmholtz, Hertz et al.)—was only partially successful, and eventually 
evolved to statistical mechanics and physics (Boltzmann, Gibbs) and quantum 
mechanics [Planck, Einstein, Bohr, Born, Heisenberg, Schrodinger, Dirac et al.; 
see also Stackel (1905, p. 453 ff)]. 

Finally, as Hamel (1917) points out, it should be remembered that AM is not 
restricted to particles: even though Lagrange himself starts with particles, that fact is 
totally unimportant to his method; he could have just as well spoken of “volume 
elements.” 

4. Theory versus experiment. The logical consequences of the principles of AM 
should not contradict experience. This, however, does not mean that these conse¬ 
quences (theorems, corollaries, etc.) should be derived from experiments; the latter 
cannot supply missing mathematics, or be used to prove and/or verify something, 
but they can be used to disprove a hypothesis. As H. R. Post puts it: 

[There are] three items of religious worship inside present-day science, the third of which is 
experiment. [l]n the main the role of experiment constitutes a harmless myth in the philo¬ 
sophy of scientists. The myth considers experiment to be a generator of theories. In fact the 
role of experiment ... is solely to decide between two or more existing theories ... 
Experiment does not generate theories but rather is suggested by them. [As quoted 
in Truesdell (1987, p. 83). And, in a similar vein, Einstein declares: “Experiment never 
responds with a ‘yes’ to theory. At best, it says ‘maybe’ and, most frequently, simply ‘no.’ 
When it agrees with theory, this means ‘maybe’ and, if it does not, the verdict is ‘no.’ ”] 

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INTRODUCTION 


But if the axioms of mechanics do not flow simply (“mechanically”) and uniquely 
out of experiments, then where do they come from? Paraphrasing Hamel, Einstein 
et al., we may say that these axioms are erected from the facts of experience (the 
object) by the human mind (the subject) as cm equal and imaginative partner, from a 
little observation, a lot of thought and eventually a rather sudden (qualitative) under¬ 
standing and insight into nature. In other words, humans are not passive at all in the 
formation of scientific theories, but because of the enormous difficulty involved, the 
creation of a successful set of axioms is the rare act of genius [e.g. (chronologically): 
Euclid, Archimedes, Newton, Euler, Lagrange, Maxwell, Gibbs, Boltzmann, Planck, 
Einstein, Heisenberg, Schrodinger], 

In CM, although open and nontrivial problems still remain, yet they are to be 
solved by the adoped principles; namely, we do not risk much in stating that this 
science is essentially closed, and that is why here the analytical/deductive method is 
possible. Otherwise, we would have to adopt a synthetic/inductive approach and 
change it slowly, depending on the new empirical facts. 

5. In addition to the Lagrangean (and Hamiltonian) analytical formulation of 
mechanics—namely, the classical tradition of Whittaker, Hamel, Lur’e, Pars, 
Gantmacher et al. followed here, and depending on the emphasis laid on their 
most significant aspects, the following complementary formulations of CM also exist: 


Variational (e.g., Lanczos, Rund). 

Vako-nomic (= Variational Axiomatic Kind; e.g., Arnold, Kozlov). 

Algebraic (= infinitesimal and finite canonical transformations. Lie algebras and groups, 
symmetries and conservation theorems; e.g., McCauley, Mittelstaedt, Saletan and 
Cromer, Sudarshan and Mukunda). 

Nonlinear dynamics (= regular and stochastic/chaotic motion; e.g., Gabor, 
Guggenheimer and Holmes, Lichtenberg and Lieberman, McCauley). 

Geometrical (= symplectic geometry, canonical structure; e.g., Arnold, Abraham and 
Marsden, MacLane). 

Statistical and thermodynamical (= Liouville’s theorem, equilibrium and nonequilibrium 
statistical mechanics, irreversible processes, entropy, etc.; e.g., Gibbs, Katz, Fiirth, 
Sommerfeld, Tolman). 

Many-body and celestial mechanics (= orbits and their stability, many-body problem; 
e.g., Charlier, Hagihara, Happel, Siegel and Moser, Szebehely, Wintner). 


All these, and other, formulations testify once more to the vitality and importance of 
CM for the entire natural science, even today. 

6. For engineering purposes, the following (nonunique) partitioning of mechanics 
seems useful: 


MECHANICS 




Kinematics (motion) 


Dynamics (forces) 



Statics (forces and equilibrium) 
Kinetics (forces and motion) 


[We consider this preferable to the following partitioning, customary in the U.S. 
undergraduate engineering education: 


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§2 HISTORY OF THEORETICAL MECHANICS: A BIRD'S-EYE VIEW 


MECHANICS 


Statics 


Dynamics 


Kinematics 


Kinetics 


In addition, we will be using the following, not so common, terms: 


Stereomechanics: mechanics of rigid bodies (and, accordingly, stereostatics, stereo¬ 
kinetics, etc.—mainly, after Maggi, late 1800s to early 1900s); 

Kinetostatics: study of internal and external reactions in rigid bodies in motion (after 
Heun, early 1900s). 


7. Finally, the problem of AM consists in the following: 

(a) . Formulation of the smallest, or minimal, number of equations of motion 
without (external and/or internal) constraint reactions; namely, the so-called kinetic 
equations; and also the ability to retrieve these reactions if needed; namely, the so- 
called kinetostatic equations. And then, 

(b) . The ability to solve these equations for the motion and unknown forces, 
respectively, either analytically (exactly or approximately) or numerically (computa¬ 
tionally or symbolically). This is aided by the possible existence of first integrals; for 
example, energy, momentum, and conservation/invariance principles; more on these 
in chapter 3. 


2 HISTORY OF THEORETICAL MECHANICS: A BIRD'S EYE VIEW 


For us believing physicists the distinction between past, present, 
and future is only an illusion, even if a stubborn one. 

(A. Einstein, Aphorisms ) 

The past is intelligible to us only in the light of the present; and 
we can fully understand the present only in the light of the past 
.... Past, present, and future are linked together in the endless 
chain of history. 

(E. H. Carr, What is History?, 1961) 

But it is from the Greeks, and not from any other ancient society, 
that we derive our interest in history and our belief that events in 
the past have relevance for the present. 

(M. Lefkowitz, 1996, p. 6) 

For in a real sense, history isn't the past—it's a posture in the 
present toward the future. 

(L. Weschler, American author/journalist, 1986) 

Rootless men and women take no more interest in the future 
than they take in the past. 

(C. Lasch, The Minimal Self, 1984) 

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INTRODUCTION 


The devaluation of history is a prerequisite for the free exercise of 
pure power. 

(J. Rifkin, Time Wars, 1988) 


The complete history of analytical mechanics, including 20th century contributions, 
has not been written yet—in English, anyway—and lack of space prevents us from 
doing so here. However, we hope that the following brief, selective, subjective, and 
unavoidably incomplete (but essentially correct and fair) summary, and references, 
will inspire others to pursue such a worthwhile and long overdue task more fully. 


Most Important Milestones in the Evolution of 

Theoretical Engineering Dynamics 

(from the Viewpoint of Analytical Mechanics) 


Unconstrained System Mechanics (Momentum mechanics of Newton-Euler) 


1638: 

Special particle motions (Galileo) 

1687: 

Physical foundations of mechanics [Newton: incomplete principles, 

no method (no equations of 
motion in his Principia)] 

1730s: 

Mechanics of a particle (Euler) 

1740s: 

Mechanics of a system of particles (Euler, late 1740s: Newtonian equa¬ 
tions of motion!) 

1750s: 

General principle of linear momentum (Euler); kinetics of rigid bodies 
(Euler) 

1770s: 

General principle of angular momentum (Euler); kinematics and geo¬ 
metry of rigid body motion (Euler) 

Constrained System Mechanics (Energetic mechanics of Lagrange) 

1743: 

Principle of d’Alembert (Jakob Bernoulli —> Jean Le Rond 
d'Alembert) 

1760: 

Principle of least action (Maupertuis —> Euler —> Lagrange) 

1764: 

Principle of Lagrange [Principle of virtual work (Johann Bernoulli, 
1717, published 1725) -f- Principle of d’Alembert] 

1780: 

Equations of Lagrange 

1788: 

Mechanique Analitique (1st ed.; note old spelling) 

1811: 

Special transitivity equations (Lagrangean derivation of rigid-body 
Eulerian equations) 

1811-1815: 

Mecanique Analytique (2nd ed.; 3rd ed.: 1853-1855; 4th ed.: 1888— 
1889; English translation, from 2nd ed.: 1997!) 

1829: 

Gauss’ Principle of least constraint (or least deviation, or extreme 
compulsion) 

1830s: 

Canonical formulation of mechanics (Hamilton) 

1840s: 

Transformation/integration theory of dynamics (Jacobi) 

1860s: 

Gyroscopic systems [Thomson (Lord Kelvin), Tait] 

1870s: 

Cyclic coordinates, steady motion, and its stability (Routh) 

1873: 

Earliest reactionless Lagrange-like equations for nonholonomic 
systems (Ferrers) 


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§2 HISTORY OF THEORETICAL MECHANICS: A BIRD'S-EYE VIEW 


1879: 


1870s-1910s: 


1903-1904: 

1910s 1930s: 


Gauss’ Principle for inequality constraints; Gibbs-Appell equations 
for unconstrained systems, but in general nonholonomic velocities 
(quasi velocities; Gibbs) 

Dynamics of nonholonomic systems , under linear (or Pfaffian) velocity 
constraints, possibly nonholonomic (Routh, Appell, Chaplygin, 
Voronets, Maggi, Heun, Hamel et al.—see below) 

Definitive and general study of nonholonomic systems (Pfaffian con¬ 
straints) in nonholonomic variables; Lagrange-Euler equations (Hamel) 

Dynamics of nonholonomic systems, under nonlinear velocity con¬ 
straints (Appell, Delassus, Chetaev, Johnsen, Hamel); Study of non¬ 
holonomic systems via general tensor calculus (Schouten, Synge, 
Vranceanu, Wundheiler, Horak, Vagner et al.) 


1970s-present: Applications of the above to multi-body dynamics, computational 
dynamics [(alphabetically): Bremer, Haug, Huston, MaifSer, 
Roberson and Schwertassek, Schiehlen, Wittenburg et al.]; Nonlinear 
dynamics (regular and stochastic/chaotic motion) 


Let us elaborate a little on the dynamics of nonholonomic systems. The original 
Lagrangean equations (1780) were limited to holonomically constrained systems. At 
that time, and for several decades afterwards, velocity constraints (holonomic or not) 
were only a theoretical possibility; though one that could be easily handled by the 
Lagrangean method (i.e., principles of Lagrange and of the relaxation of the con¬ 
straints (detailed in chap. 3)). But it was not until about a century later that such 
constraints were studied systematically. However, that necessitated a thorough re¬ 
examination of the entire edifice of Lagrangean mechanics: roughly between 1870 
and 1910, what may be accurately called the second golden age of analytical 
mechanics, a host of first-rate mathematicians (Ferrers, Lindelof, Hadamard, 
Appell , Volterra, Poincare, Klein, Jourdain, Stackel, Maurer), physicists (Gibbs, 
C. Neumann, Korteweg, Boltzmann), mechanicians (Routh, Maggi, Chaplygin, 
Voronets, Suslov, Heun , Hamel), and engineering scientists (Vierkandt, Beghin) 
developed the modern AM of constrained systems, including nonholonomic ones; 
and, also, the unified theory of differential variational principles of Lagrange, 
Jourdain, Gauss, Hertz et al. Up until then (ca. 1900), AM was used almost 
exclusively, by mathematicians and physicists, to study unconstrained systems: for 
example, celestial mechanics. The Promethean transition from heavens down to 
earth (i.e., constraints) was led by the great German mechanician Heun (1859 
1929), who can be fairly considered as the founder of modern engineering dynamics; 
and, also, by his more famous student Hamel (1877-1954), arguably the greatest 
mechaniker of the 20th century. For instance, to these two we owe the correct 
formulation and interpretation of the d’Alembert-Lagrange principle (i.e., LP), 
and its successful application (along with additional geometrical and kinematical 
concepts, already in embryonic or special forms in Lagrange’s works) to systems 
under general holonomic and/or linear velocity (or Pfaffian, possibly nonholonomic) 
constraints. Therein lie the roots of all correct treatments of the subject. [Heun also 
made important contributions to applied mathematics. For example, the well-known 
Runge-Kutta method in ordinary differential equations should be called method of 
Range—Kutta—Heun; see, for example, Renteln (1995).] 

Between the two world wars, on the basis of the so-accumulated powerful insights 
into the mathematical structure of LM (especially from the differential variational 
principles), its methods were extended to nonlinear nonholonomic constraints; first 

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INTRODUCTION 


by Appell (1911-1925) and his student Delassus (1910s) [also by Prange and Muller 
(1923)] and then by Chetaev (1920s), Johnsen (1936-1941), and Hamel (1938). 
During the post World War II era, the entire field was summarized by Hamel himself 
in his magnum opus Theoretische Mechanik (1949); and then elaborated upon by a 
new generation of Soviet mechanicians [(alphabetically) Dobronravov, Fradlin, 
Fufaev, Lobas, Lur’e, Novoselov, Neimark, Poliahov, Rumyantsev (or Rumi¬ 
antsev), et al.], whose efforts culminated in the unique and classic monograph 
Dynamics of Nonholonomic Systems by Neimark and Fufaev (1967, transl. 1972). 
Both of these works are most highly recommended to all serious dynamicists. 

[(a) On the history of the nonholonomic equations of motion, see also chapter 3, 
appendix I. (b) Nonlinear (possibly nonholonomic) constraints are an area that, 
probably, constitutes the last theoretical frontier of LM and is of potential interest 
to nonlinear control theory. Also, the differential variational principles have 
rendered important services in the numerical treatment of problems of multibody 
dynamics, and promise to do more in the future.] 


Guide Through the Literature on the History of 
Mechanics 

1. General (Mechanics and Physics): 

D’Abro (1939, 2nd ed.): Qualitative and quantitative tracing of the evolution of 
ideas from antiquity to modern quantum mechanics; excellent. Hoppe [1926(a), (b)]: 
Concise history of physics, with some quantitative detail; good place to start. Hund 
(1972): Panoramic, competent and compact history of physics, from antiquity to mod¬ 
ern quantum mechanics, cosmology, and so on; one of the best places to start. Simonyi 
(1990): Comprehensive and sufficiently quantitative history of physics from antiquity 
to modern; beautifully and richly illustrated; most highly recommended. 

2. Mechanics — General: 

Dugas (1955): Comprehensive and quantitative history of classical and modern 
mechanics, from a French physicist’s viewpoint; quite useful. Diihring (1887): 
Comprehensive treatment of the history of mechanics from antiquity to the middle 
of the 19th century; difficult to read due to its complete absence of figures and almost 
complete absence of mathematics; for specialists/scholars. Haas (1914): Detailed and 
pedagogical treatment of the principles of classical mechanics, from antiquity to the 
early 19th century; very warmly recommended, especially for undergraduates in 
science/engineering. Mach (1883-1933): Leisurely and mostly qualitative history of 
the principles of classical mechanics, from antiquity to the end of the 19th century; 
interesting and influential, but in some respects incomplete and misleading. Papastavridis 
[.Elementary Mechanics (under production)] and references cited there. Szabo (1979): 
Selective history of entire mechanics, with lots of beautiful photographs and diagrams; 
combines features of Mach, Dugas, and Truesdell. Tiolina (1979) and Vesselovskii (1974): 
General histories of mechanics, with detailed accounts of Russian contributions; very 
highly recommended for both their contents and references. 

3. Mechanics — Specialized: 

Cayley (1858, 1863): Excellent and authoritative summaries of theoretical devel¬ 
opments until the mid 19th century; by a very famous mathematician. Hankins 
(1970): Detailed account of the life and work of d’Alembert; highly recommended 
to mechanics historians/scholars. Hankins (1985): Physics during the 18th century 
(of enlightenment). Kochina (1985): Life and works of S. Kovalevskaya. Oravas and 

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§3 SUGGESTIONS TO THE READER 


McLean (1966): Detailed account of the development of energetic/variational prin¬ 
ciples, mainly of elastostatics. Polak (1959, 1960): Detailed and lively history of 
differential and integral variational principles of mechanics and classical/modern 
physics, from antiquity to the 20th century; most highly recommended. Stackel 
(1905): Excellent quantitative history of particle and rigid-body dynamics (elemen¬ 
tary to intermediate), from antiquity to the early 20th century; a must for mature 
dynamicists; complements Voss’s article. Truesdell (1968, 1984, 1987): Authoritative 
and lively detailing of the life and contributions of Euler; but invariably unfair/ 
misleading to Lagrange and to anything remotely connected to particles and physics; 
for mature mechanicians/physicists. Voss (1901): Detailed and quantitative history 
of the principles of theoretical mechanics; with extensive lists of original references, 
from antiquity to ca. 1900; very highly recommended to mechanics and physics 
specialists. Wheeler (1952): Life and works of J. W. Gibbs. Wintner (1941, pp. 
410-443): Notes and references on the history of analytical mechanics, with special 
emphasis on the mathematical aspects of celestial mechanics—the book, in general, 
is not recommended to anyone but specialists in theoretical astronomy. Ziegler 
(1985): Detailed and quantitative history of geometrical approach to rigid-body 
mechanics; primarily for kinematicians, not dynamicists. 

4. Histories of Mathematics: 

Bell (1937): Lively and enjoyable; concentrates on the lives and times of famous 
mathematicians. Bochner (1966): Informative, unconventional. Klein (1926(b), 1927): 
Detailed and authoritative. Kline (1972): Arguably, the best of its kind in English; 
encyclopedic, reliable, insightful, witty; a scholarly masterpiece. Kramer (1970): Like 
a more elementary version of Kline’s book; interesting account of the evolution of 
determinism in physics (pp. 204—245). Struik (1987): Compact, dependable; includes 
socioeconomic explanations of mathematical inventions. Also Dictionary of 
Scientific Biography (ed. Gillispie, 1970s). 


3 SUGGESTIONS TO THE READER 

A cumulative and alphabetical bibliography is located at the end of the book (see 
References and Suggested Reading, pp. 1323-1370). The following grouping of 
textbooks and treatises aims to better orient the reader relative to (some of) the 
best available international mechanics/dynamics literature, and thus obtain max¬ 
imum benefit from this work. References in bold, below, happen to be our per¬ 
sonal favorites, and have influenced us the most in the writing of this book. 

1. For Background (Elementary to Intermediate Level): 

Butenin et al. (1985), Coe (1938), Crandall et al. (1968), Easthope (1964), Fox (1967), 
Hamel (1912, 1st ed.; 1927), Loitsianskii and Lur’e (1983), Milne (1948), Nielsen 
(1935), Osgood (1937), Papastavridis (EM, in preparation), Parkus (1966), 
Rosenberg (1977), Smith (1982), Sommerfeld (1964), Spiegel (1967), Stackel 
(1905), Suslov (1946), Synge and Griffith (1959), Wells (1967). 

2. For Concurrent Reading (Intermediate to Advanced Level): 

Boltzmann (1902, 1904), Butenin (1971), Dobronravov (1970, 1976), Gantmacher 
(1970), Gray (1918), Greenwood (1977, 2000), Hamel (1949), Heil and Kitzka 
(1984), Heun (1906), Lamb (1943), Lanczos (1970), Lur’e (1968), MacMillan (1927, 
1936), Mei (1985, 1987), Neimark and Fufaev (1972), Nordheim (1927), Pars (1965), 

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INTRODUCTION 


Pasler (1968), Peres (1953), Poliahov et al. (1985), Prange (1935), Rose (1938), Synge 
(1960), Winkelman (1929, 1930). 

3. For Further Reading: 

Theoretical Physics, Nonlinear Dynamics, and so on: 

Arnold (1989), Arnold et al. (1988), Bakay and Stepanovskii (1981), Birkhoff 
(1927), Born (1927), Corben and Stehle (1960), Dittrich and Reuter (1994), 
Dobronravov (1976), Fues (1927), Hagihara (1970), Liehtenberg and Lieberman 
(1983/1992), McCauley (1997), Mittelstaedt (1970), Nordheim and Fues (1927), 
Pars (1965), Prange (1935), Santilli (1978, 1980), Straumann (1987), Synge 
(1960), Tabor (1989), van Vleck (1926), Vujanovic and Jones (1989), Whittaker 
(1937). 

Special Topics (Analytical): 

Altmann (1986), Arhangelskii (1977), Chertkov (1960), Korenev (1967, 1979), 
Koshlyakov (1985), Leimanis (1965), Lobas (1986), Lur’e (1968), Merkin (1974, 
1987), Neimark and Fufaev (1972), Novoselov (1969), Tinierding (1908). 

Applied (Multibody Dynamics/Computational/Numerical, etc.): 

Battin (1987), Bremer [1988(a)], Bremer and Pfeiffer (1992), Haug (1992), Hughes 
(1986), Huston (1990), Junkins and Turner (1986), Magnus (1971), McCarthy (1990), 
Roberson and Schwertassek (1988), Schiehlen (1986), Shabana (1989), Wittenburg 
(1977). 


4 ABBREVIATIONS, SYMBOLS, NOTATIONS, FORMULAE 

These are the customary meanings; but, of course, some, hopefully easily under¬ 
stood, exceptions are possible. The reader is urged always to keep common sense 
handy! 


Numbering of Equations, Examples, and Problems 

Chapters are divided into sections; for example, §3.4 means chapter 3, section 4. 
Equations are numbered consecutively within each section. For example, reference 
to eq. (3.4.2) means equation (2) of chapter 3, section 4. Related equations are 
indicated, further, by letters; for example, eq. (3.4.2a) follows eq. (3.4.2) and some¬ 
how complements or explains it. 

In chapters 2-8, examples and problems are placed anywhere within a section, 
and are numbered consecutively within it; for example, ex. 5.7.2 means the second 
example of chapter 5, section 7 and prob. 5.7.3 means the third problem of the same 
section. Within examples/problems, equations are numbered consecutively alpha¬ 
betically; for example, reference to (ex. 5.7.3: b) means equation (b) of the third 
example of chapter 5, section 7. Related equations in examples/problems are 
followed by numbers; for example, (ex. 5.7.2: k2) is related to or explains (ex. 
5.7.2: k). 


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§4 ABBREVIATIONS, SYMBOLS, NOTATIONS, FORMULAE 


Abbreviations 


AD Analytical dynamics 
AM Analytical mechanics 
CM Classical mechanics 
GP Gauss’ principle (§6.4, §6.6) 

H Holonomic (coordinate/constraint/ 
system) 

HP Hamilton’s principle (ch. 7) 

HZP Hertz’s principle (§6.7) 


JP Jourdain’s principle (§6.3) 

LP Lagrange’s principle (or D’Alembert’s 
principle in Lagrange’s form, §3.2) 
NP Nonholonomic (coordinate/ 
constraint/system) 

VD Virtual displacement (§2.5) 

PVW Principle of virtual work (§3.2) 


Chapter 1: Background 


Scalars in italics: for example, a, A, to, Q 
Vectors in boldface italics: for example, a. A, to, Q 

Tensors/Dyads in boldface, uppercase, italics; Matrices in boldface (always), uppercase 
(usually), roman (usually, but sometimes in italics, like tensors; should be clear from 
context, or clarified locally) 


General symbols 


N 

h 


f 

k, l, p, r, ... 
/, /', I", ... 
D, D', D", ... 

B 

E 


(...)' 





£/crs 


Number of particles of a system 

Number of holonomic constraints 
(H=l,...,h) 

Number of Lagrangean (or global) coordinates 
(= 3 N - h) 

Number of Pfaffian (holonomic and/or non- 
holonomic) constraints 

Number of (local or global) degrees of freedom 
(= n — m ) 

General (system) indices (= 1,...,«) 
Independent variable indices (= m + 1,..., n) 
Dependent variable indices (= 1,..., m < n) 

A implies, or leads to, B (A 44- B, for both 
“directions”) 

Discrete summation; usually, over a pair of 
indices (one for each such pair) 

Summation over all the material points (par¬ 
ticles) of a system, for a fixed time; a three- 
dimensional material Stieltjes integral, equiva¬ 
lent to Lagrange’s famous integration sign 
S... 

Total/inertial time derivative 

The (...) have been subjected to some kind of 

transformation 

Evaluated at some special value: for example, 
initial or equilibrium; or with some constraints 
enforced in it 

Expressed as function of the variables t , q , u> 
(quasi velocities) 

Transpose of matrix (...) 

Inverse of matrix (...) 

Kronecker delta 

Permutation symbol (—► tensor, in rectangular 
Cartesian coordinates) 


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INTRODUCTION 


r/v/a 

r A/B/ v A/Bl a A/B 

dm 

m P 

df 

dfe 






df i 





dF 





dR 

O - XYZ ( O - 

IJK) or 

Q-x k , 

( O 

- «k') 

♦ — xyz (♦ 

1 

*5 

0 

♦ 

:(♦ 

- «k) 

♦ - XYZ (♦ - 

IJK) or 

♦ - x k > 

(♦- 

- «*•') 




d(.. 

■)/dt 




d(. 

■ •)/dt 


da/dt = da/dt + wxal 
dT/dt = dT/dt + cjxT -Txto] 

A- = ( A k'k) 


A-' =A t 

R 

R(n, X ) 

R' =R-1 

X 

y = tan(x72 )n 
y = ( 71 , 2 , 3 ) = ( 7 x,y,z) 

<t>,8, *P 

to/Q = A t • (dA/dt) 
a = dco/dt/A. = dQ/dt 
o' IQ’ = (dA/dt) ■ A t 


(Instantaneous, and usually inertial) position/ 
velocity I acceleration of a particle P 
Position/velocity/acceleration vector of 
particle A relative to particle B 
Mass of a particle P (Continuum approach) 
Mass of a particle P (Discrete approach) 

Total force on particle P 
(= df e + dfp Newton-Euler) 

(= dF + dR: D’Alembert-Lagrange) 

Total external force on particle P 
Total internal or mutual, force on particle P 
Total impressed, or physical, force on 
particle P 

Total constraint reaction, or geometrical/ 
kinematical force on particle P 
Space-fixed; namely, inertial, axes (basis) at O 
Body-fixed; namely, moving, axes (basis) at 
body point ♦ 

Comoving, translating but nonrotating (or 
intermediate, or accessory) axes at body 
point ♦ 

Rate of change of vector or tensor (...) relative 
to fixed (inertial) axes 

Rate of change of vector or tensor (...) relative 
to moving (noninertial) axes 
a arbitrary vector, T arbitrary (second-order) 
tensor, o angular velocity vector of moving 
axes relative to fixed ones 
Matrix of direction cosines between moving 
(e.g., body-fixed) axes and space-fixed ones; a 
(proper) orthogonal matrix ( passive interpreta¬ 
tion): 

A k'k = A kk' = cosine [(fixed)*,, (moving)*] 

= cosine [x*,, x k ] 

= cosine [x*,x*,j 
Determinant of A = Det A = +1 
Rotation tensor ( active interpretation of A) 
Rotation tensor about a point O and axis 
through it specified by the unit vector n, by an 
angle x 

Rotator tensor 

Angle of finite rotation about a point O and 
axis specified by the unit vector n 
Gibbs vector of finite rotation 
Rodrigues parameters relative to O — XYZ 
Eulerian angles (sequence 3 —> 1 —> 3): 
precession (</>) —> nutation (0) —> proper 
(or eigen-) spin (ip) 

Angular velocity vector/tensor (moving axes 
components) 

Angular acceleration vector/tensor (moving 
axes components) 

Angular velocity vector/tensor (space-fixed 
axes components) 


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§4 ABBREVIATIONS, SYMBOLS, NOTATIONS, FORMULAE 


a ' / A! = di2'/dt 

E = A + ti ii 

G 

C 

♦ 

• 

H ..absolute = H . = S( r ~ r ■) X dm v 
H -.relative = >> . = S ^ ~ '' ■ ) X dm (v - V . ) 

= $ r /. x dm v/, 
M... 
I... 

T = (1 /2) Q v ■ v dm 

Chapter 2: Kinematics 

q = (q t ,...,q„) 
r = r(t,q) 

r(t, q + Sq)- r(t, q) & Sr = E ( dr/dq k )8q k 
e k = dr/dq k , e 0 = e„ +1 = dr/dt 

v = {dqi/dt = q\ = v u ... ,dq„/dt = = v„) 

•’ = E e k q k + e 0 

a = Y1 e kdk + No other ^ terms 
= dv/dq k = <9a/d<7/ c = • ■ ■ = e k 

ud = E a DlAk + a d = 0 

d6o = E a Dkdq k + = 0 

= E = 0 


Angular acceleration vector/tensor (space- 
fixed axes components) 

Sometimes referred to as tensor of angular 
acceleration 

Center of mass of a rigid body 
Contact point between two bodies 
Generic/arbitrary body point 
Generic/arbitrary space point 
Absolute angular momentum about • 

Relative angular momentum about • 

Moment of a force (or couple) about ... 
Moment of inertia tensor about ... 

(Usually inertial) kinetic energy of a system 


Holonomic, or global, or Lagrangean, or 
system, coordinates; otherwise known as 
generalized coordinates 
Fundamental Lagrangean representation of 
position of typical system particle P 
(First-order) virtual displacement of P 
Fundamental holonomic particle and system 
vectors (Heun’s begleitvektoren) 

Holonomic, or global, or Lagrangean, or 

system, velocities; otherwise known as 

generalized velocities 

Particle velocity expressed in holonomic 

variables 

Particle acceleration expressed in holonomic 
variables 

Basic kinematical identity (holonomic 
variables) 

Pfaffian constraints in velocity form ( a Dk , a D : 
constraint coefficients, functions of t and q; 
ur. quasi velocities) 

Pfaffian constraints in kinematically admissible, 
or possible, form (6\ quasi coordinates) 
Pfaffian constraints in virtual form 


General, kinematically admissible, variations of (...): 

d{...) = ^2{d... / dq k ) dq k + (d... / dt) dt = ^2(d... / dO k ) dO k + {d... /dd„ +l ) dt 


Virtual variation of (...): 

&{■ ■ ■) = ... /dq k )8q k = JJ(d... /d9 k )69 k 

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INTRODUCTION 


Quasi chain rules 

d.../dd k = ^2(d... /dqf) (dq,/du k ) = ^ A, k (d... / dq,) 
d... /dq, = ^2(d... /de k ){doj k /dqi) = ^ a k ,(d... /d6 k ) 
d... /dd n+l =^2(d... /dq,)(dq,/duj n+l ) + d.../dt 

= M d ■ ■ ■ l dc li ) + d.../dt = d... /d(t) + d.../dt 
d.../dt = E a k {d. . . /dQ k ) + d... /dO n+l = - ^ A k (d... /dq k ) + d... /dO n+l 

GENERAL (LOCAL) QUASI-VELOCITY 

TRANSFORMATIONS 

Velocity form 

U> D = ^2, a DkQk + <*D — Oi UJi = ^2 a Ik4k + a I ^ 0, W„ +1 = q u+ \ = t = 1 

Kinematically admissible (or possible) form 

dd D = ^2 a Dk d^k + a D dt = 0 , d6, = ^2 a ik dqk + «/ dt ^ 0 , 

dd n+i = dq n+ 1 = dt 0 

Virtual form 

S6 d = E a Dk 6 dk = 0 , Mi = E a ' k6qk ^ °’ 60 n +' ~ Sq “+ l = 6t = 0 


HOLONOMIC VELOCITIES EXPRESSED IN TERMS OF 
QUASI VELOCITIES [(«*/) and (A k ,) are inverse matrices] 
Velocity form 


<lk = v k — E Aki^i + A k — ^2 Aki^i + A k v/l 0 
Kinematically admissible (or possible) form 

dq k = E A kl d0, + A k dt = ^2 A k i dd, + A k dt ^ 0 (under d0 D = 0) 


Virtual form 


6q k = J2 Ak i 60 i = E AkI 601 ^ 0 (under 66 D = 0) 


PARTICLE KINEMATICS IN TERMS OF QUASI VARIABLES 
(9,u>, etc.) 

Virtual displacement 

Sr = E ek Sqk = E Ek 60k = E £ > 601 

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§4 ABBREVIATIONS, SYMBOLS, NOTATIONS, FORMULAE 


Velocity 


Acceleration 


v = 


a = 


W/8y + £„+l = 22 W/£/ + £o 

w^£ A - + No other w terms 
yy th/£/ + No other u> terms 


Basic kinematical identity [where / = f(t,q,q) = f*(t,q,u) = /*] 
dr/dQ k = dv*/duj k = da*/du k = ■ ■ ■ = s k 


Transformation relations between the holonomic and nonholonomic bases e , £ 

f-k = 22 (dch/duj^e, = 22 A Ik e, 

£ »+! — y^ £ ° = yz ^ /g/ e ' i+x = — yz a/ ' £,t e,i+i 

e i = 22 (duk/dq^Sk = 22 a k ,s k 

e n +1 = e 0 = a k E k + e H+l = — 22 ^l e l + £ «+l 


FROM PARTICLE TO SYSTEM VECTORS 

(i.e., vectors characterizing, or expressing, system variables) 

(particle vector) • e k = (system vector) /f (holonomic components) 

^ (particle vector) • s k = (system vector) /f (nonholonomic components) 


SPECIAL FORMS OF PFAFFIAN CONSTRAINTS 
Chaplygin 

^d = <1d- 22 b Dl Ch = 0; b D[ : functions of q I = (q m+1 q n ) 
ui = qi + 0 


Voronets 

uj d = q D — 22 b DI qi — b D = 0; b D[l b D \ functions of t and all qs 
^i = qi¥= 0 

Qd = 22 boi^r + b D , qi = w/ 

Corresponding particle virtual displacement 

Sr=22 e k Sc ik = 22 i { ' s<ir 


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20 


INTRODUCTION 


Corresponding particle velocity 


v = 


E <2lPl + Pi 


[ «+l = V o 


HAMEL COEFFICIENTS 

7*, = -7*, = EE (da kh /dq c - da kc /dq b )A hr A cs 

= EE ®kb [-Acr(dAb s /dcjc) A cs (dAbr/dq c )\ 

= EE (A br A cs A cr Ab s ) (dcikb/dq c ) 

7 r,rt+1 = ~7 «+l,r = 7 ^ EE (■ da kb /dq c - da kc /dq h )A hr A c 
+ E ( da kb/ dt ~ da k /dq h )A hr 



TRANSITIVITY EQUATIONS 


(69 k ) - 8ui k — E a kl [(6qi) - <5(<//)] + E E 7 + E 7**^4 

(%)' - = E - &>/] - EE 'y^^b - E 7*^*} 

(^«+t)' - <Wn = (^»+t)‘ - <5(?«+i) = (to)' - 6(t) = 0 


or, equivalently. 


d{80 k ) — 8(d9 k ) — E a w [<5?(^/) - <5(J<7/)] + EE 'y k bs d0 s 60 h + Y / 1 k b dt60 b 
d(8q k ) - $(<%) = E {[<w - W] ^EE 7fc dB, 89 h - £ 7 i dt 89 h } 

</($0 n+1 ) - <$(</6>„ +1 ) = </(«*) - <5(J?) = d( 0) - fi(*) = 0-0 = 0; 

or, assuming (Hamel viewpoint) 

(■ 8qk)' = 8(q k ) or d(8q k ) = 8(dq k ) 

(89 k y -&;* = £ E 7 ^ + E 7 * ^ = E hk b S9 b 

d(89 k ) - 8(dd k ) = EE 1 k hs d6 s 89 h + Y, r l k b d t 80 h 


EE' ^ bs {d9 s 89 h - 89 s d0 h ) + J2 7 \ dt 89 h 


(where Y2 ' means that the summation extends over b and s only once; say, s <b) 
Generally [with o, • = 1,... ,n; 89 n+l = 8t = 0] 


d(60.) - 8(dB,) = EE 7 \ g d9 0 89' +Y^ 7*. dt 89. 


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§4 ABBREVIATIONS, SYMBOLS, NOTATIONS, FORMULAE 


FROBENIUS’ THEOREM 

(Necessary and sufficient conditions for holonomicity = complete integrability of a 
system of m Pfaffian constraints in the n + 1 variables q { ,..., </„; q n+ \) 

l D ii'= 0, l D i,n+\ = = 0 {D= l' = m+ 


Chapter 3: Kinetics 

BASIC QUANTITIES 

r = (1/2)5 v r dm (Usually inertial) Kinetic Energy of system 

S=(l/2)5«. a dm (Usually inertial) Gibbs—Appell function of system, 
or simply Appellian 


NOTATION 


= f[t,q,q D (t,q,u)] = f*(t,q,w) = f* [arbitrary function]; 


for example, 

T(t,q,q) = T[t,q,q D (t,q,w)] = T*(t,q,w ) = T* 

=> T*{t, q, u> D = 0,w/) = T* 0 (t,q } uj) = T* 0 
T{t, q, q) = T[t, q, q D (t, q, q,),q,} = T„(t, q, q,) = T 0 (and similarly for S) 


LAGRANGE’S PRINCIPLE 


S'W R > 0 =>• 6I>6'W 


(for unilateral constraints; for bilateral constraints, > is replaced by =) 
Particle (or raw) forms 


S'W R = gdR-6r, S'W= gdF-Sr, 61= gdma-Sr 


Holonomic variable forms 


S'W R = J2 R k6q k , 
6'W=Y,QkSq k , 


R k = 5 dR • e k , 
Qk = S dF ' e *’ 


61 = J2 [(dT/dq k y - dT/dq k \ 6q k = £ (dS/dq k ) 6q k = £ E k 6q k 


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INTRODUCTION 


Nonholonomic variable forms 

6'W R = Y A- M k = Y A i 66,, A k = $ dR ■ e k , 

6'W = Y J ®kMk, 0 k =SdF-s k , 

v = Y m*/du k y - dT*/ae k - r k ] 66 k = Y (ds*/du k ) 6o k = Y 4 w k 

INERTIAL “FORCES" IN HOLONOMIC VARIABLES 
E k = S dma-e k 

= ( dT/dq k )‘ — dT/dq k (Lagrangean form) 

= dS/dq k (Appellian form) 

= dT/dq k — 2(dT/dq k ) = N k (T) = N k (Nielsen form; see chap. 5) 

INERTIAL "FORCES" IN NONHOLONOMIC VARIABLES 

4 = S dm “ ' Ek 

= (dT*/du) k )' — dT*/dd k — T k = E*(T*) — T k (Volterra—Hamel form) 

= dS*/du k (Gibbs—Appell form) 

= Y ( dq,/duj k )E, = Y A, k E, (Maggi form) 

Nonholonomic deviation 

r k = ^ dm v* • [(dv*/dui k y - dv*/dO k ] 

= £ dm v* • E k *(v*) = £ dm v* • y k (particle/raw form) 

= - E E l l ks{dT*/&j,)u s - Y 7 l k{dT*/^,) 

= - Y h'kidT*/^!) [ h 'k = Y 7 ‘ ks Us + 7 * 

TRANSFORMATION EQUATIONS 

El = Y, a k<d-k ^ A k = Yj A, k Ri 
Ql = Yj a k'®k ®k = Yj A 'kQl 
Ei = Y! ak]dk ^ 4 = Y^ A ikE, 


THE CENTRAL EQUATION 
(Lagrange-Heun-Hamel Zentralgleichung) 

First Form 


6T + 6'W + 6D = ( 6Py 


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§4 ABBREVIATIONS, SYMBOLS, NOTATIONS, FORMULAE 


Second Form 

Pk 69k +55 Pk[(6e k y - Suj k ] - 55 (dT*/de k ) se k = 55 & k se k 

where 

8T = Q dm v • hv = E [(dT/dq k ) 6q k + (dT/dq k ) 8{q k )\ 

= E K dT */ dq k ) % + ( dT*/duj k ) &*;*] 

= E K dT */dOk) 89 k + (dT*/8u k ) 6co k ] = <5 7"* 

6'W= S d F‘6r = Y,Qk % = E & k 66 k 

8D = ^ dm v • [(<5r)' — £v] 

= 55 (dT*/aj k )[(se k y - &*] - 55 ri k (dT*/^ t ) se k 

6 P= S dm v ■ 6 r = E Pk Sqk = E Pk S9k 

Pk = S dm v • e k = 97" /dv k (holonomic momentum) 

7-*/^ = ^ dm v* • z k = dT*/dui k (nonholonomic momentum) 

Pi = E ^ ^ = E y 4/A-7’/ (transformation formulae) 


EQUATIONS OF MOTION 
COUPLED 

Routh-Voss (adjoining of constraints via multipliers) 

E k = Q k + R k (multipliers; holonomic variables) 


UNCOUPLED 
Maggi (projections) 

Kinetostatic: ^Dk^D = ^di<Qd + A d (multipliers; holonomic variables) 

Kinetic: ^ik^i = J2 9 ik Qi (no multipliers; holonomic variables) 

Hamel (embedding of constraints via quasi variables) 

Kinetostatic: E D *(T*) — r D = & D + A D (multipliers; nonholonomic variables) 
Kinetic: Ef*(T*) — 7~7 = 0j (no multipliers; nonholonomic variables) 

SPECIAL FORMS (constraints of form q D = Yh boiqi + b d: boi, bo functions of t, q ) 

Maggi —> Hadamard 

Ed = Qd + An (kinetostatic) E, = Q, - E b DI X D 

=> 7t/ + 55 6 djEd = Qi + 55 ^ diQd = Qt,o = Qio (kinetic) 

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INTRODUCTION 


Hamel —> Voronets 


To = T„{t , q, q 7 ), q D = E b DI q, + b D 

(■ dT 0 /dq,)' - dT 0 /dq, - E b DI {dT 0 /dq D ) 

-EE ^ '{dT/dq D ) 0 q r - E w D fdT/dq D ) 0 = Q, + E 


A' 


db Dr /dq r + E V/(36n/'/dto')] ~ [ 9 &D// 9 ?/' + E h D'i'(db DI / dq D ') 


w°i — w D In+x = 


db D /dq! + E b D 'i{db D /dq D <) 


db DI /dt+^2 bry{db DI /dq r y) 


Voronets —> Chaplygin 

T 0 =T 0 (q I ,q I ), q D = E b D ,(q m+ y.. .,q n )q r : i.e., b D = 0 

{dTg/dcfo)' — dTg/dqj 

-EE^ '(dT/dq n ) o q r = Q, + E h DiQ D 
t D ii' = db DI /dq r - db nv jdq, 


POWER (OR ENERGY RATE) THEOREMS 
Holonomic variables 

dh/dt dL/dt + ^ ^ C/c,nonpotential Qk ^ ^ ^D^Di 

h = E (aC/5^) q k — L = T 2 + ( V 0 — T 0 ), L = T — V; 

dE/dt = —dL/dt + d(T x + 2T 0 )/dt + Ea , nonpotential Qk ^ ^ - 

E=T+v 0 , L=r-F=r-(C 0 + C 1 ), A = C-(r 1 + 27’o) 

Nonholonomic variables 

dh*/dt = -dL*/de n+x + E ®I. nonpotential ^1 ~ R : 

h* = E (dL*/du) I )ui I - L* = T* 2 + (V 0 - T* 0 ) 
dL*/dd n+l = 3L*/9t + E A k (dL*/dq k ) 

R = E E 7 r i{dL*/dui r )u> I (Rheonomic nonholonomic power) 

EXPLICIT FORMS OF THE EQUATIONS OF MOTION 
Lagrangean equations: with 

T =T 2 +T x + T ( y. 2T 2 = E E M kr q r q k , T i = E ^rQn ~Tt = M 0 , 

M k i = M /k , M kn+X = M n+lk = M k o = M ok = M k , 

M, ;+1 „ +1 = Mqq = M 0 : Inertia coefficients , 


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§4 ABBREVIATIONS, SYMBOLS, NOTATIONS, FORMULAE 


2 rk.rs = 2 r ksr = dM kr /dq s + dM ks /dq r — dM rs /dq k : 1st kind Christoffels. 
G k = Y Skr k, = Y ( 9M r /dq k ~ dM k /dq r )q r : Gyroscopic “ force ”, 

Qk = Qk, nonpotential + (dV/dq k )' - dV/dq k , 

V = Y^ E k (t,q)q k + E 0 (t,q) = V\{t,q,q) + Vo(t,q): Generalized potential, 

the Lagrangean-type equations, say E k = Q k , assume the form 
E k (T 2 ) + E k (Ti) + E k (T 0 ) = Q k , 

E k (T 2 ) = Y M krkr + E E r k,rsQrks + Y ( dM kr/9t)kn 

E k {T\) = dM k /dt — G k , 

E k {T,) = -(1/2 ){dMjdq k ). 

Hamel equations (stationary case, no constraints), with 

2 T* = 2T*2 = EE M kr ^r ^ki 

2r\ rs = 2r\ sr = dM* kr /dO s + dM\ s /do r - dM*Jde k , 

Akjp — r kjp +E 7 r k iM*. p (“nonholonomic Christoffels”) 

Hamel-type equations, say I k = 0 k , assume the form 

E M *-/^+E E Ak,i P w / u p — 0 k . 


APPELLIAN FUNCTION 
Holonomic variables 

25 = E M k,Mk + 2 E E E r k.ipkk ki k P 

+ 4 EE Ek,l,n+\kk k, + 2Y E k ,n+\,n+\kk 

Nonholonomic variables (stationary case) 

25* = E E M * kr Wfc + 2 E E E A k‘p u p 


LAGRANGEAN TREATMENT OF THE RIGID BODY 
Kinetic energy 


T — ^translation + T, 
2 ^translation = V+ 


rotation 
2 


' coupling 


(♦: arbitrary body point; nr. mass of body) 


2 ^"rotation = m ' S dm ( r /+ X v /*) 

= a> ■ ^ dm [»■/*. x («i x i^,)] = co ■ h* = co ■ • co 

^coupling = CO ■ S dm (iy* X v*) = m v«. • (<u x r G/4 ) = m r* • v G/ * 


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INTRODUCTION 


Momentum vectors 

6P = ^ dmv • 5r = p • 5r. + H * • 56 (66: virtual rotation vector) 

p = ^ dm v = m v G \ linear momentum of body 

r/+ x (dmv) = h+ + r G /+ x (m v 4 ): absolute angular momentum of body 

about ♦ 

H 0 = ^ r x (dm v) = //* + r+/ 0 x p (O: fixed point) 

Kinetic energy in terms of the momentum vectors 

2T = p-v.+H.-(o, p = dT/dv+, H+=dT/d(o 

Kinematico-inertial (KI) acceleration vectors 

51 = ^ dm a - 5r = I • 6r. + A. • 56 

I = ^ dm a = ma G : linear KI acceleration of body 

A. = ^r/+ x (dma): angular KI acceleration of body about ♦ 

Eulerian principles in Lagrangean form 

Linear momentum (£2: vector of angular velocity of moving axes) 

I = dp/dt = dp/dt + Qxp - d/dt(dT/dv .) + Q x ( dT/dv .) 

Angular momentum 

A. = dH./dt + v, xp 

= (dH./dt + Q x H.) + v, xp 
= d/dt(dT/dco) + Qx (dT/dco) + x (dT/dv.) 

(also A. = dH. /dt + v. xp\ •: any point) 

APPELLIAN FUNCTION (to within acceleration-proportional terms) 

2S = m a. 2 + 2mr G j. • (a. x a) + 2m (to x v G /+) • (a* x to) 

+ a • I. • a + 2(a x to) • /* • to 

= m a G 2 + a • I G ■ at + 2(a x a>) • I G - co 

(Appellian counterpart of Konig’s theorem) 


RELATIVE MOTION (/: inertial origin; O: moving origin) 

Positions 

i"j = r 0 (t ) + r(< 7 i,..., q„) (motion of O known, q\ noninertial coordinates) 


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§4 ABBREVIATIONS, SYMBOLS, NOTATIONS, FORMULAE 


Velocities 

V = V 0 + V relative + S2 x r, v r eiative = dr/dt = ^ (dr / dq k ) q k 

Virtual displacements 

Srj = Srg + Sr = Srg + S TC ir + 60 x r (Cl = d&/dt: frame angular velocity) 


Kinetic energy 


transport = rn v Q + 2m v Q • (Q X r G ) + Q • I 0 • Q = 2 T 0 


T — Ttransnnrt + Trplativp + T c 

27) 

2 Relative = dm V re p ve • V re p ve 2 / s 
^"coupling 7*rel’ve ' V O.retvc * H — T\ 



Prel’ve = ^ V rel’ve = m(dr G /dt) = ( tS dm ( dr / d dk)) 4 

(,noninertial linear momentum) 

#0,reEve = S X ( dr I dt ) = ^2(S ^" r X ( dr / d dkfj 4 


(,noninertial absolute angular momentum) 


LAGRANGEAN TREATMENT OF RELATIVE MOTION 
[equations of carried body; say, E k {T) — Q k ] 

Ek (-^2) “1“ Q k, transporttransl’n 

“ 1 “ transport rotat'n “ 1 “ transport rotat’n centrifugal + C4 , Coriolis 7 

transport transl’n = ^ ^translation /^Qki ^translation = W (Iq ' 

Qk ,transport rotat’n = -{da/dt) • (dH 0 , eVve /dq k ) 

= — ( dQ/dt ) • dm v x ( dr/dq k )j ; 

transport rotat’n centrifugal 

^^centrifugal = ~S (12 X f) — f2* /$ • 12, 

Qk ,Coriolis = -2$ n x (dm v reFve ) • (dr/dq k ) = ^ 
gkl = gki • *2, Hk! = dm ^ dr / d( lk) X (df/dqi)\ 


Chapter 4: Impulsive Motion 

Fundamental impulsive variational equation (impulsive principle of Lagrange — LIP) : 

6i = snv, 

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28 



INTRODUCTION 

where 


SI = ^ dm a - Sr = ^ A (dm v) • Sr: 
(first-order) virtual work of impulsive momenta, and 

tfw = $ dF-Sr = gdF -Sr: 
(first-order) virtual work of impulsive impressed “forces 

vw R = S dR ' br = Y.(S dR - e k) % = E R > 

SfV= SdF-6r=J^(SdF- e k) % = E & 6 
SI = A(dmv) • Sr = dm Av • e k j Sq k 

= E A ( S dm v ■ e,i ) 5qk - E Apk Sq ^ 

and 

Pk = S ( dm V • e k ) = dT / dq k 

^ Ap k = a(^£ dm v-ekj = g [A (dm v)- e k \: 
[holonomic (A)th component] impulsive system momentum change, 


Qk = S dF ' L ’ k = S dF ‘ ek ' 

[holonomic (A)th component] impulsive system impressed force; or, simply, 
impressed system impulse. 


R k =E dR ■ e k = gdR- e k : 

[holonomic (A)th component] impulsive system constraint reaction force, we finally 
obtain LIP in holonomic system variables: 

E Rk Sqk = °> E j ( dT l dq k) Sqk = E & k 


and similarly in quasi variables. 

Energetic theorem 

AT=T + ^T-= W_ /+ , 


2T + = Q dm r + • v + , 2T = ^ dmv • v , 

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where 


§4 ABBREVIATIONS, SYMBOLS, NOTATIONS, FORMULAE 


and 


W_ /+ = SdF-{^ + + m 2 

In words: The sudden change of the kinetic energy of a moving system, due to 
arbitrary impressed impulses, equals the sum of the dot products of these impulses 
with the mean (average) velocities of their material points of application, immedi¬ 
ately before and after their action. 

APPELLIAN CLASSIFICATION OF IMPULSIVE 
CONSTRAINTS, AND CORRESPONDING EQUATIONS OF 
IMPULSIVE MOTION 

At a given initial instant t', new constraints are suddenly introduced into the system 
and/or some old constraints are removed, or suppressed. As a result, mutual percus¬ 
sions are generated, which, in the very short time interval r = t" — t' over which they 
are supposed to act and during which the shock lasts, produce finite velocity changes, 
but, according to our “first” approximation, produce negligible position changes; 
that is, for r —> 0: Aq = 0, A(dq/dt)f0. The constraints existing at the shock 
moment are either persistent or nonpersistent. By persistent we mean constraints 
that, existing at the shock “moment,” exist also after it, so that the actual postimpact 
displacements are compatible with them; whereas by nonpersistent we mean con¬ 
straints that, existing at the shock moment, do not exist after it, so that the actual 
postimpact displacements are incompatible with them. 

The constraints that exist at the shock instant can be classified into the following 
four distinct kinds or types: 


1. Constraints that exist before , during , and after the shock ; that is, the latter neither 
introduces new constraints, nor does it change the old ones; the system, however, is 
acted on by impulsive forces. An example of such a constraint is the striking of a 
physical pendulum with a nonsticking (or nonplastic) hammer at one of its points, 
and the resulting communication to it of a specified impressed impulsive force. 

2. Constraints that exist during and after the shock, but not before it; that is, the latter 
introduces suddenly new constraints on the system. Examples: (a) A rigid bar that 
falls freely, until the two inextensible slack strings that connect its endpoints to a 
fixed ceiling become taut (during) and do not break (after), (b) The inelastic central 
collision of two solid spheres (“coefficient of restitution” = e — 0—see below), (c) In 
a ballistic pendulum, the pendulum is constrained to rotate about a fixed axis, which 
is a constraint that exists before, during, and after the percussion of the pendulum 
with a projectile (i.e., first-type constraint). The projectile, however, originally inde¬ 
pendent of the pendulum, strikes it and becomes embedded into it, which is a case of 
a new constraint whose sudden realization produces the shock, and which exists 
during and after the shock but not before it (i.e., second-type constraint). 

3. Constraints that exist before and during the shock , but not after it. For example, let us 
imagine a system that consists of two particles connected by a light and inextensible 
bar, or thread, thrown up into the air. Then, let us assume that one of these particles 
is suddenly seized (persistent constraint introduced abruptly; i.e., second type), and, 
at the same time, the bar breaks (constraint that exists before the shock but does not 
exist after it; i.e., third type). 

4. Constraints that exist only during the shock , but neither before nor after it. For 
example, when two solids collide, since their bounding surfaces come into contact, 
a constraint is abruptly introduced into this two-body system. If these bodies are 

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INTRODUCTION 


elastic (e = 1—see coefficient of restitution, below), they separate after the collision, 
which is a case of a constraint that exists during the percussion but neither before nor 
after it (i.e., fourth type); while if they are plastic (e = 0), they do not separate 
(projectile and pendulum, above; i.e., second type). If 0 < e < 1, the bodies separate; 
that is, we have a fourth kind constraint. 

Clearly, the first two types contain the persistent constraints, while the last two 
contain the nonpersistent ones. Schematically, we have the classification shown in 
table 1. 


Table 1 Appellian Classification of Impulsive Constraints 



Preshock 

(before) 

Shock 

(during) 

Postshock 

(after) 

1 (persistent) 




2 (persistent) 




3 (nonpersistent) 




4 (nonpersistent) 





In impulsive problems: the excess of the number of unknowns (postimpact velo¬ 
cities and constraint reactions) over that of the available equations [those obtained 
front Lagrange’s impulsive principle; plus preimpact velocities, impressed impulsive 
forces, constraints, and, sometimes, knowledge of the postimpact state (second type; 
e.g., e = 0)]—namely, the degree of its indeterminancy—equals the number of its 
constraints, which, having existed before or during the shock, cease to do so at the 
end of it; that is: 

Degree of indeterminacy = Number of nonpersistent constraints ; 

that is, the persistent types 1 and 2 are determinate, while the nonpersistent ones 3 
and 4 are indeterminate. 

COEFFICIENT OF RESTITUTION (e) 

(v 2 /t •n) + _ v 2/1 ,„ + Relative velocity of separation 

(r 2 /i • n )~ v 2 /i,n - Relative velocity of approach 

where 1 and 2 are the two points of bodies A and B that come into contact during the 
collision, and n is the unit vector along the common normal to their bounding 
surfaces there, say from A to B. This coefficient ranges from 0 {plastic impact, no 
separation) to 1 {elastic impact, no energy loss); that is, 0 < e < 1. 

ANALYTICAL EXPRESSION OF THE APPELLIAN 
CLASSIFICATION; PERSISTENCY VERSUS DETERMINACY 

1. In terms of elementary dynamics: Consider a system that consists of N solids, in 
contact with each other at K points, out of which C are of the nonpersistent type, 
and/or with a number of foreign solid obstacles that are either fixed or have known 
motions. Assuming frictionless collisions, we shall have a total of 6TV + K unknowns 
(6 N postshock velocities, plus K percussions at the smooth contacts, along the 
common normals), and 6N + K C equations (6 N impulsive momentum equations, 

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§4 ABBREVIATIONS, SYMBOLS, NOTATIONS, FORMULAE 

plus K — C persistent-type constraints); and therefore the degree of indeterminacy 
equals the number of nonpersistent contacts C (i.e., the kind that disappear after the 
shock). 

Hence: (a) a free (i.e., unconstrained) solid subjected to given percussions or (b) a 
system subjected only to persistent constraints are impulsively determinate. 

2. From the Lagrangean viewpoint: (a) A number of constraints, imposed on a 
system originally defined by n Lagrangean coordinates, can always be put in the 
equilibrium form: 

q x = 0 , <72 = 0 ,..., q m — 0 ( nr. number of such constraints < n). 

(b) Within our impulsive approximations, even Pfafhan constraints (including non- 
holonomic ones) can be brought to the holonomic form; that is, in impulsive motion, 
all constraints behave as holonomic; and to solve them, either we use impulsive 
multipliers, or we avoid them by choosing the above equilibrium coordinates; or 
we use quasi variables. 

Assuming, henceforth, such a choice of Lagrangean coordinates for all our impul¬ 
sive constraints (and, for convenience, re-denoting these new equilibrium coordi¬ 
nates by </i,..., < 7 ,„;..., q„), we can quantify the four Appellian types of impulsive 
constraints as follows: 

• First-type constraints (existing before , during , and after the shock). As a result of these 
constraints, let the system configurations depend on n, hitherto independent, 
Lagrangean parameters: q = {q x ,..., q„). During the shock interval ( t', t"), the cor¬ 
responding velocities q = (q x ,... ,q„) pass suddenly from the known values (q)~, at 
t ', to other values {q ) + , while the q s remain practically unchanged; that is, here we 
have 

{dk) before (gk) during 0 ) (<7fc)after 

Mk = (dkf - {dkY f 0 [fc) +: unknown, (q k )~: known], 

• Second-type constraints (additional constraints existing during and after the shock, but 
not before it). Here, with q D " = {q x ,... ,q m »), where m" < n, we have 

(to")before 7^ 0 , (to")during = 0 , (to")after = ^ 

{qD") f o, (qD") + = o => A{q D ") = -(to") f®- 

• Third-type constraints (additional constraints existing before and during the shock, but 
not after it). Here, with q D '« = {q m - ( ] ,..., q m '"), where m'" < n, we have 

(to'")before = (to")during = (to'")after f 

(<to'") - = 0 . (to"') + f~ h ~~r A{q D ") = (to'") + f °- 

• Fourth-type constraints (additional constraints existing only during the shock, but 
neither before nor after it). Here, with q D »» = {q m - + \,..., q,„""), where m"" < n, we 
have 


(to"") before ^ 

(to"") _ f 0: 


(to"")during ~ 0, (to"")after / ^ 

(to"") + f 0 => ^(to»") = (to"") + - (to'"') - f 0. 


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INTRODUCTION 


Hence, if no fourth-type constraints exist, m'" = m""; and if no third-type con¬ 
straints exist, m" = in'"; etc. 

Next, arguing as in the case of continuous motion (chap. 3), during the shock 
interval, we may view the constraints of the second, third, and fourth types as absent, 
provided that, in the spirit of the impulsive principle of relaxation (LIP), we add to the 
system the corresponding constraint reactions. All relevant equations of motion are 
contained in the LIP: 

^2 A (dT/dq k ) Sq k = ^ g* % ( k = 1,...,«). 

If the virtual displacements 6q = (6q\,... ,6q„) are arbitrary, the right side of the 
above equation contains the impulsive virtual works of the reactions stemming 
from the second, third, and fourth type constraints, and operating during the 
shock interval (t',t"). Therefore, to eliminate these “forces,” and thus produce 
n — m"" reactionless, or kinetic, impulsive equations, we choose Sq’s that are com¬ 
patible with cdl constraints holding at the shock moment ; that is, we take 

6cj \,..., Sq m , 6q m "+1 • ■ - - ? ^q m '"• m " ! +1 ■ • - -; ^q m ,m If 

6q m "" + \,...,Sq„fO. 

Corresponding two (uncoupled) sets of equations: 

Impulsive kinetostatic: A{dT / dq D ) = Q D + (D = 1,... ,m""), 

Impulsive kinetic. A(dT/dq t ) = Q k (/ = m"" + 1,..., n). 

Further, since the velocity jumps Aq are produced only by the very large impulsive 
constraint reactions, operating during the very small interval t" — t' , within our 
approximations, the Qj [since they derive only from ordinary (i.e., finite, nonimpul- 
sive) forces, like gravity] vanish: Q r = 0; and so eq. (b) reduces to Appell’s rule: 

A{dT/dq,) = 0 => (dT/dq,) + = {dT/dqf-. 

In words: The partial derivatives of the kinetic energy relative to the velocities of 
those system coordinates cf s that are not forced to vanish at the shock instant (i.e., 
(/during 7 ^ 0) have the same values before and after the impact; or, these n — m"" 
unconstrained momenta, pi = dT/dqi , are conserved. 

To make the problem determinate, in the presence of nonpersistent-type con¬ 
straints, we must make particular constitutive {i.e., physical) hypotheses: for example, 
elasticity assumptions about the postshock state. 

EXTREMUM THEOREMS OF IMPULSIVE MOTION 

All based on the following master equation (impulsive Lagrange’s principle): 

^ dm (v + — r ) ■ Sr = ^ dF • Sr 
Carnot (first part—collisions) 

Sr~v + , dF = 0^T + -T-<0 


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§4 ABBREVIATIONS, SYMBOLS, NOTATIONS, FORMULAE 


Carnot (second part—explosions) 

<5r~v”, dF = 0 T + - T > 0 
Kelvin (prescribed velocities) 

6r ~ v + , 5r~v + + 5 K v = v, v~ = 0 —> T(v) — C(v + ) > 0, 6 K T + = 0 

Bertrand-Delaunay (prescribed impulses) 

<5r~v + , 6r ~ v + + S B/D v = v -► T(v) - T(v + ) < 0, 6 B/D T + = 0 

[Taylor: r Kelvin (v) - T(v + ) > 7> + ) - C(v) Bertrand _ Delaunay ] 

Robin (prescribed impulses and constraints) 

6r ~ v + , <5r ~ v + + 6 R v = v 

—> P = (dm/2)(v — v - ) 2 — ^dF • (v — v~): stationary and minimum 

Gauss (impulsive compulsion) 

Z = ^ (dm/2)(v — v — dF/dm) 1 = P+ (dF) 2 /2dm: stationary and minimum 

Chapter 5: Nonlinear Nonholonomic Constraints 

CONSTRAINTS 

fo{t,q,q) = o 

QUASI VARIABLES 
Velocity form 

Wfl = fo(t, q, q) = 0, u,= fj(t, q, q) ^ 0, w„ =1 = q ,,+1 = i = 1 

Virtual form (by Maurer-Appell-Chetaev-Johnsen-Hamel) 

S0 D = Y ( 9f D /dq k ) Sq k = Y (9^o/dq k ) Sq k = 0, 

69, = Y ( df,/dq k ) 6q k = Y (dui/dq k ) 6q k ± 0 , 

% = Y ( d 9k/duJi) 99, = Y ( d< ik/dui) 99, 

Compatibility 

Y ( df k /dq h )(dq h /duji) = Y (du, k /dq b )(dq b /du;,) = du> k /du>, = 6 kh 

Y (9F k /dui h )(duj h /dq,) = Y ( dq k /du b )(duj b /dqi) = dq k /dq, = 6 kl 

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INTRODUCTION 


PARTICLE KINEMATICS 
Virtual displacements 

Sr = 22 (dr*/89,) SO, = 22 s, 69, = Sr*, 

where 

£ i = 22 ( dr /dq k )(dq k /duj,) = 22 ( d 9 k/duJ,)e k , 
e k = 22 (dr*/99,)(du>,/9q k ) = 22 (dui/dq k )s,\ 

that is, 

d(.. .)/89, = *22 [#(• ■ -)/dq k \(dq k /duji), 
d(.. .)/dq k = 22 [^(- • ■)/d9,](du>,/dq k ) 

[nonlinear symbolic (nonvectorial/tensorial) quasi chain rules] 

Velocities 

v = 22 ?*(*, q, u)e k +e 0 [t = q n+l } 

= 22 V) £ k + £o = r*(t, q, uj) = v*, 

where 

a 0 = dr/89 n+l = 22 (dr/8q a )(dq a /duj n+l ) [a = 1,... ,n + 1] 
= 22 (dq k /duj n+l )e k + e 0 
= 22 (Qk e k- ~ ^k £ k) + e 0 

= Co + 22 (jlk - 22 (dqk/du,)u2)e k , 

and, inversely, 

e 0 = dr/dt = 22 (dr/d9 a )(9u; a /dq n+l ) [a = 1+ 1] 

= £o + 22 { UJk ~ H (du k /dq,)q^J s k . 

For any function /* = f*(t,q,u), 

9f*/89 n+l = 22 (df*/dq k ){q k - 22 ( d q k /du,)u2) + df*/dt- 

which in the Pfaffian case reduces to the earlier 

df*/89 n+ 1 = 22 ( 5 /*/ dq k )A k + df*/dt = 9f*/d(t) + df*/dt. 

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§4 ABBREVIATIONS, SYMBOLS, NOTATIONS, FORMULAE 


In particular, for /* = q h we find 

dqb/dO s = dq h /duj s , 

dq h /dO n+l = dq h /du n+l = q h ~Y (dq b /du,)u>i 
[= q b — Yj A m uji = A b , in the Pfaffian case]; 

and, inversely, 

dO k /dt = du k /dq n+l = ui k — Y^ (a o k /dq,)q,. 

Accelerations 

a = dv/dt = Y, ( dv/dq k )q k + No other ij/d> terms 
= Y ( dv*/duj,)d>i H- 

= Y S A d-= E ( da */ ddj i)^i d- 

= a*(t,q,uj,u) = a*, 

where 

dv*/du>/ = E (dv/dq k )(dq k /duji) or £/ — ^ e k (dq k /du>i) 

(which is a vectorial transformation equation, and not some quasi chain rule). 

BASIC KINEMATIC IDENTITIES 
Holonomic variables 

dr/dq k = dv/dq k = da/dq k = • • • = e k 

Nonholonomic variables 

dr*/dO k = dv*/du k = da*/du k = • • • = s k 

System forms 

dq k /80 1 = dq k /du>i = dq k /dd>i = • • • 
dO,/dq k = dui,/ dq k = ddi,/dq k = ■■■ 

NONINTEGRABILITY RELATIONS 
Nonholonomic deviation (vector) 

7k = E k *(v*) = Y E k *(q,)e, = Y V 'k e i = ~Y 

where 

Nonlinear Voronets-Chaplygin coefficients 

V l k = ( dq,/du k y - dq,/dO k = E,*^). 

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INTRODUCTION 


Nonlinear Hamel coefficients 

H k h = Y ( d 9l/^b)[(duk/dqiY - dw k /dq,} = Y (9q,/duj h )E,(uj k ) 

=* h k b = Y 7U = Y l k bs^ s + l\,n +1 (in the Pfaffian case) 

H\ = ~Y (&>b/dq,) V' k ^V l k = ~Y (dq,/^ b )H h k 

E t {u k ) = ~YY^ b / d ^ dUk i d ^ Eh *^^ 

w = -EE (dq,/du h )(dq s /duj k )Ei(u> k ) 

For a general function /* = f*(t,q, to), the following noncommutativity relations 
hold: 

d/de,(df*/de k ) - d/de k {df*/de,) =EEE [(^ q b /dq s dw k )(dq s /duji) 

~ ( d 2 q b /dq s du,)(dqj du k )\(du p /dq h )(df* / d9 p ) 

THE NONLINEAR TRANSITIVITY EQUATIONS 

(69 k )' — 6u> k = E (du> k /dqi)[(6qi) — 6(q{)] + Y^ Ei(uj k ) 6q, 

= Y (a o k /dq,)[(6q,y - 6(qi)\ + Y ®b 
= E [(<$?/)' - (5(?,)] - E E E'bi^k/dqi) 69 h 

(6q,y - 6(q,) = £ (d?,/^)! W - &*] + £ v' k 69 k 

= Y ( d 9i/duJk)[(69 k y - 6u) k ] - E E ( dqi/du} b )H h k 69 k 

SPECIAL CHOICE OF QUASI VELOCITIES 

V D = fn(j,q,q) = 0, ujj = fiit, q, q) = q, ^0- 

=> 9d = qD(t,q,qi) = Mt,q,qi) 

System virtual displacements 

6q k : 6q D = Y ( d( Pn/dq,) 6q u 

6qi = Y ( d 9t/ d 9i') b< h' = E (^') %' = St h 

Particle virtual displacements 

6r = E 8c E = E B ' 8qi ~ 

where 

B b = dr/d(q,) = dr/dq 1 + ^ (dr/dq^idcfo/dc/j) =e, + Y (^>D/dqi)e D ] 

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§4 ABBREVIATIONS, SYMBOLS, NOTATIONS, FORMULAE 


and, in general, 

dBj/dqji ^ dBj'/dq! (i.e., the Bj are nongradient vectors) 
Particle velocities and accelerations 

v —► v 0 = E B l q l + No other q terms, 
a —+ a 0 = E Bjqj + No other q terms; 

=> dr/d{qi) = dvjdq, = da 0 /dq I = = B, 

Special transitivity relations 

6qD = '52(d(/) D /dq I )8q l and q D = q D {t,q,qi) = 4> D {t,q,qi) 
(8q D Y ~8q D = ^2 (^dI dq,)[{8qi)' - 8{qi)\ 

+ E - d(j) D /d{q,)} 6q, 7 

where 

d(bn/d{q,) = d<p n jdq ] + E (d<j) D /dq D '){d(j> D '/dqfi, 
d{- ■ -)/d{qi) = d{.. .)/dq, + E [d(.. .)ldq D ]{d<j> D /dq I ) 

Nonlinear Suslov transitivity relations 

(8q k y - 8(q k ): (6q D )' - Sq D = E W °i Sq, 0) 

(«?/)■-«?/ = 0 [= 0; i.e., W 1 '/ = 0], 

where 

W D t = E f ((j) D ) - E {d(t) D /dq D ')(d(j) D :/dq,) 

= (d^o/dq,)' - dcj> D /d{q,) = E {I) {<j> D ) 

[special nonlinear Voronets coefficients] 

Nonlinear Chaplygin system 

qd = qD{qiAi) = <t>D{qi,qi) 

W D J -> T D j = {d^ D /dq,y - dci> D /dq, = E,{(j) D ) 

[special nonlinear Chaplygin coefficients] 

KINETIC PRINCIPLES (P k = 8T*/du k ) 

Central equation 


E ( dp k/ dt ) Mk - E (9T*/d0 k ) 86 k + E Pk[(S0 k )‘ - 8u k \ = E ®k ® k 

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INTRODUCTION 


Lagrange’s principle in NNH variables 

E (dPk/dt - 8T*/dO k + Y H‘ k P, - 0 A ) S0 k = 0 


EQUATIONS OF MOTION (D = 1,... ,m; 1 = m + 1,...,«) 

Coupled 

Ek(T) = Qk + Y X D(df D /dq k ) ( Routh-Voss form) 

Uncoupled 

I D = 0 fl + /!/> (Kinetostatic) 7/ = 0 7 (Kinetic) 


where 

4 = $ fl *' 

= E ( d <ii/ duj k)Ei 
= dS*/du k 

= ( dT*/du k y - 8T*/d6 k - r k 


{Raw form) 

{Maggi form) 

{Appell form) 
(Jolmsen—Hamel form) 


r k = $dm V* . E k *(v*) = J2 V kPl = -J2 H’ k P, 

[nonholonomic correction term] 

Transformation equations between holonomic and nonholonomic components 
4 = E (dq//duJk)Ei <£> E, = Y ( duj k /dq,)I k , 

0k = E ( dqi/du k )Q, &Q, = Y (■ duj k /dq,)0 k , 

Ek'* -r k ' = J2 (&>k/fa k ')(E k * - r k ) 

Transformation equations of E k *{T*) and r k between the quasi velocities u> <-> to' 

Ek ,*{T*') = E (a^/a^o E k *(T*) + Y ( dT*/av k )E k .*(u k ), 

Ek' = E {dT*/du k )E k ,*(u; k ) + Y {dto k /dw k ,)r k 

Johnsen-Hamel forms in extenso 

4 = dP k /dt - dT*/dO k + Y H\P, 

= dP k /dt - dT*/dQ k — EE (dui/d q h ) V h k P, 

= dP k /dt - dT*/dO k - Y vb kPb* 

[ E = Pk = p k (t, q, q) = Pk*(t, q, <*>) = {dr y dq k )* 

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§4 ABBREVIATIONS, SYMBOLS, NOTATIONS, FORMULAE 


SPECIAL FORMS OF THE EQUATIONS OF MOTION FOR 
THE CHOICE 

UD = /d{U q, q)=qD- q, <li) =0, uj, = f,(t , q , q) = q, ± 0 

and its inverse 

Qd =uj d + + ^o(t,q,Wi), qi = Wj, 

and with the notation 

E k = E k {T) = ( dT/dq k Y - dT/dq k = dS/dq k 
Maggi equations =>• nonlinear Hadamard equations 
Kinetostatic: Ed — Qd + -\d 

Kinetic. Ej + W> D /dqi)E D = Qi + 55 ( d <l>n/dq,)QD 

or 

dSJdqj = Qi + 55 ( d< P D /dqi)QD (= Qi.o = Qio ), 

where 

S = S(t, q,q,q) = • • • = S 0 (t, q , qi,qi) = S 0 , constrained Appellian V 0 
Hamel equations =>■ nonlinear Voronets equations 

Ei(To) - 55 (' d<t> D /dqi)(dT 0 /dq D ) - E Io = £ (/) (r o ) - E Io = Q Io , 

H D ! - -£(/)(&,) = 

El - r /j0 = r /0 = 55 w D I {dT/dq D ) 0 = 55 

Voronets equations => Chaplygin equations 

4d = qo(qi,qi) = <t>D{qi,qi) and r 0 = r 0 (^,^) 

=> tC 11 / = E^(c/) D ) —> Ej{(j) D ) = (d^D/dqj) — c^n/cty/ = T D , 

=► e Io -+ 55 

(dT 0 /dq,y - dTjdqj - 55 T D I {dT/dq D ) 0 = Q Io 
Transformation of the nonlinear Hamel and Voronets coefficients V l k ,H ! k under 
uj b ’ = u> b '(t,q,q) <£> qi = qi(t,q,u'): 

r k = 55 ^ =55 

= 55 (duk/dvk 1 ) V l k + 55 

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INTRODUCTION 


^ = EE {du) k /du> k ')(du>i'/du>i)H l k E [{duk/duv)' - du k /dO k ']{du v /du k ), 

H l ' k , = - Y, (&>v/dqi) V l k . V l k , = -J2 {dq l /du Jl ,)H , ' k , 

Chapter 6: Differential Variational Principles 

PRINCIPLE OF LAGRANGE 

^ (dm a — dF) • Sr = 0, with St = 0 

PRINCIPLE OF JOURDAIN 

(dm a — dF) • 6v = 0, with St = 0 and Sr = 0 

PRINCIPLE OF GAUSS 

Q (dm a — dF) • Sa = 0, with St = 0, Sr = 0, and hv = 0 

PRINCIPLE OF MANGERON-DELEANU 

(dma — dF) • $r = 0, (5=1,2,...) 

with 

<5? = 0, anh hr = 0, 6(f) = 0, h(r) = 0,..., 

< 5 ( ( s 7 1) ) =0 

NIELSEN IDENTITY 

N k (T) = dT/ dq k - 2 (dT/dq k ) = (dT/dq k )' - ST/c^ = E k (T) 

TSENOV IDENTITIES 
Second kind 

E k (T) = C k {2 \T) = (1/2 )[df/dq k - 3(0r/0 ?ifc )] 

Third kind 

E k (T) = C k ®(T) = (1/3 )[dT/dq k - 4(dT/dq k )} 

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§4 ABBREVIATIONS, SYMBOLS, NOTATIONS, FORMULAE 


MANGERON-DELEANU IDENTITIES 


E k (T) = C k ®{T) = (l/s) 


« (s) 

dT/dq k -{s+l)(dT/dq k ) 


[C k {l) (T) = N k (T)} 


VARIOUS KINEMATICO-INERTIAL IDENTITIES 


(■s—1) W 

d T /dq k = dT/dq k 


= g dmr-(dr/dq k ) 


d T ] /d q k = s(dT/dq k ) =s $dmr- ( dr/dq k ) 

W ~ (-S+1) (s) (s+1) 

T = \ dm r • r + s \ dm r ■ r + no r terms, 


d/dt 


■ (i-I) (s)' 

d T 0 /dq, 


- dT/dq, 


= d/dt(dT/dqi) — dT/dq { + ^ d/dt 


W M 

{dT/dq D )(dq D /dqi) 


(s-1) W (-s— i) («) / (s-i) W \ / W W\ 

5 T„/dq I = d T jdq l + XI (j 9 T / d 9oJ [dq D /dq,j 


VIRTUAL DISPLACEMENTS NEEDED TO PRODUCE THE 
CORRECT EQUATIONS OF MOTION 
Constraints Lagrange Jourdain 

fit, q) = 0: df/dq Sf = (df/dq) Sq S'f = 0 

6'f = (df/dq) Sq 


f(t,q,q) = 0: df/dq 
f(t,q,q,q) = 0: df/dq — 


S'f = (df/dq) Sq 


Gauss 
6"f = 0, 

6"f = 0 

S"f= (df/dq) Sq 
6"f = 0 

6"f = (df/dq) Sq 
6"f = (df/dq) Sq 


CORRECT EQUATIONS OF MOTION 

[Notation: M k = E k (T) - Q k = N k (T) - Q k = dS/dq k - Q k . 
Principle: £ M k S* k = 0, 6* k = 6q k , 6q k , 6q k ,...] 

Constraints Virtual Displacements 

f D (t, q) = 0 % = £ (df D /dq k ) Sq k 

/b(<, q, q) = 0 S'f D = £ (df D /dq k ) Sq k 

f D (t,q,q. q) = 0 S"f D = £ ( df D /dq k ) Sq k 

SPECIAL FORM OF CONSTRAINTS 


Equations of Motion 
M k = £ A D (df D /dq k ) 
M k = £ A D (df D /dq k ) 
M k = £ A D (df D /dq k ) 


9d = Mfq,qi) ( D= = m+l,...,n). 

For an arbitrary differentiable function 

/ = f(t,q,q ) = f[t,q,4>D{t,q,qi)Ai] = f 0 (t,q,qi) = L, 

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INTRODUCTION 


the following identity holds: 

Ni{fo) = E/ifo) + Y {.df 0 /dq D ){d(j) D /dq I ) 

=* Nj(T 0 ) = Ej{T 0 ) + Y (dTjdq^dfo/dq,), 

Ni{q D ) = E,(q D ) + Y ( d ^D/dq D ')(9^ D '/dqi). 

NIELSEN FORM OF SPECIAL NONLINEAR VORONETS 
EQUATIONS 

N,(T 0 ) - Y ( dT/dq D ) 0 Nj{q D ) -2Y(dT/dq D ) 0 {dt D /dq I ) = Q Io 

NIELSEN FORM OF SPECIAL NONLINEAR CHAPLYGIN 
EQUATIONS 

dtjdq, - 2 {dTjdqj) 

- Y ( d T/dq D ) 0 [ d <iD/dqi ~ 2 (dq D /dq,)\ = Q lo 
Special Pfaffian —> Voronets form 

<1d = Y bm + Mb Q) 

Then the above Voronets equations assume the special Nielsen form: 
dtjdq! - 2 {dTjdqi) 

~ Y ( dT /dq D )o{Y ft 0 "' ~ 2 (dbnr/dq r )\cir + [b D [ - 2{db n /dq,)\} 

-2 Y^T/dq D ) 0 b DI = Q l0 , 

where 

b D a' = Yj [{dboi/dqD')bD'i' + ( db DI Jdq D ')b D 'j\ + (db DI /dqj' + db DI '/dqj), 
b D i = b D In+ 1 = Y^ [( db D /dq D i)b D ij + {db DI / dq D ')b D ') + ( db DI /dt + db D /dqj). 

Special Voronets —> Chaplygin form 

4d = Y b Di(9i)9i, and dT/dq D = 0 

Then the above Chaplygin equations assume the special Nielsen form: 

dtjdch - 2(dTjdq I ) ~YJ2^ dT / dq "t( db Di/dqi' - db nr Jdq,)q r = Q Io 
[b D = 0, b D i = 0, b D n i = dbojdqi' + db DI :/dqj\ 


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§4 ABBREVIATIONS, SYMBOLS, NOTATIONS, FORMULAE 


NIELSEN FORMS OF HIGHER-ORDER EQUATIONS 

Let 


,, W (-5) 

N k ^\...)^d{...)/dq k -2 


(,-1) (.5-1)' 

d(.. .)/d q k 


E k {s \...) = d/dt 


’ (*-i) W 
d(- ■ ■)/dq k 


r (5-i) (5-i)i 

3(...)/d q k 


Then, for any sufficiently differentiable function / = f(t,q,q), and any 
k= 1,25= 1,2,3,..., 


Let 


N k {s \f) = £*«(/). 




N k ^ s \...) = d(.:.)/de k - 2 
E k * {s \...) = d/dt 


■ (5-1) (5-1) 

d{...)/d e k 


' (*-i) w' 

d(...)/de k 


■ (5-1) (5-1) 

d(.. .)/d e k 


where 


(5-1) (5-1) 

d{...)/d e k = £ 


(s-l) (.5-1)' 

d{...)/8 q, 


' M W 

dq,/d9 k 


[(j)th-ordcr quasi chain rule]. 

Then, for any sufficiently differentiable function /* = f*(t,q,w), and any 
k = 1,2,... ,n; s = 1,2,3,..., 


N k * [s) (D = £a-* w (./*), 


where 


(»-i) W 

f{t,q,q) => f => ■ ■ ■ f =>f, 


5-1) 

(1) 

(5-1) (5)/ 

' (i) 

(i-i) W\1 

/ 

t, q,q = q,- 

•> 9 ; 

t,q,q, ■ ■ 

, q , 0 ) 


(5-1)/ (1) (5-1) (5) \ 

= / [t,q,q,..., q ,6 1, 


w 

w 

■ (i) 

(*-i) «/ (i) 

(5 1) (5) \ 

(S+ 1 )/ 

' ( 1 ) 

(s 1 ) (5) (5+1) \ " 

/* 

= / 

t,q,q , ■ ■ 

• , q ; q[t,q,q,.. 

., q ,0 

> q ( 

t,q,q 

q ,0, e J 


( 5 )/ (1) (5-1) ( 5 ) (5+1) 

= f[t,q,q,..., q ,9, 0 


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INTRODUCTION 


Hamel-type equations (s = 1,2,3,...) 
( 5 ) d/dt 


(■»—!) W 

d T*/d6j 


(,-i) (*-i) 

-d T*/d Oj 


E 


/ w w\ ■ w (i-iy 

si dq k /dO, ) - dq k /d 0, 


„ ( W «\ 

= Y.[ d( ikl de i)Q k = & I 


(j-l) (j) \ * 

d T / dq k 


Nielsen-type equations 

/ W W \ / I*” 1 ) (•»-!) 

(s) [dT*/d0 I j -(s+1) d T*/d 0, 


E 


(*+D w 


H d q k /dO l - (s + 1) dq k /d 0, 




(s-l) W \ * 
d T /dq k 


For s = 1, the above yield, respectively, 

d/dtid^/dO^-d^/dO, 

- E [( 5 4/<90/)' - dq k /dOj] ( dT/dq k )* = © h 
dT*/d0 I -2(dT*/d0 I ) 

~ E [ d dk/dO, - 2(dq k /d0f)\(dT/dq k )* = 0,; 

and, for s = 2, 

2(df*/d0,y -dt*/d0j 

~ E [ 2 (d4/<94)' - d^k/dOj] (dt/dq k )* = 0 r , 
2(df*/d0 I )-3(dt*/d0 I ) 

- £ [ 2 (dgyae,) - 3(^M)] ( df/dq k )* = 


GAUSS’ PRINCIPLE 
Compulsion 


Z = (1/2)^ dm [a — (dF/dm)] = (1/2) ^ (1 / dm) {dm a — dF) 

= (1/2) S(dR) 2 /dm = S (- dR) 2 /2dm = s (Lost force) 2 /2 dm > oj 

= S — ^ dF • a + terms not containing accelerations, 


where 


5= (1/2) ^ dm a- a: Appellian. 

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§4 ABBREVIATIONS, SYMBOLS, NOTATIONS, FORMULAE 


Gauss’ principle 

8"Z = 0 , 

where 

6"t = 0, 8" r = 0, <5% = 0, 8"(dF) = 0, but 8" a ± 0 

[dF = dF(t, r, v) =► 8"(dF) = 0, 6"Q k = 0] 

a = 22 e k q k + no (/-terms = 22 e,u, + no w-terms, 

=> = 22 Ck 8 ^ k = 22 £/ 

and so, explicitly, 

8" Z = ( 1 / 2 )^ dm2 [a — ( dF/dm)\ ■ 8"a 
= ^ {dm a ~ dF) ' b" a 

= ^ ( dR/dm ) • 8"(dR) = ( dR/dm ) • 8"{dm a — dF) 

= Q (dR/dm) ■ dm 8"a = ^ dR ■ 8"a = 0 . 


COMPATIBILITY BETWEEN THE PRINCIPLES OF GAUSS 
AND LAGRANGE 


% = 22 ( d Vk/duj,) 86,, 

86 1 ~ 22 ^djJ,/dq k ) 8q k = 22 (df,/dq k ) 8q k ± 0 


Also, 


8f D = 8 uj d = £ (df D /dv) ■Sr='22 ( df D /dq k ) 8q k = 0, 
instead of the formal (calculus of variations) definition 


Sf D = 22 ( d fn/ d dk) Sq k + 22 ( d fo/ d 9k) % = 0. 


The same conclusion can be reached by requiring compatibility between the 
principles of Lagrange and Jourdain. 

EQUATIONS OF MOTION 


8"Z + 22^dS"/d = 0, 


where 


8"Z = 22 \Ek(T) — Q k ] 8q k (Holonomic system variables) 

= 22 (dS*/dd) k — 0 k ) 8(u> k ) (Nonholonomic system variables) 

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INTRODUCTION 


8"{/d) = ^"{dfn/dt + £ \(df D /dr) • v + ( df D /dv ) • a]| 

= ^ (df D /dv) ■ 6a (Particle form) 

= 6"{df D /dt + Y W„/dq k )q k + (df D /dq k )q k )} 

= (dfo/dqk) 6q k (Holonomic system variables) 


MINIMALITY OF THE COMPULSION 
A”Z = Z(a + 6"a) - Z(a) 

= (1/2 )^dm [(a + 6"a) — ( dF/dm )] 2 — (\/2) ^dm [a — (dF/dm)]" 
= 6"Z+(\/2)6 nl Z > 0, 


where 


6"Z= g (dm a - dF) -6"a (= 0), 
6" 2 Z = g(dm6"a-6"a) (> 0). 


Chapter 7: Time-Integral Theorems and Variational 
Principles 

GENERALIZED HOLONOMIC VIRIAL IDENTITY 

(dT/dq k )z k + (dT/dq k + Q k + Y^ ^D a Dk) z k} dt 

= ^Y(dT/dqk)zk\ x 

[z k = z k (t): arbitrary functions, but as well behaved as needed; and integral extends 
from t\ to ? 2 (arbitrary time limits). 

Specializations 


z k —+ 6q k [virtual displacement of q k ; and assuming 6q k = (6q k )']: 

(6T + 6'W)dt = {£ PkSq k ^, 

[Hamilton’s law of virtually/vertically varying action] 
z k —» Aq k = 6q k + q k At (noncontemporaneous, or skew, or oblique, variation of q k ): 

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§4 ABBREVIATIONS, SYMBOLS, NOTATIONS, FORMULAE 


(dT/dq k )(Aq k )' + Y ( dT/dq k + Q k ) Aq k - Y A D a D A D } dt 

= {T,PkMk}\ 

[Hamilton’s law of skew-varying action] 

(. Aq k )' — A(q k ) = q k (At)' [i.e., A{.. .) and (...)' do not commute] 
z k —> q k (actual system coordinate)'. 

J{S (dT/dq k )q k + Y^ [dT/dq k + Q k + Y^ A D a Dk)<lk} dt 

[Virial theorem (of Clausius, Szily et al.)] 

Zk —► q k ( actual system velocity)', power theorem in holonomic variables. 
GENERALIZED NONHOLONOMIC VIRIAL IDENTITY 


(E (dT*/du k )z k + Y {dT*/d9 k )z k - E E hh k( dT */^b)^ 

+ Y ( &k + A k) Z k) dt = {E (dT*/du k )z k ^ 

Specializations 

z k —> 69 k (recalling that 59 D = 0, 69 n+l = St = 0, while 69 j ^ 0): 

(st*+Y 0 > 69 1) dt = {E p ‘ 6e t}\ 

[Hamilton’s law of virtual and nonholonomic action], 
z k —> 9 k = uj k (recalling that uj d = 0): power theorem in nonholonomic variables. 
z k —> 9 k : This case is meaningless because there is no such thing as 9 k . 
z k —> Lu k : This case does not seem to lead to any readily useful and identifiable result. 
z k -> A9 k \ 


= 69 h + 9 h At =69 h + u} h At 

(A0 h ) — Aio h = ( 69 h ) — 6{9 h ) + to b {At) = Y, h b k 69 k + to h (At ) 

(. 3T*/d9 k ) A9 k + Y (dT*/du k ) Aco k 
+ El {dT*/duj k ) u> k (At) ~Y^l k b^bAt + E + A-k) A9 k | 

= {E {dT*/duj k ) A6 k } 

[Hamilton’s law of skew-varying action in nonholonomic variables]. 


dt 

2 

1 


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INTRODUCTION 


NONLINEAR NONHOLONOMIC CONSTRAINTS; 
HOLONOMIC VARIABLES 


(dT / dq k )z k + Y [dT/dq k + Q k + Y X n(9fn/9q k )^z k } dt 


= {Y ( < dT i d ^ Zk } l 


Specializations 

z k —> q k (Virial theorem): 


J{£ ( dT/dq k )q k + Y \ dT / dc lk + Qk + Y X ni d f d! dq k )\ q k } dt 

= {Y^ dT / d ^ qk ] x 

z k —> q k (Nonlinear (nonpotential) generalized power equation): 


d/dt (E {dT/dq k )q k - T} 

= -dT/dt + Y Qkdk + Y E x o(df D /dq k )q k 

z k —> 8q k (Hamilton’s law of varying action); 
z* —> Aq k (Hamilton’s law of skew-varying action): 


J{£ (dT/dq k )(Aq k )' + Y ( dT/dq k + Q k ) Aq k 

+(EE ^D(df D /dq k )q^J At} dt 

= {Y^ dT l d ^ Zk } i 


NONLINEAR NONHOLONOMIC CONSTRAINTS; 
NONHOLONOMIC VARIABLES 

{8T* + 8'W)dt={YPkM k } 2 ^ 


where 
8T* = 

=> 


• = (E ( dT */^k) S0 k )' - Y (dT*/du k y 86 k 

-EE*** (dT*/du, k ) 86 h + ^ (dT*/dO k ) 86 k , 

8T* dt 

Y [(dT*/du k y - + £ H h k {dT*/ du> h ) 

+ {Y(dT* 1^)86^, 


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§4 ABBREVIATIONS, SYMBOLS, NOTATIONS, FORMULAE 


( 89 b )' — 8u> b = £ E s {u b ) = £ £ E s {u b ){dqJdiJ k ) 89 k = J2 h\ se k 
= -££ E k {q,){duj h /dq,)89 k = - ££ V' k (duj h /dq,) 89 k , 
Ct = - £ H b k (dT*/ckv h ) = V h k (dT/dq h r 

[assuming again that (6q k )' = 6(q k )]. 

GENERAL INTEGRAL EQUATIONS 

{8T + ( dT/dq k )[(8q k y - 8(q k )] + 8'w) dt 

= {bT + 8'W + £ Pk[{50k) - t>u k \ + £ £ v \p k SOb } dt 

= {ST + S'W+Y, p k\W - 6u k ] - ££ n k bPk^b\dt 

= {£( a7 7 9 ?*) Sq k 


where 


(%)' - S(q k ) = ( dq k /du h )[(69 h y - Su b ] + £ V k h 69, b 

= £ (dq k /duj b )[{6e b y - 8u b ] - ^ £ ( dq k /duj,)H' h 69 h 
T = T[t,q,q(t,q,uj)\ = T*(t,q,u) = T*. 


The above yield the “equation of motion forms” [without the assumption 

(%)' = £(?*)]: 


£ [(dT*/du k y -dT*/de k 

^{dT/dq^k-Qk 

£ [(dT*/fa k y-dT*/do k 

+ Y J {dT*/du Jh )H h k -& k 


89 k dt = 0, 


69 k dt = 0. 


HOLDER-VORONETS-HAMEL VIEWPOINT 

(6q k )' = 5q k , whether the 6q k are further constrained or not. Then, with: 
6'W* = : 89 k , the above yield 

(8T + 8' W) dt = I ( 8T * + 8 1 W*) dt 

= {£(9r*/au,)«5d,}j 

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INTRODUCTION 


CONSTRAINED INTEGRAL FORMS 
[i.e., in terms of T * -A T* a = T*(t, q , W;)] 

Generally: 

<5^* = 6T\ + J2 (dT*/du; D ) 0 6 u d , 

6T* a = (dT* 0 /dOj) 69, + E (dT*J&,,) 6co, 

Under the Holder-Voronets-Hamel viewpoint: 

6{qk) = (6q k ) > = 0, d(66 D ) = 0 => (#?/>) = 0; 

but 6(d9 D ) ^ 0, &ud = EE rVaWafc)»/ = -£ »,#0. 

and 6luj = (<5$/) — ^ ^ ^v 5 


we obtain the constrained integral equation 

+ E E ^T/dq k ) 0 V k , 68, + 6' W* 0 


dt 


6T* 0 — EE (dT*/du D ) o H D ,66, + 6'W* 0 


dt 


(E(sr/%) %}j- 


Special form of the constraints: 

?z> = 4>D(t,q,qi) 

=> u n = Rd - 0x>(G q, qi) =0, U), = q,i- 0, 

Qd = + </>/)[?, q, q,(t , q, uj,)\ = ui D + <j) D (t , q,uj,), 

69 n = 6q n - E {dfbn/dq,) 6q, = 0, <Sd 7 = % ^ 0. 
Suslov transitivity assumptions and integral equation: 

{6q D y l L 6{q D ), (6q,)'- 6(q,) = 0; 


but 


<W) = 0 


— S(q D — <!>d) — 6{q D ) — 6<p D — 0 [and (50/)) — 0] 

=>■ = ^0/) [definition of <*)(#£,)]; 

=> = (E ^o/dq,) 6q,^j - 6(f> D 

= ■ ■ ■ = E E^{(j> D ) 6q, = E 6qj ^ 0; 

6T = 6T a . 


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§4 ABBREVIATIONS, SYMBOLS, NOTATIONS, FORMULAE 


Suslov principle: 

{6T 0 + J2 0 dT/dq D ) 0 [(6q D y - S<f> D ] + 6'W 0 )dt 

= {&T 0 + EE (dr/dq D )W D ,6q 1 + 6'W 0 }dt 

= {Ew^)^}, 


Holder-Voronets-Hamel transitivity assumptions: 

S(qk) = {Sq k )', 66 d = 0, d(60 D ) = 0 => (60 D )' = 0; 


but 


«(</*/>) ^ 0 


or 


— S(q£) — — <5(4o) — — (^n) — S(f>j) 

= E e (iMd) Sqi = Y W °I 5c li ± 0 [definition of d(?n)]; 
^6T = 6T 0 + YY (' dT / d 9 d) W°, Sqj. 


Voronets principle: 


ST 0 + YH o dT / d 9D )«?/ + -5' 






In both cases: 


T —> T 0 (t,q,qi) —> (variation of constrained T), 

dT/dq D -> (dT/dq D ) a = p D [t, q, q,, (j) D (t, q, q,)\ = p Dfi {t,q,qi) = p Do , 

{Y( dT / d 4k) tq*} i = ••• = {E (■■•)/««/},. 

6'W 0 = Y QioSqj- 


NONCONTEMPORANEOUS VARIATIONS AND RELATED 
THEOREMS 

Definition: 

zl(...) = 6 (...) + [rf(.. .)/<&] zlt: noncontemporaneous variation operator 
=Z + qk A t, zl? = + ( dt/dt) At = 0 + (1) zlt = zlt. 

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INTRODUCTION 


Basic identities: 


(...) dt — 


A(...)dt = 


8(. ..)dt+ {(...) At} { 


{A(. ..) + (■■ .)[d(At)/dt]} dt 


[A(. ..)dt +(...) d(At)\, 


{8(...)-(...)[d(At)/dt]}dt+{(...)At}\ 


[8(...)dt-(..)d(At)] + {(...)At}\; 


(■••)*- 


A (...) dt = 


(...) d(At) 


(...) dt =••• = — 


{(.. .)[d(At)/dt]} dt; 

^2 E k(---)Mkdt 
+ | [dh(.. .)/dt + d(...)/dt] At dt 

+ ( d - ■ ■ / d 9k) Mk ~ h(...) At 

- | Ek (- ■ ■) hqk dt + ■ • • / dq k) Ac lk ~ fl (- --) At } 

Y J E k (...)8q k dt + (d... /dq k ) 8q k + (...) At} 


where 


With 


/>(•••) = ^ [<9(.. -)/dq k ]q k — (...): generalized energy operator. 


h = /?(L) = ^ p k q k — L = h(t, q, q): generalized energy, 


yl// = 


Ai 


I V) dt = 


Ldt: Hamiltonian action (functional), 


22" Jt: Lagrangean action (functional), 


2? = 7" + V: total energy of the system; 
we have the following mechanical integral theorems: 

A \ Tdt+ \ kw dt = + ( T - p - pk ) Al } 


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§4 ABBREVIATIONS, SYMBOLS, NOTATIONS, FORMULAE 


AA 


AA l - 


6'W„ p dt= p k Aq k - h At^ 

= {^PkSqk + LAt} 1, 

(6E - S' W np ) dt = {55 Pk Mk ~ (55 PkVk - 2r) At} 

= {Y^P k8c lk + 2T At } l ’ 

SE dt + {E At}\, 

I (6E — S' W„ p ) dt 

+ {£ PhAqk + (lT — 55 Pkdkj At \^ 

\{SE + S'W np )dt 

+ {- 55 ^ Mk + ( 2v +55 Pk ^ k ) At ]p 

[.AT+ 2T(At)'+ t At\dt+ S'Wdt 

= | [ATdt + 2Td{At) + dT At + S'Wdt] 

= {^2PkS<lk + {2T)At} i 

= {51 Pk Ac lk - (55 Pkdk -2 r'j At^ 


E dt = 
2 Tdt = 

2 Vdt = 


SECOND (VIRTUAL) VARIATION OF A H 
Total (virtual) variation: 


S t A h = A H (q + Sq) — A H (q) — SA H + (1/2) 6 A H + ■ 


First (virtual) variation: 


SA H — 


SLdt = ••• = — 


E(q) Sqdt + {pSq}\ 


Second (virtual) variation (one Lagrangean coordinate): 


6 A H = 8{8A H ) — 


8 2 Ldt 


= ••• = — J J{8q) Sqdt + {8p8q}\, 

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INTRODUCTION 


where 

S 2 L = 6{6L) = [(d/dq) 6q + (d/dq) 8(q)] 2 L 

= ■ ■ ■ = ( d 2 L/dq 2 )(6q ) 2 + 2(d 2 L/dqdq) 6qSq + (d 2 L/dq 2 )(8q ) 2 

Jacobi’s variational equation: 

J(8q) = {d/dt[d/d(6q)] - [d / d(8q)]}(\/2) S 2 L 

= (d 2 L/dq 2 ) 6q + (d 2 L/dq 2 )' 6q + [(d 2 L/dq dq)' — (d 2 L/dq 2 )] 8q 
= d/dt[(d 2 L/dq 2 ) 8q\ — \d 2 L/dq 2 — d/dt(d 2 L/dq dq )] 6q = 0 


Equivalently: 

E[L(t, q + 8q,q + 8q)\ - E[L(t, q , q)} « 8E(q, 6q) (to first-order) 

= J{Sq;q) = J{8q) 


Chapter 8: Hamiltonian/Canonical Methods 

CONJUGATE (HAMILTONIAN) KINETIC ENERGY 


T' = (Y Pkik -T) =Y1 Pk9k{U CLP) - T (qp) = T\t , q,p) 

= Y ( dT/dq k )q k - T = (2T 2 + T x ) - (T 2 + 7) + T 0 ) = T 2 - T 0 : 
i.e., if T = T 2 (e.g., stationary constraints), then T 1 = T 


CANONICAL, OR HAMILTONIAN, CENTRAL EQUATION 

Y ( d Pk/dt + dT'/dq k - Q k ) 8q k + Y (' dq k /dt - dT'/dp k ) 6p k = 0 

CANONICAL, OR HAMILTONIAN, EQUATIONS OF MOTION 
(for unconstrained variations) 

dpk/dt = ~(dT'/dq k ) + Q k (= dT/dq k + Q k => dT/dq k = -dT'/dq k ), 
dq k /dt = dT'/dp k 

If Q k = —dV(t,q)/dq k , the above assume the antisymmetrical form: 

dp k /dt = — dH/dq k , dq k /dt = dH jdp k , 


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§4 ABBREVIATIONS, SYMBOLS, NOTATIONS, FORMULAE 


where 


H=T'+V= (J2 PkQk -T+V) = 55 PkQk{t, <LP) ~ ( T ( qP ) ~ V) 

/ q=q{t,qj>) ' 

= (51 PkQk ~ L ) 


q=q{t,q,p) 

= 55 Pk4k(t, <hP) ~ L (qp) 


q=q(t,q,p) 

= H(t,q,p): Hamiltonian of system (function of 2n + 1 arguments). 
If both potential and nonpotential forces (Q k ) are present, the above are replaced by 
dpk/dt = - dH/dq k + Q k , dq k /dt = dH/dp k ; 


also, 

dH/dq k = —dL/dq k and dL/dt = —dH/dt. 

For stationary (holonomic) constraints, 

H = T(t,q,p) + V Q {t,q) = E(t,q,p) = total energy , in FLamiltonian variables. 
In all cases, the following kinematico-inertial identities hold: 

dT'/dt = —dT/d t, dT'/dq k = —dT/dq k , dT'/dp k = dq k /dt ; 

dH/dt = —dL/dt, dH/dq k = —dL/dq k , dH / dp k = dq k /dt. 


LEGENDRE TRANSFORMATION (LT) 

An LT transforms a function Y(..., y ,...) into its conjugate function Z(... ,z ,...), 
where z = dY/dy, so that dZ/dz = y. Here in dynamics we have the following 
identifications: 

Y(...)->L, q,t, y^q, z = dY / dy-> p = dL/dq, 

Z(...) —> H, dZ/dz = y —> dH/dp = q. 

POWER THEOREM 

dH/dt = dH/dt + 55 QtAk 

If dH/dt = 0 (e.g., stationary constraints) and Q k = 0 (e.g., potential forces), then 
the Hamiltonian energy of the system is conserved: 

H = H(q,p) = constant. 

CANONICAL ROUTH-VOSS EQUATIONS 
Under the m Pfaffian constraints 


55 a Dk fiqk = o, 

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INTRODUCTION 


the canonical equations are 

dpk/dt dT /dq k + Q k ^ ^ ^D®Dk 3H /dq k “h Qk ,nonpotential “h ^ ^ ^D^Dki 
dq k /dt = dT'/dp k (= dH/dp k ). 


ROUTHS EQUATIONS 

Ignorable (or cyclic) coordinates and momenta 

(q u ...,q M ) = (V’u-mV’m) = {ipi) = ip, (Pu---,Pm) = (Pi ,...,P M ) = ('Pi) = P 
Positional (or palpable) coordinates and velocities 

(qm+ i , • • •, q„) = (q P ) = q (qm +i ,---,q n ) = (q P ) = q 

Kinetic energy 

T = T(t; ip M \ q M+ i,... 7 q n \ Vh,..., V’m; Qm+ u • • • > 4n) 

= T(t,ij,q;ip 7 q) = T[t, ip,q;ip(t,'il;,q;'F,q), q] 

= T(t,ip,q ; ?',4) = 

Modified (Routhian) kinetic energy 


t " = (^ - E ^ 


'ip='ip(t;ip,q;'F ,q) 


= T"(t, ip, q\ <P,q) 


Routhian central equation 

E (' d Pk/dt - dT"/dq k - Q k ) Sq k + E (/V - dT"/dq p ) 6q p 
- E (dipi/dt + dT"/dT/) ST, = 0 

Routh’s equations (for unconstrained variations) 

dp k /dt = 3T"/dq k + Q k : d'Pi/dt = dT"/dpi + Q t (i = 1,..., M), 

dp p /dt = dT"/dq p + Q p (p = M + 1,..., n); 
d^/dt = -dT"/dV t (/ = 1,..., M), 

Pp = dT"/dq p (p = M+l,...,n) 

Hamilton-like Routh’s equations 

d'Pi/dt = -d(-T")/di/ji + Q h dipjdt = d(-T")/dPi 

Lagrange-like Routh’s equations 

dp p /dt = dT"/3q p + Q p , p p = dT"/dq p (= dT/dq p ) 

=► (dT"/dq p y-dT”/dq p = Q p 

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§4 ABBREVIATIONS, SYMBOLS, NOTATIONS, FORMULAE 


Additional Routhian kinematico-inertial identities 

dT/dq k = dT"/dq k : dT/ty, = dT"/dtp t (i = 1,..., M), 

dT/dq p = dT"/dq p (p = M + 1,..., n) 

In sum, we have the following two groups of such kinematico-inertial identities: 

dT"/dip t = dT/chbi and dT"/dWi = -dip t /dP, 

dT"/dq p = dT/dq p and dT"/dq p = dT/dq p (= p p ). 


If p k = 8L/dq k , the above are replaced by the following: 
Hamilton-like Routh’s equations 


dTj/dt = dR/dipf + Q h dipjdt = —dR/d'Pj 

and Lagrange-like Routh’s equations 

dp p /dt = dR/dq p + Q p , p p = dR/dq p (= dL/dq p ) 
=> E p (R) = ( dR/dq p Y - dR/dq p = Q p ; 

where 


R = (l - J2 


ip=ip(t^,q-, *F,q) 


R{t]i>,q-, v,q) 


= Routhian function , or modified Lagrangean , 

L = L(t ; ip, q; T, q) = T^ w - V = L^ 

= Lagrangean expressed in Routhian variables; 


that is, the Routhian is a Hamiltonian [times (—1)] for the ip h and a Lagrangean for 
the q p . 

Relation between Routhian and Hamiltonian 


H = Y] PkVk -L, R = ^Z Ppdp - H = L - ' ¥ >^ i 


STRUCTURE OF THE ROUTHIAN 
Decomposition of T (scleronomic system): 

T=T qq+ T qi, + T w> = T M 9\ 9), 

where 

^ Rqq ~ q p q q = homogeneous quadratic in the q’s 

(a pq = a qp : positive definite), 

r # E EE b p jq p ipj = homogeneous bilinear in the q’ s and ip’s 
(in general: b pi f b ip , sign indefinite), 

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INTRODUCTION 


— EE Cyipiipj = homogeneous quadratic in the ip's 
(cjj = Cji\ positive definite) 

[/. j = 1 q = M + 1and the coefficients are functions of all n ^ fc ’s]. 

Next, 

'F, = dT-/dipt = Y Cji^j + E b pMp ^ E c i^i = l[, < ~ E 

dipj/dt = Y C ji( ,f/ i ~ Y b P‘dp) 

(since is positive definite =>■ Cy is nonsingular), where 

Cjj = [cofactor of element c /7 in Det(c / -,)]/Det(c /V ) = Cy 
(= known function of the cf s and ip' s). 

Then 

T = T 2 f) + Tq 2 = T(ip, q\ W, q), 

where 

2r 2,o ^EE( a «-EE wjw. 2r o,2 = E EE 

that is, T = 7’('0, q\ T, 4) does not contain any bilinear terms in the </’s and T’s; and 
so 

T" = T-Y = r “E ^(E C ^/-E *„)) 

r . rrill rr r _ rr r // . rriH . rri / / 

2,0 “r 1,1 — -*0,2 = -* 2,0 “t" J- 1,1 ~r 7 0,2 
= T"{iP,q- W,q), 

where 

2^"2,o = E E («* - E E c fi b pj b ^d P q q = Y E r P^)q P q q 

= 2T 2 o (= positive definite in the g’s), 

r "u = E E (E c a<) ^ = E »>(«> 

[No counterpart in T = T(ip,q\ 'F,q),i.e., T xl = 0; sign indefinite], 

it\ 2 = - E E ow = 2r V(?, y) 

= —27) ) 2 (= negative definite in the T’s). 


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§4 ABBREVIATIONS, SYMBOLS, NOTATIONS, FORMULAE 


Conversely, 

T=T" + J2 w iTpi = T " > Fi(dT"/d'Pj) 

= (T\ o + T" u + T\ 2 ) - (T\ x + 2T\ 2 ) 

= T" 2fi -T\ 2 = T{^,q- W,q). 

Hence, 

L=T-V= (T v0 + T 0i2 ) -V=T 2fi -(V- r 0i2 ) 

= (r" 2 , 0 - r" 0j2 ) — f = r" 2 , 0 - (f + rV) = ?; 'P,g) 

=>R = L~Y1 = L + Y, (' dT"/d'F i )'F i 

= (T \o - n, 2 - f) + (2r" 0j2 + r" M ) 

= iC + = ^(V'i?)> 

where 

^2 = n,o = r 2 , 0l /?, = r" u , *0 = r " 0t2 -v = -t 0i2 - v. 

Additional results 

(i) With 

T=T tt+ «'> ?) 

^t"=t-J 2 vdi = t -J 2 (dT/fyirit = T m - T u = T "^ ?; i>, ?); 

(ii) d^i/dt = —dT"/d'F i = ■■■ = dT Q2 /d'F i - dK 22 /dir n 
where 

2 T 0i2 = - 2 r " 0]2 = ££) Cji'Pj'Pj, 

and 

2K 22 = ^ ^ (y, b p j c/p'j V'4?) = Cji' K i 7 'i- 

Matrix form of these results: 

q T = (to+i,-•■,?„), = T t = (•?!,..., -Pm), 

^ Ppq) i®qp) ® 5 ^ (Pip) P {bpi) ^ 3 Pij) ( ttji ) C , 

IT = q T aq + 2vj/'b T q + vj/ T c vj/, 

<977chj/ = b T q + cvj/ = T 

=> \j/ = c(T - b T q) = COP - b T q) => ij/ T = (»P T - q T b)C 


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INTRODUCTION 


[since c is symmetric, so is its inverse C = (C /7 ): C = c 1 = (c *) T = C T ], 

T = ■ ■ ■ = (l/2)q T (a - bCb T )q+ (1/2)«F T C*P = T 2fi + T , 0 , 2 = T\ 0 - T\ 2 
[since T t C b T q = q T b C4* ] 

Y T vj/ = = Y T C Y - Y T C b T q = -2 T\ 2 - Y t C b T q, 

R= (T — V) — *P T v|/ = ... = R 2 + R l +R 0 
R 2 = T\ o = r 2 ,o = (1/2) q T (a - bCb T )q, 

R, = T \i = T t C b T q, 

Ro = T'\ 0,2 — V = —(V + T 02 ) = —(1/2)'P t C T - V. 

If b = 0 (i.e., q s and i/>’s uncoupled in the original T ), 7? reduces to 
R= (l/2)q T aq-(l/2)»P T CT- V. 


CYCLIC (OR GYROSTATIC) SYSTEMS 

(i) (qi,. ■ •, q M ) = {tpu---,ipM) = (V'i) = ^ 

do 770/ appear explicitly, neither in its kinetic energy nor in its nonvanishing 
impressed forces; only the corresponding Lagrangean velocities 

{q u ...,q M ) = = {'•Pi) = i> 

appear there, and, of course, time / and the remaining coordinates and/or velocities 
{Qm+u ■ ■ ■, c ln) = {q P ) = q and (q M+l ,..., q n ) = (,q p ) = q 
respectively; that is. 


dr/a^i = o 


but, in general, 


dT/dipi ± 0 =>• T = T(t;q,ip,q). 

(ii) The corresponding impressed forces vanish; that is, 

Qi = 0, but Q p = Q p (q) ± 0. 

If all impressed forces are wholly potential , the above requirements are replaced, 
respectively, by 


dL/dipj = 0 and dL/dipi ^ 0 => L = L(t\ q , ip, q). 

The coordinates ip, and corresponding velocities ip, are called cyclic (Helmholtz), or 
absent (Routh), or kinosthenic, or speed (J. J. Thomson), or ignorable (Whittaker). 
The remaining coordinates q, and corresponding velocities q , are called palpable, or 

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§4 ABBREVIATIONS, SYMBOLS, NOTATIONS, FORMULAE 


positional. Then the Lagrangean equations corresponding to the cyclic coordinates/ 
variables, become 

( dT/d’tpi)' - dT/dip, = Qf. ( dT/dxpj )' = 0 =>■ dT/dip, = T,- = constant = C,-; 


that is, the momenta T, corresponding to the cylic coordinates ipi are constants of the 
motion. [Conversely, however, if dT/dipj = 0, then dT/dip t = 0, and, as a result, 
T = T(t;q,q); that is, the evolution of the ip’s does not affect that of the q's.] 
Hence, the Routhian of a cyclic system is a function of t, q 7 q and = (T,); that 
is, with C = (Q), 


R = (l - Y, Vii’. 


ip=ip(t;q;q,C) 


[after solving dT/dipj = T, = C, : for the ip in terms of t , q, q 7 C] 
= L[t 7 q 7 q 7 ip(p 7 q- 7 C 7 q)- 7 C]~^2 ' F pfit; q 7 q: C) 

= R{t-,q,q ; Q 

=> L = <T T C) + R{t- 7 q 7 q ; C)]; 


that is, the system has been reduced to one with only n — M Lagrangean coordinates, 
new “reduced Lagrangean” R , and, therefore, Lagrange-type Routhian equations 
for the positional coordinates and the “palpable motion” q p (t): 

(dR/dqp) dR/dcjp C//anonpoLcntial impressed positional forces ■ 


Then, 


R = known function of time 
=>■ dR/dCj = known function of time = —/[■(?; C), 


fi = - 


( dR/dT,) dt + constant = 


f(t: C) dt + constant 


= ipi(t 7 C) + constant. 


EQUATIONS OF KELVIN-TAIT 
Let 


T = T(q 7 q 7 ip) = homogeneous quadratic in the ip and q 7 
=$• R = R 2 T R\ + R 07 


where 

(i/2) r pq{q)q p (lq{= Tift) 

= R 2 (q 7 q) = homogeneous quadratic in the nonignorable velocities q 7 
R\ = T" 1,1 = ^2 r p (q,C)q p 

= Rfq 7 q 7 C ) = homogeneous linear in the nonignorable velocities q 7 
[apparent kinetic energy T'\{\\ 

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INTRODUCTION 


and 


fp ^ , PpiCj ppi — ^ ' Cjjbpj fpiiip ) J ) 

r 0 = t'\ B, 2 — v = —(v — t\ 2 ) = -(i/2) ^£ qqc, - V 


[— -(^+ To,2)] 


= R,,((/■ C) = homogeneous quadratic in the constant ignorable momenta = C 
[ 1 apparent potential energy T" 02 = —T Q 2 (< 0)]. 


Hence, the situation is mathematically identical to that of relative motion (§3.16) 
Lagrangean equations of palpable motion: 

(dR/dq p ) dR/dcjp Op.nonpolcntial impressed positional forces ■ 

From the above we obtain the following. 

Kelvin-Tait equations (with p, p' = M + 1 

E p (R) = E p (R 2 + R[ + Rq) = E p {R 2 ) + £p(2?]) + E p (R 0 ) = Q p , 


or 


E P (R 2 ) — Qp — E p (R\) — E p (R 0 ), 


or, explicitly, 

(dR 2 /dq p y - dR 2 /dq p = Q p + dR 0 /dq p - [(dRi/dq p )' - dRfidq p ] 

= Qp-d(V- T" 02 )/dq p + Y {dr p ,/dq p - dr p /dq p ,)q p , 

= Qp-d(V-T\ 2 )/dq p + G p , 

where 

G p = -[(dRjdcjpY - dRjdq p \ 

= Y ( dr p'/ dc ip - dr p/ dc ip')<ip' = Y G pp'Qp' 

[Gyroscopic Routhian “force,” since G pp ' = — G p ' p = G pp \q ; C)]. 

These are the equations of motion of a fictitious scleronomic system (sometimes 
referred to as “conjugate” to the original, or reduced, system) with n — M positional 
coordinates q, and subject, in addition to the impressed forces Q p (nonpotential) and 
~dV/dq p (potential), to two special constraint forces: a centrifugal-like dT" 02 /dq p , 
and a gyroscopic one G p . 

Ignorable motion, once the palpable motion has been determined: 

q p (t) => dfii/dt = —dR/dCj = -dRQdCj - dR^/dCt = -dT" op _/dC i - Y PpA 
Gyroscopic uncoupling G pp > = 0 

=> E p {Rf) = ( dR 2 /dq p Y - dR 2 /dq p = Q p + dR 0 /dq p 


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§4 ABBREVIATIONS, SYMBOLS, NOTATIONS, FORMULAE 


A system is gyroscopically uncoupled if, and only if, R x dt=^f r p (q; C) dq p is an 
exact differential. [A similar uncoupling occurs if all the C, vanish: r p = 0 => R t = 0; 
and Rq = — V(q ).] 

A cyclic power theorem 


dh R /dt = ^2 Q p d P , 


where 


h R = R 2 — Rq — T\ o + (V — T" 02 ) 

= T 2fi + (V + T 0j2 ) = T(q, q , C) + V{q) = E(q , q, C) 

= Modified (or cyclic) generalized energy; 

if E Q P dp = 0: 

h R = T" 20 + (V - T" oz ) = T(q,q, C ) + V(q) = constant. 
Alternatively, 

H = ^2 ( dL/dq k )q k — L (= constant, if Q p = 0 and dL/dt = dR/dt = 0) 
= - r +'52 ( dR / d d P )dp 

= — (R 2 + R\ + Rq) + ( 2R 2 + R\) 

= R 2 -R 0 = H(q,q,C) (= h R ). 

For rheonomic cyclic systems; that is, L = L(t, q , q, C) 

=> R = L(t,q,q,C) - Cii>i(t,q,q,C) = R(t,q,q,C). 


STEADY MOTION (OR CYCLIC SYSTEMS) 

ipi = constant = c,- (in addition to 1 = constant = C,), 

and q p = constant = s P (=> q p = 0) 

(with i = 1,..., M; p = M + 1,..., n); 

that is, all velocities are constant (and, hence, all accelerations vanish); and, for 
scleronomic such systems, the Lagrangean has the form L = L(cj,s p ). 

Conditions for steady motion [necessary and sufficient conditions for the steady 
motion of an originally (scleronomic and holonomic) system; or, equivalently, for 
the equilibrium of the corresponding reduced ^-system]: 

Q P + dR 0 /dq p = Q P + (dT" 02 /dq p - dV/dq p ) = 0, 


or, if the forces are wholly potential: 


dR 0 /dq p = 0, or dT" ()2 /dq p = dV/dq p . 

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INTRODUCTION 


Equivalently, since 

R = R 2 (homogeneous quadratic in the q’s) 

+ R\ (homogeneous bilinear in the Ws and q s) 

+ Rq (homogeneous quadratic in the IP’s) 

and 

dR/dq p = dL/dq p , 

the above equations can be rewritten as 

{dR/dq p ) 0 = (,dL/dq p ) 0 = 0 [(.. .)„ = (• ■ •)I^= C , 9 =J> 

expressing q’s = s’s in terms of the arbitrarily chosen IP’s = C’s. The ip's can then be 
found from the second (Hamiltonian) group of Routh’s equations: 

dA/dt = -(dR/d'FA = ~(dR o /d'F i ) 0 = -(dT\ 1 /d'P i ) 0 

= ^ CjjCj = constant = c t [with q p = 0] 

= Function of the s’ s and the (arbitrarily chosen) C’s, 

^ Ipi(t') = Ci(t initial) T A, initial 

= Function of the s’s and the (now) arbitrarily chosen c,’s and VWiai’ s > 
i.e., in steady motion, the cyclic coordinates vary linearly with time. 


If we initially choose arbitrarily the IP’s, then the above equations relate them to the 
cf s. If, on the other hand, we choose the ip’s = c’s, then, to relate them directly 
to the cf s: first, we take T" 02 , and, using *P, = c jiA > change it to a homo¬ 
geneous quadratic function in the ip’s (with i, j, j', j": 1,..., M): 


2T" 0 2 = 2 T"qnp = — EEw [recalling that ^ Cji c j'j — by 
= faj, = —2T^; 


or, since 

dT\ w /dq p = ~(dT"^/dq p ) = dT^/dq p , 

we can, finally, replace the steady motion conditions by 

~( dT "^p/ d ^ = dV / d ^ or dT u/ d( lp = dv /dq p , 

relating the cf s to the ip’s; and, using *P, = c jiA > we can re l ate both to the IP’s. 

VARIATION OF CONSTANTS (OR PARAMETERS) 

Theorem of Lagrange-Poisson: 

Equations of motion: 

dp k ldt = f k (t, q.p) and dq k /dt = g k {t,q,p), 

[fk = —dH/dq k + Q k and g k = dH/dp k , for a Hamiltonian system]; 


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§4 ABBREVIATIONS, SYMBOLS, NOTATIONS, FORMULAE 


general solutions: 

Pk = Pk ( t\c) and q k = q k ( t;c), 

where 

c = (cj,..., c 2n ) = (c y ; v = 1,..., 2n\. constants of integration. 

Adjacent trajectory, II = I + 6(1), 

6p k = Y (dp k /dc v ) 6c v and 6q k = Y {dq k /dc v ) 6c v . 

Linear variational, or perturbational, equations: 

(6p k y = 6(p k ) = Y K d fk/dp t ) 6p, + (df k /dq,) 6q,\, 

(%)' = S(q k ) = Y [(dgk/dpi) SPi + (■ dgk/dqj) 6q,}. 

Then, for a Hamiltonian system, 

d/dt (E (diPk 6 2 q k — 6 2 p k 6\q k )^ — Y^ (^i Qk & 2 Ik ~ & iQk )• 

Theorem of Lagrange-Poisson: In a holonomic and potential (i.e., Q k = 0, or 
dQk/dqi = dQi/dq k , for all k, 1= but possibly rheonomic, system, the 

bilinear expression 

I = Y Pk 6 2 Qk - S 2 p k 6 x q k ) 

is time-independent, that is, it is a constant of the motion. 

Lagrange’s brackets (LB): 

I ^ ^ ' [tfji Cv] Cfi 

where 

= Y [( d Pk/dCe){dqkldc v ) - (dp k /dc v )(dq k /dc ll )\ 

= Lagrange an bracket of c^, c v . 

Properties of LB: 

[tfn Cfi] [C/n t/y] [^1/: , 

d[c^ c v ]/dc x + d[c v , c a ]/<9c m + d[c x , c li ]/dc v = 0, 

M = d/dc v (Y q k (dp k /dc,Sj -d/dc^ (U <lk{dp k /dcS) ■ 

PERTURBATION EQUATIONS 
Unperturbed problem and its solution 

dpk/dt = —dH / dq k , dq k /dt = dH/dp k ; p k = p k (t; c), q k = q k (t ; c) 

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66 



INTRODUCTION 

Slightly perturbed problem 

dpk/dt = —dH/dq k + X k , dq k /dt = dH/dp k , 

where 

X k = X k (t,q,p) = given function of its arguments 

« X k ^\t-,c) [first-order approximation, upon substitution of unperturbed 
solution in it] 

2 n first-order differential equations for the c^ = constant —> c M (t): 

X! {dPk/dc^dcJdt) = X k (x \ 22 (dqk/dcjidcp/dt) = 0. 
Lagrangean form of the perturbation equations: 

22 [c v ,c^{dc v /dt) = 22 x k W {dq k /dc fJ ). 

If the perturbations are potential —that is, if X k = —dQ/dq k —then, since 
q k = q k {t\ c), the above specializes to 

22 [ c "’ c v\( dc »/ dt ) = - dQ/dc 

Inverting, we obtain 

= hJt,q,p ) = first integral (constant) of the unperturbed problem, 
dc^/dt = E (dhJdp^Xk = ^ (dcJdp k )X k w . 

Poisson’s brackets. If the perturbations are potential —that is, if 

X k = -dQ/dq k = ~22 ( dQ l dc v)( dc vl d( lk), 

then 

dcjdt =~22 (dQ/dc v )(c^,c v ), 

where 

(c M ,c„) = 22 [( dc »/ d Pk)(dc v /dq k ) - {dc li /dq k ){dc v /dp k )\ 

= Poisson bracket of c^c,,. 

Compatibility with LB: 

'y ^ [Cj/J C^] (Cy, C^) 

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§4 ABBREVIATIONS, SYMBOLS, NOTATIONS, FORMULAE 


First-order corrections. Setting in c M = c, M) + c„i, where c^ = unperturbed values 
and c„| = corresponding first-order corrections, we have 

dcpi/dt = - 55 ( dQ o/dCvJiC'to, c v0 ) [where Q a = Q{c 0 )]. 

Lagrange’s result. Let 

1 Ik — c lkO + c lk\ t + fet 2 H-, Pk = PkO + Pkl t + Pk2 {2 H-■ 

Then, with 

Ck=qid> and c n+! =pio (k,l = 1,... ,n), 

the perturbation equations assume the canonical form: 

dc k /dt = dQ/dc n+k , dc n+k /dt = —dQ/dc k (k = 1,... ,n). 


CANONICAL TRANSFORMATIONS 
Transformations 

q = q(t,q',p') <->■ q' = q'(t,q,p)-, p=p(t,q',p') p' = p'{t,q,p), 

[with nonvanishing Jacobian \d(q' ,p')/d(q,p)\] that leave Hamilton’s equations 
form invariant. 

Requirements: 


L dt = L' dt + dF 

=> 55 p k dq k — Ft dt = 55 p k ' dq k : — Ft' dt + dF, 

=> 55 Pk d dk ~ Pk' dqk' = (H - H') dt + dF, 

where F is the generating function of the transformation (an arbitrary differentiable 
function of the coordinates, momenta, and time); and H' satisfies the Hamiltonian 
equations in the new variables. 

Alternatively, 

55 Pk dqk -Hdt= df(t,q,p) and ^ p k , dq k , - H' dt = df\t,q',p'), 

=> 55 Pk dqk ~ 55 Pk ' dqk ' ~( H ~ H ') dt 

= df(t, q,p) - df\t, q',p') = dF. 


Virtual form of a canonical transformation: 

55 Pk s qk - 55 Pk' 6 qk' = SF - 

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INTRODUCTION 


Forms of F and their relations with the corresponding conjugate variables: 


F = Fft, q, q')\ 

Pk = dF x /dq k , 

Pk' 

= —dF x j dq k '\ 

H' = H+dFjdt; 

F = F 2 {t, q,p')\ 

Pk = 9F 2 /dq k , 

qk' 

= dF 2 /dp k r, 

H' =H + dF 2 /dt, 

F = F 3 {t,p,q'): 

q k = -dF 3 /dp h , 

Pk' 

= -dF 3 /dq k r, 

H' = H + dF 3 /dt; 

F = F 4 {t,p,p'y. 

q k = -dF 4 /dp k , 

qk' 

= dF 4 /dp k r, 

H' = H + dFjdt ; 


Fi = F\ + Y Pk'Qk', 
p 3 = Fi~Y^ PkQki 

F 4 = F\+^2 Pk'Qk' ~ Y P k q k = f 2 ~ Y PkQk = F i + Y Pk^k'- 


POISSON'S BRACKETS (PB) AND CANONICITY 
CONDITIONS 

The PB of f g (where f g. h are arbitrary differentiable dynamical quantities) is 
(/,#) = Y i( d f/ d Pk)(9g/dq k ) - {df/dq k ){dg/dp k )\ = Y d{f,g)/d{p k ,q k ). 
Then 

df /dt = df /dt + (H, f) + Y ( d f ! d Pk)Qk\ 

and so for / to be an integral of the motion, we must have 

df/dt + Y, ( df/d Pk )Q k +(H,f)=0 => {H, f) =0, if / = f(q,p) and Q k = 0, 


that is, its PB with the Hamiltonian of its variables must be zero. 

[Remarks on notation: A number of authors define PBs as the opposite of ours; that 
is, as 

{f,g) = Y [i d f/ dq k)(dg/dp k ) - {df /dp k ){dg/dq k )]. 


Therefore, a certain caution should be exercised when comparing references. Also, 
others denote our Lagrangean brackets, [...], by {...}; and our Poisson brackets, 
(■ ■ •). by [• • .]■] 

Properties/theorems of PBs 


(f,g) = -(g,f) = (-g,f) =>/,/) = o 
(f,c) = 0 

(/i + fi,g) = {fug) = (fug) 

{flfug) = f\{fl,g) +fl{fug) 

=>• {cf,g) = c{f,g) 

=> !r/ = Y C kfk - then {f,g) = Y C k{fk> g) 


(anti-symmetry) 
(c = a constant) 
(distributivity) 

(c = a constant) 
(c k = constants) 


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§4 ABBREVIATIONS, SYMBOLS, NOTATIONS, FORMULAE 


9/dt(f,g) = ( df/dt,g) + (f,dg/dt) (“Leibniz rule”) 

[Actually, d/dx{f,g) = (df /dx,g) + (/, dg/dx); x = any variable] 

(/, <lk) = df'/dp k , 

(f,Pk) = ~df / dq k , 

0 <ik,qi ) = 0 , 

(Pk,Pi) = 0, 

( Pk,cj /) = 8 k i (= Kronecker delta). 

[The last three types of brackets are called fundamental , or basic , PB] 
(/, (g,h)) + {g, ( h,f)) + ( h , (f,g)) = 0, 

{{f,g),h) + ((g, /?),/) + ((/?,/), g) = 0 (Poisson-Jacobi identity) 


Theorem of Poisson-Jacobi: If/and g are any two integrals of the motion, so is their 
PB; that is, iff = c x and g = c 2 , then (f,g) = c 3 {c X 2 , 3 = constants). 

Theorem: The PBs are invariant under CT\ that is, ( f,g) p = {f,g) q > p ' = • • •; where 
/ and g keep their value, but not necessarily their form, in the various canonical 
coordinates involved. 

Canonicity conditions via PB 


L pv,Pk'] = 0, 

( Pl',Pk 0 = 0 , 


Vli'iCik'} = 0 , 

fli’i c lk') = 0 , 


[Pk',<}r\ = Ski, 
(. Pr,qk') = 8, k , 


since both Poisson and Lagrange brackets are canonically invariant. 
Theorem of Jacobi 

(i) The integration of the canonical equations 

dq k /dt = dHldp k , dp k /dt = -dH/dq k , 
is reduced to the integration of the Hamilton-Jacobi equation (H — J): 

H(t , q, dA/dq) + dA/dt = 0, 

A = A(t,q,p'): generating function (Hamiltonian action). 


(ii) If we have a complete solution of H — J\ that is, a solution of the form 

A = A(t; q 1 ,...,q„;j3 1 ,...,0„)=A(t;q,0), 

where (3= ..., /3„) = n essential arbitrary constants, and \d 1 A/dqd/3\ f 0 (non¬ 

vanishing Jacobian), then the solution of the algebraic system: 


dA/dfk = a k 

[Finite equations of motion, a: new arbitrary constants => q k = q k (t,a,/3)\, 

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INTRODUCTION 


dA/dq k = p k 

[=>■ p k = p k (t,a,/3 ): canonically conjugate (finite) equations of motion], 

constitutes a complete solution of the canonical equations. Schematically, these are 
as follows. 

Hamilton: Differential equations of motion: 

dq/dt = dH/dp , dp/dt = —dH/dq 
(If these equations can be integrated, an action function can be obtained) 

Hamilton-J acobi: 

H(t,q , dA/dq ) + dA/dt = 0 =>■ A = A(t, q,/3) 

Jacobi: Finite equations of motion: 

dA/d/3 = a —> q = q(t , a, /?); 
dA/dq = p —> p = p{t, a, /?) 

(If an action function can be obtained, then Hamilton’s equations can be integrated.) 

No significant new notations are involved in the remaining sections §8.12—§8.16 
(i.e. special topics on Hamiltonian mechanics). 


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Background 

Basic Concepts and Equations of 
Particle and Rigid-Body 
Mechanics 


Therefore it would seem right that any systematic treatment of 
classical dynamics should start with axioms carefully laid down, 
on which the whole structure would rest as a house rests on its 
foundations. The analogy to a house is, however, a false one. 
Theories are created in mid-air, so to speak, and develop both 
upward and downward. Neither process is ever completed. 
Upward, the ramifications can extend indefinitely, downward, 
the axiomatic base must be rebuilt continually as our views 
change as to what constitutes logical precision. Indeed, there is 
little promise of finality here, as we seem to be moving towards 
the idea that logic is a man-made thing, a game played 
according to rules to some extent arbitrary. 

(Synge, 1960, p. 5, emphasis added) 


In this chapter we summarize, without detailed proofs and/or elaborate discussions, 
in a handbook (not textbook) fashion, like a first-aid kit, but in a hopefully accurate 
and serviceable form, the basic concepts, definitions, axioms, and theorems of 
“elementary” (or momentum/Newton-Euler, or general) theoretical mechanics. 
This compact, highly selective, perhaps nonhomogeneous, and unavoidably incom¬ 
plete account should help to establish a common background with readers, and thus 
enhance their understanding of the rest of this relatively self-contained book. 

For complementary reading, we recommend (alphabetically): 

Fox (1967): one of the best, and most economically written, U.S. texts on elementary- 
intermediate general mechanics. 

Hamel (1909), (1912, 1st ed., 1922, 2nd ed.): arguably the best text on elementary- 
intermediate general mechanics written to date, (1927), (1949). 

Hund (1972): concise, insightful. 

Langner (1996-1997): dense, clear; “best buy.” 

Loitsianskii and Lur’e (1982, 1983): excellent. 

Marcolongo (1905, 1911/1912): rigorous, comprehensive. 

Milne (1948): interesting vectorial treatment of rigid dynamics. 

Papastavridis: Elementary Mechanics (EM for short), under production: encyclopedia/ 
handbook of Newton-Euler momentum mechanics, from an advanced and unified viewpoint; 
includes the elements of continuum mechanics. 


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CHAPTER 1: BACKGROUND 


Parkus (1966): an educational classic. 

Synge and Griffith (1959): clear, reliable. 

Synge (1960): comprehensive, encyclopedic, mature. 

Winkelmann (1929, 1930): concise, comprehensive. 

Additional references, at particular sections, and so on, will also be given, as deemed 
beneficial. 


1.1 VECTOR AND (CARTESIAN) TENSOR ALGEBRA 

Vectors: Basic Concepts/Definitions and Algebra 

Geometrically, vectors are straight line segments that, in the most general case, have 
the following five characteristics: (i) length, (ii) direction, (iii) sense, (iv) line of action 
(or carrier), and (v) origin (or point of application) on carrier; (iv) and (v) can be 
replaced with spatial origin. Also, vectors obey the well-known parallelogram law 
of addition (=> commutativity); that is, not all line segments with characteristics 
(i)-(v) are vectors (e.g., finite rotations, §1.10). Next, if only characteristics (i) (iii) 
matter, but (iv) and (v) do not, the vector is called free', if characteristics (i)—(iv) 
matter, but (v) does not, the vector is called line bound or sliding', and if all five 
characteristics matter, the vector is called point bound. As a rule, the vectors of 
continuum mechanics and the system vectors of analytical mechanics (chap. 2 ff.) are 
point bound', while those of rigid-body mechanics are line bound. 

Notation for vectors: a, b, ... (bold italic). 

Length, or magnitude, or modulus, or intensity, or norm, of a. \a\ = a > 0. If 
a = 0, the vector is called null', if a = 1 , the vector is called unit (or normalized). 

The physical space of classical mechanics is a three-dimensional Euclidean point 
space, denoted by E 3 or E', while the associated (also Euclidean) vector space is 
denoted by E 3 or E. 

An orthonormal basis (i.e., one whose vectors are unit and mutually orthogonal — 
see below) 


{«i,H 2 ,» 3 } = {" 1 , 2 , 3 } = {«*;£= 1,2,3} = {«/J 

= {u x , u y . u z } = \u xyz } = k}, (1-1-1) 

together with an “origin,” O, make up a (local) rectangular Cartesian frame: {O , u k .}. 
If the origin is not important, we simply write {«/{. 

[Since E is flat (noncurved), a single such frame, and associated rectilinear and 
mutually rectangular axes of coordinates <9—123 = O-xyz, can be extended to cover, 
or represent, the entire space: local frame —> global frame. For details, see, for exam¬ 
ple, Papastavridis (1999, pp. 84-91, 211-218), or Lur’e (1968, p. 807).] 

In such a basis, a vector a can be represented by its rectangular Cartesian com¬ 
ponents 


or 


{a\,a 2 ,afj = {a u , 3 } = {a k ',k = 1,2,3} = {a k } = {a x ,a y ,a z } = {a w }, (1.1.2a) 

a = a 1111 + a 2 u 2 + a 2 u 2 = a x u x + a y u v + a : ii z = ci k u k . 

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(1.1.2b) 


§1.1 VECTOR AND (CARTESIAN) TENSOR ALGEBRA 


In terms of the famous Einsteinian summation convention [= lone, or free, sub¬ 
scripts range over the integers 1, 2, 3, or x, y, z, while summation is implied over 
repeated (i.e., pairs) of subscripts ], we can simply write a = a k u k . In this book, how¬ 
ever, and for reasons that will gradually become clear (chap. 2), we shall NOT use 
this convention! 

Dotting (1.1.2b) with u k , and noting the six orthonormality (metric!) conditions or 
constraints: 

u k ■ u{. scalar , or dot , or inner, product of u k , Uj = &kl — £>lk (Kronecker delta) 

= 1 if k = l, =0 if kfl (k,l= 1,2,3, or x,y,z), (1.1.3) 

in extenso: 

i ■ j = j ■ j = k • k = 1 (normality), i -j = i • k = j • k = 0 (orthogonality). 

(1.1.3a,b) 


we obtain the following expression for the a-components: 


a k = a-u k . (1.1.2c) 

In such a basis, the dot product of two vectors a and b is expressed as 

a b=b a= a k u^j • b,u^j = ••• = ^ a k b k . (1.1.4) 

For a = b, the above yields the length, or norm, or magnitude, of a: 

N(a) = a = \a\ = (a ■ a) 1/2 = (E a k a k ^j > 0 (this book). (1.1.5) 

The basis {« 1A3 } is called O rtho N ormal D extral (i . e ., rig ht-handed) = OND, if, in addition to 
(1.1.3), it satisfies 

u k • (i/,. x «j) = (u k ,u r ,u s ) = e krs (permutation symbol, or alternator, of Levi—Civita) 

= +1/—1/0 according as k,r,s are an even/odd/no permutation 
of 1,2,3" 

[i.e., £123 = £23 1 = £312 = + 1 ; e 132 = £213 = £321 = — 1) 

£112 = £122 = £313 = £222 = • • • = 0 (two or more indices equal)], (1.1.6) 

or, equivalently, if 

u r xu s = Y^ £rVc«/c = e krs u k «/< = (1/2) ^ e krs {u r x uf: (1.1.6a) 

that is, («,. x u s ) k = e rsk , otherwise {iq 2 , 3 } is left-handed, or sinister, in which case 
(u k ,u r ,u s ) = —£ krs . Flenceforth, only OND bases will be used. 


• The symbols of Kronecker and Levi-Civita are connected by the following “ed iden¬ 
tity”: 


f ] Ckrs^hns ^ C skr £ s l m ^kfrm ^km fl: 


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(1.1.6b) 


CHAPTER 1: BACKGROUND 


which, for r = m (and then summation over repeated subscripts), produces 

E £ £krs£ > rs = 2SkI ; (1.1.6c) 


and this, for k = /, etc., yields 

£ E E £k " £k " = 2 (E 6kk ) = 2(3 >= 6 - (i 16d) 

• The dextrality of the orthonormal basis ( i,j,k ) (i.e., i x i =j xj = k x k = 0 ), is 
expressed by 

i x./ = ~{j x ') = k, ./ x k = -(k xj) = i, kxi = -(i x k) =j. (1.1.6e) 


With the help of the above, we express the vector , or cross, or outer, product of a 
and b as 


«xA = -(Axa) = EEE e klr a kblU r , 

that is, 

(«x *), = ££ ZklrO-kbl — EE ZrklUkbl- 

It can be shown that 

|«1 X U 2 1 2 = |«2 X M 3 I 2 = |«3 X U\ | 2 = (» ll « 2 , l / 3) 2 = +1, 

where 

(a, b, c) = a ■ (b x c) = h • (c x a) = c • (a x b) 
= (a x b) • c = (b x c) • a = (c x a) • b 

^ ^ ^ ^ ^ ^ ~'krs^l: b r C,■ 


(1.1.7a) 


(1.1.7b) 


(1.1.8a) 


(1.1.8b) 


[+, if (a, h , c) is ng7zt; —, if (a, A, c) is left', 0, if (a, b, c ) are coplanar or zero]: scalar 
triple product of a, b, c = signed volume of parallelepiped having a, b, c as sides; 
also 


[a, b, c\ = a x (b x c) = (a ■ c)b — (a • b)c 

If (a x b) x c = —c x (a x b) = (a • c)b — ( b • c)a\: (1.1.8c) 

vector triple product of a, b, c. 

The dyadic, or direct, or open, or tensor product of two vectors a and h, 

ab = a®b (fb®a, in general), (1.1.9a) 

is defined as (the tensor — see below): 

a h = a ® b = (E «*»*) ® (E b > u >) = E E a k b I (u k ®u,). (1.1.9b) 

• This product can also be defined as the tensor that assigns to each vector x the vector 

a (b ■ x): 


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§1.1 VECTOR AND (CARTESIAN) TENSOR ALGEBRA 


(a ® A) ■ x = a (A ■ x) = (A ■ x)a, (1.1.9c) 

and also 

x ■ (a® b) = (x ■ a)h = b [x ■ a). (1.1.9d) 

In components, these read, respectively, 

(a®b)-x = J2J2(a k b l Xi)u k , x-(a® b) = ^ j 'Y^(x l a 1 b k )u k . (1.1.9e) 

• It can be shown that 

[a, A, c] = [(A ® c) — (c® A)] • a. (1.1.8d) 


Tensors: Basic Concepts/Definitions and Algebra 

[For a detailed classical mostly indicial treatment of general tensors, see, for example, 
Papastavridis (1999), and owe Elementary Mechanics.} 

A second-order (or rank) tensor (or dyadic, from the Greek A YO = two ) or, here, 
simply tensor T (bold, in italics or roman) is defined as a linear transformation from 
V to F; or as a linear mapping assigning to each vector a another vector A: 

b=T-a , (1.1.10a) 

or in components 

^2 b k u k = T ki a i u k => b k = ^2 T ki a h (1.1.10b) 

or as 

h =“- T = J2J2 a k T kl u, => b) — 'y ' T ki a k , 

where 

T k i = u k -{T-ui ) = ( T-u,)-u k = T ■ (u k ®u,), (1.1.10c) 

are the Cartesian components of T (see tensor products, below). Alternatively, a 
vector/tensor//(n)th order tensor associates a scalar/vector //(it — l)th order tensor 
with each spatial direction ii d = (u^)k- direction cosines of unit vector u d ), via a 
linear and homogeneous expression in the u^ k ; that is, for a (second-order) tensor: 

T —> v d = T ■ u d (direct notation), v^ k = ^ Tki u (d)i (component notation). 

Thus (and in addition to the well-known 3x3 matrix form), T has the following 
representations: 

T = ^ 2 ^2 Tkt u k ® u i ( Dyadic or nonion representation) 

= 11 k ® h, where t k = ^ T kl u h 

= '^2t,®ui, where r, = ^ T kl u k . 

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(1.1.lOd) 
(1.1.lOe) 


CHAPTER 1: BACKGROUND 


The nine tensors {u k ® «/} span the set of all (second-order) tensors; they form an 
orthonormal “tensor basis” there. If T l2 = T 1X {= —T 2 \), etc., then T is called sym¬ 
metric {anti-, or skew- symmetric). Generally [see definition of transpose, (.. .) T , 
below]: 

Symmetric tensor: T = T J , T u = (1.1.11a) 

Antisymmetric tensor: T = — J T , T k i = —T lk {^T kk = 0, no sum!) (1.1.11b) 


Algebra of Tensors: Basic Operations 

• Sum/difference of tensors T and S: 

T ± S = E E ( r « ± -Sjfc/)»jfc ® »/■ (1.1.12a) 

• Product of T with a scalar (number) A, XT: 

XT = YY1 ( A7 */W ® «/ (1.1.12b) 

• Tensor product of T and S, T ■ S, is defined by 

™=EEEr* S ri u k ®i// S • T, in general); 

that is, 

(T- S) kl = Y T k ,S rI * {S ■ T) u = Y S kr T rl . (1.1.12c) 

• Inner, or dot, scalar product of T and S, T : S, is defined by (see trace below) 

= EE = Tr{S-T T ) = S:T, (1.1.12d) 

where Tr means “trace of.” If T = S, 

T = \ T\ = {T : T) l/2 : magnitude of T{> 0, unless J=0). (1.1.12e) 

If either of T, S is symmetric (as is almost always the case in mechanics), then, 


t -s = yy = EE TkiSik 

= Tr(T ■ S) = T•• S 
= EZ S, k T kl =Tr(S-T) = S-- T 


(1.1.12f) 


In sum, we have defined the following three tensorial products: 

( r - s )ki = Y TkrSri ( Tens ° r ). 

T ■ s = E E T » s » ( Scalar )> T • • 5 = E E Tk > Sik ( Scalar ) • 


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(1.1.12g) 




§1.1 VECTOR AND (CARTESIAN) TENSOR ALGEBRA 


The reader should be warned that these notations are by no means uniform, and so 
caution should be exercised in comparing various references. 

• Transpose of T, T J , is defined uniquely by 

(T • a) ■ b = a • (T J ■ b) , for all a,b. (1.1.12h) 

• Trace of T is defined by 

Trace of T = Tr(T) = T U + T 22 + T 33 = £ T kk . (1.1.12i) 

• Determinant of T is defined by 

Determinant of T = Det(T) = Det(T ki ) = |TJy|. (1.1.12j) 

It can be shown that: 

(i) Tr(T)= Tr(T J ) : Det(T) = Det(T T ), (1.1.12k) 

(ii) For any two vectors a and b\ 

(a ® b) T = b ig) a, 7>(a 0 b) = a • b = a k b k , Det(a® h) = 0, (1.1.121) 


(iii) For any two tensors T and S. 

(T-S) t = S t -T t , Tr(T-S) = Tr(S ■ T) = T - S, (1.1.12m) 

Det(T ■ S) = Det(T) Det(S); (1.1.12n) 

also (in three dimensions): 

Det(tT) = P Det(T), for any real number t. (1.1.12o) 

• Inverse of T, T 1 , is defined uniquely by: 

T ■ T 1 = T 1 ■ T = 1 (unit tensor), [Det(T) f 0], (1.1.12p) 

From the above, we can easily deduce that 

(i) Det(T~ x ) = (Det T)~ x (1.1.12q) 

(ii) (T-S)~ l = S~ l -T~ l (T,S: invertible) (1.1.12r) 

(iii) d/dx(Det T) = (Det T)Tr[(dT/dx) • T~% (1.1.12s) 


where T = T(x) = invertible, x = real parameter, and dT/dx = ( dT kt /dx ). 

• A tensor can be built from two vectors; but, in general, it cannot be decom¬ 
posed into two vectors. 

• Every tensor can be decomposed uniquely into a sum of a symmetric part (T' ki ) 
and an antisymmetric part ( T" k /): 

T kl = T’ kl + T" kh 

2T’ kI = T kl + T lk = 2T\ k , 2T" kl = T kl - T lk = -2T" lk - (1.1.13a) 

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CHAPTER 1: BACKGROUND 


that is, 

T=T' + T", r' = (J') T , r" = -(J") T . (1.1.13b) 

• For any tensor T and any three vectors a, b, c, the following identities hold: 

(i) a- (T ■ b) = T: (a 0 b), ^ «*(E T kl b^j = ^ E Tki{a k bj) (in components). 

(1.1.14a) 

(ii) Since 

r—EE ( T k ia,)u k , « J = EE (■ a,Ti k )u k , 

we will have T • a = a ■ T, only if T is symmetric, from which we also conclude that 

(u k ® «/) : (« r 0 u s ) = 8 kr 8 h . (1.1.14b) 

(iii) (a x T) • b = a x (J • b), (T x a) • b = T • (a x h), (1.1.14c) 

where 


Jx «=EEEE ( T kr a s e rs i)u k ® </,; 

that is, 

(J x a) fc/ = EE £ irs T k,a s , (1.1.14d) 

and 


fl X J=EEEE ( T sl a r£rsk)» k ® «/, (« X 7)*/ = E E £ krs a r T sh (l-l-14e) 

(iv) (T ■ a, T • h, T ■ c) = (Det T)(a,b,c). (1.1.14f) 

(v) T t -(T- ax T-b) = (DetT)(ax b). (1.1.14g) 


Special Tensors 

Zero tensor O. 

0-a = 0, for every vector a. (1.1.15a) 

Unit, or identity, tensor 1 : 

1 • a - a, for every vector a, (1.1.15b) 

1 = ^ ^2 8 k ju k ® «/ = ui ® a i + «2 0 a 2 + «3 0 a 3 (Dyadic form) 

= (8 k/ ) = diagonal^, 1,1) (Matrix form), 

=>Detl = +l. (1.1.15c) 


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§1.1 VECTOR AND (CARTESIAN) TENSOR ALGEBRA 


Diagonal tensor D. 

D = D n iii (g> ui + Z>22«2 ® «2 + ^ 33 m 3 ® u 3 (Dyadic form) 

= diagonal (£> n , D 22 ,D 33 ) (Matrix form). (1.1.15d) 

If _D n = £> 22 , D reduces to 

D — D\\l + (D 33 — £>ii)w 3 ® H 3 , (1.1.15e) 

a result that is useful in the representation of moments of inertia of bodies of revolu¬ 


tion. 

Alternator tensor 

E = E E £ klm U k ®u l ®u m . (1.1.15f) 

It can be shown that 

( 1 ) Det J=|7’,H = EEEEEE( 1 / 6 ) e «^AW-- ( LU5 g) 

(ii) If S is symmetric, then T.S = J T : S = (l/2)( T + J T ): S', (1.1.15h) 

If S is antisymmetric , then T:S = —(J T :S) = (1/2)(T— J T ) : 5', (1.1.15i) 

If S is symmetric and T is antisymmetric , then T:S = 0. (1.1.15j) 

(iii) If T:S = 0 for every tensor S', then T = 0, (1.1.15k) 

If T:S = 0 for every symmetric tensor S, then T = antisymmetric , (1.1.151) 


If T : S = 0 for every antisymmetric tensor .S', then J = symmetric. (1.1.15m) 


Axial Vectors 

There exists a one-to-one correspondence between antisymmetric tensors and 
vectors: given a (any) antisymmetric tensor W —that is, W = —W 1 —there exists 
a unique vector w, its axial (or dual) vector or axis, such that for every vector a: 

W - a = w x a, (1.1.16a) 

that is, (recalling the earlier definitions of products, etc.) 

W •(...) = (w x 7) •(...) = w x (...) [=$■ a• (W• a) = 0]. (1.1.16b) 

And, conversely, given a vector yv, there exists a unique antisymmetric tensor W, 
such that (1.1.16a,b) hold. In components, the above read: 

w k = -(1/2) EE £ktmW/ m — (1/2) EE km Wlm t (1.1.16c) 

^ ^ &Imk W k ^ ^ ^Ikm W k ? (1.1.16d) 

or, in matrix form: 



( 0 

W\ 2 — — W '3 

IFi 3 — W2 

W = {W lm ) = 

w 2 1 = w 3 

0 

w 2 3 = —H 7 ! 


^ PT 31 = -w 2 

^32 = W’, 

0 


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CHAPTER 1: BACKGROUND 

[Sometimes (especially in general indicial tensorial treatments) w k is defined as the 
negative of the above; that is, 

Wk = (1 /2) £ £klm W[m ^ W ‘ m = £ £ > mkWk ’ (1.1.16f) 

or 

W • a = — w x a = a x w>, (1.1.16g) 

and so, here too, the reader should be careful when comparing references.] 

It can be shown that: 

(i) The axial vector of a general nonsymmetric tensor equals the axial vector of its 
antisymmetric part; that is, the axial vector of its symmetric part (and, generally, of 
any symmetric tensor) vanishes; and, conversely, the vanishing of that vector shows 
that that tensor is symmetric. 

(ii) The axial vector of T, t (or T x , or f x ), can be expressed as 

—2t = (Ty — 7 32 )tfi + (7 31 — 7 13 )h 2 + (T u — T 2 \ )w 3 

= »i xf|+i/ 2 x( 2 +« 3 xt 3 . (1.1.16h) 

(iii) The axial vector of 

/ 2 ) w ki( u k ®»/ -«/ ® «k ), 

w has the following dyadic representation (note k, l order): 

* = £££ [-(1/2 )e rkl W kl ]u r = ■■■ = ££ (1/2)^/(«/ x «,). (1.1.16i) 

(iv) Let w = wu\. Then, 

W = w(w 3 <8 « 2 — « 2 ® « 3 ), w = « 3 • (IF-« 2 ); (1.1.16j) 

and cyclically for w = vv « 2 , w = wk 3 . 

(v) The antisymmetric part of the tensor a 8 h equals (in matrix form) 

( 0 — w 3 vv 2 \ 

vr 3 0 —w\ J, (1.1.16k) 

-w 2 wi 0 / 

where 

hi=(1/2)Ax« (note order). (1.1.161) 

(vi) The tensor a®b — b® a, where «, b are arbitrary vectors, is antisymmetric; 
and, by the preceding, its axial vector is b x a (note order). 

(vii) Let W \, w 2 be the axial vectors of the antisymmetric tensors W ,, W 2 , respec¬ 
tively. Then, 


W\ • W 2 = w 2 ® W\ - (wq • w 2 )l, Tr(W x • W 2 ) = — 2(wq • w 2 ). (1.1.16m) 

=> W -W = w ® w - (w • w)l, or W 1 -w ® w - w 2 l. (1.1.16n) 


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§1.1 VECTOR AND (CARTESIAN) TENSOR ALGEBRA 


Spectral Theory of Tensors 


DEFINITION 

A scalar A is a principal , or characteristic, or proper value, or eigenvalue, of T if there 
exists a unit vector n = (n l ,n 2 ,n 2 ) such that 

T-n = Xn, or in components ^ T kl n/ = Xn k . (1.1.17a) 

Then n is called a principal, or characteristic, or proper, or eigen-direction of T 
corresponding to that value of A. 

DEFINITION 

The principal, or characteristic , or proper, or eigen-space of T corresponding to A is 
the subspace of V consisting of cdl vectors a satisfying (1.1.17a): T ■ a = A a, that is, 
the subspace of all the eigenvectors of T. 

If T is positive definite — that is, if a - (T - a) >0 for all a f 0 — then its eigen¬ 
values are strictly positive. 

THEOREM OF SPECTRAL DECOMPOSITION (of T ) 

If T = T t (i.e., symmetric ), there exists an orthonormal basis \n i,n 2 ,n 3 } for V 
and three real, but not necessarily distinct, eigenvalues A ls A 2 , A 3 of T such that 

T n k = X k n k (k = 1,2,3; no sum), (1.1.17b) 

and 

T = T ■ 1 = T • (^2 "k ® «a ) = ^2 ( T ' "*) 18 Hk 

= ^2 X k (n k <g> n k ) (Dyadic representation) 

= diagonal (Aj, A 2 , A 3 ) (Matrix representation); (1.1.17c) 

=>■ n k • (T • it/) = T • n k • ii t = X k 6 k i (= X k or 0, according as k = l or k fi l)] 
that is, with 


n k = («(£)/: components of n k ), 

T k i = X\n^ k n^i + X 2 n( 2 )k n ( 2 )i + ^ 3 w ( 3 )/t w ( 3 )/- ( 1 . 1 . 1 7d) 

Conversely, if J = A k (n k ® n k ), with {n k } = orthonormal, then T •n k = X k n k (no 
sum). 

Depending on the relative sizes of the three eigenvalues, we distinguish the follow¬ 
ing three cases: 

(i) If A[, X 2 , A 3 = distinct, then the eigendirections of T are the three mutually 
orthogonal lines, through the origin, spanned by ti\, n 2 , « 3 . 

(ii) If A| X 2 = A 3 (i.e., two distinct eigenvalues), then the spectral decomposition 
(1.1.17c) reduces to the following (with |/f 3 1 = 1): 

T = Ai(«i ® Hi) + A 2 (7 - Hi ® «i). (1.1.17e) 

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CHAPTER 1: BACKGROUND 


Conversely, if (1.1.17e) holds with A| f A 2 = A 3 , then Ai and A 2 are the sole distinct 
eigenvalues of J; which, in this case, has the two distinct eigenspaces: (a) the line 
spanned by n u and (b) the plane perpendicular to //,. 

(iii) If A[ = A 2 = A 3 = A, in which case 


T=X1 = A(«i ® n i + n 2 <g> n 2 + n 3 <g) n 3 ) (Dyadic representation) 

= diagonal( A, A, A) (Matrix representation), (1.1.17f) 


then the eigenspace of T is the entire space V. Conversely, if V is the eigenspace of J, 
then Thas the form (1.1.17f). [For extensions of the theorem to polynomial functions 
of T see books on linear algebra; also Bradbury (1968, pp. 113-116).] The require¬ 
ment of nontrivial solutions for n, in (1.1.17a), leads, in well-known ways, to the 
characteristic (polynomial) equation for T: 

- Det(T-XI) = Det(\l — T) = D( A) = A 3 - fX 2 + I 2 X - h = 0, (1.1.18a) 


where the coefficients, or principal invariants of T (i.e., quantities independent of the 
choice of the basis used for the representation of T), are given by 

I\(T) =h = Tr(T) = Y, T kk = Aj + A 2 + A 3 , 

h(T) = I 2 = (1/2)[(Tr T) 2 — Tr(T 2 )} 

= (1/2) Taj Tu'j — TkiTik'j = A[A 2 + AjA 3 + A 2 A 3 , 

/ 3 (J) = I 2 = Det T = \T kl \ e kim T ki T, 2 T m3 = A[A 2 A 3 


= (1/6) (TrTy - 3(TrT)(TrT 2 ) + 2Tr(T i ) 


also 


If - 21 2 = X x 2 + A 2 2 + A 3 2 = Tr(T 2 ). 
[(a) It is shown in linear algebra/matrix theory that: 


(1.1.18b) 


(1.1.18c) 


• In general, that is, T = nonsymmetric , eq. (1.1.18a) has either three real roots', or one 
real and two complex ( conjugate ) roots. 

• Every tensor T satisfies its own characteristic equation; that is, eq. (1.1.18a) with A 
replaced by T: T 2 - I\T 2 + I 2 T - I 3 1 = 0 ( Cayley-Hamilton theorem). And, more 
generally, if/(A) = real polynomial in an eigenvalue A of T , then/(A) is an eigenvalue 
of f(T ); and, an eigenvector of T corresponding to A is also an eigenvector of f(T) 
corresponding to /(A). 


(b) The above show that Tr T, Tr(T 2 ), Tr(T 3 ) may also be considered as princi¬ 
pal invariants of T.] 

Further, it can be shown, that: 

(i) If TV) 2 3 are the antisymmetric tensors whose axial vectors are, respectively, the 
three orthonormal eigenvectors of (the symmetric tensor) T. /t 123 , then T has, in 
addition to (1.1.17c), the following spectral decomposition: 

T= XfNi ■N 1 ) + X 2 (N 2 -N 2 ) + X 3 (N 3 -N 3 ) + Tr(T)l; (1.1.18d) 


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§1.1 VECTOR AND (CARTESIAN) TENSOR ALGEBRA 


and, therefore, for an arbitrary vector a, 

T-a = X l (N l ■N l )-a+X 2 (N 2 -N 2 )-a+X 3 (N 3 -N 3 )-a+Tr(T)a-, (1.1.18el) 

also, 

Tr(N\ • (V,) = Tr(N 2 -N 2 ) = Tr(N 3 -N 3 ) = -2. (1.1.18e2) 

(ii) If a = axial vector of A, then 

T • a = axial vector of [— ( T • A + A • T) + Tr(T)A], (1.1.18f) 

(iii) The principal invariants of 

T = Y.Y, T kl U k ® »/ = "k 0 t k , where t k = Y^ T kt u h (1.1.18g) 

can be expressed as 

h = ll \ • t\ + u 2 • h + m 3 • hi (1.1.18h) 

I 2 = M| • (t 2 X t 3 ) + «2 • (f 3 x fj) +83 • (#! x t 2 )j (1.1.18i) 

h = t\' (h x * 3 )- (1.1.18j) 

(iv) The principal invariants of an antisymmetric tensor W are 

/, = TrW = 0, (1.1.18k) 

I 2 = W 23 2 + W 3l 2 + W {2 2 

= (—Hq) T (—vtq) + (—W 3 ) - = vv] T- nq T W 3 - , (1.1.181) 

/ 3 = Z>e? IT = 0 |m’| 2 = w 2 = (axial vector of IF) 2 ; (1.1.18m) 


from which, and from (1.1.18a), we can deduce that W has a single real eigenvalue 
A = 0. 

(v) If T is a symmetric and positive definite tensor with (=> positive) eigenvalues, 
then 

Det T > 0 (i.e., T is invertible), T l = ^ X k ~ l (n k <g> «*). (1.1.18n) 

(vi) If T is an invertible tensor, and the characteristic equation of T~ l is 

Det(T~ l - /ii) =()=>• /z 3 -/' 1 /i 2 +/'2M-/'3 = 0, (1.1.18o) 

then 

/z = 1/A; i.e., the eigenvalues of T _1 are the inverse of those of T, (1.1.18p) 

I\=h/hi I'i = hlhi r ' 3 = 1//3, (1-l.lSq) 

T 1 = (T 2 -I x T + I 2 1)/I 3 . (1.1.18r) 

Orthogonal Transformations 

A tensor T is called orthogonal (or length-preserving ) if it satisfies 

T- T t = T t ■ T = 1 => T l =T t ; (1.1.19a) 

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CHAPTER 1: BACKGROUND 


or, in components, 

E T ki( T \ = E T u T 'i = (1.1.19b) 

E ( T \l T ‘r = E T * T * = 4rl (1.1.19 C ) 

from which, since Det T = Det T t (always), and Det(T • T t ) = (Det T)(Det T J ) 
and Det 1 = 1, it follows that 


(DetT) 2 = 1 => Det T = ±1. 


(1.1.19d) 


THEOREM 

The set of all orthogonal tensors forms the (full) orthogonal group ; and the set of all 
orthogonal tensors with Det T = +1 forms the proper orthogonal (sub) group. 


THEOREM (transformation of bases and preservation of their dextrality) 

If A = (A k . k = A kk i) is a proper orthogonal tensor, or a rotation, and the basis 
{u k \ k= 1,2,3} is ortho-normal-dextral (OND), the new basis {u k e, k' = 1,2,3} 
defined by 


U k < = E A k’k«k <=> Hfc = E A kk'«k' (1.1.19e) 

is also OND. Conversely, if both {u k } and {u k f are OND, then there exists a unique 
proper orthogonal tensor such that (1.1.19e) holds. It is not hard to see that 

A k'k = COS (life-, Hfc) = COS (u k ,u k >) = A kk r, (1.1.19f) 

and in this commutativity of the indices lies one of the advantages of the non- 
accented/accented index notation: one does not have to worry about their order. 
[In a matrix representation: A = (A k ' k ), k'\ rows, k : columns; A T = (A kk >), k: rows, k'\ 
columns; where (in general): Ap 2 — A 2 g ^ A 2 p = A n > etc.] Also, in view of the earlier 
orthonormality conditions (or constraints): 

Uk' • u r = 6 kr and u k ■ u, = 6 kh (1.1.19g) 

[which, due to (1.1.19e) are none other than (1.1.19a): A • A T = A T • A = 1] only three 
of the nine elements (direction cosines) of A are independent. 

• For a vector a, we have the following component representations in {u k }, {u k '}\ 

a = E a k u k = E a k' u k'\ (1.1.19h) 

and from this, using the basis transformation equations (1.1.19e), we readily obtain 
the corresponding component transformation equations'. 

a k' = E A k'ka k = E A kk' a k & a k = E A kk>a k ' = E A k'kUk'- (1119i) 

• Polar versus axial vectors : In general tensor algebra, the word axial (vector, 
tensor) is frequently used in the following broader sense: 

(a) Vectors that transform as (1.1.19i) under any/all orthogonal transformations 
{u k } {u k i} proper or not, are called polar (or genuine ); whereas, 

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§1.1 VECTOR AND (CARTESIAN) TENSOR ALGEBRA 


(b) Vectors that, under such transformations, transform as 

a k ' = ( DetAy 1 A k' k a k = ( DetA ) ^ A k , k a k <=> 

a k = (Det ^ A kk m k , = ( DetA J y 1 ^ A kk ,a k , = (DetA) ^ A kk ,a k f, 

are called axial (or pseudo-) vectors. Hence, under a change from a right-hand system 
to a left-hand system (a reflection), in which case DetA = Det(A k ' k ) = — 1, the com¬ 
ponents of the axial vectors are unaffected; while those of polar vectors are multi¬ 
plied by — 1. Since only proper orthogonal transformations are used in this book, this 
difference disappears — all our vectors will be polar, in that sense. This polar/axial 
distinction is of importance in other areas of physics; for example, relativity, electro¬ 
dynamics (see, e.g., Bergmann, 1942, p. 56; Malvern, 1969, pp. 25-29). 

• Every orthogonal tensor is either a rotation, A —> R, or the product of a rotation 
with — 1; that is, R or — 1 • R (1: 3 x 3 unit tensor). 

• The eigenvectors of R — that is, the set of vectors satisfying R ■ x = x (R f 1) — 
build a one-dimensional subspace of V called the axis (of rotation) of R. 

• Under {u k } {h^} transformations, the components of a tensor T = (T k/ ) = 
( T k n /) transform as follows: 

T'k'l' = EE A k'k A l'lT kl = A kk' A U'Tkl, (1.1.19j) 

T ki = E E A kk' A u' Tk'V = E E A k'k A viTk'V, (1 • 1 ■ 19k) 

or, in matrix form (also shown, frequently, in bold but roman), 

(l.l.lSj): (T k ' V ) = (A k > k )(T kl )(A lv ) or T' = A T A t , (1.1.191) 

(1.1.19k) :(T kl ) = (A kk ,)(T k , r )(A n ) or T = A 1 • T' • A. (1.1.19m) 

[(a) Here, T' should not be confused with the symmetrical part of T. (1.1.13a, b). The 
precise meaning should be clear from the context. 

(b) We do not see much advantage of (1.1.191,m) over (1.1.19j,k), especially as a 
working tool in new and nontrivial situations. However, (1.1.191,m) could be useful 
once the general theory has been thoroughly understood and is about to be applied 
to a concrete/numerical problem.] 

It can be shown that: 

(i) If W is antisymmetric, then 

(a) 1 + TV is nonsingular, that is, Det(l + IT) f 0; and 

(b) (1 — W) • (1 + W) “ 1 is orthogonal 

(a result useful in rigid-body rotations). (1.1.19n) 

(ii) If 0-u \ 2 3 and 0-u\: 2 'y originally coincide, then the rotation tensor of a counter¬ 
clockwise (positive) rotation of 0-u m through an angle <j> about » 3 = uy has the 
matrix form (with ctj> = cos cj>,stj> = sin f)\ 

/ ctj) —s(j> 0\ 

A —* R = I c(j> 0 . 

V o 0 1 ) 

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(1.1.19o) 


CHAPTER 1: BACKGROUND 


Moving Axes Theorems for Vectors and Tensors 

Let us consider the following representation of a vector a and a tensor T , measured 
relative to inertial, or fixed, OND axes {u k i}, but expressed in terms of their com¬ 
ponents along (also OND) moving axes {u k } rotating with angular velocity m relative 
to {u k :}-. 

« = E a k»k, T = Tk ' Uk ( 1 ■ 1 - 20a ) 

Let us calculate their inertial rates of change [i.e., relative to the fixed axes, da/dt, 
dT/dt (t = t'\ time)], but in terms of their moving axes representations (1.1.20a) and 
their rates of change. 

(i) By d{. . .)/dt-differentiating the first of (1.1.20a) and invoking the fundamental 
kinematical result (most likely known from undergraduate dynamics)—a result 
which, along with the concept of angular velocity, is detailed in §1.7: 

du k /dt = a x u k , (1.1.20b) 

we obtain 

da/dt = ^ [(da k /dt)u k + afiio x u k )] = da/dt + to xa, (1.1.20c) 


where 


da/dt = ^ ( da k /dt)u k : rate of change of a relative to the moving axes. 

(1.1.20d) 

(ii) Repeating this process for the second of (1.1.20a) we obtain 

dT/dt = EE {( dT k j / dt)u k ® iij + T k /[(a> x u k ) ® U; + u k ® (<n x u/)\ } 

= dT/dt + coxT-Txa), (1.1.20e) 


where 

dT/dt = ^ ( dT kI /dt)u k ® uy. rate of change of T relative to the moving axes 
(or Jaumann. or corotational, derivative of T). (1.1.20f) 

Recalling the earlier results on the algebra of vectors/tensors and axial vectors [eqs 
(1.1.12), (1.1.14), (1.1.16)] we can rewrite (1.1.20c,e) in %-components as follows: 

(i) ( da/dt) k = da k /dt + (a) x a) k (fida k /dt) 

= da k /dt + E E £krs UJrCls = da kl dt + E Qk > a >- ( 1 • 1 - 2 °g) 

(ii) ( dT/dt) kl = ( dT/dt) kl + (to x T) k , - (T x (o) k , 

dT k i/dt -f 'y' y' s krs tu r T s/ y ( y' si rs u> s T kr 
= dT k i/dt + ^ Q ks T sl + ^ Q r T kr 

[after some index renaming in the last (third) group of terms, and noting that 
17/, — 12 s /] 

= dT kt /dt + y ' Q ks T s i — y ^ T ks Q s i. 

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(1.1.20h) 


§1.1 VECTOR AND (CARTESIAN) TENSOR ALGEBRA 


where 

^=EE Qki u k <S> uf. moving axes representation of angular velocity tensor 
(of these axes relative to the fixed ones); 

i.e., antisymmetric tensor whose axial vector is a>: Q • a = coxa. 


in components: 

w* = -(l/2) EE ^krs^rs ^rs — E Ekrs^k • (1.1.20i) 

Thus, in dyadic/matrix notation (see table 1.1), eq. (1.1.20e) reads 

dT/ dt = 8T/ dt + £2 ■ T - T • £2 (1.1.20j) 

[ = dT/dt + £2 • T + (£2 • T) t , if T=T J ]. (1.1.20k) 


REMARKS 

(i) Overdots, like (...)', are unambiguous only when applied to well-defined com¬ 
ponents of vectors/tensors; that is, a k , a k ', T kh T k n',...\ not when applied to their 
direct or dyadic , and/or matrix representations; that is, does a mean da/dt or da/dtl 
This is a common source of confusion in rigid-body dynamics. 

(ii) We hope that this has convinced the reader of the superiority of the indicial 
notation over the (currently popular but nevertheless cumbersome and after-the- 
factish) dyadic/matrix notations. 


Coordinate Transformations versus Frame of Reference Transformations 

See also §1.2, §1.5. Let a' and a be the values of a vector as measured, respectively, in the 
fixed (inertial) and moving (noninertial) frames. Then [recalling (1.1.19e i)], we have 

Inertial: «' = ^ a' k u k = ^ a' k ai k t\ (1.1.201) 

=> a'k’ = E A k'k a 'k ^ a'k = E A kk' a 'k' (definition of a' k , , a' k ) (1.1.20m) 

Noninertial: « = E a k»k = E °k' u k'\ (1 • 1 -On) 

=> a k i = 'y ) A k ' k a k a k = E A kk ,a k : (definition of a k i,a k ). (1.1.20o) 


Tabled Common Tensor Notations 


Direct/Dyadic 

Matrix 

Indicial/Component 

a- h = h-a (Dot product) 

a T • b = b T • a 

E a kbk 

T = a® b (Outer product) 

T = a • b T 

II 

E-i 

h= Ta 

b = Ta 

b k — E T k i a i 

h = a-T 

b T = a T • T or b = T t • a 

b k = E a iT, k 

a - T ■ h (Bilinear form) 

a T • T ■ b 

E E T kl a k bi 

T ■S (Tensor product) 

TS 

E T kr S rl 

T ■ S T (Tensor product) 

T-S t 

E T kr S) r 

T:S = S:T (Dot product) 

Tr(T-S T ) = Tr(S • T t ) 

E E TuSki 

T ■ ■ S = S ■ ■ T (Dot product) 

Tr(T-S) = Tr(S-T) 

E E T kl s lk 


Note: In matrix notation, the product dot is, frequently, omitted. 


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CHAPTER 1: BACKGROUND 


However, to relate the noninertial components a k >, a k to the inertial components a' k i, 
a' k , say, to be able to write something like 


a k = ci'k ^ a k' = a'k’, (1.1.20p) 

we need additional assumptions (postulates) or derivations—eqs. (1.1.20p) express 
frame of reference ( physical) transformations; that is, they do not follow from 
eqs. (1.1.20m,o), which are simply coordinate system (geometrical[projection) trans¬ 
formations; (1.1.20p) have to be either postulated or derived from these postulates! 
Mathematically, a frame of reference transformation is equivalent to an explicitly 
time-dependent transformation between coordinate systems representing the two 
frames: x k i = x k '(x k , t) <^> x k = x k (x k ', t ), while an ordinary coordinate transforma¬ 
tion is explicitly time-independent: x k i = x k t(x k ) x k = x k (x k i). 

For example, let us consider an inertial frame represented by the (fixed) axes 
0-x k i and a noninertial one represented by the (moving) axes 0-x k , related by the 
homogeneous transformation (common origin!) 

x k> = X A k'kX k ^ x k = X A kk’*k', (1.1.20q) 

where 

A k'k = A kk' = A k'k(t)- 

Clearly, from geometry [i.e., (1.1.20p)-type postulates]: 

x' k ’=x k t, x' k = x k . (1.1.20r) 

By (...)'-differentiating the first of (1.1.20q), and since dx' k >/dt = dx k '/dt = v' k >: 
inertial velocity of particle (with inertial coordinates x k ') resolved along inertial 
axes, dx' k /dt = dx k /dt = v k : noninertial velocity of same particle (with noninertial 
coordinates x k ) resolved along noninertial axes, we get 

v'k’ = X A k'kVk + X (' dA k'k/dt)x k = v k ' + X ( dA k'k/dt)x k , (1.1.20s) 

[invoking (1.1.20o)], where dA k > k /dt = J2^k't' A i'k = E A k'i-hk (see §1.7); that 
is, v'k 1 7 ^ v k ', even if the x k and x k > are, instantaneously, aligned (i.e., A k i k = 6 k ' k —see 
§1.7); and, similarly, from the second of (1.1.20q), v' k ^ v k , where v' k = J2 A kk’ v 'k'- 
As eq. (1.1.20s) shows, v' k t depends on both the relative orientation between x k and 
x k ' (term ^ A k ' k v k = v k '\ noninertial particle velocity, but resolved along inertial 
axes — a geometrical effect) as well as on their relative motion [term 
J2(dA k ' k /dt)x k — a kinematical effect]. There is more on moving axes theorems/ 
applications in §1.7. Vectors transforming between frames as (1.1.20p) are called 
objective — namely, frame-independent; otherwise they are called nonobjective. 
Similarly for tensors: if T' kr = T kr , or T' kl = T kh where T kr = A k'k A nT' k i 
and T k 'i’ =E 22 A k'k A nTkh that tensor is called objective. 

These concepts are important in continuum mechanics: the constitutive (physical) 
equations — namely, those relating stresses with strains/deformations and their time 
rates of change — must be objective. They also constitute the fundamental, or guid¬ 
ing, philosophical principle of the “Theory of Relativity” [A. Einstein, 1905 (special 
theory); 1916 (general theory)]. Classical mechanics does not admit of a fully phy¬ 
sically invariant formulation (although its geometrically invariant formulation is 
easy via tensor calculus), and the reason is that it is based on Euclidean geometry 

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§1.2 SPACE-TIME AXIOMS; PARTICLE KINEMATICS 


and on a sharp separation between space and ( absolute , or Newtonian) time. Hence, to 
obtain such a physically invariant mechanics, one had to change these concepts— 
and this was the great achievement of relativity: The latter replaced classical space 
and time with a more general non-Euclidean “space-time,” a fusion of both space 
and time (and gravity). In this new “space,” physical invariance is again expressed as 
geometrical invariance, via a “physical tensor calculus.” (See, e.g., Bergmann, 1942.) 

Table 1.1 summarizes, for the readers’ convenience, common vector and tensor 
operations in all three notations. [We are reminded that in matrix notation, vectors 
are displayed as 3 x 1 column matrices, so that, in order to save space, we write 
a —> a T = (a\,a 1 ,a- i ) T ] 

Differential Operators (Field Theory) 

The most important differential operators of scalar (/)/vector («)/tensor (T) field 
theory, needed not so much in analytical mechanics as in continuum mechanics/ 


physics, are 

(■ d/dr)f = grad f = df /dr = ^ (df '/dx k )u k - (1.1,21a) 

(d/dr) ® a = grad a = da/dr = E ( da l/ ax k) (u k ® U/) , (1.1.21b) 

(d/dr) • a = Tr(grada) = diva = (da k /dx k ), (1.1.21c) 

(d/dr) x a = curl a = £ krs(9 a s/dx r )u k ; (1.1.21d) 

(d/dr) 0 T = grad T = ^ ^ E (9T rs /dx k )[u k 0 (u r ®u s )\, (1.1.21 e) 

(d/dr) • T = Tr(grad T) = div T -EE (dT ks /dx k )u s - (1.1.21f ) 


where r = (x. y. z): position vector, from some origin O , on which/, a , T depend; and 
(a k ), (T kl ) are the respective components of «, T relative to an OND basis { O. u k }. 


1.2 SPACE-TIME AXIOMS; PARTICLE KINEMATICS 

Space, Time, Events 

Classical mechanics (CM), the only kind of mechanics studied here, and that of 
which analytical mechanics is the most illustrious exponent, studies the motions of 
material bodies, or systems, under the action of mechanical loads (forces, moments). 
Hence, bodies, forces, and motions are its fundamental ingredients. Before examining 
them, however, we must postulate a certain space-time, or stage, where these phe¬ 
nomena occur, so that we may describe them via numbers assigned to elements of 
length/area/volume/time interval. In CM: (i) space is assumed to be three-dimensional 
and Euclidean (C 3 ); that is, in good experimental agreement with the Pythagorean 
theorem, both locally and globally, and (ii) there is a definite method for assigning 
numbers to time intervals, which is based on the existence of perfect clocks’, that is, on 
completely periodic physical systems (i.e., such that a certain of their configurations 
is repeated indefinitely; e.g., an oscillating pendulum in vacuo, or our Earth in its 
daily rotation about its axis). Further, we assume that space and time are homoge¬ 
neous (i.e., no preferred positions), and that space is also isotropic (i.e., no preferred 
directions). A physical phenomenon that is more or less sharply localized spatially 

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CHAPTER 1: BACKGROUND 


and temporally (i.e., one that is occurring in the immediate neighborhood of a space 
point at a definite time: e.g., the arrival of a train at a certain station at a certain 
time) is called an event. Geometrically, events can be viewed as points in space-time, 
or event space ; that is, in a four-dimensional mathematical space formed jointly by 
three-dimensional space and time. There, the four coordinates of an event, three for 
space and one for time, are measured by observers using geometrically invariant, or 
rigid, yardsticks (space) and the earlier postulated perfect clocks (time). [Fuller 
understanding of this measurement process requires elaboration of the concepts 
of immediate ( spatial) neighborhood and ( temporal) simultaneity. This is done in 
relativistic physics. Here, we take them with their intuitive meaning.] 


Frame of Reference 

A frame of reference is a rigid material framework, or rigid body, relative to which 
spatial and temporal measurements of events are made, by a team of (equivalent) 
observers, distributed over that body (at rest relative to it), equipped with rigid 
yardsticks and mutually synchronized perfect clocks. Clearly, some, if not all, of 
these measurements will depend on the state of motion of the frame (relative to some 
other frame!); that is, this “coordinate-ization of events” is, generally, nonunique. 
The relation between the measurements of the same event(s), as registered in two 
such frames, in relative motion to each other, is called a frame of reference transfor¬ 
mation-, and the latter is expressed, mathematically, by an explicitly time-dependent 
coordinate transformation — one coordinate system rigidly embedded to each frame 
and “representing” it. 


Inertial Frame of Reference 

This is a frame determined by the center of mass (“origin”) of our Sun and the so- 
called fixed stars (directions of axes of frame). This primary, or astronomical, frame is 
Newton’s absolute space', and, like a cosmic substratum, is assumed to exist (in 
Newton’s words) “in its own nature, and without reference to anything external, 
remains always similar and immovable.” Similarly, Newton assumes the existence of 
absolute time, which is measured by standard clocks, and flows uniformly and inde¬ 
pendently of any physical phenomena or processes —something that, today, is con¬ 
sidered physically absurd: “[I]t is contrary to the mode of thinking in science to 
conceive of a thing (the space-time continuum) which acts itself, but which cannot 
be acted upon” (Einstein, 1956, pp. 55-56). In spite of its logically/epistemologically 
crude and no longer tenable foundations, CM is astonishingly accurate in several 
areas. For example, the planet Mercury in its motion around our Sun sweeps out a 
total angle of 150,000°/century; which is only 43" more than the Newtonian predic¬ 
tion! In this sense, of Machean Denkokonomie (~ Principle of economy, in the for¬ 
mation of concepts), CM is an extremely economical intellectual and practical 
investment. 

As the mathematical structure of the Newton-Euler laws of motion shows 
(§1.4,5), any other frame moving with (vectorially) constant velocity, relative to 
the primary frame, is also inertial; so we have a family, or group, of secondary 
inertial frames. In inertial frames, the laws of motion have their simplest form [the 
familiar “force equals mass times acceleration (relative to that frame)”]. 

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§1.2 SPACE-TIME AXIOMS; PARTICLE KINEMATICS 


Particle Kinematics 

The instantaneous position , or place , of a particle P relative to an origin, or reference 
point, O, fixed in a, say, inertial frame F in E 3 , is given by its position vector 
r = (x,y,z); where x, y. z are at least twice (piecewise) continuously differentiable 
functions of time t. Clearly, r depends on O while x, y, z depend on the kind of 
coordinates used in F. (Also, we are reminded that in kinematics, the frame does not 
really matter; any frame is as good as any other.) At time t, a collection of particles, 
or body B , occupies in E 3 a certain shape, or configuration, described by the single¬ 
valued and invertible mapping 

r = f(P,t ): Place of P, in F, at time t: (1.2.1a) 

from which, inverting (conceptually), we obtain 

p=r\v,t). (i. 2 .ib) 

A motion of B is a change of its configuration with time; that is, it is the locus of r 
of each and every P of B, for all time in a certain interval. Formally, this is a one- 
parameter family / of configurations with time as the (real) parameter. 

Often, especially in continuum mechanics, the motion of P is described as 

r=f(r 0 ,t)=r(r 0 ,t), (1-2.2) 


where r a is the position of P at some “initial or reference” time; that is, a reference 
configuration used as the name of P (see also §2.2 ff.). The above representation 
— in addition to being single-valued, continuous, and twice (piecewise) continuously 
differentiable in t — must also be single-valued and invertible in r 0 ; that is, one-to-one in 
both directions. (In mathematicians’ jargon: a configuration is a smooth homeomorphism 
of B onto a region ofE 3 .) 

The velocity and acceleration of P , relative to a frame F, are defined, respectively, 
by (assuming rectangular Cartesian coordinates) 

v = dr/dt = (dx/dt,dy / dt,dz / dt), 

a= dv/dt = d 2 r/dt 2 = ( d 2 x/dt 2 ,d 2 y/dt 2 ,d 2 z/dt 2 ). (1.2.3) 

Clearly, v and a depend on the frame, but not on its chosen fixed origin O. The 
representation of the velocity and acceleration of P, relative to F, moving on a general 
space, or skew, (F-fixed) curve C, along its natural, or intrinsic, ortho-normal-dextral 
moving trihedron/triad {u t , u n , uf\ = {t,n,b} (see fig. 1.1 for definitions, etc.) is 

v = dr/dt = {dr/ds)(ds/dt) = (ds/dt)t = v,t (= v t t + 0n + 0b), (1.2.3a) 

a = dv/dt = ( d 2 s/dt 2 )t + [(ds/dt) 2 /p]n = (dv,/dt)t + ( v, 2 / p)n 

= (dv,/dt)t + (v 2 / p)n (= a t t+ a„n + Ob, see below). (1.2.3b) 

• The speed of P is defined as the magnitude of its velocity: 

Speed = v = |v| = |v,| = \ds/dt\ = + [(dx/dt) 2 + ( dy/dt ) 2 + (dz/dt) 2 ~\ V ~ f 0; 

i.e., v t = v ■ t = s = ±v. (1.2.3c) 

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CHAPTER 1: BACKGROUND 


Center of 
Curvature 




Figure 1.1 Natural, or intrinsic, triad representation in particle kinematics, s: arc coordinate 
along C, measured (positive or negative) from some origin A on C\ p\ radius of curvature of C at 
P(0 < p < oo); orthonormal and dextral (OND) triad: {Uf, u n , u b } = {f, n, to); (oriented) tangent: 
ut = t = dr/ds) [always pointing toward (algebraically) increasing values of s (= positive 
C-sense)]; (first, or principal) normal: u n = n = p(dt/ds ) (always in sense of concavity, toward 
center of curvature); (second) normal, or binormal: u b = to = f x n; osculating plane: plane 
spanned by f and n (locus of tip of acceleration vector); rectifying plane: plane spanned by fand to; 
normal plane: plane spanned by n and to; f • (nx b) = (f, n, b) = +1 > 0. [More in §1.7: (1,7.18a)ff.[ 


Hence, in general, since v 2 = v t v t = (,sj 2 , 

a=\a\ = \dv/dt\ = [{dv/dt) 1 + (v 4 /p 2 )] 1/2 ^ \dv,/dt\ = \d 2 s/di 2 \ = \dv/dt\; 
i.e., a, = s (tangential accel’n), a„ = ( s) 2 /p (normal a.), a b = 0 (binormal a.) 

=> v a = ss = (i 2 /2)‘ , vxa = (s 3 /p)b. (1.2.3.d) 


REMARKS 

(i) The difference between speed v = \dr/dt\ = |v| = \ds/dt\ > 0 and the ( tangential ) 
velocity component v, = v-t = ds/dt = ± v [i.e. by equation (1.2.3c): v, = +v > 0 if 
ds > 0, and v, = — v < 0 if ds < 0] results from the oriented-ness of the curve C [i.e. that 
it is equipped with (a) an origin A, and (b) a positive /negative sense of traverse =T ± ,s]; 
i.e. in any motion of P along it, the unit tangent vector t = dr/ds (f 0) points always 
towards the increasing arcs s (just like i = dr/dx always points towards the positive/ 
increasing x- see below). Fortunately, this v, versus v difference (almost never noticed in 
the literature) rarely results in fatal errors. 

(ii) Thus, it becomes clear that s [= (intrinsic) arc/path/trajectory curvilinear 
coordinate/abscissa, of P relative to a chosen C-origin A] is the “natural” curvilinear 
generalization of the rectilinear position (-al) coordinates x, y, z (and [t. n.b) are of 
[i. j, k), respectively). 

(iii) The equation s = s(t), resulting by integrating ds = ±.\v{t)\dt = ± v(t)dt [say, 
from t(A) to f(P)], is referred to as the equation / law of motion of P on C. 

(iv) Last, (a) the length of the arc AP is defined as the absolute value of ,v, |,v > 0, 
while (b) the (total) distance traveled by our particle P, along C, from an origin A to its 
current /final position (i.e. what a car odometer shows) is defined by: 

I \ds\ (> |s| >0). (1.2.3e) 

J origin —f current C-position 

For details on arc length, admissible curve parametrizations etc, see works on differential 
geometry. 


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§1.2 SPACE-TIME AXIOMS; PARTICLE KINEMATICS 


It can be shown that [with the additional notation (...)' = d{- ■ •)/ du \: 


(i) t = r' /s' = r' / (r' • r') 1 ! 2 = ( dx/ds)i + ( dy/ds)j + ( dz/ds)k, (1.2.4a) 

(ii) n = p{dt/ds) = p{d 2 r/ds 2 ) = p[{d 2 x/ds 2 )i + (d 2 y/ds 2 )j + ( d 2 z/ds 2 )k\ 

= P{t'/s') = pt '/(/'' -r') 1/2 

= [p/{y' -r'f 2 W •r') x ^ 2 v" — (r 1 • r')~ x ^ 2 {r' •r")r'] 

= [p/( r ' -r')r" -(/•/>']; (1.2.4b) 

(iii) k = \/p = \dt/ds\ = \d 2 r/ds 2 \ (^ 0 , 0 ^ p < + 00 ): 

(first) curvature of C, at P, (1.2.4c) 


(iv) 


(v) 


K 2 = l/p 2 = (/ X r") 2 /{r'-r'f = [{/■ r'){r" ■ r") - (/*' • r") 2 ]/[r' • r ') 3 


[= {d 2 x/ds 2 ) 2 + {d~y / ds 2 ) 2 + {d 2 z/ds 2 ) 2 , if u = s]\ 


(1.2.4d) 


/ /. 
r =st, 


r w = s"t + s V = s"t + s'[{dt / ds){ds / du)] [ = s"t + (s') 2 (dt/ds)] 

= s"t + k(s') 2 ii = s"t + [{s') 2 / p]n (1.2.4e) 

[= (dv,/dt)t + {v 2 / p)n, v,v, = vv = {ds/dt) 2 ; if u = t]; 


(vi) t = v/(ds/dt) = v/v „ (1.2.4f) 

n - p[r 2 a - (v • a)v]/v 4 = - • ( 1 - 2 .4g) 

b = t x « = p[(r/r/cfe) x (J 2 r/cf.r)] = p(r' x r")/{r' -r ') 3 ^ 2 

= p(v x a)/v , 3 = p(v x a)/v t v 2 ; (1.2.4h) 

(vii) k 2 = 1/p 2 = (v x a) 2 /v 6 = [v 2 a 2 — (v-a) 2 ]/v 6 ; (1.2.4i) 

(viii) a, = a-t= [v x (dv x /dt) + v y (dv y /dt) + v z {dv z /dt)]/vp (1.2.4j) 


(ix) = |a x #| = {[v x (</v ; ,/t/t) — v J ,(t/v x /t/?)] 2 + [v y (</v z /t/t) — v,(iiv v /Jt)] 2 

+ [v.-f^M) - v x {dv z /dt)] 2 } l/2 / (v Y 2 + v y 2 + v z 2 ) 1/2 . (1.2.4k) 


• In plane polar coordinates, the position/velocity/acceleration of a particle P are 
(where a,, u,,;. unit vectors along OP and perpendicular to it, in the sense of increasing 
r, </> respectively): 


du r /dt = {dfi/dfju^ and du^/dt = —{d<j>/dt)up 
or du r = d<j)u ( p and du^ = —d<j>u r , 
r = r u r [= (r)u r + (0)ity], (1.2.5a) 

v = ( 1 dr/dt)u r + r{d</>/dt)u^ = v r u r + v^Ua, (1.2.5b) 

a = [d 2 r/dt 2 — r{d</>/dt) 2 ]u r + {r~ x d / dt[r 2 (d<f> / dt)]}u^, 

= [d 2 r/dt 2 — r{d(f>/dt) 2 ]u r + [2{dr / dt){d(/> / dt ) + r{d 2 (j)/dt 2 )\u$ 

= a (r) u r + a w « 0 . (1.2.5c) 


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CHAPTER 1: BACKGROUND 


The vectors ( d(j>/dt)k and ( d 2 cpldt 2 )k are, respectively, the angular velocity and 
angular acceleration of the radius OP = r relative to O—xy. It can be shown that 

(i) a, = a-t= ±(v ,.a r + v^af) / ( v r 2 + v 0 2 ) 1/2 [+ if v, > 0, - if v, < 0], 

(1.2.5d) 

(ii) The rectangular Cartesian components of the velocity and acceleration are, 
respectively, 

dx/dt = ( dr/dt ) cos <f> — [r(d(/)/dt)\ sin p, dy/dt = ( dr/dt ) sin <p + [r(d(j)/dt)] cos </; 

(1.2.5e) 

d 2 x/dt 2 = [ d 2 r/dt 2 — r(dcf>/dt)~} cosp — [2 {dr / dt){d<p / dt) + r{d 2 <f>/dt 2 )\ simp, 
d 2 y/dt 2 = [ d 2 r/dt 2 — r{d<p/dt) 2 ] sin p + [2 (dr/dt)(d<f>/dt) + r(d 2 (/)/dt 2 )\ sin p, 

(1.2.5f) 


and, inversely, 

dr/dt = (xv x +yVy)/(x 1 + y 2 ) 1 ^ 2 , dp/dt = (xv y ~yv x )/{x 2 +y 2 ), etc. (1.2.5g) 

[A more precise notation of vector components along various bases of orthogonal 
curvilinear (i.e., nonrectangular Cartesian) coordinates is introduced below.] 

• In general (i.e., not necessarily plane) motion, the areal velocity dA/dt of a 
particle equals 

dA/dt = (1/2) |r x v| = (1/2) | angular momentum of particle about origin , 

per unit mass\. (1.2.6a) 

It can be shown that (assuming r f 0) 

d 2 A/dt 2 = (r x v) • (r x «)/2|r x v|. (1.2.6b) 


Velocity and Acceleration in Orthogonal Curvilinear Coordinates 

(A certain familiarity with the latter is assumed—otherwise, this topic can be omitted 
at this point.) In such coordinates, say q = (q \, q 2 -qfj = (< 71 , 2 , 3 ) [see fig. 1.2(a)] the 
position vector r, of a particle P , is expressed as: 

r = x{q)i + y{q)j + z(q)k = r(q), (1.2.7a) 

and so the corresponding unit tangent vectors along the coordinate lines < 71 , 2,35 w i, 2 , 3 > 
are 

M, = (\/h x ){dr/dq x ) = e x /h u 
« 2 = ( 1/^2 ){.dr/dq 2 ) = e 2 /h 2 , 

«3 = ( 1/^3 )(.dr/dq 2 ) = e 3 //? 3 , 

where 


«*•«/ = 4/ (/c,/= 1,2,3); 


(1.2.7b) 


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§1.2 SPACE-TIME AXIOMS; PARTICLE KINEMATICS 



Figure 1.2 (a) General orthogonal curvilinear coordinates; 

(b) cylindrical (polar) coordinates: x = rcostfi, y = rsin<^>, z — z, r = \OP'\; 2 ,i = h r ^ z = 1,r, 1; 

(c) spherical coordinates: x = (rsin 6) cos (f>, y = (rcosff) sin <j>, z = r cosd, r = \OP\; 
h i,2,3 = h rM = 1, Arsine. 


and since 

dr/dcji = (dx/dqffi + ( dy/dq x )j + ( dz / dq { )k, dr/ dq 2 = ■ ■ ■, 

the (normalizing) Lame coefficients h\ 22 are given by 

hi = \dr/dqi \ = [( dx/dq ,) 2 + [dy/dq { ) 2 + (&/<9t/i) 2 ] 1/2 , h 2 = 


dr/dq 3 = ■■■, 
(1.2.7c) 


• ,h 2 — ■■■. 
(1.2.7d) 


We notice that 

cos (u k ,x) = u k • i = [(1 /h k )(dr/dq k )\ •/= (1 /h k ){dx/dq k ), etc., 


or, generally, with x = x\, y = x 2 , z = x 3 , and i = i\, j = i 2 , k = / 3 , 


cos (u k ,x,) = u k -i, = (1 /h k )(dxi/dq k ). (1.2.7e) 

As a result of the above, and since dr/dq k = e k = /ia-h^, the arc length element ds, 
velocity v, and speed |v| of P are given, respectively, by 


(i) 

(>i) 

(hi) 


ds = \dr\ = 1^ 0 dr/dq k ) dq k = (/;f dqf + h 2 2 dq 2 2 + h 3 2 dq 3 2 )' /2 , (1.2.7f) 

v = dr/dt = Y ( dr/dq k )(dq k /dt) = Y v k e k = Y v k(h k u k ) = Y v (k) u k, 

(l-2.7g) 

|v| = v=(h l 2 v l 2 + h 2 2 v 2 2 +h 3 W) l/2 , (1.2.7h) 


where 

dq k /dt = v k : “contravariant” or generalized component of v along q k , (1.2.7i) 

V(A) = h k (dq k /dt) = h k v k : corresponding physical component 

(with units of length/ time ), 


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( 1 . 2.79 













CHAPTER 1: BACKGROUND 

Next, we define the generalized and physical components of the particle acceleration 
a as 

a k = a-e k = ( dv/dt ) • (■ dr/dq k ), a [k) =a-u k = a- (e k /h k ) = a k /h k . (1.2.7k) 


REMARKS 

(i) For an arbitrary vector h, in general orthogonal curvilinear coordinates, we 
have the following representations: 

h = Y b k e k = Y b k e k = Y b k (e k /h k ) = Y ( b k /h k )(e k /h k ) = Y b {k)U k 
where 


e k • e/ 

= gkl ~ 

= 0 , if kfl- = 

h k 2 if k = /; 

e k • ry = 

II 

k 1 kl Ik 

, e -e =g =g 


=> 

1 / b k , g kk 

1 1 i 2 

1 /g = h , 

Det(g k i) 

= hfh 2 

2 hi 2 ', e k = e k /h k 2 -, 

b k = 

b ■ e k = 

b ■( h k u k ) = h k (b 

• «fc) = hb {k y, 

b k =b- 

e k = b- 

II 

^5 

that is. 







b (k) 

= b k h k : 

= b k /h k \ physical 

l components 

of b, b k = 

= b k /h k 2 : 

3 

II 

II 


(ii) Strictly speaking, q k should have been written as q k ; and, consequently, v k 
as v k \ 

(iii) In rectangular Cartesian coordinates/axes (this book), clearly, h k = 1 =>■ b( k ) = 
b k = b k . 

(iv) For the extension of the above to general curvilinear coordinates, see books 
on tensor calculus; for example, Papastavridis (1999, chap. 2, especially §2.10). 

From the first of (1.2.7k) we obtain successively (what are, in essence, the famous 
Lagrangean kinematico-inertial transformations, to be generalized and detailed in 
chaps. 2 and 3): 

a k = a- e k = (dv/dt) • ( dr/dq k ) = d/dt [v • (dr/dq k )\ — v • d / dt(dv / dq k ) 

| and, using the basic kinematical identities: 

(a) dr/dq k = dv/dv k [from (1.2.7g)] 

(b) d/dt(dr/dq k ) 

= Y d/dq,(dr/dq k )(dq,/dt ) + d/dt(dr/dq k ) 
= d/dq k (Y (' dr/dqi)(dq,/dt) + dr/dtj 
= dv/dq k ; 

i.e., d/dt(dv/dv k ) — dv/dq k = 0 

= d/dt\v • (dv/dv k )] - v • (dv/dq k ) 

= d/dt[d/dv k (v 2 /2)\ - d/dq k (v 2 /2) 

(since v • v = v 2 ); 

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(1.2.71) 


§1.2 SPACE-TIME AXIOMS; PARTICLE KINEMATICS 


and, invoking the second of (1.2.7k), we get, finally, the Lagrange an form: 

a (k) = a k /h k = (1/A*) [d/dt{dT/dv k ) - dT/dq k ], (1.2.7m) 

where 

T = v 2 /2 = {\/2)[h l 1 {dq x /dtf + h 2 2 {dq 2 /dt) 2 + h 3 2 (dq 3 /dt) 2 ] l/1 
= (1/2)(A 1 2 v 1 2 + /i 2 2 v 2 2 + A 3 2 v 3 2 ) 1/2 : 

kinetic energy of a particle of unit mass (i.e., m = 1). (1.2.7n) 


Application 

(i) Cylindrical {polar ) coordinates [fig. 1.2(b)]. Here, x = rcosf, y = r sin z = z, 
and, therefore, 

ds 2 = ds 2 + dsf 2 + ds 2 = dr 2 + r 2 dcj> 2 + dz 2 , (1.2.8a) 

from which we immediately read off the following Lame coefficients: 

h i —> h r = 1, h 2 —> h ( j > = r, /? 3 —> A- = 1. (1.2.8b) 

Hence, the “unit kinetic energy’’ equals 


2 T = (ds/dt) 2 = v 2 = [{dr/dt) 2 + r {df/dt) + (dr/tA) 1 ] = v, 2 + r 2 v^ + v z 2 , 




(1.2.8c) 

and so, 

by (1.2.71), the (physical) components of the acceleration are 


a W 

—> «(,.) = d/dt{dT/dv r ) — dT/dr = d 2 r/dt 2 — r{df/dt) 2 , 

(1.2.8d) 

<3(2) 

->■ % i) = (l/p)[rfM(9r/av^) - dT/d(j>\ 



= (l/r){A/<A[r 2 (A<(>/Ar)]} = r{d 2 cj)/dt 2 ) + 2{dr / dt){df / dt), 

(1.2.8e) 

fl (3) 

—> «( z ) = d/dt{dT/dv z ) — /dz = d 2 z/dt 2 . 

(1.2.8f) 


(ii) Spherical coordinates. Here, x= (r sin 6) cos </>, y= (r cos 9) sin <j>, z = rcos9 
[fig. 1.2(c)]. Using similar steps, we can show that 

—> a( r ) = d/dt{dT/dv r ) — dT/dr = d 2 r/dt 2 — r{d6/dt) 2 — r{d<f>/dt) 2 sin 2 9\ 

(1-2.8g) 

«(2) -► 0(9) = (i/r)[5r/5v e - ar/96»] 

= (1 /r){d/dt[r 2 (d9/dt)\ — r 2 {dcj)/dt) 2 sindcos d}; (1.2.8h) 

«( 3 ) —> = (1 /r sin 9) [d/dt{dT/dvf) — dT/d<j>\ 

= {l/r sm9){d/dt[r 2 (d(/)/dt) sin 2 0]}; (1.2.8i) 

v x = dx/dt = {dr/dt) sin 9 cos <f> + r{d9/dt) cos0cos</> — r{d<j>/dt ) sin 9 sin f, 

(1.2.8J) 


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CHAPTER 1: BACKGROUND 


v y = dy/dt = ( dr/dt) sinf?sin</> + r(dO/dt) cost? sin <(> + r(d<p/dt) sin t?cos<(>, 

(1.2.8k) 

v z = dz/dt = (dr/dt) cos 9 — r(dO/dt) sinf?; (1.2.81) 

a x = d 2 x/dt 2 = [d 2 r/dt 2 — r(dd/dt) 2 — r(d<f>/dt)~] sin(9cos<(> 

+ [r(d 2 0/dt 2 ) + 2(dr/dt)(d9/dt)\ cos 9 cos cj) 

— [r(d 2 (j)/dt 2 ) + 2 (dr/dt)(d(j>/dt)\ sin 8 sin cj) 

— 2r(d<j>/dt)(d9/dt) cost? sin ((>; (1.2.8m) 

a y = d 2 y/dt~ = [ d 2 r/dt~ — r(d0/dt) 2 — r(d<j>/dt)~] sint?sin</> 

+ [r(rf 2 t?/iir) + 2(dr/dt)(d6/dt)\ cos0sin0 
+ [r(6? 2 </>/<ft 2 ) + 2 (dr/dt)(dcj)/dt)\ sin t? cos</> 

+ 2 r(dcj>/dt)(d9/dt) cosOcosrf), (1.2.8n) 

a z = d 2 z/dt 2 = [d 2 r/dt 2 — r(dO/dt ) 2 ] cos (9 

— [r(d 2 0/dt 2 ) + 2(dr/dt)(d9/dt)\ sin#; (1.2.8o) 

and, inversely, 

dr/dt = [x(dx/dt) + y(dy/dt) + z(dz/dt)]/(x 2 + y 2 + r 2 ) 1 ^ 2 , (1.2.8p) 

d8/dt = { \x(dx/dt) + y(dy/dt)\z — (x 2 + y 2 ) (dz/dt)} / (x 2 +y 2 )*^ 2 (x 2 + y 2 + z 2 ), 

(1.2.8q) 

d(f)/dt= [x(dy/dt) — y(dx/dt)]/(x 2 + y 2 ); (1.2.8r) 

and 

d 2 r/dt 2 = • • •, d 2 0/dt 2 = • • ■, d 2 (f)/dt 2 = • • ■, 

in complete agreement with (1.2.5). 

REMARK 

From now on, parentheses around subscripts (employed to denote physical compo¬ 
nents) will, normally, be omitted; that is, unless absolutely necessary, we shall simply 
write a r , ag, for U( r) , a^, a^s, respectively, etc. 


1.3 BODIES AND THEIR MASSES 

Body or System 

A body or system is an ordinary three-dimensional material object whose points fill a 
spatial region completely; or a continuous connected three-dimensional set of mate¬ 
rial points, or mass points, or particles, such that any part of it, no matter how small, 
possesses the same physical properties as the entire object. The interactions of 
bodies, under the action of forces/fields, produces the various physical phenomena. 

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§1.3 BODIES AND THEIR MASSES 


Bodies are usually classified as solids, fluids, and gases. 


• The rigid body is a special solid whose deformation (or strain), relative to its other 
motions, can be neglected; and whose geometric form/shape and spatial material 
distribution are invariable. 

• The particle is a special rigid body whose rotation, relative to its other motions, can be 
neglected; it is small relative to its distance from other bodies, and its motion as a 
whole is virtually unaffected by its internal motion. It is a special localized continuum 
of infinite material density (see below). 


The complete characterization of a particle requires specification of its spatial posi¬ 
tion and of the values of its associated parameters (e.g., mass, electric charge). The 
former varies with time but the latter, since they describe the internal constitution of 
our particle, do not; if they did, we would have a more complex system. 

Whether one and the same body or system will be modeled as deformable con¬ 
tinuum, or rigid, or particle, etc., depends on the problem at hand. Below, we show 
such a problem to model correspondence for the system Earth: 


Problem 

Orbit around the Sun 
Tides and/or lunar eclipses 
Precession of the equinoxes 
Earthquakes 
etc. 


Mathematical Model 
Particle 
Rigid sphere 
Rigid ellipsoid 
Elastic sphere 


Mass 

To each body, B, that instantaneously occupies continuously a spatial region of 
volume V, we assign, or order, a real, positive and time-independent number expres¬ 
sing the quantity of matter in B, its mass m: a primitive concept with dimensions 
independent of the (also primitives) length and time. Symbolically, we have 


B —> m(B) = m = dm 
where (continuity hypothesis) 


(dm/dV)dV = pdV > 0, 
v Jv 


(1.3.1) 


p = [lim(d?n/zl V)\ Av ^ 0 = dm/dV\ mass density, or specific mass, of B 

(a piecewise continuous function of t and /•) (1.3.2) 


and m = constant, for a given body ( conservation of mass). 

The above imply that the mass is additive : the mass of a body, or system, equals 
the sum of the masses of its parts; with some intuitively obvious notation: 

m(B) = m(B\ + Bf) = m{B{) + m(B 2 ) = m { + m 2 ■ (1.3.3) 


REMARKS 

(i) For so-called “variable mass problems” (clearly, a misleading term); for exam¬ 
ple, rockets, chemical reactions, see Fox (1967, pp. 321-326) and, particularly, 
Novoselov (1969). 

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CHAPTER 1: BACKGROUND 


(ii) To describe several bodies, including possible gaps, via (1.3.1) and (1.3.2), we 
may have to assume that in some regions p = 0. 

(iii) Mathematically, mass additivity can be expressed as follows: Consider an 
arbitrary subset of the body B , b. If we can associate with b a nonnegative real 
number m(b), with physical dimensions independent of those of time and length, 
and such that 

m{b\ U bf) = m(bi) + mlbf) [U = union of two sets] 
for all pairs b\ and b 2 of disjoint subsets of b; and 

m(b) —> 0, 

as the volume occupied by b goes to zero; then we call B a material body with mass 
function m. The additive set function m(b ) is the mass of b; or the mass content of the 
corresponding set of points occupied by b. The above properties of m {...) imply the 
existence of a scalar field p = mass density of B. defined over the configuration of B. 
such that (1.3.1) holds. 


Impenetrability Axiom (and One-to-One Event Occurrence) 

Not more than one particle may occupy any position in space, at any given time. 
More generally (continuum form), if, during its motion, the material system initially 
occupies the spatial region V a , and later the region V, then the relation between V 0 
and V is mutually one-to-one, and piecewise continuously differentiable (for the 
associated field functions). Discontinuities (e.g., rupture, impact) and accompanying 
loss of uniqueness can occur only across certain (two-dimensional) boundary sur¬ 
faces. 


Remarks on Particles, Bodies, Mathematical Modeling, and so on 

(i) A finite, or extended , body B or system 5 can be treated exactly, or approxi¬ 
mately, as a particle in the following three cases: 


(a) If B undergoes pure translation; that is, all its points describe congruent paths with 
(vectorially) equal velocities and accelerations. In this case, any point of B can play 
the role of that particle. 

(b) If the description of the kinetic properties of B requires only the investigation of the 
motion of its center of mass (§1.4). 

(c) If B is such that its dimensions are so small (or its distances from other bodies, its 
environment, are so large) that its size can be neglected; and its motion can be 
represented satisfactorily by the motion of either its mass center or any other inter¬ 
nal point of it. Such bodies we call small. 

• In cases (b) (always) and (c) (usually) that particle is the mass center. 

• Cases (a, b) are exact, while (c) is only approximate. 

• In case (a), that particle describes the motion of B completely, in (b) only partially 
(the motion about the mass center is neglected), and in (c) with an error depending on 
the neglected dimensions of B. 


From such a continuum viewpoint, a particle is viewed not as the building block 
of matter, but as a rigid and rotationless body ! As Hamel (1909, p. 351) aptly 
summarizes: “What one understands, in practice, by particle mechanics 

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§1.4 FORCE; LAW OF NEWTON-EULER 


(Punktmechanik) is none other than the theorem of the center of mass 
(Schwerpunktsatz). ’ ’ 

(ii) Both models of a body—that is, the one based on the atomistic hypothesis 
(body as a finite, discrete, set of material points, or particles; namely, small hard balls 
with no rotational characteristics) and the other based on the continuity hypothesis 
(body as a family of measurable sets, with associated additive set functions represent¬ 
ing the mass of that set)—have advantages and disadvantages; and both are useful 
for various purposes. The sometimes (in some engineering circles) fierce debate for/ 
against one or the other viewpoint, we consider counterproductive and petty hair¬ 
splitting; and so we will use both models as needed. Such dualisms are no strangers 
to physics (e.g., particles/corpuscules vs. waves/fields in atomic phenomena) and 
constitute a creative, dialectical, stress in it. 

Thus, we will view the rigid body (§1.8 If.) either as a (finite or infinite) set of particles 
whose mutual distances are constrained to remain invariable (i.e., fixed in time); or, 
more conveniently, as a rigid continuum, and accept the Newton-Euler law of motion 
for its differential mass elements as for a particle (§1.4, §1.6). In the discrete model, the 
building block is the single “sizeless,” but possibly quite “massive,” particle of mass 
m k > 0 (k = 1,2,...); while, in the continuum model, it is the differential element with 
mass dm = pdV > 0. In sum, we shall adopt the logically unorthodox, but quite fertile 
and successful, dialectical compromise: particle language and continuum notation ; and 
eventually (chap. 3 ff.) we will end up with ordinary differential equations. 

[In general, it is extremely difficult, if not impossible, to go by a limiting process 
from a statement about particles to one about continua; whereas, conversely, con¬ 
tinuum statements formulated in terms of Stieltjes’ integrals, like our earlier S (■■■)'■ 


St--) dm - XX- -)k m k (discrete), 


or 


(...) dm (continuum), 


lead to the same statements for discrete systems without much difficulty, almost 
automatically. See, for example, Kilmister and Reeve (1966, pp. 129-131).] 


1.4 FORCE; LAW OF NEWTON-EULER 

[l]n the concept of force lies the chief difficulty in the whole of 
mechanics. 

(Hamel, 1952; as quoted in Truesdell, 1984, p. 527) 

Jeder weip aus der Erfahrung, was Schwerkraft ist; jede gerichtete 
Physikalische Grope, die sich mit der Schwerkraft in 
Gleichgewicht befinden kann, ist eine Kraft! 

[Approximate translation: Everyone knows from experience what 
gravity is; every directed physical quantity that can be in 
equilibrium with gravity is a force! (emphasis added).] 

(How Hamel used to begin his mechanics lectures; quoted in 

Szabo, 1954, p. 26) 

The fundamental law of mechanics [i.e. mass x acceleration = 
force] is a blank form which acquires a concrete content only 
when the conception of force occurring in it is filled in by 
physics. 

(Weyl, 1922, pp. 66-67) 

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CHAPTER 1: BACKGROUND 


Local Form of Newton-Euler Law 

To each and every material particle P of elementary mass dm and inertial accelera¬ 
tion a, of a body B or system S, we associate a total elementary force vector df acting 
on it, such that 


dma = df , (1.4.1) 

where df itself is the resultant of other “partial” elementary forces of various origins 
(to be examined later); that is, 

df = Y. d fk (* = 1 , 2 ,...). (1-4.2) 

Equation (1.4.1) is not simply a definition of one vector (df) in terms of another 
(dma), but is an equality of two physically very different vectors: one, the effect or 
kinetic reaction (dma), depending only on the properties of the particle P itself, and 
another, the cause (df), depending on the interaction between P and the rest of the 
universe — that is, on the action of the external world on the moving system, and the 
mutual, or internal, actions of the body parts on each other. Paraphrasing Elamel 
(1927, p. 3) slightly, we may state: The forces are determined by their “causes”; that 
is, by variables that represent the geometrical, kinematical, and physical state of the 
matter surrounding P (local causes) and away from it (global causes). This depen¬ 
dence is single-valued and, in general, continuous and differentiable; and, in addi¬ 
tion, these forces are objective — that is, independent of the frame of reference (see 
also Hamel, 1949, pp. 509-512). In practice, this leads to constitutive equations for 
the forces (stresses) that, when combined with the field, or ponderomotive, equations 
(1.4.1) lead to relations of the form: 

a = a(t,r,v; physical constants); (1.4.3) 

where a may also depend on the rs and v’s of other system (and even external) 
particles, but not on accelerations or other higher (than the first) d/dt(.. ^-deriva¬ 
tives. Such an (/-dependence would introduce an additional constitutive, or con¬ 
straint, equation of the form: dm a = df (...,«,...). However, and this does not 
contradict (1.4.1), such equations can occur as part of the solution process', namely, 
through elimination of variables from the complete set of equations of the problem; 
that is, elimination of forces related to the accelerations of other parts of the body, so 
that the acceleration of point P depends on, among other things, the accelerations of 

points Q, R .On this delicate and sometimes confusing point, see Hamel (1949, 

p. 49). In view of such difficulties in defining the force, a number of authors (mostly 
continuum mechanicians) consider it as a primitive concept — along with space, time, 
and mass. 


Force Classification 

[This also includes moments; and, in analytical mechanics, both forces and moments 
are replaced by system, or generalized, forces (§3.4).] 

The most important such classifications are as follows: 

Newton-Euler (or momentum) mechanics: 

Internal', originating wholly from within the system; in pairs. They depend on the 
spatial limits of the system. 

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§1.4 FORCE; LAW OF NEWTON-EULER 


External, originating, even partially, from outside the system. Only such forces appear 
in the corresponding equations of equilibrium/motion. 

Lagrangean (or energetic) mechanics: 

Impressed, depending, even partially, on physical (material) coefficients (chap. 3). 
Constraint reactions', depending exclusively on the constraints; geometrical and/or 
kinematical forces (chap. 3). 

Continuum mechanics: 

Surface , or contact : continuously distributed over material surfaces (and/or lines and 
points). 

Volume , or body : continuously distributed over material volumes. 

Usually, a given force is a combination of the above, and more. For example: 

Gravity: external, impressed, body; 

Stresses in rigid bodies: internal, reactions, surface; 

Stresses in elastic bodies: internal, impressed, surface; 

Dry rolling friction: internal or external, reaction, surface; 

Dry sliding friction: internal or external, impressed, surface. 

Other, more specialized force classifications are the following: potential/nonpotential, 
conservative/ nonconservative, gyroscopic / nongyroseopic, circulatory/noncirculatory, 
autonomous/nonautonomous, etc. They will be introduced later, if and when needed. 
Occasionally, forces are classified with the help of the momentum principles as 
follows: 

Linear or translatory loads: forces; 

Angular or rotatory loads: moments of forces and moments of couples; 

but such terminology is not uniform. For example, the authoritative Truesdell and 
Toupin (1960, p. 531) states that, in the general case, the ( total) torque consists of two 
parts: the moment of the force(s) and the couple', also, virtually alone among 
mechanics works, it refuses to use the term internal forces, opting instead for the 
term mutual (loc. cit., pp. 533-535). 

On Centers of Gravity and Mass, and Centroid 

The center of gravity ( CG ) of a material system in a parallel gravitational field is a 
point defined uniquely by 

v cg = S r dG / S dG ’ (1-4-4) 

where dG = elementary gravity force = g dm = pgdV = 7 d V: g = acceleration of 
gravity; p = density of matter; 7 = specific weight; dm = element of mass; 
dV = element of volume; and CG is independent of the orientation of the system , 
and through it passes the resultant gravity force, or weight, of the system, and: 
S {■■■)'• material summation, for a fixed time, and valid for discrete and/or contin¬ 
uous distributions (Stieltjes’ integral). This helpful notation, originated informally by 
Lagrange, is used a lot in the main body of this work. 

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CHAPTER 1: BACKGROUND 


The center of mass , or inertial center , (CM) of a material distribution is defined 
uniquely by 


i'cm = r G = S r dm / S dm ' (1.4.5) 

The centroid (or geometrical center, or geometrical center of gravity) (C) of a figure is 
defined uniquely by 


r c = S rdV /S dV ' (1-4.6) 

• If g = constant, the gravitational field is uniform. Then, g = gu = constant, 
it = vertical unit vector (positive downward). 

• If p = constant, the body (matter) is homogeneous. 


In a uniform field: 

r CG = r CM = r G> 

For a homogeneous body: 

•‘cm =r G = r c \ 

For a homogeneous body in a uniform field: 

r CG = r CM = r c- 


(1.4.7a) 

(1.4.7b) 

(1.4.7c) 


REMARK 

In nonuniform fields, eq. (1.4.7a) is no longer true: the parts of the body closer to the 
attracting earth experience stronger gravity forces than those farther from it; and, 
therefore, upon rotation of the body, the point of application of the resultant of such 
forces changes relative to the body, that is, the center of gravity is no longer definable 
as a unique body-fixed point, independent of the orientation of the body relative to the 
field. The center of mass and centroid, however, are still defined uniquely by (1.4.5) 
and (1.4.6), respectively. Such complications may arise in problems of astronautics/ 
spacecraft dynamics; there, we replace the constant g with a central-symmetric grav¬ 
itational field. 


1.5 SPACE-TIME AND THE PRINCIPLE OF GALILEAN RELATIVITY 

Galilean Transformations (GT) 

These are frame of reference transformations that leave the Newton-Euler law 
(1.4.1) form invariant. The most general such transformations have the following 
form (fig. 1.3): 


r' = A • r + bt + c 

{Direct/matrix notation) 

(1.5.1a) 

x k' = Y A k'k x k + b k ’t + c k t 

{Component notation), 

(1.5.1b) 


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§1.5 SPACE-TIME AND THE PRINCIPLE OF GALILEAN RELATIVITY 



Figure 1.3 On the geometry of Galilean transformations. 


where A = (A^f) is a proper orthogonal tensor with constant components — that is, 
A~ l = A t ; Det A = +1; and h = [h^) and c = (c^) are constant vectors — that is, F 
and F' are in nonrotating uniform motion ( uniform translation ) relative to each other, 
with velocity h, and 

t' = at + (3, (1.5.1c) 

where t is measured in F and t' in F', and a, (3 are constant scalars; a depends on the 
units of time, while (3 depends on its origin in the two systems of time measurement. 
Hence, if these units are taken to be the same, and these origins are made to coincide, 
then a = 1 and (3 = 0; in which case (henceforth assumed in this book), 

t' = t\ (1.5.Id) 

that is, in classical (Newtonian) mechanics there is, essentially, only one time scale. 

From the transformation equations (1.5.1a-d) we immediately obtain the follow¬ 
ing: 

d 2 r' / dt 2 = A • (d 2 r/dt 2 ) or a' = A-a, (1.5.2a) 

or, explicitly, with some easily understood notation, 

d 2 x'/dt 2 = cos (x 1 ,x)(d 2 x/dt 2 ) + cos {x' ,y)(d 2 y / dt 2 ) + cos(x', z){d 2 z/dt 2 ), etc.; 

(1.5.2b) 

that is, the accelerations of a particle P as measured in F and F' differ only by an 
ordinary (time-independent) geometrical transformation due to the, possibly, differ¬ 
ent orientation of their axes; and, therefore, they are physically equal: that is, un¬ 
affected by the relative motion of F and F 1 . Hence, we may take, with no loss in 
physical generality, the corresponding axes of F and F' to be ever parallel, in which 
case A = 1 (unit tensor), in which case (1.5.1a) simplifies to 

r' = r + bt + c => a'= a. (1.5.2c) 

Since dm\ F = dm\ F , = dm, and assuming that from dm a = df(t,r,v) and (1.5.2c) it 
follows that 

dma = df (t,r' — bt — c,dr/dt — b) = df ' (t,r ,dr / dt = v') = df (1.5.3) 

that is, df is also invariant under GT, and, therefore, as far as the law of motion 
(1.4.1) is concerned, there is no one ( absolute ) frame in which it holds, but, in fact, once 

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CHAPTER 1: BACKGROUND 


one such “inertial" frame is established, there is a whole family of them dynamically 
equivalent to it. More precisely, there is a (continuous linear) group that depends on 
ten (10) parameters: three for A [out of its nine components (direction cosines), due 
to the six orthonormality constraints, only three are independent], three for h, three 
for c, and one for (3 [equations (1.5.1c, d), a = 1, with no loss in generality]. This 
Galilean, or Newtonian, principle of relativity can be summed up as follows: an 
inertial frame—that is, one in which dm(d 2 r/dt 2 ) = df holds — is determined only 
to within a Galilean transformation (1.5.1a—d). 

REMARKS 

(i) The linear transformation (1.5.1c) can also be obtained by requiring that if 

a = d 2 r/dt 2 = 0 , (1.5.4a) 

then also 

d 2 r/d(f) 2 = 0 , (1.5.4b) 

for arbitrary values of r and dr/dt. Indeed, using chain rule, we find: 
dr/dt' = (dr / dt) / (dt' / dt) 

=> d 2 r/d(f) 2 = [(dt'/dt)(d 2 r/dt 2 ) - (dr/dt)(d 2 f/dt 2 )}/(dt'/dt)\ (1.5.4c) 

and so, due to (1.5.4a), the requirement (1.5.4b) translates to 

(dr/dt)(d 2 1'/dt 2 ) = 0 , for arbitrary dr/dt\ (1.5.4d) 

that is, 

d 2 t'/dt 2 = 0 =>■ /' = at + /3, a,f3\ integration constants; Q.E.D. (1.5.4e) 

(ii) The logical circularity involved in the classical mechanics definition of inertial 
frames (i.e., “if dm a = df holds, the frame is inertial" and “if the frame is inertial 
frame then dm a = df holds”) can be resolved only by relativistic physics. Here, we 
are content to postulate the existence of frames in which dm a = df holds exactly 
(or, equivalently, of frames in which forceless motions are also unaccelerated motions', 
i.e., the position vectors are linear functions of time, and vice versa); and to call such 
frames inertial. For detailed discussions of this important topic, see any good text on 
the physical foundations of relativity; e.g., Bergmann, 1942; Nevanlina, 1968. 


1.6 THE FUNDAMENTAL PRINCIPLES (OR BALANCE LAWS) OF 
GENERAL SYSTEM MECHANICS 


An Axiom is a proposition, the truth of which must be admitted 
as soon as the terms in which it is expressed are clearly 
understood ... physical axioms are axiomatic to those only who 
have sufficient knowledge of the action of physical causes to 
enable them to see their truth. 

(Thomson and Tait, 1912, part 1, section 243, p. 240) 

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§1.6 THE FUNDAMENTAL PRINCIPLES (OR BALANCE LAWS) OF GENERAL SYSTEM MECHANICS 


Conservation of Mass (Euler, Early 1 760s) 


dm(B)/dt = dm/dt = d/dt(^J dmj = d/dt 


pdV 


d/dt(pdV) = 0. 

(1.6.1a) 


(Henceforth, we shall, usually, omit the subscripts V, dV, etc., in the various 
integrals.) 

In the absence of discontinuities, the above leads to the local (differential) form: 


d/dt(p dV) = 0 =>■ pdV = constant = p 0 dV 0 
[Material , or Lagrangean , or referential , equation of continuity ] (1.6.1b) 


where p 0 (dV 0 ) = density (element of volume) in some initial or reference configura¬ 
tion. 


Principle of Linear Momentum [Euler, 1750 (publ. 1752)] 


d/dt(^S v d n ^j = $ df or dp/dt=f , 

where 


p(B, t) = p = ^ v dm 


pv dV : Linear momentum of B , 


(1.6.2a) 


(1.6.2b) 


a system vector that depends on the frame, but not on the (fixed) origin in it; 
equivalent to Newton’s “quantitas motus”; and $ df =/. From the above, and 
invoking mass conservation [§1.3:(1.3.1)ff.), (1.6.1a, b)] and the definition of mass 
center (§1.4), we obtain 

p = mv G => ma G =f, (1.6.2c) 

where »” G /v G /a G are, respectively, the position/velocity/acceleration vectors of the 
center of mass of B, G. Equation (1.6.2c) shows that the motion of the center of 
mass G, of a body (or any material system, rigid or not), B, is identical to that of a 
fictitious particle of mass m located at G and acted upon by the body resultant on B,f; 
that is, by the vector sum of all (—> external) forces transported parallel to themselves 
to G. Thus, the motion of G is taken care of by this simple principle —> theorem. But 
the remaining problem of the motion of B about G (and, generally, of the motion of 
other body points) is far more difficult, and, unlike the motion of G, does depend on 
the specific material constitution of B (e.g., rigid, elastic), as well as on its motion 
(i.e., 1-, 2-, 3-dimensional); and, therefore, that problem necessitates additional con¬ 
siderations, such as the following. 


Principle of Angular Momentum [Euler, 1775 (publ. 1776)] 

d/dt(S(r> <v dm))= S( rxd f) or dH 0 /dt = M 0 , 


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(1.6.3a) 





CHAPTER 1: BACKGROUND 



where 

Ho{B , t) = Ho = $ ( r x vdm ): absolute angular momentum (or moment of momentum, 

or kinetic moment), about the fixed point 0, (1.6.3b) 


and 


M 0 = s (r x df): total moment about 0 (fig. 1.4). (1.6.3c) 

Other angular momenta, and their interrelations, are detailed in “Additional Forms 
of the Angular Momentum,” below. 


External and Internal Loads 

In the Newton-Euler approach to system mechanics, whether discrete or continuous, 
we classify body and/or surface forces and moments as internal or mutual (i.e., those 
due exclusively to internal causes) and external [i.e., those whose cause(s) lie, even 
partially , outside of the body or system]. Stresses are caused by one or more of the 
following: (i) deformations (solids); (ii) flows (gases, liquids); (iii) geometrical/kine- 
matical constraints [e.g., incompressibility, inextensibility (= incompressibility in one 
or two dimensions)]. 

Analytical mechanics necessitates a different force/moment classification (chap. 3). 


Principle of Action-Reaction 

(i) Discrete version. Let us consider a system of N particles {P k y k = 1,..., N}. 
Each particle P k is acted upon by a total external (to that system) force / k ext and a 
total internal force / ( r. inl due to the other N — 1 particles: 

fk.un = fkh with Iflk- i.e., f kk is, as yet, undefined(l) (1.6.4a) 

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§1.6 THE FUNDAMENTAL PRINCIPLES (OR BALANCE LAWS) OF GENERAL SYSTEM MECHANICS 


Now, by Newton’s third law of motion ( action-reaction ) we shall understand the 
constitutive (i.e., physical) postulate; 

(a) f kl = —f lk and f kk = 0 (i.e., the particle cannot act on itself!) (1.6.4b) 

and 

(b) (r k — r/) x f kl = 0 (i.e., the internal forces are central and opposite', or 

oppositely directed pair by pair and collinear). (1.6.4c) 

[The second of (1.6.4b) is not included in the original Newtonian formulation. We 
follow Hamel (1949, p. 51).] 

In the discrete/particle model, so popular among physicists and such an anathema 
among certain mechanicians, this postulate, plus the principle of linear momentum, 
lead to the theorem of angular momentum for the external loads only. However, the 
converse is not necessarily true; that is, the angular momentum equation for a finite 
body dH 0 /dt = T / 0 externa i does not necessarily lead to (1.6.4b, c); other combinations 
of the internal forces may lead to the same effect (e.g., a sum of terms may vanish in a 
number of different ways). The converse may hold if we assume the validity of the 
angular momentum equation for any part of the system, or for any size subsystem. 

(ii) Continuum version. For every pair of particles P\ and P 2 , with respective 
positions iq and r 2 , the mutual forces and moments satisfy the following constitutive 
postulate: 

df (ti,t 2 ) = ~df (r 2 , fi) and dM(r\,r 2 ) = —dM{r 2 ,r\). (1.6.4d) 

Without (1.6.4d), or something equivalent that supplies knowledge of the internal 
loads, the problem of mechanics would, in general, be indeterminate (i.e., the adopted 
model would produce more unknowns than the number of scalar equations 
furnished by its laws). 


Additional Forms of the Angular Momentum 

Although the results derived below hold for any body or system, they become useful 
only for rigid ones. We define the following two kinds of (inertial) angular momen¬ 
tum (fig. 1.4): 

H. absolute = H. = £ (r - r.) x dm v = £ r /m x dm v: [v = dr/dt] 

Absolute angular momentum of body B, about the arbitrarily moving point • 

[because it involves the absolute (inertial) velocity v = dr/dt), (1.6.5a) 

and 

.relative = h » = S ^ ~ ^ X dm ( V “ = S * l* X dm V /•'' 

Relative angular momentum of body B, about the arbitrarily moving point • 

[because it involves the relative (inertial) velocity v — v. = v/.]. (1.6.5b) 

REMARKS 

(i) Although these kinematico-inertial definitions hold for any frame of reference 
(with r, r., v, v. denoting the positions and velocities relative to points fixed or 
moving with respect to that frame—see §1.7), they will normally be understood to 
refer to a specific inertial frame, unless explicitly stated to the contrary. 

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CHAPTER 1: BACKGROUND 


(ii) Some authors define absolute angular momentum as in our (1.6.5a), but only 
for fixed points (i.e., v. = 0); in which case, clearly, (1.6.5a) and (1.6.5b) coincide. 
Unfortunately, here too, there is no uniformity of terminology and or notation in the 
literature; but, as will be seen in kinetics, some angular momenta are more useful 
than others. The connection between the above two angular momenta is given by the 
following basic theorem. 

THEOREM 

The angular momenta H. and h„ defined by equations (1.6.5a, b), are related by 
H. — It, = m(r G — r.) x v. = mr G /, x v.. (1.6.5c) 


PROOF 

Subtracting (1.6.5b) from (1.6.5a) side by side, and then utilizing the properties of 
the center of mass of B , G, we obtain 

H. - h. = g x (v - v/.) dm = £ {r /m x v.) dm 

= ^rx {dm v.) — ^ (r. x v.) dm = ( mr G ) x v. — r. x (mv t ), Q.E.D. 

(1.6.5d) 

Equations (1.6.5c, d) show immediately that, in the following three cases, the differ¬ 
ence between absolute and relative angular momentum disappears: 

(i) r G/ . = 0, i.e., • = G: H G = h G = gr /G x (, dmv /G ), (1.6.5e) 

(ii) v. = 0, i.e., • = fixed origin, say O: H 0 = h 0 = ^r x (dmv), (1.6.5f) 

(iii) r G /. parallel to v.. (1.6.5g) 

The first and second cases, (1.6.5e, f), are, by far, the most important; (1.6.5g) may 
be hard to check before solving the (kinetic) problem. 

Next, let us relate H. and It. with H a (which appears in the basic Eulerian form of 
the angular momentum principle). We have, successively, 

H o‘S' x {dmv) (introducing positions/velocities relative to •) 

= S [ 0 . + r/,) x dm (v. + v/.)] 

= ••• = /». + m{r. x v G ) + m{r G/ . x v.), (1.6.5h) 

— H. + m{r. x v G ) [thanks to (1.6.5c)]. (1.6.5i) 

The above leads easily to the following corollaries: 

(i) If • = fixed => r. — 0. then 

H 0 = H. + r. x {mv G ) = h. + m{r. x v G ) [r. = r./ 0 , H. = h,]; (1.6.5j) 

a slight generalization over (1.6.5f). 

(ii) If • = G, then 


H 0 = H G + r G x {mv G ) = h G + m{r G x v G ) 

[r G = r G/Oi V G = dr G /dt ; H G = h G ]. 

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(1.6.5k) 


§1.6 THE FUNDAMENTAL PRINCIPLES (OR BALANCE LAWS) OF GENERAL SYSTEM MECHANICS 


By comparing (1.6.5h,i) with (1.6.5k), it can be seen that 

H. = H G + r G/ . x (mv G ), h. = H G + r G/ . x (mv G/ .). (1.6.51) 

(Interpret these “transfer” equations geometrically. What happens if • is fixed ; say, 
an origin 01) Finally, by applying the transfer equations (1.6.5h, i) between O and 
the arbitrarily moving points 1 and 2 , and then comparing, we can obtain the rela¬ 
tion between the absolute, relative, and absolute-relative angular momenta of a 
body! H\ *■—► H 2 5 *—* hi, h\ <—> hi- 


Additional Forms of the Principle of Angular Momentum 

With the help of the preceding kinematico-inertial identities/results, and the purely 
geometrical theorem of transfer of moments (hopefully well known from elementary 
statics) 

M. = M g + r G /, x / [where the force resultant / goes through G] 

= M g + r G j, x ( ma G ) [by the principle of linear momentum], (1.6.6a) 

the Eulerian principle of angular momentum 

^rx {dm a) = d/dt[ Q r x {dm v)^ = ^ r x df\ 

that is, 

dH 0 /dt=M 0 (1.6.6b) 

[=> M 0 externa |, by action—reaction (plus, in the continuum version, 

of Boltzmann's axiom => symmetry of the stress tensor)], 

assumes the following forms: 

Center of Mass Form 
By (1.6.5k): 

dH 0 /dt = d/dt[H G + r G x {mv G )] = dH G /dt + m{r G x a G ), (1.6.6c) 

and by (1.6.6a), for • —> O: 

M 0 = M (j + r G x {ma G )\ (1.6.6d) 

and comparing these expressions with (1.6.6b), we obtain the fundamental form 

M g = dH G /dt (= dh G /dt). (1.6.6e) 


Absolute Form 

Using the above, and (1.6.51), we obtain, successively, 

M. = M g + r G/ . x {ma G ) = dH G /dt + r G/ , x {ma G ) 

= d/dt [H. - r G/ . x {mv G )\ + r G/ . x {ma G ) 

= dH./dt - v G/ . x (mv c ) - r G/ , x {ma G ) + r G/ . x (ma G ); 

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CHAPTER 1: BACKGROUND 


that is, finally, 


M. = dH./dt - v G/ . x ( mv G ) (using v G/ . = v G -v.) 

= dH./dt + v. x (mv G ) = dH./dt + v. x (mv G /.). (1.6.6f) 


Relative Form 

Similarly, using the above, and (1.6.51), we obtain, successively, 


M. = M g + r G/ , x ( ma G ) = dH G /dt + r G/ . x ( ma G ) 

= d/dt (h. - r G/ . x (mv G /.)) + r G/ . x (ma G ) 

= dh./dt - v G/ . x (mv G/ .) - r G/ . x (ma G/ .) + r G/ . x (ma G ); 




(1.6.6g) 





(1.6.6h) 


which, since it holds for any fixed point, is a slight generalization of (1.6.6b). These 
forms show clearly the importance of fixed points and of the center of mass, above all 
other points, in rotational dynamics, especially rigid-body dynamics. All these forms 
of the principle of angular momentum, and many more flowing from them, can be 
quite confusing, they are almost impossible to remember, and may be error-prone in 
concrete applications. They are stated here only for comparison purposes with the 
existing literature. From them, the most useful in both theoretical and practical 
situations, are, by far, (1.6.6b,e), and, secondarily, (1.6.6a) with (1.6.6e). We 
summarize them here: 



O'. fixed origin ; (1.6.6i) 



G\ center of mass; (1.6.6j) 


M. = dH G /dt + r G j, x (. ma G ), • : arbitrarily moving spatial point , (1.6.6k) 

or, compactly, 


Kinetic vectors (“torsor”) at G: ( ma Gl dH G /dt) 

~Kinetic torsor at • : {ma G ,dH G /dt + r G /. x (. ma G )); 


and we are reminded that their left sides, by action-reaction (plus Boltzmann’s 
axiom, i.e., symmetry of stress tensor), include only external moments and couples. 

By comparing the absolute and relative forms of the principle of angular momen¬ 
tum, eqs. (1.6.6f, g) [or by d/dt(.. .), eq. (1.6.5c)], we can show that 

dH./dt — dh./dt = r G i, x (ma.) + v G /. x (mv.) 


r G/ . x (ma.) + v G x (mv.) = r G/ . x (ma.) + v G x (mv. /G ). 

(1.6.61) 


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§1.7 ACCELERATED (NONINERTIAL) FRAMES OF REFERENCE 


Finally, crossing the local law of motion dma = df with r/. = r — r„ and then 
integrating over the body, etc., we obtain the following additional form of the 
principle of angular momentum: 

M. = dH 0 /dt — r. x ( ma G ) (= M 0 — r. x /, with / applied at •). (1.6.6m) 


1.7 ACCELERATED (NONINERTIAL) FRAMES OF REFERENCE 
(OR RELATIVE MOTION, OR MOVING AXES); 

ANGULAR VELOCITY AND ACCELERATION 

The theory of moving axes, a subject indispensable to rigid-body dynamics and other 
key areas of mechanics (including the transition to relativity), is based on the follow¬ 
ing fundamental kinematical theorem. 


Theorem (of Moving Axes) 

Let us consider two frames of reference in arbitrary relative motion, each represented 
by an ortho-normal-dextral (OND) basis and associated coordinate axes, rigidly 
attached to the frame; say, for concreteness but no loss in generality, one fixed or 
inertial F: 

(0 F - /, J , K/X, Y , Z) = (0 F - «r, u Y ,u z /X, Y, Z) = (0 F - u x ^ z /X, Y, Z) 

= (0 F - u k '/x k >; k' = 1,2,3 /X, Y,Z), (1.7.1a) 

and one moving or noninertial M : 

(0 M ~ h j,k/x,y,z) = (0 M - u x ,u y ,u z /x,y,z) = (0 M - u x ^ z /x,y,z) 

= (0 M ~u k /x k -, A-=1,2,3 /x,y,z), (1.7.1b) 

and an arbitrary (say free) vector p [fig. 1.5(a)]. Then its rate of change in F and M, 
dp/di and dp/dt, respectively, are related by 

dp/dt = dp/dt + a>xp, (1.7.2a) 


M (bj 



Figure 1.5 (a) Geometry of moving frames; (b) geometrical proof of (1.7.3c). 

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CHAPTER 1: BACKGROUND 


where (recalling the moving axes theory, §1.1) 

P = PxUx + PY U Y + Pz u z = Y Pk' U k' = PxUx+Pylty + Pz“z = Y PkU ^ 

[assumed instantaneous representation of p in F and M\; (1.7.2b) 

dp/dt = ( dp x /dt)u x + (dp Y / dt)u Y + (clp z /dt)u z = E 0 dPk'/dt)ti k '■ 

Absolute rate of change of p (or time flux)-, i.e., relative to F; (1.7.2c) 

dpldt = ( dp x /dt)u x + (i dpy/dt)u y + ( dp : /dt)u z = Y^ (dp k /dt)u k : 

Relative rate of change of p; i.e., relative to M; (1.7.2d) 

01 = ! jJyUy UJyUy L0 Z U Z ~ ^ — CO v li x Ulyliy i0 Z U- = ^ ~ OJyUy 

= [(du r /dt) • u z \u x + [( du : /dt ) • u x ]u y + [(du x /dt) • u v ]u z : 

Angular velocity ( vector) of M movingframe relative to F Rxe d frame ; 

i.e., of ( 0 M — u k ') relative to ( 0 F — u k ); (1.7.2e) 
co x p = Transport rate of change of p relative to F. (1.7.2f) 


NOTATIONAL CLARIFICATION 

Here, partial derivatives, d(...)/dt, are, normally, associated with moving frame(s); while, 
for simplicity, primed subscripts signify fixed axes/components. 

To express this theorem in components, which is the best way to understand it, the 
simplest way is to choose the axes O f —XYZ and O m —xyz so that, instantaneously, 
either they coincide or are parallel. Then, since in such a case, 

(i dp/dt) x = {dp/dt) • Uy = dpx/dt = (dp/dt) ■ u x = ( dp/dt) x , etc., cyclically, 

(1.7.3a) 

(dp/dt) x = (dp/dt) • u x = dp x /dt = (dp/dt) • u x = (dp/dt) x , etc., cyclically, 

(1.7.3b) 


the theorem assumes the component form: 


dpx/dt = dp x /dt + u> r p- — ui-p Y , 

dpy/dt = dp y ,/dt + u> : p x — ui x p z , 
dp z /dt = dp : /dt + oj x p y — oj y p x ; 

(1.7.3c) 

and gives inertial rates of change, but expressed in terms of noninertial (relative) and 
transport rates. The above show clearly that 

(dp/dt) k f dp k /dt ( k = x, y, z ); 

(1.7.3d) 

even though, instantaneously, 


Px=Px, etc., cyclically, 

(1.7.3e) 


unless co x p = 0 (=> co = 0, or p = 0, or co parallel to p). 

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§1.7 ACCELERATED (NONINERTIAL) FRAMES OF REFERENCE 


A geometrical interpretation of (1.7.3c) is shown in fig. 1.5(b): the moving axes 
Om— xyz momentarily coincide with the axes O m —XYZ\ the latter are always trans¬ 
lating relative to Op—XYZ —that is, they are “rotationally equivalent” to them. 

PROOF OF EQUATION (1.7.2a) 

By d/dt(.. .(-differentiating (1.7.2b), we obtain 

dp/dt = ( dp x /dt)u x + p x {du x /dt) + • • • = dp/dt + 5Z Pk{du k /dt). (1.7.4a) 

To transform the key second term in the above, we begin by d/dt(.. .(-differentiating 
the six geometrical orthonormality (=>• rigidity) constraints of these basis vectors 
u k ■ U/ = d ki ( k,l = x,y,z ), thus translating them into the following six kinematical 
constraints: 


(i du k /dt ) •/// + u k • ( duj/dt) = 0; (1.7.4b) 

that is, from the nine components of {du k /dt} only 9 — 6=3 are independent. 

Let us find them. By (1.7.4b) for k, l = x, du x /dt is perpendicular to u x , that is, it 
must lie in the plane of u r , Therefore, we can write 

du x /dt = l\u v + l 2 u z \ (1.7.4c) 

and, cyclically, 

du v /dt = l 2 u, + U u x , du z /dt = l^u x + l(,u y ; (1.7.4d) 

where /] 6 are scalar functions of time. Substituting these representations back into 

(1.7.4b) for k = x, l = y, and taking into account the geometrical constraints, we 
obtain 

{du x /dt) • u y + u x • (du v /dt) = 0 => ly + / 4 = 0; (1.7.4e) 

and, cyclically, 

(dUy/dt) • u : + u r • (du : /dt) = 0 => / 3 + l 6 = 0, (1.7.4f) 

(du z /dt) • u x + u z • (du x /dt) = 0 =>- l 5 + l 2 =0. (1.7.4g) 

Hence, (1.7.4c,d) can be rewritten in terms of the following three independent 
(unconstrained) /’s, or in terms of the three equivalent parameters w x , cj y , lj z : 

l\ = —I 4 = lo z , l 2 = —If, = u x , (5 = —U = w„ (1.7.4h) 

as 

du x /dt = Lu z u y — u> y u z = a) x u x , 
dUy/dt = ui x u z — u> z u x = to x u v , 
du z /dt = ui y u x — u) x u v = co x 

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(1.7.4i) 


CHAPTER 1: BACKGROUND 


where 

(O = U) X U X + UJyUy + jJ U- 

= u x [(diiy/df) • w z ] + u r [(du : /dt ) • h v ] + u : [(du x /dt) • /#,.] 

[a form that shows the cyclicity of the subscripts x, y, z] 

= —u x [(du z /dt) •«„] — u y [{du x /dt ) •«.] — u : [(du v /dt) •«*]. (1.7.4j) 

Finally, substituting these results into (1.7.4a), we obtain (1.7.2a): 

dp/dt = dp/dt + $>( 0 , x M 0 = dp/dt + uj x p k u^\ 

= dp/dt + uxp. (1.7.4k) 


REMARKS 

(i) Frequently, and with some good reason, the notation Sp/bt is employed for our 
dp/dt. Here, however, we chose the latter because in analytical mechanics 5{...) is 
reserved for virtual changes, under which St = 0 (chap. 2ff.). Other popular notations 
for the relative rate of change are d*p/dt (British authors; but some German authors use 
dp/dt for our u x p), ( dp/dt)M or (dp/dt) re i or d*p/df, or with a tilde over d 
(Soviet/Russian authors) d. Also recall remarks made regarding eq. (1.1.20i) about the 
overdot notation. 

(ii) The vector equation (1.7.2a) can be expressed in component form (i.e., it can 
be projected) along any axes, fixed or moving, by eqs. (1.7.3c), if 0 M —xyz and 
O m —XYZ momentarily coincide; and, if they do not, by 

( dp/dt) x = cos(a;, X){dp x /dt) + cos(x, Y)(dp y /dt) + cos(x , Z)(dp z /dt) dp x /dt) 

= cos(x, X)(dpi/dt + W2P3 ~ ^ 37 * 2 ) + • • •, (1.7.5a) 

where the new axes <9^—123 coincide momentarily with Om—XYZ , but, in general, 
have an angular velocity m' = (w 1 ; 022 , 023 ) relative to them. 

(iii) The above show that as long as no rates of change are involved, the compo¬ 
nents of a vector along the various axes (fixed or moving) are related by ordinary 
coordinate transformations, with possibly time-dependent coefficients — that is, like 
the first line of (1.7.5a), or (1.7.5b), below; all such axes are mechanically (though not 
mathematically) equivalent. But when rates of change between such moving axes 
(—> frames) are compared, then, in general, a component of a vector derivative 
( dp/dt) x does not equal the derivative of that component dp x /dt [(1.7.3d, e)]; these 
quantities are related by a frame of reference transformation — that is, like the second 
line of (1.7.5a). Mathematically, this is equivalent to an explicitly time-dependent 
coordinate transformation: a; = x(X, Y,Z;t),... X = X(x,y, z; t),... (recall dis¬ 
cussion following eq. (1.1.20k)). In such cases, to obtain equations like (1.7.3c), we 
begin with O m —XYZ and 0 M —xyz in arbitrary relative orientations, then we 
d/dt(.. ^-differentiate the component transformations, like 

p x = cos (x,X)p x + cos(a:, Y)py + cos(x, Z)p z , etc., cyclically, (1.7.5b) 

{not like p x = p x ) and then we make O m —XYZ and 0 M —xvz coincide. 

(iv) In kinematics, all frames are theoretically equivalent; and thus during the 17th 
century both Galileo and the Catholic church were ... kinematically correct! This is 

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§1.7 ACCELERATED (NONINERTIAL) FRAMES OF REFERENCE 


expressed by the following geometrical, or Euclidean, and kinematical principle of 
relativity, any system of rectangular Cartesian coordinates can be replaced by any 
other such system that moves in an arbitrary fashion relative to the first; or, 
alternatively, the form of geometrical relationships must he invariant under the proper 
orthogonal group of rotations —and this, in effect, constitutes a definition of Euclidean 
geometry—that is, any two such sets of coordinates x k > and x k are related by 

x k' = A k'k(t)x k + (1.7.6a) 

where 

A k'k( 0 A k'l (0 = t>kh 


and 


Det(^ fc (0) = +1, (1.7.6b) 

and A k ' k (t), A k >(t ) are continuous functions of time, with first and second time 
derivatives. Such transformations include all frames/motions produced from the 
moving frame M by a continuous rigid-body movement (translations and rotations, 
but not mirror reflections). 

(v) If the moving triad u xyz is non-OND, then its inertial angular velocity is, 
instead of (1.7.4j), 

o) = [u x \{du y /dt) • u : ] + u v [{du~/dt) • u x \ + u : [{du x /dt ) • u y ]} J[u x • (u y x «_)]. 

(1.7.6c) 

[See, for example, Truesdell and Toupin (1960, p. 437). In case such angular velocity 
vector definitions seem unmotivated, another more natural one, based on the linear¬ 
ization of the finite rotation equation, is detailed in §1.10.] 


Corollaries of the Moving Axes Theorem 

Applying (1.7.2a) for a>, we get 

duo /dt = dus/dt + u) x u) = du>/dt = cx. 

Angular acceleration of moving axes relative to fixed axes. (1.7.7a) 

This result shows the special position of to in moving axes theory. 

From eq. (1.7.2a) and its derivation, we easily obtain the following general opera¬ 
tor form: 

d{.. .)/dt = d(.. -)/dt + u; x (...), (...): any vector. (1.7.7b) 

Applying (1.7.7b) to (1.7.2a), and invoking (1.7.7a), we obtain the following expres¬ 
sion for the second absolute rate of p, d/dt{dp/dt) = d 2 p/dt 2 : 

d 2 p/dt 2 = d(.. .)/ dt(dp/dt + u> x p) 

= [<9(. ..)/dt + us x (.. ,)\(dp/dt) + ( du/dt) xp + ux ( dp/dt) 

= ... = d 2 p/dt 2 + [a xp + co x (n> x/?)] + 2co x (dp/dt), 

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(1.7.7c) 


CHAPTER 1: BACKGROUND 


where 

d 2 p/dt 2 = ( d 2 p x /dt 2 )u x + ( d 2 p y /dt 2 )u y + ( d 2 p z /dt 2 )u z . (1.7.7d) 

In general, if a —> b = da/dt —> c = db/dt = d^a/dt 2 then we shall have for their 

components: 

b x = b x = da x /dt + cu y a z — u> z a y , (1.7.7e) 

c x = c x = db x /dt + ui v b z — u z b v 

= d/dt(da x /dt + u> v a z — co z a y ) + u y {da z /dt + u> x a y — u> y a x ) 

— LO z (da v /dt + u z a x — to x a z ), etc., cyclically. 

(1.7.7f) 

For example, application of (1.7.7c, d) to the moving basis vectors u xy z yields 

d 2 u x /dt 2 = d 2 u x /dt 2 + [a x u x + oj x (oj x u x )\ +2 uj x ( du x /dt ) 

= 0 + [a x u x + co x (e> x u x )\ + 0 

= atxu x + a)x((ox u x ), etc., cyclically. (1.7.7g) 

Since (1.7.2a) is a purely kinematical result, the roles of the frames F and M can be 
interchanged. Indeed, from it, we immediately obtain 

dp/dt = dp/dt + (— cj) x p , (1.7.7h) 

where —co is the angular velocity of F relative to M. 

In particular, if p remains constant (i.e., fixed) relative to F, (1.7.2a) and (1.7.7h) 
yield 

dp/dt = (- u) x p ; (1.7.7i) 

that is, an observer, stationed in M, sees the tip of p rotate relative to that frame with 
an angular velocity —co. Application of (1.7.7i) to the fixed basis u X j^z gives 

dux/dt = (-u>) x u x = -(oj x ,ojy,u} Z ) x (1, 0, 0) 

= • • • = (0)«x + (—wz)«r + (ojy)u z , 

du Y jdt = (-iv) X Uy = ■ ■ ■ = ( UJ Z )u X + (0 )u V + (-UJ X )u Z , 

du z /dt = (-uj) xu z = ■■■ = (-w r )«x+ (wx)«r + (0 )u z ; (1.7.7j) 

and, therefore, 

(dux/dt) -Uy = —uj z (= —w z , for coinciding axes) 

( dux/dt ) • u z = +uj y (= +uJ y , for coinciding axes), etc., cyclically. (1.7.7k) 

Alternative Definition of Angular Velocity 

(i) Below, we show that 

co = (1/2) \uk x (diik/dt)] (where k = 1,2,3 —> x,y,z), (1.7.8a) 

which can be viewed as an alternative to (1.7.2e, 6c) definition of angular velocity. 

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§1.7 ACCELERATED (NONINERTIAL) FRAMES OF REFERENCE 

Indeed, using the fundamental equations (1.7.4i), we obtain, successively, 

[»fc x ( du k /dt)\ = ^2 [u k x (ox u k )\ = ^2 [(u k ■ u k )(o - (. u k • (o)u k )\ 

= (“fc • «*)) - 51 = ro ( 3 ) - ffl = 2<u, Q.E.D. 

(1.7.8b) 

From the above, and using the results of §1.1, we can show that the (inertial) angular 
velocity tensor of the moving frame to [i.e., the antisymmetric tensor whose axial 
vector is the (inertial) angular velocity of that frame to] can be expressed as 

to = (l/2)]T \(du k /dt) (g) u k — u k 0 (du k /dt )]. (1.7.8c) 

(ii) Next, if the (orthonormal) basis vectors u k are functions of the curvilinear 
coordinates q = (</|, q 2 , qfi —that is, u k = u k (q) — then, applying (1.7.8a), we find, 
successively (with all Latin subscripts running from 1 to 3; i.e., x,y,z), 

= 0/2){«, x (,du k /dqi){dqi/dt )) ^ c k (dq,/dt ), (1.7.9a) 

where 


ci = ^2 (1/2) [life x (du k /dqi)\ (“Eulerian basis” for to); (1.7.9b) 

that is, the dqj/dt are the (contravariant) components of to in the (covariant) basis ty. 

By formally comparing (1.7.8a) and the earlier equations (1.7.4i), (1.7.2e, 4j), with 
(1.7.9a, b) [i.e., to —> c t and du k /dt —> du k /dqj\, it is easy to conclude that 


duk/dqi = ci x u k , (1.7.9c) 

c, = u 1 [(du 2 /dqi)-u 3 ] + u 2 [(du 3 /dq,) ■ i/j] + u 3 [(du x /dq,)-u 2 \. (1.7.9d) 

We leave it to the reader to extend the above to the “rheonomic” case: u k = u k (q , t). 

EXAMPLES 

1. The absolute (i.e., inertial) components of the angular acceleration of a rigid 
body rotating with angular velocity co B are (with the hitherto used notations) 

duB,x/dt = du Bx /dt + uiy lo Bz — ui z u> B v , etc., cyclically. (1.7.10a) 

What happens if a> B = wl 

2. The conditions for a straight line with direction cosines (relative to moving 
axes) 4, /,,, l : to have a fixed inertial direction are 

dl x /dt + uj y l z — 'jj 7 l y = 0, etc., cyclically. (1.7.10b) 

How many of these three conditions are independent? Hint: l x 2 + l 2 + l 2 = 1. 

3. The moving axis theorem (1.7.2a), applied to the generic vector p expressed in 
plane polar coordinates'. 

p=p r u r +pfill'/,, (1.7.10c) 


yields 

dp/dt = [dp r /dt — pfidtj)/dt)\u r + [dp^/dt + p r (d(f>/dt)\u rj> . (1.7.lOd) 

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119 


CHAPTER 1: BACKGROUND 


Apply (1.7.10d) for p = position vector of a particle r, and velocity vector of a particle v. 

Hint: The angular velocity of the moving polar ortho-normal-dextral triad 
u r,<j>.:=z > relative to the inertial one u XY7 , is 

at = (d(j)/dt)u : = ( d<f>/dt)u z . (1.7.lOe) 


Particle Kinematics in Moving Frames 

Velocities 

Application of the fundamental formula (1.7.2a) to the motion of a particle P, of 
inertial position vector 9? = v 0 + r (fig- 1-6) (i.e., for p —> r), yields 

v = dR/dt = d{r 0 + r)/dt = dr Q /dt + dr/dt 

= dr 0 /dt + (dr/dt + oxr), (1.7.11a) 

(since, in general, r is known only along the moving axes) or, rearranging, 

v = (dr 0 /dt + (ox r) + dr/dt (1.7.11b) 

or 

*’abs = Ttrans “F r re l, (1-7.11 c) 

where 

v abs = v = d^R/dt = (dX/dt)u x + ( dY/dt)u Y + ( dZ/dt)u z : 

Absolute velocity of P, (1.7.lid) 

v re i = dr/dt= ( dx/dt)u x + ( dy/dt)u y + ( dz/dt)u : : 

Relative velocity of P, (1.7.1 le) 

vtrans = dr 0 /dt + a) x r = dr 0 /dt + \x{du x /dt) + y(du r /dt) + z(du z /dt)]: 

Transport velocity of P. (1.7.Ilf) 



Figure 1.6 (a) Relative kinematics of particle P in two dimensions; (b) geometry of centripetal 
acceleration. 


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§1.7 ACCELERATED (NONINERTIAL) FRAMES OF REFERENCE 


Clearly, if P is rigidly attached to M moving frame (e.g., if it is one of the particles of 
the rigid body M), then v rel = 0 and v = v trans ; that is, generally, v trans is the velocity 
of a particle rigidly attached to M and instantaneously coinciding with P. 


Accelerations 

Application of (1.7.2a) to (1.7.1 la-f) yields 

Aibs Ad A Arans A Aon 


where 


(1.7.12a) 


Abs = « = d 2< H/dt 2 = {d 2 X/dt 2 )u x + ( d 1 Y/dt)u Y + (d^ Z / dt 2 )u z : 

Absolute acceleration of P, (1.7.12b) 

Aei = dv K \/dt = d 2 r/dt 2 = ( d 2 x/dt 2 )u x + ( d 2 y/dt 2 )u y + (d 2 z / dt 2 )u z \ 

Relative acceleration of P, (1.7.12c) 

fl trans = d 2 r 0 / dt 2 + X X r + CO X (co x r) 

= d 2 r 0 /dt 2 + [. x(d 2 u x /dt 2 ) + y(d 2 u v /dt 2 ) + z(d 2 u : /dt 2 )\'. 

Transport (or drag) acceleration of P 

[ = Inertial acceleration of a particle fixed relative to M, and momentarily 
coinciding with P; its first term, 

d 2 r 0 /dt 2 = dv 0 ldt = dvo/dt + ojxvo, 

is due to the inertial acceleration of the origin of M\ its second , a x r, to 
the inertial angular acceleration of M; and its last term, 

co x (co x r) = (co ■ r)co — ui 2 r = —u) 2 r pi 

where r p = vector of perpendicular distance from co — axis 
(through O) to P, (fig. 1.6(b)),is called centripetal acceleration of T 5 ], 

(1.7.12d) 

A or = 2 to x v re i = 2a) x ( dr/dt ) = 2 \fdx/dt)(du x /dt) + (dy / dt)(du y /dt) 

+ {dz / dt){du z / dtf): 

Coriolis (or complementary) acceleration of P 

[ = Acceleration due to the coupling between the relative motion of the 

particle P, v rel , and the absolute rotation (transport motion) of the frame 

M, co; it vanishes if v rel = 0, or if co is parallel to v re j]. (1.7.12e) 

If co = 0 and a = 0 — that is, if M translates relative to F — these equations reduce to 

v = v re i + Vo = dr/dt + dro/dt = dr/dt + dro/dt, (1.7.12f ) 

a = Ae l +«o = d 2 r/dt 2 + d 2 r 0 /dt 2 = d 2 r/dt 2 + d 2 r 0 /di 1 , (1.7.12g) 


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122 CHAPTER 1: BACKGROUND 

Component Forms 

To appreciate eqs. (1.7.11) and (1.7.12) better, and prepare the reader for the key 
concept of nonholonomic coordinates , and so on (§2.9 If.), we present them below in 
terms of their components. In the general case of nonaligned axes we can project 
them on an arbitrary, fixed, or moving axis; that is, each of their terms can be 
resolved along any set of axes. 

(i) The position relation 9? = r a + r, with r 0 = (X 0 , Y 0l Z 0 ), reads 

X = X Q + cos(X, x)x + cos(3f, y)y + cos)!", r)z, etc., cyclically. (1.7.13a) 

(ii) The velocity equations (1.7.11a ff.) assume the following forms, along the fixed 
axes: 

dX/dt = dX 0 /dt + cos(T", x)(dx/dt + uj y z — uj z y) 

+ cos (X,y)(dy/dt + u : x — ui x z) + cos(3f, z)(dz/dt + u) x y — u v x) 

= dX 0 /dt + d / dt(X — X 0 ), etc., cyclically; (1.7.13b) 

and, along the moving axes: 

v • u x = v x = v 0x + dx/dt + u) y z — oj z y, etc., cyclically, (1.7.13c) 

where 

v Qx = v Q • u x = cos (x : X)(dX Q /dt) + cos(x, Y)(dY 0 /dt) + cos (x,Z)(dZ Q /dt): 

component of inertial velocity of moving origin O , along the moving axis 
Ox [in general, not equal to the d/dt{.. .)- derivative of a coordinate , like 
dX Q /dt or dx/dt , and hence a quasi velocity (§2.9ff.)], etc., cyclically. 

(1.7.13d) 

(iii ) The acceleration equations (1.7.12a IT.) read, along the fixed axes: 
d 2 X/dt = d 2 X 0 / dt 2 + cos(X, x) [d/dt{dx/dt + uj y z — u> z y) 

+ uj y (dz/dt + yco x — xco y ) — uj z (dy/dt + xu> z — zuj x )\ + ••• 

= d 2 X 0 /df + cos(3f, x){(d 2 x/dt 2 ) + [z(du y /dt) - y{dui z /dt)] 

+ uj y (u x y — LO y x) — oj z (lo z x — oj x z) 

+ 2[co y (dz/dt) — u) z (dy/dt)]} + ■■■ 

= d 2 X 0 /dt 2 + d 2 /dt 2 (X — X 0 ), etc., cyclically; (1.7.13e) 

= (d 2 X/dt\ el + (d 2 X/dt\ ans + (d 2 X/dt 2 ) cm , (1.7.13f) 

where 

( d 2 X/dt 2 ) tel = cos {X, x){d 2 x / dt 2 ) + cos (X,y)(d 2 y/dt 2 ) +cos (X,z)(d 2 z/dt 2 ), 

(d 2 X/dt 2 ) tans = d 2 X 0 /dt 2 + cos(X,x) {[z{du] y /dt) - y(dio z /dt)\ 

+ tu y (u} x y — oj y x) — cj z {uj z x — oj x z )} + • • •, 
(d 2 X/dt 2 ) cor = cos(X, x){2[u} y {dz/dt) - ut z (dy/dt)\ } + • • •, etc., cyclically; 

(1.7.13g) 


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§1.7 ACCELERATED (NONINERTIAL) FRAMES OF REFERENCE 


and, along the moving axes: 

a • u x = a x = a Qx + [d/dt(dx/dt + u> y z — u z y) 

+ ai y {dz/dt + vw , — xuj y ) — u> z {dy/dt + x uj z — zuj x )\ 

FC.rel “b dv.trails “1” dr.cor■ (1.7.13h) 

where 

ci x j*0j d jc jdt , 

«x,trans = «o,.x-+ [z(du y /dt) -y(du z /dt)\ +U)y(u x y-UyX) -uj z (cj z x-uj x z), 

cor — 2 [uj y (dz/dt) — u> : (dy/dt)], and (1.7.13i) 

a 0 x = ao • u x = cos (x,X)(d 2 Xo/dt 2 ) + cos(x, Y)(d 2 Y 0 /dt 2 ) + cos (x,Z)(d 2 Z 0 /dt 2 ), 

(in general, a quasi acceleration ), etc., cyclically. (1.7.13j) 


EXAMPLES 

1. It is not hard to show that the conditions for a particle, with coordinates x, y, 
z, relative to moving axes, to be stationary relative to absolute space are 

u + dx/dt + zujy — yuj z = 0, etc., cyclically, (1.7.14) 

where («, v, w) = inertial components of velocity of origin of moving frame. 

2. Plane Rotation Case. Let us find the components of velocity and acceleration of 
a particle P in motion on a plane described by the two sets of momentarily coincident 
rectangular Cartesian axes, a fixed O XY and a second O xy rotating relative to the 
first so that always OZ = Oz, with angular velocity to = (0,0, u> z = u> z = u). Here, 
momentarily, 


X = x, Y=y. (1.7.15a,b) 

Application of the moving axes theorem (1.7.2a), or (1.7.3c), (1.7.7e), with co XJ = 0 
and c o z = w, yields the velocity components: 

dX/dt = dx/dt — yu>, dY/dt = dy/dt + jcw; (1.7.15c, d) 

and application of that theorem, or (1.7.3c), (1.7.7f), to the above gives the accel¬ 
eration components: 

d 2 X/dt 2 = d / dt(dx / dt — y u) — (dy / dt + x uj)u 

= d 2 x/dt 2 — y(duj/dt) — xui 2 — 2 {dy/dt)co 
(= relative + transport + Coriolis), (1.7.15e) 

d 2 Y/dt 2 = d/dt(dv/dt + xco) + ( dx/dt — yu)ui 

= d 2 y/dt 1 + x(du)/clt) — yu 2 + 2 {dx/dt)cj 
( = relative + transport + Coriolis)', (1.7.15f) 

and similarly for higher d/dt{.. ^-derivatives. 

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CHAPTER 1: BACKGROUND 


[Alternatively, we may start from the geometrical O-XY/O-xy relationship for a 
generic angle of orientation 0 = 0(t): 

X = (cos 4>)x + (— sin f)y, Y = (sin0)x + (cos0)_v, (1.7.15g) 

d/dt{.. .)-differentiate it, and then set 0 = 0 (df/dt = u f 0), thus obtaining 
(1.7.15c, d); then d/dt(.. .)-differentiate once more, for general 0, and then set 
0—0 (to f 0, duj/dt = a f 0), thus obtaining (1.7.15e, f). The details of this straight¬ 
forward calculation are left to the reader. In this way we do not have to remember 
any kinematical theorems—differential calculus does it for us!] 

3. Velocity and Acceleration in Plane Polar Coordinates via the Moving Axes 
Theorem [continued from (1.7.10c—e)]. Here, with the usual notations, 

r = ru,. and m = (d(/>/dt)u z = (d(f>/dt)u z , (1.7.16a) 

and, therefore, by direct d/dt (...)-differentiation and then use of (1.7.4i) — that is, 
treating the corresponding OND basis/axes through P , P — u r ii Ci) /r, as the moving 
frame — we obtain 

(i) v = dr/dt = ( dr/dt)u r + r(du r /dt) = ( dr/dt)u r + r(m x */,.) 

= ( dr/dt)u+ r{d(/)/dt)(u z x u r ) = ( dr/dt)u r + r{d(/>/dt/u^ = v r u r + rv^u^\ 

(1.7.16b) 

(ii) a = dv/dt = ( dv r /dt)u r + v r (du,./dt) + [d/rv^/dt/ju^, + (rv^/du^/dt) 

= ( dv,./dt)u r + vfco x u r ) + [d(rv <j> )/dt)]u (j> + (rv 0 )(<n x uf) 

= ( dv r /dt)u r + v r [{d(j)/dt)uf\ + [dirvfj / 'dt))u^ + {rv^-df/ dt)u,] 

= [ dv,./dt — {df/df/rvffu,. + [vfdf/dt) + d{rv^)/dt\u^ 

= [d 2 r/dt 2 — r{d(/>/dt) 2 ]ii r + {(dr / dt)/d(f>/dt ) + d / dt[r(d(f>/dt)}} u^, 

= [ d 2 r/dt 2 — r(d(f>/dt) 2 ]n r + \2{dr / dt){d(j) / dt) + r(d 2 (j)/dt 2 )\u^ 

= a [r) u r + (1.7.16c) 

4. Velocity and Acceleration in Spherical Coordinates via the Moving Axes 
Theorem. Proceeding as in the preceding example, and since here r = ru,. (not 
the r of the polar cylindrical case) and to = (dc/>/dt)u z + (dO/dt/u^, 
u z = — sin 9u g + cos #«,., we can show that the velocity and acceleration are given, 
respectively, by 

v = ( dr/dt)u r + [r/dd/dt)]u g + [r/df/dt) sin^Ji/^ = v,.u r + rv g u g + v^u^, (1.7.17a) 

a = [ d 2 r/dt 2 — r(d9/dt)~ — r/df/dt)" sin 2 9\u r 

+ [2/dr/dt){d9/dt) + r{d 2 9/dt 2 ) — r/dfi/dt) 2 sin#cos#]M fl 
+ [ 2{dr/dt){d(/)/dt ) sin# + r{d 2 cj)/dt 2 ) sin# + 2r{dcj) / dt){d9 / dt) cos#]h^, 
= a {r) u r + a m u g + a^u^. (1.7.17b) 

The above are, naturally, in agreement with (1.2.8a ff.) 

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§1.7 ACCELERATED (NONINERTIAL) FRAMES OF REFERENCE 


5. Inertial Angular Velocity of the Natural, or Intrinsic, OND Triad 0 M —u t u„Ub = 
0 M —tnb\ Fre.net Secret Equations (fig. 1.7). We have already seen (§1.2) that 

dt/ds = n/p => dt/dt = (dt/ds)(ds/dt) = [(ds/dt)/p)n = (v t /p)n , (1.7.18a) 

also 

b = t x n. (1.7.18b) 

Next, d/dt( (. . .(-differentiating b-t = 0, we obtain 

0 = ( db/dt ) • t + b • (dt/dt) = (db/dt) •t + b • [(v t /p)n] = ( db/dt ) • t\ (1.7.18c) 

and, similarly, d/dt(.. .(-differentiating b ■ b = 1 we readily conclude that 

(db/dt) -h = 0. (1.7.18d) 

Equations (1.7.18c, d) show that db/dt must be perpendicular to both t and b. Hence, 
we can set 

db/ds = —(1 /r)« =>■ db/dt = (db/ds) (ds/dt) = —(v t /r)n, (1.7.18e) 

where r = radius of torsion (or second curvature) of the curve C, traced by the moving 
origin 0 M = O, at O: positive (negative) whenever the tip of db/dt turns around t 
positively (negatively); that is, like a right- (left-)hand screw; or, according as db/dt 
has the opposite (same) direction as n. [Some authors use r for our 7/r; others use p K and 
p T for our p and r, respectively.] 

Now, the angular velocity of O tnb, relative to some background fixed triad 
Op—UxUyUz, is found by application of the basic formulae (1.7.4j), with the identi¬ 
fication 0 M —u x u y u z = O—tnb , and eqs. (1.7.18a-e). Thus, we find 

Tangent, u, —> u> x = u z • (du y /dt) = —u r • (du z /dt) = —n • (db/dt) = v ( /t; (1.7.1 8f ) 

Normal: u> n —> „ = u x • (du z /dt) = —• (du x /dt) = —b • (dt/dt) = 0; (1.7.18g) 

Binormal: u h —> co z = u y • (du x /dt) = — u x • (du y /dt) = n • (dt/dt) = v t /p. (1.7.18h) 


Binomial (b, z) 



Figure 1.7 On the geometry and kinematics of the Frenet-Serret 
triad O-tnb. 


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CHAPTER 1: BACKGROUND 


In sum, the triad O-tnb rotates with inertial angular velocity: 

( 0 = (v,/t )t + (0 )n + ( v,/p)b = v,(t /t + b/p). (1.7.18i) 

In general, a> ^ d6/dt, where 9 is some vector expressing angular displacement/rota- 
tion; that is, 6 is a quasi vector (§1.10, chap. 2). Further, from (1.7.18a-e) we also conclude 
that 

dn/ds = d/ds(b x/) = ( db/ds ) x t + h x ( dt/ds ) 

= “(I M(n Xt) + (1 /p)(b x n) 

= -(1 M(-b) + (l/p)(-f) = (-l/p)/+ (l/r)4. (1.7.18j) 

Equations (1.7.18a, e, j) (where 0 ^ p ^ +oo and —oo ^ r ^ +oo, r ^ 0) are the famous 
Frenet-Serret formulae for a space (or skew, or twisted ) curve. It is shown in differential 
geometry that: the “natural/intrinsic” curve equations p = p{s) and r = r(s) determine 
the spatial position of that curve to within a rigid displacement (i.e., a translation and a 
rotation). 

The F-S equations can also be written in the following memorable “antisymmetric 
form”: 

dt/dt= (0)f+(v f /p)n+(0)A, (1.7.18k) 

dn/dt = (— v t ] p)t + (0 )n + ( v t /r)b , (1.7.181) 

db/dt = (0)f + (— v t /r)n + (0 )b. (1.7.18m) 

The above allow us to calculate the torsion, l/r. From (1.7.18j, k, 1), with 
(...)' = d(.. .)/ds, we get 

b/T = t/p+ ( pt ')' = t/p + p't' + pt" = t/p + p'(n/p) + pr"', (1.7.18n) 

and so, dotting this equation with b , we find 

1 j T = p(b ■ r"') = p[(t x n) • r'"] = p 2 [(r' x r") ■ /"], (1.7.18o) 

or, since [recalling (1.2.4c)] 

1/p 2 = r" • r" = \r r '\ 2 , (1.7.18p) 

finally, 

Torsion = l/r= [(/ x r") • r"']/\r"\ 2 . (1.7.18q) 

With the help of the above, we can easily show that 

(i) The Frenet-Serret equations can be put in the following kinematical form: 

dt/dt = a) x t, dn/dt = a> x n, db/dt = a> x h 

(tu: kinematical Darboux vector, (1.7.18i)). (1.7.19a) 

(ii) If t, n , b can be expressed, in terms of their direction cosines along a fixed OND 
triad, as 

t={t\,t 2 ,h), n= (n u n 2 ,n 3 ), b = (b u b 2 ,b 3 ), (1.7.19b) 

then 

dt\/ds = n { /p , dni/ds = b\/r — t\/p, db\/ds = — «i/t; 

and similarly for the other components. 

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(1.7.19c) 


§1.7 ACCELERATED (NONINERTIAL) FRAMES OF REFERENCE 


(iii) The (inertial) angular acceleration of the F renet Serret triad, a = dm/dt, is 
given by 

a = [(dv t /dt)/r — (v 2 /T 2 )(dr/ds))t + [( dv t /dt)/p — (v 2 / p 2 )(dp/ds)\b. (1.7.19d) 

(iv) The (inertial) jerk vector of a particle, j = da/dt (or hyperacceleration, or 
velocity of the acceleration) is expressed along the Frenet-Serret triad as 

j = [d 2 v,/dt 2 - ( v, 3 /p 2 )\t + {v, 2 [d/dt{\/p)} + 3 v,{dv,/dt)/p)]}n + (v 3 / pr)b 

= [d 2 v,/dt 2 - (v 3 /p 2 )\t + [d/dt(v 3 /p)/v,]n+(v 3 / P T)\h 

= [d 2 v,/dt 2 - {v, 3 /p 2 )\t + [(3v 2 /p){dv,/ds) - (v, 3 / p 2 ){dp/ds)]n + (v 3 / pr)]b, 

(1.7.19e) 

where 

d(.. .)/dt = [d(...) / ds](ds / dt) = . .)/ds]\ 

that is, contrary to the acceleration, a = (dv,/dt)t + (v 2 /p) n, the jerk vector has t, n, 
and b components, and involves both p and r. 

(v) The following kinematic formulae hold for the curvature and torsion: 

k = l/p = |v x «|/|v| 3 = [v 2 a 2 — (v-a) 2 ] X 2 /v 3 , 

7/r = [v (a xj)\/{a xjf = (v, a,j)/n 2 v 6 . (1.7.19f) 


HISTORICAL 

The theory of accelerations of any order (along general curvilinear coordinates) is 
due to the Russian mathematician/mechanician Somov (1860s), who also gave recur¬ 
rence formulae, from the (n — l)th order to the («)th order; and to the French 
mathematician Bouquet (1879). The second order shown above is due to the 
French mechanician Resal (1862), although the earliest such investigations seem to 
be due to a certain Transon (1845) (see, e.g., Schonflies and Griibler, 1902: 1901 
1908). The jerk vector is called “acceleration du second ordre” (Resal), or 
"Beschleunigung a (2 *” (Schonflies/Grubler), where the ordinary acceleration (of 
the first order) is denoted by a : 1J = a. Clearly, such derivations are enormously 
aided with the use of vectors. These results allowed Mobius (1846, 1848) to give a 
geometrical interpretation to Taylor’s expansion (with some standard notations): 

Ar = r(t) - r(0) = vt + a (,) (t 2 /2) + « (2) (t 3 /1.2.3) + • • • 

= chord of particle trajectory between the times 0 and t. 


Particle Kinetics in Moving Frames 

Substituting the inertial acceleration a of a particle P of mass m, in terms of its 
moving axes representation, into its Newton-Euler equation of motion 

ma= f (= toted noninertial , or real, or objective, force on P ), (1.7.20a) 

and, rearranging slightly, we obtain its fundamental equation of relative motion (fig. 

1 . 8 ): 

ma re \ f T ./'trails + ./cor ' 

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(1.7.20b) 


CHAPTER 1: BACKGROUND 



in words: 

mass x relative acceleration (« re i) = total real (/) plus inertial (/ lrans + /cor) f orce i 

(1.7.20c) 


where 

a re i = dv re \/dt = d 2 rldt 2 : apparent or Relative acceleration ofP, (1.7.20d) 

/ trans = — wa tra ns = — m[d 2 r 0 /dt 2 + os x r+ a> x (to x r)]: 

total iniertial force of Transport on P 

= —m(d 2 i’ 0 /dt 2 ) [due to the inertial acceleration of the origin of the moving 

—m(<x x r) [due to the inertial angular acceleration of frame M\ 

— m[co x (to x r)] = —m[(a> • r)a> — ufr] = ■ ■ ■ = mafrp 

[centrifugal force on P, due to the inertial angular velocity of frame M; 
always perpendicular to the instantaneous axis of to, in the plane of P 
and that axis, and directed away from it (fig. 1.6(b))], 

/cor = ~ ma coT = -2 m(co x v re i) = -2 m[u x (dr/dt)\ : 

inertial force of Coriolis (or composite centrifugal force ) on P [due to 
the interaction of the relative motion of P (v re i = dr/di) with the 
absolute rotation of the moving frame (to); normal to both v rel , to, and 
such that v re i, to, and/ cor = 2 m(v rt \ x to), in that order, form a 
right-hand system]. (1.7.20f) 


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§1.7 ACCELERATED (NONINERTIAL) FRAMES OF REFERENCE 


REMARKS 

(i) In classical mechanics, only / is a frame independent, or objective (or absolute) 
force; / trans and / cor are relative (i.e., frame dependent). At most, / can depend 
explicitly on relative positions (displacements), relative velocities, and time; but 
not on relative accelerations (as an independent constitutive equation). [In relativity 
all forces are relative, and hence can be eliminated by proper frame choice. On the 
classical objectivity requirements for /, see, for example, Pars (1965, pp. 11-12), 
Rosenberg (1977, pp. 12-16).] In addition, in general, the relative forces are not 
additive ; for example, the total force acting on a particle P due to two or more 
attracting masses, each exerting separately on it the absolute forces f x and/ 2 , equals 

if 1 ~b fl) A ( f trans ~b f cor) > if 1 ~b ./'trails “b f cor) ~b ( f 2 “b f trans “b ./cor) ■ -^S for the 

Coriolis “force”/ C0r = —2m(to x v rel ), even for the same problem (i.e., same m and/ ) 
that term obviously does depend on the particular noninertial frame used. This, how¬ 
ever, does not mean that its effects on people, property, and so on, are any less physi¬ 
cally/technically real than those of the real force/. [In fact, the study of such similarities 
between these forces led to the general theory of relativity (mid-1910s).] 

For the comoving (noninertial) observer, both / trans and/ cor are very real! Some of 
the most spectacular Coriolis effects occur in the atmospheric sciences (meteorology, 
etc.); that is, in phenomena involving the coupling between the motion of large liquid 
and/or gas masses and the Earth’s rotation about its axis. A prime such example is 
Baer’s law of river displacements: The inertia force on the northbound flowing water, 
along a meridian, presses against the right (left) bank in the northern (southern) hemi¬ 
sphere. The effects of this pressure are a stronger erosion of the right embankment; and 
a slightly but measurably higher water level at the right shore of the river. [In view of 
these realities, statements like the following cannot be taken seriously: “From the 
foregoing it is clear that the Coriolis-acceleration term arises from the description 
adopted, namely, via moving observers, and hence, contrary to popular belief it 
bears no physical significance ” [Angeles, 1988, p. 74 (the italics are that author’s)].] 

Finally, since / cor is perpendicular to v rel , its “relative power” / cor • v rel vanishes 
(more on such “gyroscopic forces” in §3.9). 

(ii) In the case of a finite body, v rel (a re i) in (1.7.20b) refers to the relative velocity 
(acceleration) of its center of mass G; and r is the position of G relative to the origin 
of the moving frame. 


Power Theorem in Relative Motion 

This constitutes the vector/particle form of theorems treated in detail in §3.9. Let us 
consider a system S in motion relative to the noninertial axes O xyz. To find its 
power equation in relative variables, we start with the equation of relative motion of 


a generic particle P of S, of mass dm [recall (1.7.20b fif.)] 

dm tf,. e | df 4“ r//'trans “b df co r> (1.7.21a) 

where 

0 ,-ei = dv re i/dt = d 2 r/dt 2 , (1.7.21b) 

df = df + dR ( impressed + constraint reaction —see §3.2) (1.7.21c) 

‘//'trans = —dm « tran s — —dm [d 2 r 0 /dt 2 + at x r + co x (tax r)], (1.7.21 d) 

df COI = —dma cor = —2 dm(yo x v re j) = —2 dm[u) x (dr/dt)\ (1.7.21e) 


129 


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CHAPTER 1: BACKGROUND 


Now, the system power equation corresponding to the particle equation (1.7.21a) is 



S dm fl rel • Cel = S ^ ' Vrel + S ^rel ' Cel + S rf /cor ' Cel- 

(1.7.21f) 

Let us transform each of its terms: 


(i) 

S dm «rel • Vrei = S dm Cel ‘ (^Cel / dt) 



= d/ <9r( S ( 1/2 ) dm Cel • Vrei) = dT rel /dt, 

(1.7.21 g) 

or, since 



Vrei • {dv K \/dt) = Vrei 1 (dv K \/dt + UJ X V re l) = V re l • (<9v re l/ dt) , 


finally, 

S dm Orel • Vrel = dT rc] /dt = dT le \/dt. 

(1.7.21 h) 

(ii) We define 



S df‘ Cel = S d f ■ (dr/dt) = d'W/dt ; 

(1.7.21i) 

where, 

in general, no W exists (i.e., W is a quasi variable — more on this 

in §2.9). If 


$ dR • v re i = 0 , then d'W/ dt = $ dF • v re i. 


(iii) Clearly, 



S d f cot • Vrel = S [ - 2 dm (® X V re l)] • V re l = 0. 

(1 -7-2 lj) 

(iv) 

S d ftel ■ Vrel = ~S dm [“0 + * X r + m X (fi> X v)] • V rel . 

(1.7.21k) 

(a) 

- £ dm a 0 • v rel = -a Q ■ yfij dm v re i) = -m v Gjre i • «o (vc,rei = 

e drc /dt) . 


(1.7.211) 

(b) 

- $ dm [v rel • (a x r)] = -a • dm ( r x Cel)) = • #o,rei- 

(1.7.21m) 

(c) 

We have, successively, 



v re i • [w x (w x r)] = (w x r) ■ (v re i x ca) = —(uj x r) • [uj x (dr/dt)\ 

— —d/dt[(uj x r) 2 /2] = — d/dt[\uj x r| 2 /2] = —d/dt [\uj x r| 2 /2] , 

(1.7.21n) 

(i.e., as if during d/dt the vector uj remains constant) and, therefore, 

— $ dmv Te \ • [uj x (uj x r)] = d/dt(^Q dm[\uj x /*| 2 /2]^ 

= d/dt(^ $ dm [|cu x r\/ /2\^j. 

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(I. 7 . 210 ) 


§1.7 ACCELERATED (NONINERTIAL) FRAMES OF REFERENCE 


In view of (1.7.21g-o), eq. (1.7.21f) takes the following definitive form: 

dT K \/dt = d'W/dt - (mv G ,rei ) -«o —H 0 , re i • <x + d/dt(^^ dm [|u; x r| 2 /2]^. (1.7.21p) 


Specializations 

If O-xyz spins about a fixed axis through O, Ol, then a Q = 0 and eq. (1.7.21p) 
reduces to 

dT re \/dt = d'W/dt — Ho,re\ • tx + d/dt(Iuj 2 /2), (1.7.21q) 

where 

I = ^dmv 2 = moment of inertia of S about Ol. (1.7.21 r) 

If, further, uj = constant , then (1.7.21q) simplifies to 

dT Kl /dt = d'W/dt + ( dl/dt)uj 2 /2 . (1.7.21s) 

Finally, if d'W/dt = —dVo(r)/t, where Vo = Vo(f) = potential of impressed forces, then 
(1.7.21s) yields the conservation theorem: 

d/dt[T rei + (V 0 - Iuj 2 /2)\ = 0 => r rel + (V 0 ~ Iu 2 / 2) = constant. (1.7.21t) 

The above is a special case of the Jacobi-Painleve integral (§3.9). As with the equa¬ 
tions of motion, the “Newton-Euler” power equation (1.7.2lp) may be physically 
clearer than its Lagrangean counterparts, but the latter have the sam q form in both 
inertial and noninertial frames, and hence are easier to remember and apply. For 
further details and insights, see Hamel (1912, pp. 44(M143). 


The Angular Velocity Tensor 

Moving Axes Components 

Let us consider two OND frames/axes with common origin 0 F = O m = O (no loss in 
generality here), in arbitrary relative motion (rotation): one fixed 0 —u x u y u z I—XYZ 
and another moving 0—u x u y u z /—xyz ; or, compactly (in view of the heavy indicial 
notation that follows), O—u^j—x^ and 0—u k /—Xk, respectively. 

Now, d/dt(.. ^-differentiating their transformation relations, 

u k = ^2 A kk 'U k ', A kk , = u k • u k i = cos(x k , x k >) = cos(x k ,, x k ) = A k - k , (1.7.22a) 

and then employing their inverses, we find (since du k fdt = 0): 

du k /dt = ( dA kk fdt)u k - = Y ( dA kk fdt ) ( Y A di u ' ) = H , (1.7.22b) 

where 

O/k = Y1 A k 't(dA kk '/dt) = Y/, Aik’(dA kk '/dt) = Y^ {dA kk f/dt)A lk ' = ■■■ 

= Y { cos(x,, x h >) d/dt[cos(x k , x k ,)\} 

= I// • ( du k /dt ) = (du k /dt) • ui [= (l)th component of du k /dt\: 

Tensor of angular velocity of moving axes relative to the fixed axes ; 

but resolved along the moving axes. (1.7.22c) 

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132 CHAPTER 1: BACKGROUND 

[As already pointed out (§1.1), this commutativity of subscripts in A _ constitutes one 
of the big advantages of the accented indices over other notations, such as A kh A ' k[ .\ 
Below we show that this tensor is antisymmetric : f2i k = —£2 k i- Indeed, d/dt(...)- 
differentiating the orthonormality conditions (constraints!), 

u k • u/ = ( £ A kk iu k i ^ • ^£ +//'«/') = • • • = £ A kk 'A, k t = S k/ , (1.7.22d) 
and then invoking the definition (1.7.22c) we obtain 

0 = £ {dA kk ,/dt)A/ k J + £ A kk '(dAi k i/dt) [= u/ • ( du k /dt) + u k • ( du//dt)\ 

= fii k + ftki => f l/ k = —£lki , Q.E.D.; (1.7.22e) 

that is, due to the six constraints (1.7.22d), only three of the nine components of 14/ 
are independent. Hence, we can replace this tensor by its axial vector (1.1.16a ff.) 

W* = -££ (l/2)e krs n rs = - £££(1/2 )e krs [A rp ,{dA sp ,/dt)], (1.7.22f) 

and, inversely, 

Mrs = - £ Zkrs U k = - £ e rsk iO k . ( 1.7.22g) 

In extenso, and recalling the properties of e krs (§1.1), eqs. (1.7.22f) yield 

= Wx = — (l/2)(ei23^23 + £ 132 ^ 32 ) = f^23 = f?32 

= — A 2k fdA/, k '/dt) [= — « 2 • (du 2 /dt) = —u y • ( du z /dt)\ 

= £ + 3 k i ( dA lk '/dt ) [= 1/3 • ( du 2 /dt ) = u : • ( du v /dt)\ 

{with 1,2,3 —> x,y,z; l',2',3' -> X, Y,Z: 

= ~[A Xy (dA Xz /dt) + A Yy (dA Yz /dt) + A Zy (dA Zz /dt)\ 

— A Xz {dA Xy /dt) + A Yz (dA Yy /dt') -y A Zz {ydA Zy fdify\ 

UJ 2 =Ul y = —(l/2) (£2311^31 +£213f2i3) = —1731 = I2l3 

= — £ A 3k i (dA\ k '/dt ) [= —m 3 • ( dii[/dt ) = — u : • ( du x /dt)\ 

= Y^ A lk t (dA 2k fdt) [= U\ • ( du 2 /dt ) = u x • ( du./dt)\ 

{ = ~[Axz(dA Xx /dt) + A Yz (dY Yx /dt) + A Zz (dA Zx /dt)] 

= A Xx (dA Xz /dt) + A Yx (dA Yz /dt) + A Zx (dA Zz /dt)}- : 

W 3 = UJ Z = —(l/2)(£3i2l7i2 + £3211721) = — I7l2 = 1721 

= — £ A xk i{dA 2k i/di) [= —i/j • ( du 2 /dt ) = — u x • ( du y /dt)\ 

= Y^ A 2k fdA lk i/dt) [= «2 • ( du\/dt ) = u r • ( du x /dt)\ 

{ = —\Axx{dA Xy /dt) + A Yx (dA Yy /dt) + A Zx (dA Zy /dt)\ 

= A Xy (dA Xx /dt) + A Yy (dA Yx /dt) + A Zy ( dA Zx /dt )}; 


(1.7.23a) 


(1.7.23b) 


(1.7.23c) 


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§1.7 ACCELERATED (NONINERTIAL) FRAMES OF REFERENCE 


which are in complete agreement with (1.7.4j), and justify the name angular velocity 
tensor for (1.7.22c). In terms of matrices, the above assume the following memorable 
form: 


(12*/) 


7 0 —CU3 0J2 \ 

ui 3 0 —cuj 


y — CU2 CJ\ 0 J 


( 0 

E A \k'{dA 2 k'/dt) 

E A \k'{dA2 k fdt)\ 

(1.7.23d) 

E A ik'{dA\ k f dt) 

0 

E A 2k' {dAyfr-r/dt) 

\E A 3k’{dA lk f/dt) 

E A 2k'{dA 2 k'/dt) 

o ) 


REMARKS 

(i) The formulae (1.7.22f ff.) can be combined into the following useful form: 

to k = ( du,./dt) •*/,, (1.7.24) 

where 

k,r,s= cyclic {even) permutation of 1,2, 3 (= x,y, r). 

(ii) The final expressions (1.7.23d) would have resulted if we had employed the 
following common angular velocity tensor definitions: 

1 = ( du k /dt) ■ u, = ~{du,/dt) ■ u k = ^ {dA kk , / dt)A Ik , = - ^ {dA Ik ,/dt)A kk ,, 

(1.7.25a) 

but in connection with the also common axial vector definition-. 

UJ k = EE (1/2 )e krs f2 rs (1.7.25b) 

Then, we would have 

U 1 = (l/2) (£1231723 +£1321732) = 1723 = —1732 

= E ( dA 2 k'ldt)A 2 k , = -^2 (, dA 3k fdt)A 2k etc. (1.7.25c) 


Fixed Axes Components 

Let us express the above inertial angular velocity tensor in terms of their components 
along the fixed axes 0—u x u y u z /—XYZ = 0—u k f—x k '. Dotting the representations of 
the position vector of a typical particle P, 

r = Y^ x k u k = E (1 -7.26a) 

with «/ and uy, respectively, and taking into account the orthonormality constraints 
of their basis vectors: 

I'k ■ Ui = (E A kk'«k') • (E A ii' u r) = E A kk ,A ik' = 6 k h (1.7.26b) 

Ilk' -U/' = (E A k'k«k) ■ (E A n U l) = E A k'k A l'k = 4'/', (1.7.26c) 

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CHAPTER 1: BACKGROUND 


we easily obtain the component transformation equation 

x k' = Y A k'k*k *>x k = Y A kk' x k'- (1.7.26d) 

If the two sets of axes do not have a common origin, but [recalling fig. 1.6(a)] 

9? = /* 0 + r, (1.7.26e) 

where 

SR = E **'«*', (1.7.26f) 

To = T m0 ving origin/fixed origin ^ ' b k U k ^ , b k ' ^k ' 

=>• h' = Y A k ' k b k b k = Y A kk fb k ', (1.7.26g) 

r = ^2 x k u k , (1.7.26h) 

then (1.7.26d) are replaced by 

X k' = E A k'k x k + b k ' = E ^ k ' k (x k + b k ) 
x k = Y A kk'{x k ' - b k 0 = E ^ k k lX k' - b k . (1-7.26i) 

Now, let us consider P to be rigidly attached to the moving axes. Then d/dt(.. .)- 
differentiating the x k ', while recalling that in this case x k = constant =>• dx k /dt = 0, 
we obtain, successively, 

dxu/dt = E {dA k , k /di)x k = E (dA k . k /dt) ( Y A ki' x i ') = Y Q k'i' x f > 

[which is none other than the familiar v = <o x r, resolved along the fixed axes] 

(1.7.26]) 


where 


Q k 'i' = Yj ( dA k ' k /dt)A kt i = Y, ( dA k : k /dt)A rk 

= Y{ cos ( x r,Xk) d/dt[cos(x k ',x k )]}: 

Tensor of angular velocity of moving axes relative to the fixed axes ; 
but resolved along the fixed axes [Note order of accented indices, and 
compare with order of unaccented indices in expression (1.7.22c, 25a).](1.7.26k) 

The components flgy, just like the f2 k i, are antisymmetric. Indeed, d/dt(.. .)-differ¬ 
entiating (1.7.26c), we obtain 

0 = Y, {dA k ' k /dt)A/f k + Y, A k'k{dAi' k /dt) = Qf# + 

=> n (k! = -f4y , Q.E.D. (1.7.261) 

u* = “EE (l/ 2 )^w4v = (1.7.26m) 

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§1.7 ACCELERATED (NONINERTIAL) FRAMES OF REFERENCE 

and, inversely, 

Q's' = ~ £ k'r's' u k' = — £ r's'k lU} k '! (1.7.26n) 

or, in extenso, 

w l' = w V = = “ (l/2)( e l'2 , 3'^2'3' + Ei'3'2'^3'2') = — ^2'3' = ^3'2' 

{with 1,2,3 —> x,y,z; l',2',3' -► X, Y,Z: 

= — [A Zx {dA Yx /dt) + A Zy {dA Y y / dt) + A Zz (dA Yz /dt)\ 

— A Yx (dA Zx /dt) A Yy {dA Zy j dt) -\- A Yz (dA Zz /dt)'^', (1.7.27a) 

w 2' = u y' = w r= — (l/2)( £ 2'3'l'^3'l' + e2'l'3'^l'3') = ~^3'l' = ^l'3' 

= ~Yz A\'k(dA Vk /dt) = Y / Ay k (dAy k /dt) 

{ = — + A Xy (dA Z y/dt) + /l X; (r/y4 z „/(7r)] 

= A Zx {dA Xx /dt) + A Zy (dA Xy ldt) + A Zz {dA Xz /dt )}; (1.7.27b) 

w 3 ' = w z' — W Z = ^(l/ 2 )( e 3 'l' 2 '^l' 2 ' + e 3 ' 2 'l'^ 2 'l') = — ^l' 2 ' = ^ 2 'l' 

= ~Y A Tk{dA Vk /dt) = Y A Vk {dA Vk /dt) 

{ = [A Yx (dA Xx / dt) + A Y y(dA Xy /dt) + A Yz (dA Xz /dt)] 

= A Xx (dA Yx /dt) + A Xy (dA Yy / dt) + A Xz (dA Yz /dt)}\ (1.7.27c) 

or, finally, in the following memorable matrix form: 

^ 0 —LUy 0J 2 ' \ 


Mt'/O = 


UJy 0 — UJ\ 

\— UJ2' (jJ\> 0 J 


0 

E {dA\' k /dt)A 2 ' k 

E {dAy k /dt)Ay k 

E {dA Vk /dt)A Vk 

0 

E {dA 2 ’k/dt)Ay k 

\E {dAy k /dt)Ay k 

E {dAy k /dt)A 2 ' k 

0 


(1.7.27d) 


or 


-h'l' — Y, Y A k 'k A-i'i^ki ^ ^ 4 / — A kk >A n ' fly. 


(1.7.27e) 


A Special Case 

If the axes and x k > coincide momentarily —that is, if instantaneously A k ' k = 6 k ' k 
(Kronecker delta), then eqs. (1.7.23) and (1.7.27) yield 

to x = dA Zv /dt = —dA Yz /dt , ui y = dA Xz /dt = —dA^/dt, 

to z = dA Yx /dt = —dA Xy /dt ; (1.7.28a) 

coy = dA Zy /dt — — dA Yz jdt , cuy : dA x ~/dt — — dA Zx /dt : 

cj z — dA Yx /dt = — dA Xy /dt. (1.7.28b) 

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135 















CHAPTER 1: BACKGROUND 


Rates of Change of Direction Cosines 

Let us calculate dAp k /dt in term of f2 k i, fi k 'f. 

(i) Fixed axes representation-. Multiplying both sides of (1.7.22c) with A/// and 
summing over /, we obtain 

E fiikA-n = E (' dA k i k / dt) (E A k i / A r/ 'j = E {dA k > k /dt){8 k >i>) = dA rk /dt. 

(1.7.29a) 

(ii) Moving axes representation-. Multiplying both sides of (1.7.26k) with A ks and 
summing over /', we obtain 

E f2 k ' t ' A t ' s = E (' dA k t k / dt) (E A kr A k ^ = E ( dA k i k /dt)(6 ks ) = dA k ' s /dt\ 

(1.7.29b) 


dA, 

II 

YAk'Ak = 

E ^4'/' i 


(1.7.29c) 

— A^/j + A 

k '2^2k + 

A k > 2 f2 3k 






=>■ dA k >\/dt = 

A k ' 2 uj 2 

— A k ' 2 u 2 -, 

i.e., 

dA k > 

x/dt = 

= d. k 'yU z ~ 

A k ' Z Uy, 

dA k i 2 /dt = 

A k f 3 uji 

- A k :\LO y 

i.e., 

dA k 

'y/dt = 

= A t z z w Y - 

■ A k i x u> z , 

dA k i 3 /dt = 

A k '\U> 2 

- A k ' 2 u> x \ 

i.e., 

dA k 

z z /d? = 

= A k t x u) y - 

- A k 'yU x 






(*' = 

Z, y,Z); 

(1.7.29d) 

+ 

dr 

.id 

ii 

^2' k-^k'2 

' + A'k^k'l' 






=>• dAy k /dt = 

A Vk u 2 ' 

— A 2 ' k L0 2 i; 

i.e. 

•; dA xk /dt 

= d zk u Y 

- A Yk u z , 

dA 2 ' k l dt = 

Ai' k Uy 

— A 3 z jt w 1 z; 

i.e. 

dA 

n-/ 

= A xk (x>z 

_ 

dA yk /dt = 

A 2 'kU\' 

— A^u^z; 

i.e 

., dA 

Zkl dt 

= ^U' W X 

- dxk^Y 






(k 

= -UZA). 

(1.7.29e) 


Additional Useful Results 

(i) By d/dt{.. ^-differentiating the fixed basis vectors: 

0 = du k i/dt = E [( dA k . k /dt)u k + A k , k {du k /dt)\ = • • •, (1.7.30a) 

it can be shown that 

du k /dt = E Q k' k u k ' , (1.7.30b) 

where 

O k > k = = yy (9x ; z / = • • ■ = dA k i k /dt (mixed “tensor”) 

(1.7.30c) 

Similarly, we can define the following mixed angular velocity “tensor”-. 

-<V = E = ^2(dx,/dx k ')f2 k , = E4(EEV A W Q pw') 

= • • • = e ^i'k‘ ■ 


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§1.7 ACCELERATED (NONINERTIAL) FRAMES OF REFERENCE 

(ii) By d/dt{. . ^-differentiating x' k < = x k > = A k ' k x k , and noticing that 
v 'k = E Akk'V k ', it can be shown that the inertial velocity of a particle permanently 


fixed in the moving frame (i.e., dx k /dt = v k = 0 =>- v k * = 0) equals: 

v k = Y, f4/X/ (along the moving axes), (1.7.30d) 

dx k '/dt = v' k i = Y ftk'i'Xi’ (along the fixed axes). (1.7.30e) 

(iii) Let us define the following matrices : 

ft = (, fi k i ): matrix of angular velocity tensor, along the moving axes, (1.7.30f) 

ft' = matrix of angular velocity tensor, along the fixed axes, (1.7.30g) 

A = (A k > k ): matrix of direction cosines between moving and fixed axes. (1.7.30h) 


It can be shown that the earlier relations among them (i.e., among their elements) 
can be put in the following matrix forms [recalling that A -1 = A T and 
(.. .) T = Transpose of (...)]: 

ft = A t • (dA/dt) = — (dA/dt) T • A dA/dt = A • ft , (1.7.30i) 

JV = (dA/dt)-A T = -A-(dA/dt) T <*=> dA/dt = fl' - A, (1.7.30j) 
W = A • • A t O n = A t • Cl' ■ A . (1.7.30k) 

[(a) Equation (1.7.30j) expresses the following important general theorem: for an 
arbitrary (differentiable) orthogonal matrix (or tensor) A = A(t), 

dA/dt = (matrix of second-order antisymmetric tensor) • A; (1.7.301) 

and similarly for equation (1.7.30i). 

(b) Recall remarks on p. 84, below (1.1.19f), e.g. Ap 2 = A 2 y fA 2 y = A 12 /.] 


Angular Velocity Vector in General Orthogonal 
Curvilinear Coordinates 

[This section may be omitted in a first reading. For background, see (1.2.7a ft.).] 

In such coordinates, say q = [q \, q 2 , qf] = (( 71 , 2 , 3 ), the inertial position vector of a 
particle r becomes 

r = X(q)u x + Y(q)u Y + Z(q)u z = ^ x k fq)u e , (1.7.31a) 

and so the corresponding moving OND basis along q { 2 ,3 (i-e., the earlier x k ) is 

„ k = (dr/dq k )/\dr/dq k \ = (\/ h k )(dr / dq k ) (k = x,y,z), (1.7.31b) 

= (k, l = x,y,z). (1.7.31c) 

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with 


CHAPTER 1: BACKGROUND 


Next, d/dt(.. ^-differentiating (1.7.31b), we obtain, successively, 
d/dt(dr/dq r ) = d/dt(h r u r ) = (dh,./dt)u r + h r (du r /dt) 

= ( dh r /dt)u r + h r (o) x u r ) [by (1.7.4i)], (1.7.3Id) 

and dotting this with dr/dq s = h s u s (= e s , where r f s), in order to isolate u k , we get 

[d/dt(dr/dq r )] • ( dr/dq s ) = ( dh r /dt)hfu r ■ u s ) + h r h s [(a> x u r ) • «J 

= 0 + h r h s [(co • («,. x «,)] = h r h s (m ■ u k ) = h r h s u) k 

[dehnition of io k &\ where k, r, s = even (cyclic) permutation of 1, 2, 3 = x, y, z], 


that is, finally, 

Uk = (1 /h,.h s ) [d/dt(dr/dq r ) • (dr/dq s )\ 

{ = ( du r /dt ) • u s = d/dt[(l/h r )(dr/dq,.)] • [(1 /h s )(dr/dq s )]}. (1.7.3le) 

Additional forms for these components exist in the literature; for example, with the 
help of the differential-geometric identities: 

du r /dq s = (1 /h r )(dh s /dq r )u s (r f 5 ), (1.7.31f) 

du,./dq r = -{\/h s ){dh r /dq s )u s - (1 /h k ){dh r /dq k )u k (rfsfkfr), (1.7.31g) 

and applying the second line of (1.7.3 le), we can easily show that 

wi = (l/h 2 )(dh 3 /dq 2 )(dq 3 /dt) - {l/h 3 )(dh 2 /dq 3 )(dq 2 /dt), (1.7.31h) 

w 2 = {1 / h 3 )(dh\ / dq 3 )(dqi / dt) - (l/h x )(dh 3 /dq x )(dq 3 /dt), (1.7.31i) 

w 3 = (l/h x )(dh 2 /dq x )(dq 2 /dt) - {\/h 2 ){dh x /dq 2 ){dq x /dt). (1.7.31J) 


[See Richardson (1992), also Ames and Murnaghan (1929, pp. 26-34, 94-98), for an 
alternative derivation based on the direction cosines between the moving and fixed 
axes: 

A k 'k = A kk t = u k '-u k = ( dr/dx k t) ■ [(1 /h k )(dr/dq k )\ 

= (1 /h)[{dr/dx k ’) ■ (^2 ( dr/dx r )(dx r /dq k 2j j 

= (1 /h) (X! (“*' • u i')i. dx i'/ dc lS) (since u k > ■ u v = 6 kr ) 

= (1 /h k )(dx k fdq k ), (1.7.31k) 

and their d/dt^.. ^-derivatives.] 


1.8 THE RIGID BODY: INTRODUCTION 

The following material relies heavily on the preceding theory of moving axes (§1.7). 
The reason for this is that every set of such axes can be thought of as a moving rigid 
body, and, conversely, every rigid body in motion carries along with it one or more sets 
of axes rigidly attached to it, or embedded in it. To describe the translatory and 

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§1.8 THE RIGID BODY: INTRODUCTION 



Figure 1.9 Axes used to describe rigid-body motion. 

O-XYZ/IJK: fixed axes/basis; +-xyz/ijk: moving (body-fixed) axes/basis; 
+-XYZ/IJK: moving, translating but nonrotating axes/basis. 


angular motion of a rigid body B, we consider (at least) two sets of rectangular 
Cartesian axes and associated bases: 

(i) a fixed: that is, inertial, O XYZ/1JK or compactly 0-x k fiu k r, and 

(ii) a moving: that is, noninertial, and body-fixed set +-xyz/ijk or compactly ♦ -x k /u k , 
at the arbitrary body point ♦ (fig. 1.9). 

In the language of constraints (chap. 2), a free rigid body in space is a mechanical 
system with six degrees of global freedom; that is, six independent possibilities of 
finite spatial mobility: (i) three for the location of its body point ♦, say its O—XYZ 
coordinates 

X.=fi{i), Y.=f 2 (t), Z*=/ 3 (0; (1.8.1a) 

and (ii) three for its orientation —that is, of ♦ xyz relative to either O XYZ or ♦- 
XYZ\ where the latter are a translating frame at ♦ ever parallel to O XYZ —that is, 
one that is nonrotating but translating and hence is, generally, noninertial. Such 
“rotational freedoms” can be described via the nine direction cosines of ♦-xyz 
relative to +-XYZ (of which, as explained in §1.7, only three are independent); or 
via their three attitude angles: for example, their Eulerian or Cardanian angles 

4>=m, e=f 5 {t), *p=f 6 (ty, (l.s.ib) 

or via a directed line segment called rotation “ vector ” [or via four parameter form¬ 
alisms (plus one constraint among them); for example, Hamiltonian quaternions, 
Euler-Rodrigues parameters, or complex numbers; detailed in kinematics treatises, also 
our Elementary Mechanics, ch. 16 (under production)]. With the help of the six positional 
system parameters, or system coordinates, fi 6 (t), the location/motion of any other body 
point P can be determined: 

'' = r{P, t) = r{P ; ,/ 6 ) = r.(f ], / 2 ,/ 3 ) + r /4 (P; / 4 , f 5 ,f 6 ), 

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139 


(1.8.2a) 







CHAPTER 1: BACKGROUND 


or, in components, 

X = Z* + cos(A, x)x/* + cos(X,y)y/+ + cos(X,z)z/,*, etc., cyclically, (1.8.2b) 
where 

»■/♦ = 0 >> In¬ 

constant rectangular Cartesian coordinates of P relative to +—xyz. (1.8.2c) 

or, in compact (self-explanatory) indicial notation, 

Xk' = *♦,*' + A k'k x k- (1.8.2d) 

In addition to ♦ -xyz and +-XYZ, we occasionally use other intermediate axes (or 
accessory axes, in Routh’s terminology) that, like +-XYZ, are neither space- nor 
body-fixed, but have their own special translatory and/or rotatory motion. 


1.9 THE RIGID BODY: GEOMETRY OF MOTION AND KINEMATICS 
(SUMMARY OF BASIC THEOREMS) 

Sections §1.9-1.13 cover material that is due to Euler, Mozzi, Cauchy, Chasles, 
Poinsot, Rodrigues, Cayley et al. (late 18th to mid-19th century). For detailed dis¬ 
cussions, proofs, insights, and so on, see for example (alphabetically): Alt (1927), 
Altmann (1986), Beyer (1929, 1963), Bottema and Roth (1979), Coe (1938), Gamier 
(1951, 1956, 1960), Hunt (1978), McCarthy (1990), Schonflies and Griibler (1902), 
Timerding (1902, 1908). 

The position, or configuration, of a rigid body B is known when the positions of 
any three noncollinear of its points are known; hence, six independent parameters are 
needed to specify it [e.g., 3x3 = 9 rectangular Cartesian coordinates of these points, 
minus the three independent constraints of distance invariance (i.e., rigidity) among 
them; or six coordinates for two of its points defining an axis of rotation, minus one 
invariance constraint between them, plus the angle of rotation of a body-fixed plane 
with a space-fixed plane, both through that axis]. If the body is further constrained, 
this number is less than six. It follows that the most general change of position, or 
displacement, of B is determined by the displacements of any three noncollinear of its 
points; that is, given their initial and final positions and the initial (final) position of a 
fourth, fifth, and so on, we can find their final (initial) positions with no additional 
data. 


Special Rigid-Body Displacements 

(i) Plane, or planar, displacement'. One in which the paths of all body points are 
plane curves on planes parallel to each other and to a fixed plane / [fig. 1.10(a)]: the 
body fiber P'PP" remains perpendicular to/, and the distance P+ remains constant, 
so that we need to study only the motion of a typical body section, or rigid lamina, b 
imagined superimposed on f and sliding on it. 

THEOREM 

Every displacement of a rigid lamina in its plane is equivalent to a rotation about 
some plane point / [fig. 1.10(b)], 

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§1.9 THE RIGID BODY: GEOMETRY OF MOTION AND KINEMATICS 



iJ 



(...) = midpoint of (...) 


Figure 1.10 (a) Plane displacement of a rigid body, (b) The plane displacement of a rigid lamina 
on its plane is equivalent to a rotation about /; if / — > oo, that displacement degenerates to a 
translation. 


(ii) Translational displacement: One in which all body points have vectorially equal 
velocities. Translations can be either rectilinear or curvilinear, and can be represented 
by a free vector (three components). 

(iii) Rotational displacement: One in which at least two points remain /zxer/. These 
points define the axis of rotation-, and either they are actual body points, or belong to 
its appropriate fictitious rigid extensions. Rotations are, by far, the more complex 
and interesting part of rigid-body displacements/motions. 

The rotation is specified by its axis (i.e., its line of action) and by its angle of 
rotation-, and since a line is specified by, say, its two points of intersection with two 
coordinate planes — that is, four coordinates — and an angle is specified by one 
coordinate, the complete characterization of rotation requires 4+1 = 5 positional 
parameters. 

THEOREM 

Every translation can be decomposed into rotations. 

COROLLARY 

All rigid displacements can be reduced to rotations. The above special displacements 
(plane, translations, rotations) are all examples of constrained motions; that is, they 
result from special geometrical [or finite, or holonomic (chap. 2)] restrictions on the 
globed mobility of the body, as contrasted with local restrictions of its mobility [by 
nonholonomic constraints (chap. 2)]. 

EULER’S THEOREM (1775-1776) 

Any displacement of a rigid body, one point of which is fixed but is otherwise free to 
move, can be achieved by a single rotation, of 180° or less, about some axis through 
that point; that is, any displacement of such a system is equivalent to a rotation. Or: 
any rigid displacement of a spherical surface into itself leaves two (diametrically 
opposed) points of that surface fixed; and hence, in such a displacement, an infinite 
number of points, lying on the axis of rotation defined by the preceding two points, 

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CHAPTER 1: BACKGROUND 


remain fixed. (Under certain conditions this theorem extends to deformable bodies: 
one body-fixed line remains invariant.) 

To understand this fundamental theorem, let us consider a body-fixed unit sphere 
S B with center the fixed point ♦, representing the body, and let us follow its motion 
as it slides over another unit sphere S s concentric to S B but space-fixed and repre¬ 
senting fixed space. (This is the spatial equivalent of the earlier plane motion pro¬ 
blem where a representative rigid lamina slides over another fixed lamina.) Now, 
since this is a three degree-of-freedom system, its position can be specified by the 
coordinates of two of its points on S B , P , and Q [fig. 1.11(a)]: 2x2 = 4 coordinates 
[of which, since the distance between P and Q (= length of arc of great circle joining 
P and Q) remains invariable, only three can be varied independently]. Hence, to 
study two positions of the body—that is, a displacement of it—it suffices to study 
two positions of an arbitrary pair of surface points of it: an initial PQ and a final 
P'Q' [fig. 1.11(b)]. Then we join P and P', and Q and Q' by great arcs and draw the 
two symmetry planes of the arcs PP' and QQ'; that is, the two great circle planes 
that halve these two arcs. Their intersection, ♦C (which, contrary to the plane 




C-*~C, P^P', Q-* Q' 

A A 

CPQ, CP'Q': Congruent Triangles 
[CP = CP', CQ = CQ\ PQ = P'Q'] 


Figure 1.11 (a, b) The motion of a rigid body about a fixed point ♦ can be found by 

studying the motion of a pair of its points on the unit sphere with center ♦: from PQ to P'Q'; 
(c) special case of (b) where the planes of symmetry of the arcs PP' and QQ' coincide. 


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§1.9 THE RIGID BODY: GEOMETRY OF MOTION AND KINEMATICS 


motion case, always lies & finite distance away), defines the axis of rotation; and their 
angle, x, defines the angle of rotation (around ♦C) that brings the spherical triangle 
CPQ into coincidence with its congruent triangle CP'Q'; and hence arc (PP 1 ) into 
coincidence with arc ( QQ'); and +PQ into coincidence with +P'Q ', and similarly for 
any other point of S B . In the special case where these two symmetry planes coincide 
[fig. 1.11(c)], the rotation axis is the intersection of the planes defined by ♦PQ and 

♦P'6'- 

FUNDAMENTAL THEOREM OF GEOMETRY OF RIGID-BODY MOTION 
Any rigid-body displacement can be reduced to a succession of translations and 
rotations. Specifically, any such displacement can be produced by the translation 
of an arbitrary “base point,” or “pole,” of the body, from its initial to its final 
position, followed by a rotation about an axis through the final position of the 
chosen pole—and this is the most general rigid-body displacement. The translatory 
part varies with the pole, but the rotatory part (i.e., the axis direction and angle of 
rotation) is independent of it (fig. 1.12). 

COROLLARY FOR PLANE MOTION 

Any rigid planar displacement can be produced by a single rotation about a certain 
axis perpendicular to the plane of the motion; in the translation case, that axis 
recedes to infinity [fig. 1.10(b)]. 

THEOREMS OF CHASLES (1830) AND POINSOT (1830s, 1850s) 

Any rigid-body displacement can be reduced, by a certain choice of pole, to a screw 
displacement; that is, to a rotation about an axis and a translation along that axis. In 
special cases, either of these two displacements may be missing. 

In a screw displacement: (a) The axis of rotation is called central axis, and (for 
given initial and final body positions) it is unique, except when the displacement is a 
pure translation; (b) The ratio of the translation (/) to the rotation angle (x), which 



Figure 1.12 The most general displacement of the rigid body ♦PQ can be effected by a 
translation of the pole ♦, from ♦PQ to ♦ , P"Q"; followed by a rotation about an axis 
through ♦', from ♦ , P"Q" to ♦'P'Q'. 


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CHAPTER 1: BACKGROUND 


equals the advance (p) per revolution (2ir), is called pitch of the screw: 
p/2ir = l/x => p = 2n(I/x); and (c) The translation and rotation commute. 

EXTENSION TO DEFORMABLE BODIES 
(Chasles’ Theorem + Deformation = Cauchy’s Theorem) 

The total displacement of a generic point of a continuous medium, say a small 
deformable sphere (fig. 1.13), is the result of a translation , a rigid rotation [of the 
local principal axes (or directions) of strain], and stretches along these axes; that is, 
the sphere becomes a general ellipsoid. Hence, rigid-body kinematics is of interest to 
continuum mechanics too; the latter, however, will not be pursued any further here. 


Rigid-Body Kinematics 

Thus far, no restrictions have been placed on the size of the displacements; the above 
theorems hold whether the translations and rotations are finite or infinitesimal. The 
finite case is detailed quantitatively in the following sections. 

Next, let us examine the important case of sequence of rigid infinitesimal displace¬ 
ments in time, namely, rigid motion. In particular, let us return to the motion about a 
fixed point (Euler’s theorem) and consider the case where the initial and final posi¬ 
tions of the arcs PQ (at time t) and P'Q' (at time t' = t + At) are very close to each 
other. Now, as At —> 0 the earlier (great circle) planes that halve the arcs PP' and 
QQ' coincide with the normal planes to the directions of motion of P and Q, respec¬ 
tively, at time t; and their intersection yields the instantaneous axis of rotation. Then 
the velocity of the generic body point P equals 

v P = v= /At)\ At ^ 0 = a) x r P/4t = m x r, (1-9.1) 

since Vp = v = | v| equals the magnitude of the angular velocity of that rotation, |o»|, times 
the perpendicular distance of P from the rotation axis. [Euler (1750s), Poisson (1831); 
of course, in components.] Hence, the instantaneous rotation of the body B about the 
fixed point ♦ is described by the single vector a>, which combines all three character¬ 
istics of rotation: axis, magnitude, and sense. As the motion proceeds, and since only 
the point ♦ is fixed, the axis of rotation (carrier of a >) traces, or generates, two general 
and generally open conical surfaces with common center ♦: one fixed on the body, the 


INITIAL INTERMEDIATE FINAL 



Figure 1.1 3 General displacement of a small deformable sphere: 
Translation —> Rotation —> Strain (Sphere —> Ellipsoid). 


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§1.9 THE RIGID BODY: GEOMETRY OF MOTION AND KINEMATICS 



Figure 1.14 Rolling of body cone (P Po ih 0 de) on space cone (H er P Po | hode ). 


polhode cone; and one fixed in space, the herpolhode cone (fig. 1.14). Hence, the 
following theorem: 

THEOREM 

Every finite motion of a rigid body, having one of its points ♦ fixed, can be described 
by the pure (or slippingless) rolling of the polhode cone on the herpolhode cone; and, 
at every moment, their common generator (through ♦) gives the direction of the 
instantaneous axis of rotation/angular velocity. If ♦ recedes to infinity, these two 
cones reduce to cylinders and their normal sections become, respectively, the body 
and space centrodes. 


Velocity Field (Mozzi, 1 763) 

Since, for the first-order geometrical changes involved here (“infinitesimal displace¬ 
ments”) superposition holds, we conclude that the velocity of a generic body point P 
in general motion, v P = v, is given by the following fundamental formula of rigid 
body kinematics: 


v = v* + ft) x (r — r*) = r* + co x Y 1 + = v* + V/* 

[r ; /> = velocity of P relative to ♦ (both measured in the same frame)] (1-9.2) 

where ♦ is any body point (pole) (fig. 1.15); or, in terms of components (fig. 1.9) as 
follows: 


Space-Fixed Axes 

dX/dt = dX+/dt + wy(Z — Z*) — u z (Y — Y+), etc., cyclically, (1.9.2a) 

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CHAPTER 1: BACKGROUND 



Figure 1.15 Geometrical interpretation of eq. (1.9.2). 


or, equivalently, 

v x = v+j + - u> z Y/+, etc.,cyclically. (1.9.2b) 


Body-Fixed Axes 

v x = v«x + '^y z i* - w-V/«, etc., cyclically; (1,9.2c) 

where 

v+ x = cos(x,X)v* x + cos(x, Y)v+ Y + cos(x,Z)v 4z , etc.,cyclically; (1.9.2dl) 
and, inversely, 

v* x = cos(A, x)v+ iX + cos(A,y)v* :> , + cos(A,c)v* etc., cyclically. (1.9.2d2) 

The six functions of time v« ;w ,w w (or v* ; x,y,Zi w x,y,z ) characterize the rigid- 
body motion completely. The line-bound vectors co and r* constitute the torsor of 
motion, or velocity torsor, at ♦, from which the rigid-body velocity field can be 
determined uniquely. [Just as, in elementary statics, the resultant force / (or R) 
and moment M * of a system of forces constitute the force system torsor at ♦ (see 
“Formal Analogies ...” section that follows.] In the case of motion about a fixed 
point ♦, that torsor reduces there to (co, 0 ). 

Now, from the displacement viewpoint, the velocity transfer equation (1.9.2) states 
that: 

(i) The state of motion of the body consists of an elementary translation 
(di\ = r* dt) of a base point (or pole) ♦, and an elementary rotation (r// = oidt) 
about that point. Therefore, applying the earlier theorem of Chasles, we deduce that: 

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§1.9 THE RIGID BODY: GEOMETRY OF MOTION AND KINEMATICS 


(ii) Any infinitesimal rigid (nontranslatory) displacement can be reduced uniquely 
to an infinitesimal screw ; that is, an infinitesimal translation plus an infinitesimal 
rotation about a (central) axis parallel to the translation. (The location of that axis 
and the pitch of the screw are given in the “Formal Analogies ..section below.) As 
the motion proceeds, that axis traces two (ruled) surfaces with it as common gen¬ 
erator: one fixed in space (F s ) and another fixed in the body (F B )—which constitute 
the “no fixed point” generalization of the herpolhode and polhode, respectively. 
Flence, the following theorem: 

(iii) The general finite motion of a rigid body can be produced by the rolling and 
sliding of r B over F s . (In plane motion, sliding is absent.) Next, we prove that 

(iv) The angular velocity vector to is independent of the choice of the pole. Applying 
the fundamental formula (1.9.2) for the two arbitrary and distinct poles ♦ and 

we have 

v = v* + (D x (r — r*) = v* + co x 

= »v + co' x (r — iv) = v*/ +co' x r/+i, (1.9.2e) 

where initially, we assume that co and co 1 are different and go through ♦ and ♦ 1 , 
respectively. We shall show that 

to = co'. (1.9.2f) 

Indeed, since 

»•/♦ = »•/♦'+*•♦'/♦ and tv = v* + to x iv/*, (1.9.2g) 

equating the right sides of (1.9.2e) we obtain 

to x r/* = to x /*♦</♦ + co' x rt+i =>■ to x »*/«.' = a>' x r/+i, (1.9.2h) 

from which, since r/+> is arbitrary, (1.9.2f) follows. 

[Since to is a body quantity (a system vector), it carries no body point subscripts 
(like v ), just like a force resultant. The only “insignia” it may carry are those needed 
to specify a particular body and/or frame of reference. Perhaps this supposed “base 
point invariance” of it may have given rise to the false notion that “to [of a body- 
fixed basis relative to a space-fixed basis] is a free vector, not bound to any point or 
line in space” (Likins, 1973, p. 105, near page bottom); emphasis added. A correct 
interpretation of (1.9.2e,f), however, shows that to is a line-bound, or sliding, vector, 
not a free one (just like the force on a rigid body); hence, to in eq. (1.9.2), is under¬ 
stood to be going through point ♦.] 

A USEFUL RESULT 

Let i"i and r 2 be the position vectors of two arbitrary points of a rigid body. Then, its 
angular velocity to equals 

to = (vj x v 2 )/(v[ • v 2 ), where v =dr felt. (1.9.2i) 


Formal Analogies Between Forces/Moments and 
Linear/Angular Velocities 

Comparing (1.9.2), rewritten as v 2 = v\ + r\/ 2 x co (1, 2: two arbitrary body points) 
with the well-known moment transfer theorem of elementary statics (with some, 
hopefully, self-explanatory notation): M 2 = M x + r 1 /2 x f - we may say that the 

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CHAPTER 1: BACKGROUND 


velocity v 2 is the moment of the motion, or velocity torsor (to , V[) about point 2; that 
is, m is the kinematic counterpart of the force resultant (/ or R), and hence is a line- 
hound, or sliding vector; while v ... is the counterpart of the point—dependent moment of 
the torsor M. Hence, recalling the (presumably, well-known) theorems of elementary 
statics, we can safely state the following: 

• An elementary rotation d% = m dt about an axis can always be replaced with an 
elementary rotation of equal angle about another arbitrary but parallel axis, plus 
an elementary translation dr = v dt , where v = at x /• is perpendicular to (the plane 
of) both axes of rotation, and r is the vector from an arbitrary point of the original 
axis to an arbitrary point of the second axis; that is, an elementary rotation here is 
equivalent to an equal rotation plus an elementary perpendicular translation there. 

• Several elementary rotations about a number of arbitrary axes can be replaced by a 
resultant motion as follows: (a) We choose a reference point 0 , and transport all these 
elementary rotations parallel to themselves to 0, and then add them geometrically 
there. Then, (b) We combine the corresponding translational velocities, created by the 
parallel transport of the rotations in (a) (according to the preceding statement), to a 
single translational velocity at 0. For example, two equal and opposite elementary 
rotations about parallel axes can be replaced by a single elementary translation per¬ 
pendicular to (the plane of) both axes. These formal analogies between forces/ 
moments and linear/angular velocities (also, linear/angular momenta), which are 
quite useful from the viewpoint of economy of thought (elimination of unnecessary 
duplication of proofs), are summarized in table 1.2. 


Table 1.2 Formal Analogies Among Vectors/Forces/Rigid-Body Velocities 

Forces/Moments Rigid-Body Velocities 

Vector Systems (On Rigid Bodies) (Instantaneous Geometry) 


Single vector a 

Moment of a about point O 
Vector couple (a, . a 2 = -Oi) 
=> Constant moment 

Vector resultant R 
Vector torsor ( R, M Q ) 


Invariants: R ■ R, R ■ M... 


Vector wrench (or screw) 

( RMc ) 


r = \R+ (R x M a )/R 2 
[A = (r-R)/R 2 } 

Pitch = p = M c /R = R ■ Mo/R' 

• p = 0: 

Vector resultant R 

• p = oo: 

Couple 


Single force f 
(along line of action) 
Moment of f about 0 
Force couple (/), f 2 = — f)) 
=> Constant moment; or 
couple 

Force resultant R 
Force torsor (R, M 0 ) 


Invariants: R • R, R • M... 

Simplest Representation of Torsor 

Force wrench 

(R M c ) 

Central Axis of Wrench/Screw 

r = \R+ (R x M 0 )/R 2 

p = M c /R= R-Mo/R 2 
Pure force (resultant) R 
Pure couple 


Angular velocity m 

(about axis of rotation) 

Linear velocity of body point Ov 0 
Rotational pair (», ,<o 2 = -o»i) 

=> Constant translational 
velocity 

Rotation resultant to 
Motion torsor (to, v 0 ) 

0'(R,o> at O) 

v o’ = v o + ro/o' x 

Invariants: a> ■ to, co-v... 


Motion screw 

(ox v c ) 


r = pro + (o x v 0 )/io 2 
\p,= {r-(o)/ui 2 } 
p = v c /a> = co ■ Vo/u 2 

Pure rotation oj 

Pure translation* 


Spatial Variation (or Transfer) Theorem: 0 

Mo' = M 0 + ro/o' x R M 0 ' = M 0 + r 0 /o' x R 


*See also Hunt (1974). 

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§1.9 THE RIGID BODY: GEOMETRY OF MOTION AND KINEMATICS 


Acceleration Field 

By d/dt(. . ./differentiating (1.9.2), we readily obtain the acceleration field of a rigid 
body in general motion : 

fl = «♦ + a x r/+ + co x (to x r/+) = «* + a x r/* + [(o> • r/+)co — 0J 2 r/+] 

= «♦ + («/*) lange „, + (^normal (= «♦ + «/♦)! O' 9 ' 3 ) 

or in terms of components (figure 1.9): 

Space-Fixed Axes 

a x — a *,x + { a Y^i* ~ a zY/+) 

T x xz+'j — uj Xj^ J, etc., cyclically. (1.9.3a) 


Body-Fixed Axes 


a x = a. rX + {ot y z/+ - a z y/.) 



+ \f>x(. u x x /+ + Wyy /♦ + io z z/+) — u?x/f\ 

, etc., cyclically; 

(1.9.3b) 

where 





a*, x = 

cos(x, X)a+ x + cos(x, 

Y)a+ Y + cos(x, Z)o* z 



= 

cos(x, X)(d 2 X+/dt 2 ) + 

cos(x, Y){d 2 Y+ /dt 1 ) + 

cos (x,Z)(d 2 Z+/dt 2 ), 





etc., cyclically; 

(1.9.3c) 

and, inversely. 




a+ x = cos (X,x)a+ X + cos 

(X,y)a+ }y + cos{X,z)a+ 

z , etc., cyclically; 

(1.9.3d) 

and 






a x = ( dw/dt ) y 

■ = dux/dt, etc., cyclically, 

(1.9.3e) 


a x = ( dao/dt ) • 

3 

1 

5/ 

"^3 

II 

‘ •»» 

/dt) 



= duj x /dt — 

co • (co x /) = dui x /dt , 

etc., cyclically. 

(1.9.3f) 


Plane Motion 

The distances of all body points from a fixed, say inertial, plane /' remain constant; 
and so the body B moves parallel to/' (fig. 1.10a). [For extensive discussions of this 
pedagogically and technically important topic, see, for example. Pars (1953, pp. 336— 
356), Loitsianskii and Lur’e (1982, pp. 227-261).] A rigid body in plane (but other¬ 
wise free) motion is a system with three global, or finite, degrees of freedom. As 
such, we choose (fig. 1.16): (a) The two positional coordinates of an arbitrary body 
point (pole) ♦ (that is, of a point belonging to the cross section of B with a generic 

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CHAPTER 1: BACKGROUND 



Figure 1.16 Plane motion of a rigid body 6. 


plane / ever parallel to/') relative to arbitrary but /-fixed rectangular Cartesian 
coordinates O—XY, (X*, 7*); and (b) The angle between an arbitrary /-fixed line, 
say the axis OX, and an arbitrary 5-fixed line, say ♦/*, where P is a generic body 
point. 


(i) The velocity field (i.e., the instantaneous spatial 
distribution of velocity) 

Here, 

on = u-k = lo z K = uj K = (d(j)/dt)K (i.e., m is perpendicular to v), (1.9.4a) 

and so the general velocity formula (1.9.2) becomes 
v P = dvpjo = dr/dt 

= v = v* + v P /+ = v* + v/* = v* + oj x vpj+ = v* + oj x r/*, (1.9.4b) 

or, in components [along space-fixed (inertial) axes] 

{dX/dt,dY/dt, 0) = {dX./dt,dY./dt, 0) + (0,0, u) x (X /4 , T /4 ,0), 

=> dX/dt = dX+/dt — ujY/+, dY/dt = dY+/dt + loX/+. (1.9.4c) 

The above show that, in plane motion, there exists—in every configuration—a point, 
either belonging to the body or to its fictitious rigid extension, called instantaneous 
center of zero velocity, or velocity pole (IC , or I, for short), whose velocity, at least 
momentarily, vanishes; that is, locally, at least, the motion can be viewed as an 
elementary rotation about that point (local version of fig. 1.10b). Indeed, setting in 
(1.9.4b,c)’ 

v —» vj = 0, i.e., choosing P = I, (1.9.4d) 


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§1.9 THE RIGID BODY: GEOMETRY OF MOTION AND KINEMATICS 


we obtain its inertial instantaneous coordinates relative to our originally chosen 
pole ♦: 

X tj . = X,-X.= -( dY./dt)/u, Yj/. = Y, — Y. = +(dX./dt)/to. (1.9.4e) 

From these equations we conclude that, as long as to 0, / is located at a finite 
distance from the body and is unique; if lj = 0, then / recedes to infinity, and the 
motion becomes a translation-, and if we choose / as our pole — that is, ♦ = I —then 
(1.9.4b, c) yield 

dX/dt = —uiY/i, dY/dt = (jjX/j, or v — uirji [v 2 = (dX/dt) 2 + (dY/dt) 2 ]. 

(1.9.4f) 


[In the case of translation, eq. (1.9.4f) can be written qualitatively/symbolically as 
finite velocity = (zero angular velocity ) x (infinite radius of rotation)]. 

As the body moves, / traces two curves: one fixed on the body (space centrode) and 
one fixed in the plane (space centrode)-, so that the general plane motion can be 
described as the slippingless rolling of the body centrode on the space centrode, with 
angular velocity ui. 


(ii) The acceleration field 
Flere, 

a = da>/dt = ( duj/dt)k = ak = a K, (1.9.4g) 

and «•/'/♦ = 0, and so the general acceleration formula (1.9.3) becomes 
a P = a = «* + «/♦ = + a x r + to x (o x r/f) 

= «♦ + a x »■/♦ — lo 2 v/+, (1.9.4h) 

or, in components [along space-fixed (inertial) axes], 

(d 2 X/dt 2 ,d 2 Y/dt 2 , 0) = ( d 2 X./dt 2 , d 2 Y./dt 2 , 0) 

+ (o,o,«) x (*■,♦, y/*,o) - w 2 (*■/♦, r/*,o), 

=> d 1 X/dt 2 = d 2 X * /dt 2 — aY/+ — u?X/+, d 2 Y/dt 2 = d 2 T* /dt 2 + aI / ,-w 2 f / ». 

(1.9.4i) 

Along body-fixed axis ♦— xy, eq. (1.9.4h) yields the components (with some easily 
understood notation): 

a x = («*) x - ay/. - c?x/., a y = (a.) y + ax f . - cry/.-, (1.9.4j) 

where 

(a.) x = a. -i = cos (x,X)(d 2 X./dt 2 ) +cos(x, Y)(d 2 Y./dt 2 ), etc.; 
and similarly for the velocity field (1.9.4b), if needed. 

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CHAPTER 1: BACKGROUND 


Here, too, there exists an instantaneous center of zero acceleration , or acceleration 
pole, /', whose coordinates are found by setting in (1.9.4i) d 2 X/dt 1 = 0, d 2 Y/dt 2 = 0 
and then solving for X/+, Y/+(P —> /') : 

X v/ + = X v -X 4f = [cv 2 {d 2 X./dt 2 ) - a(d 2 Y./dt 2 )\/{a 2 + w 4 ), 

Y rh = Y v - 7* = [w 2 (d 2 7,/* 2 ) + a(d 2 X./dt 2 )\/(a 2 + w 4 ). (1.9.4k) 

These equations show that as long as a 2 + w 4 0 (i.e., not both ui and a vanish), the 
acceleration pole /' exists and is unique. If ui, a = 0 (i.e., if the body translates), then 
I' (as well as I) recedes to infinity. Finally, with the choice ♦ = /' eqs. (1.9.4h,i) 
specialize to 

a = a* + <*/♦ = a x r/p + oj x (to x »•///) = a x r/ji — u> 2 r/p, (1.9.41) 

or, in components 

d 2 X/dt 2 = -olY/p - u?X / V , d 2 Y/dt 2 = +aX /7 , - w 2 T /7 ,. (1.9.4m) 

For the geometrical properties of I', the reader is referred to texts on kinematics. 


Additional Useful Results 

(i) Crossing 0 = v* + co x (/, — r 4 ) with <u, expanding, and so on, it can be 
shown that the position of the instantaneous velocity center is given by 

'"//♦ = r, - r* = (t» x t’*)/tu 2 ; (1.9.4n) 

and similarly for the location of the acceleration pole 

(ii) The location of the instantaneous center of zero velocity I, and zero accelera¬ 
tion in body-fixed coordinates ♦— xy, are given, respectively, by (fig. 1.17) 



Figure 1.17 Body-fixed axes in plane motion. 

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§1.9 THE RIGID BODY: GEOMETRY OF MOTION AND KINEMATICS 


Xj = (1/w) \(dX+/dt) sin0 — (dY^/dt) cos<^>] = — {y+) y /w, (1.9.4o) 

yj = (1/w) [(dX+/dt) cos</> + (dY^/dt) sin<^>] =(v+) x /u, (1.9.4p) 

*/' = [ W 2 (« 4 ), - a(a.) y ]/( U * + a 2 ), = [«(«♦), + u; 2 («*),]/(</ + a 2 ), 

(1.9.4q) 

where 

(v*) x = v* •/ = cos (x,X)(dX+/dt) +cos(x, Y)(dY+/dt ), etc. 


Contact of Two Rigid Bodies; 

Slipping, Rolling, Pivoting 

Let us consider a system of rigid bodies forced to remain in mutual contact at points, 
or along curves or surfaces of their boundaries. For simplicity and concreteness, we 
restrict the discussion to two rigid bodies, B' (fixed) and B (moving), in contact at a 
space point C; that is, a certain point P of the bounding surface of B, S, is in contact 
with a point P' of the bounding surface of B' , S 1 '; that is, then, C = P = P' (fig. 
1 . 18 ). 

Now: (i) If C is fixed on both bodies, we call such a “bilateral constraint” (i.e., one 
expressible by equalities) a hinge, and we say that the bodies are pivoting about it. 

(ii) If, on the other hand, C is not fixed on one (both) of the bodies, we say that it 
is wandering on it (them). In this case, we call the relative velocity of P and P', which 
are instantaneously at C, the slip velocity there: 

Vp/pi = v P — v P f = v s . (1.9.5a) 

If we view the motion of C relative to B', C/B' , as the resultant of C/B and B/B ', 
then, since the velocities of the latter are tangent to the surfaces S and S', respec¬ 
tively, at C we conclude that lies on their common tangent plane there, p. 
Analytically, 

= y s ,T + v Sl N = Vs.T, (1.9.5b) 



Figure 1.18 Two rigid bodies in contact at a space point C. 

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CHAPTER 1: BACKGROUND 


where 

v s t = component of v s along p 1 v sN = component of v, normal to p 

(= 0 ; i.e., contact is preserved; the two bodies cannot penetrate each other); 

(1.9.5c) 


and if, at that instant, B and B' separate , then v s N lies on the side of B 1 . 

Next, if the angular velocity of B relative to B\ at C, is to with components along 
and normal to p: a> T , oj n , respectively; that is, 


(o = (o T + a) N , (1.9.5d) 

then we can say that the most general infinitesimal motion of B relative to B\ B/B'. 
is a superposition of the following special motions: 


a pure slipping : 

v, f 0 , 

o)f — 0 , 

co N = 0 ; 

(1.9.5e) 

a pure rolling : 

= 0 , 

a> T f 0 , 

% = 0 ; 

(1.9.5f) 

a pure pivoting : 

v s = 0 , 

(Of —— 0 , 

oj n f 0 . 

(1-9.5g) 


If v s = 0 and m f 0 , the motion B/B' is an instantaneous rotation called rolling and 
pivoting ; which results in two (scalar) equations of constraint. In this case, the point 
C has identical velocities relative to both B and B'; and hence its trajectories, or loci, 
on the bounding surfaces of B and B', 7 and 7 ' respectively, are continuously tangent, 
and are traced at the same pace ; that is, if, starting from C, we grade them in, say cen¬ 
timeters, then the points that will come into contact during the subsequent motion will 
have the same arc-coordinates numerically. Such a B/B' rolling is expressed by saying 
that P and P' , both at C at the moment under consideration, have equal velocities 
relative to a (third) arbitrary body, or frame or reference; and the velocities of B 
about B' are the same as if B had only a rotation m about an axis through the 
“instantaneous hinge” C. If the locus of a> on B is the ruled surface Z, and on B' the 
also ruled surface Z 1 , then the slippingless motion B/B' can be obtained by rolling Z 
on Z' [The earlier curve 7 ( 7 ') is the intersection of Z with S(Z' with S 1 ')]. 

If B and B' are in contact at two points, say C and C' , and if 17 = 17 ./ = 0 , then the 
motion B/B' is an instantaneous rotation about the line CC'\ that is, co is along it. 
And if B , B' contact each other at several points C, C', C", ..., then slipping cannot 
vanish at all of them unless they all lie on a straight line. If, in addition, a> N = 0 (or 
co T = 0 ), we have pure rolling (or pure pivoting). In sum, slippingless rolling along a 
curve can happen only if that curve is a straight line carrying co (like a long hinge). 


Some Analytical Remarks on Rolling 

(i) The contact among rigid bodies is expressed analytically by one or more 
equations of the form 


f{t;qi,q 2 ,---,q n ) = 0, (1.9.6a) 

where q = (q u ..., q„) are geometrical parameters that determine the position, or 
configuration, of the bodies of the system; hence, their alternative name: system 
coordinates. Equation (1.9.6a) is called a holonomic constraint. 

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§1.10 THE RIGID BODY: GEOMETRY OF ROTATIONAL MOTION 


(ii) If, in addition to contact, there is also slippingless rolling, and possibly pivot¬ 
ing, then equating the velocities of the two (or more pairs of) material points in 
contact, we obtain constraints of the form 


a\dq x + a 2 dq 2 + • • • + a n dq n + a n+ \dt = 0, (1.9.6b) 

or, (roughly) equivalently, 

ci\{dq\/dt) + a 2 (dq 2 /dt) + • • • + a„(dq„/dt) + a n+l = 0, (1.9.6c) 

where a /c = cik{t,q) (k = 1,...,«). If (1.9.6b,c) is not integrable [i.e., if it cannot be 
replaced, through mathematical manipulations, by a finite (1.9.6a)-like equation], it 
is called nonholonomic. In mechanical terms, holonomic constraints restrict the mobi¬ 
lity of a system in the large (i.e., globally)', whereas nonholonomic constraints restrict 
its mobility in the small (i.e., locally). The systematic study of both these types of 
constraints (chap. 2) and their fusion with the general principles and equations of 
motion (chap. 3 ff.) is the object of Lagrangean analytical mechanics. 


1.10 THE RIGID BODY: GEOMETRY OF ROTATIONAL MOTION; 
FINITE ROTATION 


The peculiarities of the algebra of finite rotations are just the 
peculiarities of matrix multiplication. 

(Crandall et al., 1968, p. 58) 

Recommended for concurrent reading with this section are (alphabetically): Bahar 
(1987), Coe (1938, pp. 157 ff.), Hamel (1949, pp. 103-117), Shuster (1993), Timerding 
(1908). 


The Fundamental Equation of Finite Rotation 

Since, by the fundamental theorem of the preceding section, the rotatory part of a 
general displacement of a rigid body is independent of the base point (pole), let us 
examine first, with no loss in generality, the finite rotation of a rigid body B about the 
( body- and space-) fixed point O', and later we will add to it the translatory displace¬ 
ment of O. Specifically, let us examine the finite rotation of B about an axis through 
O, with positive direction (unit) vector it, by an angle \ that is counted positive in 
accordance with the right-hand (screw) rule (fig. 1.19). 

As a result of such an angular displacement, a generic body point P moves from 
an initial position P, to a final position Pj] or, symbolically, 

(*i, Pi) -► (''/>/’/)> ( 1 . 10 . 1 a) 

where p is the projection, or component, of the actual position vector of P, r, on the 
plane through it normal to the axis of rotation; that is, to n. Our objective here is to 
express »y in terms of ly, n, and x- To this end, we decompose the displacement 
Ar = ly — rj = pj — p t = Ap, which lies on the plane of the triangle APfif, into two 
components: one along p h P,B = Ar l , and one perpendicular to it, BP, = Ary. 

Ar = Ar\ + Ar 2 . (1.10.1b) 


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CHAPTER 1: BACKGROUND 




Figure 1.19 Finite rigid rotation about a fixed point O (axis n, angle x)- 


Now, from fig. 1.19 and some simple geometry, we find, successively, 

(i) Av x = —(APj - AB) = -0,. — Pi cos x) = ~Pi{ 1 - cos*) = -2p,-sin 2 (x/2); or, 
since Av x is perpendicular to both n x /y and n, and 

n X (« x v t ) = (n • i‘i)n — (it • «)r ; = OA — r t = PjA = —p h 


finally, 


Av\ = n x (n x r ,)2 sin 2 (x/ 2 ). 


( 1 . 10 . 1 c) 


(ii) The component Ar 2 is perpendicular to the plane OAP h and lies along n x r,; 
and since the length of the latter equals 


|n x Vj | = |n||f,-| sincr = |r,-| sincr = [p ; |, 


and 

|p, | sinx = |CiP | = \BP f ] = \Ar 2 \ (the triangle APjP f being isosceles!), 
finally 

Ay 2 = (/i x Vj) sin X- (1.10.Id) 

Substituting the expressions (1.10.1c, d) into (1.10.1b), we obtain the following 
fundamental equation of finite rotation : 

Av = Vf — Vj = (n x Vj) sin x + nx (// x r,)2 sin 2 (x/2). (1.10. le) 

All subsequent results on this topic are based on it. 

Alternative Forms of the Fundamental Equation 

(i) With the help of the so-called “ Gibbs vector of finite rotation” 

7 = tan(x/2)« = (71,72,73) = ( 7 a -, 7 y , 7 z ) = Rodrigues parameters, (l.i0.2a) 


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§1.10 THE RIGID BODY: GEOMETRY OF ROTATIONAL MOTION 


relative to some background axes, say O—XYZ [Rodrigues (1840) — Gibbs ( late 
1800s) ‘vector’] and, since by simple trigonometry, 

sinx = 2 sin(x/ 2 ) cos(x/ 2 ) = 2 tan(x/ 2 )/[l + tan 2 (x/ 2 )] 

= 27/(l+7 2 ), where 7= |y| = |tan(x/2)|, (1.10.2b) 

sin 2 (x/ 2 ) = tan 2 (x/ 2 )/[l +tan 2 (x/ 2 )] = (1 - cosx )/2 = 7 2 /(l + 7 2 ), ( 1 . 10 . 2 c) 

we can easily rewrite (l.lO.le) as 

Ar= [2/(1 +7 2 )]b xr, 4 yx(yx /*,•)]; (1.10.2d) 

and from this, since y x (y x r,) = — 7 2 r,- + (y • 17 ) y, we obtain the additional form 

r f = [2/(1 +7 2 )][y X *•/+ (r »-i)y] + [(l - 7 2 )/(l + 7 2 )h; (1.10.2e) 

which, clearly, has a singularity at 7 = ±/. 

Further, in terms of the normal projection of r, to the rotation axis n, r in , defined 
by 

i'i,„ = r i - (y • F)y/ 7 2 = n - [(y ® y) • n]h 2 , (i.io. 2 f) 

we can rewrite ( 1 . 10 . 2 e) successively as 

r f = n + [2/(1 + 7 2 )](y X r in - 7 \„) 

= r i + [2/(1 + 7 2 )] [y X r,- - rfri + (y ® y) • rj 
= n + P/(l + 7 2 )][yxr,- + yx(yx r,-)] 

= r i + [ 2 y/(l + 7 2 )] x (r,- + y x r,-); ( 1 . 10 . 2 g) 

that is, express 17 in terms of r, and the single vector y. 

{It is not hard to show that the components, or projections, of a vector a along 
("along = «/) and perpendicular to ("perpendicuiar/normai = a n) another vector b (of com¬ 
mon origin) are 

«/ = («• b)b/b 2 , «„ = a — = a — (a • b)b/b 2 = [b x (a x b)\/b 2 }. 


Inversion of Eqs. (1.10.2e,g) 

Since a rotation —y should bring 17 back to r h if in (1.10.2g) we swap the roles of r, 
and rf and replace y with —y, we obtain the initial position in terms of the final one 
and its rotation: 

n = r f - [2y/(l + 7 2 )] x (iy - y x iy); (1.10.3) 

and thus avoid complicated vector-algebraic inversions. 

Rodrigues’ Formula (1840) 

Adding r, to both sides of (1.10.2e), we obtain 

n + r f = [2/(1 + 7 2 )][r,. + y x r t + (y • r,-)y], (1.10.4a) 

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CHAPTER 1: BACKGROUND 


and crossing both sides of the above with y, and then using simple vector identities 
and (1.10.2g) [or, adding (1.10.2g) and (1.10.3) and setting the coefficient of 
2y/(l + 7 2 ) equal to zero, since it cannot be nonzero and parallel to y], we arrive 
at the formula of Rodrigues'. 


r f - c, = yx (r, + rf) = 2y x r m = 2n x r m tan(x/2), 

(1.10.4b) 


where 


2r m = )',■ + iv = 2 {position vector of midpoint of PjPf ); 

(1.10.4c) 

or, rearranging, 


r f + r f x y = + y x r,. 

(1.10.4d) 

Finally, dotting both sides of this equation with y (or n), we obtain 


y»y = yr h 

(1.10.4e) 

as expected. 

(ii) With the help of the finite rotation vector 


l = X», 

(1.10.5a) 

which is, obviously, related to the earlier Gibbs vector y by 


y = tan(x/2)(z/x), 

(1.10.5b) 


and since 


1 + 7 2 = l/ cos 2 (x/2), 1 - 7 2 = cos x/ cos 2 (x/2), 

(1.10.5c) 


the preceding rotation equations yield 

r f = 2cos 2 (x/2) [tan(x/2)(y x r,-)(l/x) + tan 2 (x/2)(y • r,-)(x/x 2 )] + cos x*t, 

(1.10.5d) 


or finally, 


r f = r/ cos x + (z x r,)(sin x/x) + (z * »h)z[(l " cosx)/x 2 ], 

(1.10.5e) 


a form that is symmetrical and (integral) transcendental function of y • y = \ 
The above can also be rewritten as 


Yf - r ; . = (sin x)(« x r,) + (1 - cosx)[« x (// x r,-)] 

= (sinx)(« x r,) + (1 - cos x) [(«•»•;)« - {n 2 )rfi 

(1.10.5f) 

or, slightly rearranged (since « 2 = 1), 


rf = r t cos x + [n x r,-) sin x + («• **,■)«( 1 — cos x) 



= r, + sin x(n x r,) + (cos x - 1) ['*/ - (»*,■ • n)n] 

= r t + sinx(n x r,-)+(cosx — 1) ( component of c,- perpendicular to n)\. (1.10.5g) 


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§1.10 THE RIGID BODY: GEOMETRY OF ROTATIONAL MOTION 


REMARK 

The preceding rotation equations give the final position vector iy in terms of the 
initial position vector /*, and the various rotation vectors y, /, n (and x)- It is shown 
later in this section that, despite appearances, y is not a vector in all respects, but 
simply a directed line segment ; that is, it has some but not all of the vector character¬ 
istics (§1.1). This is a crucial point in the theory of finite rotations. 


Additional Useful Results 

(i) In the preceding rotation formulae: 

(a) For x = Firm (n = 1,2,3,...) they yield 

ly = i',-, (1.10.6a) 

that is, the body point returns to its initial position, as it should; and 

(b) If r,- • n = 0, and \ = 7t/ 2, then 

iy = n x iy, (1.10.6b) 

that is, n, r h /y form an orthogonal and dextral triad at O. 

(ii) By swapping the roles of /y and r, and replacing x with — x in (1.10.5g) (i.e., 
inverting it), we get 

e, = tycosx — (n x iy) sin\-+ (»*iy)n(l — cosx)- (1.10.6c) 

(iii) For small x, eqs. (1.10.5d, e) linearize to the earlier “Euler-Mozzi” formula: 

Vf = r t + i x r, => Ar = iy — r,- = / x r,-. (1.10.6d) 


Finite Rotation of a Line 

By using the rotation formulae, one can show that the final position of a body-fixed 
straight fiber joining two arbitrary such points P\ and P 2 , or 1 and 2 (fig. 1.20), is 
given by 

{'2/i) f = r 2 j ~ rij 

= ■■■ = (sin x)« x (r 2/1 ) ; + (cosx)(»-2/i); + (1 - cosx)[»- (^/OJb, (1.10.7a) 

where 

Initial position = (r 2 /i),- —> Final position = (r 2 / t h, (1.10.7b) 

and 

1 * 2/1 = r 2 — r u for both i and/. (1.10.7c) 


Finite Rotation of an Orthonormal Basis 

By employing the finite rotation equations, let us find the relations between the two 
ortho-normal-dextral (OND) bases of common origin, O—uy (space-fixed) and 

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CHAPTER 1: BACKGROUND 



Figure 1.20 Finite rotation of straight segment 12 , 
from ( 12 ), to ( 12 ) f . 


0—u k (body-fixed), if the latter results from the former by a rotation x about an axis 
«; that is, symbolically, 

u k > u k . (1.10.8a) 

Applying the earlier rotation equations to this transformation, with /*, = u k ' and 
vj = u k , we obtain the following equivalent expressions: 

(i) u k = u k ' + (sinx)(« x u k <) + (cosx- !)«*',«, (1.10.8b) 

where 

Up n = u k i — ( u k t • n)n = u k f — (n 0 n) -u k i = (1 — n®n) • u k i 
= P • u k i = Component of u k t normal to n 

[P = projection operator , 1 = unit tensor (§1.1)]. (1.10.8c) 

(ii) u k = (cos x)«k' + (sinx)(» x u k ') + (1 - cosx)(« • u k <)n 

= [(cos x)l + (sin x)( n X 7 ) + (1 - cos x)(« ® »)] • u k ' 

= ( rotation tensor ) • u k < [examined in detail below] 

= up + (x«) x up (to the first order in x) 

~ «k' + (X«) x u k , 

= up + / x up [Euler—Mozzi formula for small rotations], (1.10.8d) 


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§1.10 THE RIGID BODY: GEOMETRY OF ROTATIONAL MOTION 


(iii) u k = u k ' + [2/(1+ 7 2 )] [y x u k , - 7 2 u k > + (y <8 7) • u k i] 

= » k ' + P/(i + 7 2 )][y x «*:' + yx(yx «*/)] 

= «*./ + [2y/(l + 7 2 )] x (u k > + y x «*/). (1.10.8e) 

To express the initial basis vectors u k ' in terms of the final ones u k , we simply replace 
in any of the above, say (1.10.8e), y with —y. The result is 

Uk' = u k - Py/(1 + 7 2 )] x {u k f - y x «*,). (1.10.8f) 


From the above, we can easily deduce that 


yu k = yu k 


(1.10.8g) 


as expected; or setting 


7 = ^ kUk = Ik’Uk', 


(1.10.8h) 


in component form 


Ik = Ik'- 


(1.10.8i) 


The Tensor of Finite Rotation 

Let us express the earlier rotation equations in direct/matrix and component forms. 
Along the rectangular Cartesian axes O—XYZ = 0—X k , common to all vectors and 
tensors involved here, and with the component notations (k = X , Y,Z): 

r i = (X k ), r f = ( Y k ) , 

7 = ilk- Rodrigues parameters) => 7 2 = ^ lk 2 = ( 7z ) 2 + ( 7y ) 2 + ( 7z ) 2 , 
n = ( n k : direction cosines of unit vector defining the axis of rotation), (1.10.9a) 

our rotation equations become 

r f = Rr i , Y k = Y J R ki x i = Yy*l( 1+7 2 )]*/, (1.10.9b) 

where, recalling (1.10.2efif.) and the simple tensor algebra of §1.1, the (nonsym- 
metrical but proper orthogonal) tensor of finite rotation , 


R = R{n, x) = ( Rki ) = {r k i /(1 + 7 2 )), 


has the following equivalent representations. 

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(1.10.9c) 


CHAPTER 1: BACKGROUND 


(i) Direct/matrix form (with N\ antisymmetric tensor of vector n ): 


R =1 

cos 

XHf 

- N sin x + 

n <g 

n( 1 — 

cosx) 






0 




( 

0 

-«Z 

n Y \ 



= 

0 

1 

0 

cosx 

+ 


«z 

0 

-n x 

sinx 



0 

J 



\ 

~n Y 

n x 

o ) 










"x 2 

n x n Y 

n x n z N 







+ 


n Y n x 

n Y 2 

Hy« Z 

(i 







\ 

n z n x 

n z n Y 

2 

«z 

) 



/ C X + "x 2 (1 ~ cx) ~n z sx + n x n Y (1 - cx) n Y s\ + n x n z (1 - cx) \ 
= nzsx + n x n Y ( 1 - cx) cx + «y 2 ( 1 - cx) -«x^X + «y«zO - cx) 

V ~ n y s X + »y«z(l - cx) + «y«z(l - cx) cx + « z 2 (l - cx) / 

= R(n x ,n Y ,n z ;x), under n x 2 + n Y 2 + n z 2 = 1, (1.10.10a) 

where, as usual, c(...) = cos(...), s(.. .) = sin(...). 

(ii) Indicial (Cartesian tensor) form [with IV = (Nki), n = (n k )]: 

R k i = R k i(n r , x) = (Ski) cos x + (N k i) sin x + n k n,{ 1 - cos x) 

= (Ski) cosx + (5^ e w«,-) sinx + n*H/(l ~ cosx). ( 1 . 10 . 10 b) 

Occasionally, the rotation formula is written as 

Ar = R'-r, where Ar = rf — r,-, r = r (1.10.10c) 

and 


R' = (R' kl ) =R — 1 = ( R k i — Ski ); rotator tensor, 


R'ki = Rki-S k i = ---= (51 e krin r ) sinx + (n k n t - $ w )(l - cosx)- (1-lO.lOd) 

We notice that the representation (l.lO.lOd) coincides with the decomposition of R' kt 
into its antisymmetric part: 


(ekri n r ) sin x = N k i sin x , 


and symmetric part: 


(n k n, - 6 k ,)( 1 - cosx); 

of which, the former is of the first order in x, while the latter is of the second order, a 
result that explains the antisymmetry of the angular velocity tensor [(1.7.22e)]. 

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§1.10 THE RIGID BODY: GEOMETRY OF ROTATIONAL MOTION 


(iii) In terms of the Rodrigues parameters (a form, most likely, due to G. Darboux): 


('«) 


/ 1+7 X 2 - ( 7a 2 + lz 2 ) 
!{pixlY + lz) 

V ^{ixlz - 7 y) 


-{iXlY - lz) 

1 + 7y 2 - {.lz + 7a- 2 ) 
2(7a7z + 7a) 


2(7a7z + 7a) \ 

2(7a7z - 7a) 

1 + 7z 2 - (7 a 2 + 7a 2 ) / 

(l.lO.lOe) 


The properties of R can be summarized as follows: 


(i) 


lim /?(«, x) 


= R(n, 0) = 1 , for all n ; 


x—>o 

that is, R(n, x) is a continuous function of X- 

(ii) R{n, x) • n = 11 ! n = axis of rotation. 


(1.10.11a) 


(1.10.11b) 


(iii) 

R(n,X i) -R(n,X 2 ) =R{n,X 1 + X 2 ) ■ 

(1.10.11c) 

(iv) 

R(n,x) -tf T («,x) = 1 , 

(1.10.lid) 


Sc 

1! 

So 

1 

s' 

II 

5 

a 

1 

(1.10.lie) 


Also, since the elements of R, Ry, depend continuously and differentiably on three 
independent parameters — for example, Euler’s angles (§1.12) — we can say that the 



Figure 1.21 Plane rotation about Oz, through an angle x- 

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164 CHAPTER 1: BACKGROUND 

rotation group is a continuous one; or a Lie group ; see, for example, Argyris and 
Poterasu (1993). 

Plane Rotation 

This is a special rotation in which 

y= (lx = 0,7r = 0,7z = tan(x/2)) = tan(x/2)« => n = K (1.10.12a) 


Then, with X k = X, Y and Y k = X', Y' (fig. 
(1.10.2g), and so on, specialize to 

1.21), the rotational equations. 

A' = [(1 - 7 2 )/(l + r)]X - [27/(1 + 7 2 )] Y = -- 

= (cos x)X + (- sin x) Y, 


(1.10.12b) 

Y' = [2 7 /(l + r)]X + [(1 - 7 2 )/(l + 7 2 )] Y = ~. 

= (sin x)A + (cos x) Y, 


(1.10.12c) 

z' = z. 

(1.10.12d) 


Additional Useful Results 

(i) Alternative expressions of the rotation tensor: 

(a) Indicial notation: 


Rkl = Ski + ( £ krl n r) sin x + ( n k ni - 5 a ){ 1 - cos x) 

= Ski + N/d sin x + y~]Af fa lV ri (l - cosx) (1.10.13a) 

(b) Direct/matrix form [N = (N/f) antisymmetric tensor of vector n = (nf)\. 

R = 1 +Asinx + 21V-iVsin 2 (x/2) (1.10.13b) 

= 1 + (sinx)iV+ [2sin 2 (x/2)]iV 2 (1.10.13c) 

= 1 + (sin x)N + (1 - cosx)N 2 (1.10.13d) 

= J +2iVsin(x/2)[f cos(x/2) +iVsin(x/2)] (1.10.13e) 

[Notice that 1 — cos x = 2 sin 2 (x/2) and = n^n/ — Sy, or, in direct notation, 

N-N = n <S> n — 1. See also Bahar (1970)]. 


(ii) By swapping the roles of r , and /y and setting \ —>• — Xi nl the preceding 
rotation formulae, one can show that 

n=R l r f , (1.10.14a) 

where 

R 1 =1 —Asinx + 21V"IVsin 2 (x/2) = R J = R(n, ~x) ; (1.10.14b) 

that is, the rotation tensor is indeed orthogonal. 


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§1.10 THE RIGID BODY: GEOMETRY OF ROTATIONAL MOTION 


(in) Let P = ;Vtan(x/2): antisymmetric tensor of the Gibbs vector 7. By applying 
the Cayley-Hamilton theorem to P [i.e., every tensor satisfies its own characteristic 
equation (§1.1)], 

A( A) = |r- A7| = 0 => A{r) = -r 3 - [tan 2 (x/2)]r = 0, (1.10.15a) 

(since 7> P = 0 and Det P = 0), one can show that 

/? = 7 + 2cos 2 (x/2)(r + r 2 ), r = (i - ry 1 -(i + r) . (l.io.isb) 

Next, expanding (1.10.15b) symbolically in powers of P, we obtain the representa¬ 
tion 


R = (7 + P + ■■■)•(! + P) =1 +2r, to first r-order; (1.10.15c) 

=> R'=R-l=2r , to first r-order. (1.10.15d) 

[Equations (1.10.15c, d) shed some light into the meaning of 7 and P, and prepare us 
for the treatment of angular velocity later in this section.] Similar results can be 
obtained in terms of N. 


The Mathematical Problem of Finite Rotation 

Usually, this takes one of the following two forms: (i) given x an d «, find R: or 
(ii) given R. find x an d n - Now, from the preceding indicial forms, we easily obtain 
(with k = X,Y, Z): 


(i) TrR = y R kk = cosxf+ sin x (EE £krk n r 

+ (1 — cosx) (EE n k n^j 

= cosx(3) + sinx(0) + (1 - cosx)(l) = 2cosx + 1. (1.10.16a) 


^■skl^krl^r 


(-) EE £ ski^kl — cos X (E E e skAl) + sin x (EEE 

+ (1 - cosx) (EE £ skl n k n / S j 

= cosx(O) + sinx(E (~ 2 S rs)n r ) + (1 - cosx)(« x n) s 

= — 2(sinx)«s [Thanks to the e-identities (1.1.6b ff.)]. 

(1.10.16b) 

In sum, 

7] = TrR = Rki- = 1 + 2cosx = First invariant of R , (1.10.16c) 

-EE £ skiRki = 2R S = 2{Axial vector ofR) s = 2(sin x) n s => Rk = (sin x) n k , 

(1.10.16d) 


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CHAPTER 1: BACKGROUND 


or, explicitly, 

R { = (— 1 / 2 ){e 122 R 22 , + e 132^32) = (^32 — ^23,)/^- = ( s i n X) w l> ( 1 . 10 . 16 e) 

R 2 = (—l/ 2 )(e 2 3 i^ 3 i + £213-^13) = (^13 ~ ^ 3 i )/2 = (sin x) ,z 2? ( 1 . 10 . 16 f) 

Rt, = (—l/ 2 )(e 312 i?i 2 + £321^21) = (^21 — *i 2 )/2 = (sin x) n 3- ( 1 . 10 . 16 g) 

Now, the first problem of rotation is, clearly, answered by the earlier rotation for¬ 

mulae (1.10.10 ff.); while the second is answered by solving the system of the four 
equations (1.10.16c, e-g) for the four unknowns x; « 123 . Indeed, 

(i) From (1.10.16c), we obtain 

cos x = (Ii ~ l)/2 = (Tr /? — l)/2 . (1.10.17a) 

(a) From (1.10.16e g), if sinx ^ 0, 

«i = (^32 - ^23)/2sinx, n 2 = {R u ~ R 3 i)/2sinx, « 3 = (R 21 - ^i2)/2sinx, 

(1.10.17b) 

or, vectorially, 

n = (1 /n') [(/? 32 — /? 23 )/ + {Ru — Rji)J + (R21 ~ ^12)^"] i 

where 


«' = 2 sin x = • • • = [(1 + Tr/?) • (3 — Tr/?)] 1 / 2 : normalizing factor , (1.10.17c) 


(b) If sinx = 0^ then X = 0 or ±7r (or some integral multiple thereof); 

(b.l) If x = 0, then, as (1.10.11a) shows, R = {Ru) = {Ski) = 1\ that is, n becomes 

undetermined: no rotation occurs ; while 

(b.2) If x = ±tt =>• cosx = —1, then, as (1.10.10 ff.) show, 


R = {Ru) = (2 n k n, - 8u) (a symmetric tensor) 

/2»i 2 — 1 2/?| n 2 2«|H 3 \ 

= 2n 2 n\ 2n 2 2 — 1 2 n 2 n 2 , 

\ 2n 3 »i 2n 3 « 2 2 n 2 — 1 / 

or, explicitly, 

R n =W-\ =* m =±[{l + R n )/2}' /2 , 
7? 22 = 2 n 2 — 1 =>■ n 2 = ±[(1 + /? 22 )/ 2 ] ^ , 
/? 33 = 2« 3 2 — 1 => « 3 = ±[(1 + /? 33 )/ 2 ] 1/,_ , 


(1.10.17d) 


(1.10.17e) 
(1.10.17f) 
(1.10.17g) 


and the ultimate signs of n l 2 3 are chosen so that (1.10.17e-g) are consistent with the 
rest of (1.10.17d): 


«1*2 - *12/2 - *2l/2, n l n 3 — /? 13 /2 — Rji/2, « 2 « 3 —/? 23 /2 —/? 32 /2. 


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§1.10 THE RIGID BODY: GEOMETRY OF ROTATIONAL MOTION 


The angle x can also be obtained from the off-diagonal elements of R as follows: 
multiplying (1.10.17b) with /q, n 2 , n 3 , respectively, adding together, and invoking the 
normalization constraint tp 1 + n 2 1 + n 3 = 1, we find 

si n X = (1/-)[ w i(-^32 " ^ 23 ) + ” 2(^13 ~ -^ 31 ) + ^ 3(^21 — ^ 12 )]■ (1.10.17h) 


Rotation as an Eigenvalue Problem 

(This subsection relies heavily on the spectral theory of § 1.1.) In view of the rotation 
formula 

r f = R-n, (1.10.18a) 

the earlier fundamental Eulerian theorem (§1.9: The most general displacement of a 
rigid body about a fixed point can be effected by a rotation about an axis through 
that point => that axis is carried onto itself: R • n = n) translates to the following 
algebraic statement: The real proper orthogonal tensor of rotation R has always the 
eigenvalue +1; that is, at least one of the eigenvalues of the eigenvalue problem 

(r f =) R- n = Ar,, (1.10.18b) 

equals +1; or, every rotation has an invariant vector, which is Euler’s theorem. 

Let us examine these eigenvalues more systematically. The latter are the three 
roots of 

|*-A2|=0 (A:Ai, 2 . 3 ), (1.10.18c) 

and it is shown in linear algebra that: 

(a) They all have unit magnitude [Since rp-rp = ( R ■ 77) ■ ( R ■r, ) = (77 • R T ) • ( R ■ 77) = 
r, ■ri = rp, the eigenvalue equation (1.10.18b) becomes 

r f . r f = = A 2 r, • 17 =>■ A 2 = 1 (for 17 7^ 0 )]; 

(b) At least one of them is real [From the corresponding characteristic equation: 

/1(A) = \R-X1\ = (-1) 3 A 3 + ••• + (DetR)\° = 0 , 
we readily see that 


limzl(A) 


= + 00 , 


and lim A (A) 


A—>+00 


Hence, d(A) crosses the A axis at least once', that is, zl(A) = 0 has at least one real 
roof, and, by (i), that root is either +1 or —1.] 

(c) Complex eigenvalues occur in pairs of complex conjugate numbers [since the coeffi¬ 
cients of d(A) = 0 are real]; 

(d) I 3 (R) = / 3 — : Det R = lA^/l = R = AiA 3 A 3 = +1. [Initially, that is before 
the rotation, rj = R ■ ri = 77 => R = 1 => Det 1 = + 1, and since thereafter R 
evolves continuously from 1, it must be a proper orthogonal tensor, that is, 
\R\ = Det R = +1 — ^1(0). This expresses the “obvious” kinematical fact that, as 
long as we remain inside our Euclidean three-dimensional space, a right-handed 
coordinate system cannot change to a left-handed one by a continuous rigid-body 


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CHAPTER 1: BACKGROUND 


motion of its axes', such “polarity” changes, called inversions or reflections , require 
continuous transformations in a higher dimensional space; for example, right- 
handed two-dimensional axes can be changed to left-handed two-dimensional axes 
by a continuous rotation inside the surrounding t/tree-dimensional space.] 

Combining these results, we conclude that either: (i) All three eigenvalues of R are 
real and equal to +1; which is the trivial case of the identity transformation-, or, and 
this is the case of main interest (Euler’s theorem), (ii) Only one of these eigenvalues is 
real and equals +1 [ => zf(l) = \R — 1\ = 0]; while the other two are the complex 
conjugate numbers: cosx± / sin \ = exp(±;'x)- As a result of the above: 

(a) The direction cosines of the axis of rotation n = (n x ,n Y ,n z ) can be obtained by 
setting in eq. (1.10.18b) A = 1, r, = n: 

(R-\l)-n = 0^R-nmn, (1.10.19a) 


and then solving for n x Y z under the constraint n x 2 + n y 2 + n z 2 = 1; and 
(b) The invariants of R can be summarized as follows: 


I\ (R) = TrR = + R 22 + R 33 

= Ai + A 2 + A 3 = 1 + exp(-H'x) + exp(—i'x) = 1 + 2cosx ; (1.10.19b) 

I 2 (R) = [(TrR) 2 - Tr(R 2 )}/2 = ( DetR)(TrR ~ l ) 

= (+l)(TrR T ) = (+l)(Tr *)=/,(*) 

[ = A| A 2 + A[A 3 + A 2 A 3 

= (l)[exp(i'x)] + (l)[exp(-/x)] + exp(/x) exp(-ix) = 2cosx + l]; (1.10.19c) 
h(R) = DetR = AiA 2 A 3 = +1; (1.10.19d) 


that is, R has only two independent invariants. 

Composition of Finite Rotations 

Here we show that finite rotations are noncommutative; specifically, that two or 
more successive finite rotations of a rigid body with a fixed point O (or, generally, 
about axes intersecting at the real or fictitious rigid extension of the body) can be 
reproduced by a single rotation about an axis through O ; but that resultant or 
equivalent single rotation does depend on the order of the component or constituent 
rotations. 

Quantitatively, let the rotation vector carry the generic body point position 
vector from iq to r 2 ; and, similarly, let y 2 carry r 2 to r 3 . We are seeking to express the 
vector of the resultant rotation yj 2 (i.e., of the one carrying iq to r 3 ) in terms of its 
“components” yj and y 2 . Schematically, 

7i 72 


_ / 

y 12 (1.10.20a) 



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§1.10 THE RIGID BODY: GEOMETRY OF ROTATIONAL MOTION 


By Rodrigues’ formula (1.10.4b), applied to iq —> r 2 and r 2 —> r 3 , we obtain 

= 7 i x ( i -2 + i-i), 1-3 ~t 2 = y 2 x (r 3 + r 2 ), ( 1 . 10 . 20 b) 

respectively. Now, on these two basic equations we perform the following opera¬ 

tions: 

(i) We dot the first of the above with y { and the second with y 2 : 

71 • (r 2 - n) = yi • [yj x (r 2 + Tj)]= 0 =>• Vi -r 2 = yi -r u (1.10.20c) 

72 • Ob - r 2 ) = y 2 • [y 2 X (r 3 + f 2 )] = 0 =>• y 2 • t 3 = y 2 • t 2 . (1.10.20d) 

(ii) We cross the first of (1.10.20b) with y 2 and the second with yj and subtract side 
by side: 

y 2 x (r 2 - i-i) - y 3 X (r 3 - r 2 ) = (y 3 + y 2 ) xr 2 -y 2 x c,-y, xr 3 

= y 2 x [ yi x (t 2 + n)] - y { x [y 2 x (r 3 + r 2 )] 

= {yita-fo + 'i)] - (yi • y 2 )( r 2 + r i)} 

- biiyi • fa + r 2 )] - fa • 7 2 )fa + r 2 )} 
[expanding, and then rearranging while taking into account ( 1 . 10 . 20 c, d)] 

= [(72 • '*2 + 72 • 'h fa - fa • 7i) r 2 - fa • y 2 fa] 

- [(yj • r 3 + y x • r 2 )y 2 - fa • y 2 )r 3 - fa • y 2 )r 2 ] 

= [(72 • 'h + 72 • )7i - (7i • 72>2 - (7i • 72)**l] 

- [( 7 i • *3 + 7 i • **i )72 - ( 7 i • 72)^3 - ( 7 i • 72)^2] 

= [(72-'*3 + 72-'*i)7i] " [(71 •''3+71 -''1)72] ^ (71 *72)('"l -'h) 

= [(72 • r 1)71 - (7i •'h)72] + [(72-''3)71 - (7i •''3)72] 

- ( 7 i * 72)(''l -**3) 

= (72 x 71 ) x r x + (72 x 71 ) x r 3 - (y, -y 2 )fa - r 3 ) 

= (72 x 7 i) x fa +r 3 ) + (71 • 7 2 )fa - fa, ( 1 . 10 . 20 e) 

or, equating the right side of the first line with the last line of ( 1 . 10 . 20 e) and rearrang¬ 
ing, 

(7i + 72 ) xr 2 = y 2 xr 1 +7 1 x»'3 

+ (72 x 7 i) x fa +r 3 ) + fa • y 2 )fa - fa. ( 1 . 10 . 20 f) 

(iii) We add (1.10.20b) side by side and rearrange to obtain 

r 3 - n = yj x (r 2 + fa + y 2 x fa + r 2 ) = y x x r 2 + y x x r, +y 2 xc 3 + y 2 xr 2 

=> (7i + 72 ) x r 2 = r 3 - r x -y { x r, - y 2 x r 3 . (1.10.20g) 

(iv) Finally, equating the two expressions for fa + y 2 ) x r 2 , right sides of 
(1.10.20f) and (1.10.20g), and rearranging, we obtain the Rodrigues-like formula 
[i.e., a la (1.10.4b)] 

*•3 “''I = 7t,2-fa + fa, (1.10.20h) 

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CHAPTER 1: BACKGROUND 


where 

ri ,2 = 7i— >2 = bi + 72 + 72 x 7i]/(1 — 7i * 72 ) 

= Resultant single rotation “vector,” that brings f] to r 3 . (1.10.20i) 


This is the sought fundamental formula for the composition of finite rigid rotations. 
[For additional derivations of (1.10.20h, i) see, for example, Hamel (1949, pp. 107 
117; via complex number representations and quaternions), Lur’e (1968, pp. 101 
104; via spherical trigonometry); also, Ames and Murnaghan (1929, pp. 82-85). The 
above vectorial proof seems to be due to Coe (1938, p. 170); see also Fox (1967, p. 8); 
and, for a simpler proof, Chester (1979, pp. 246-248).] 

In terms of the corresponding rotation tensors, we would have (with some ad hoc 
notations). 


r t —t rf\ rf = R\ • r,, (1.10.21a) 

rp —>• r f : r f = R 2 - r f , = R 2 (R\ ■ rf) = R IX ■ r l , (1.10.21b) 

where 

R\ 2= R-2 * R\ ( 7 ^ R\ • i ?2 = /? 2,1 ) : resultant rotation tensor. ( 1 . 10 . 21 c) 


REMARKS ON 7! 2 

(i) Equation (1.10.20i) readily shows that the y’s are not genuine vectors; as the 
presence of y 2 x 7 ] there makes clear [or the noncommutativity in ( 1 . 10 . 21 c)], in 
general, finite rotations are noncommutative. Indeed, had we applied y 2 first, and y x 
second, the resultant would have been [swap the order of 7 ! and 7 2 in ( 1 . 10 . 20 i)] 

(72 + 7i + 7 i x 72)/(1 — 72 * 7i) = 72,1 = 72^1 ^ 7i,2 = 7i->2- (1.10.22a) 

For rotations to commute, like genuine vectors, the term 7 2 x 7 , must vanish, either 
exactly or approximately. The former happens for rotations about the same axis; 
and the latter for infinitesimal (i.e., linear) rotations: there, y 2 x y x = second-order 
quantity « 0. 

(ii) If 7 i *72 = 1, the composition formula ( 1 . 1 0.20i), obviously, fails. Then, the 
corresponding “resultant angle” 2 is an integral multiple of 7 r. 

(iii) From (1.10.20i) it is not hard to show that 

1/(1 + 71, 2 2 ) 172 = (1 — 72 • 7 i)/[(l + 7 i 2 ) 1/,_ (l + T 2 “) 1/_ ], ( 1 . 10 . 22 b) 

and combining this, again, with ( 1 . 10 . 20 i) we readily obtain 

71 , 2 /(1 + 7i,2 2 )' A = [7i + 72 + 72 x 7i]/[(l + 7C) 172 (1 + 72 2 )* 7 "] (1.10.22c) 

[which is the formula for the vector part of a product of two (unit) quaternions; see 
Papastavridis ( Elementary Mechanics, under production)]. 

Finite rotations may not be commutative, but they are associative: the sequence of 
rotations, expressed in terms of their 7 vectors—for example, 7 t — 7 2 — 73 —can be 
achieved either by combining the resultant of y x —*• y 2 with y 3 , or by combining y x 
with the resultant of y 2 ^ y 3 . In view of this, the sequence —y x ——> y 2 is equiva- 

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§1.10 THE RIGID BODY: GEOMETRY OF ROTATIONAL MOTION 


lent to the rotation y 2 , and also to the sequence —y x —> 2 . Therefore, if in the 
fundamental “addition” formula (1.10.20i) we make the following replacements: 

72-^71,2, 71,2^ 72. (1.10.23a) 

we obtain the “subtraction” formula: 

72 = [-7i + 7i,2 + 7i,2 x (— 7i)]/[l - (— 7i) ' 71 , 2 ]) (1.10.23b) 


or, finally, 

72 = [ 71,2 — 7i +7i x 71 , 2 ]/(! +7i - 71 , 2 ): (1.10.23c) 

which allows us to find the second rotation “vector” from a knowledge of the first 
and the compounded rotation “vectors.” Similarly, to find y, from y 2 and y 12 , we 
consider the rotation sequence y X2 —> —y 2 , which, clearly, is equivalent to the rota¬ 
tion y x . Hence, with the following replacements: 

7i~ > 7i,2. 72^-72: 7i,2 —* 7d (1.10.23d) 

in (1.10.20i) we obtain the “subtraction” formula: 

7i = (7i,2 - 72 + 7 i,2 x 72 )/(1 + 72' 71 , 2 )- (1.10.23e) 

With such simple (and obviously nonunique ) geometrical arguments, we can avoid 
solving (1.10.20i) for y x , y 2 . (These results prove useful in relating y to the angular 
velocity to.) 


Infinitesimal (Linearized) Rotations Commute 

First, let us apply the infinitesimal rotation y x to r t [recalling (1.10.6d)]: 

~*r\' = r, + dr t = r, + Zi x f,. (1.10.24a) 

Next, let us apply z 2 to rfi: 

'■| >y = r 1 + dr 1 = r, + Z 2 X r j 

= (c + Zi X V,) + 1 2 x (r,- + *i x r,-) 

= r i + (Zi + Z 2 ) x n + x 2 x (z, x r,). (1.10.24b) 

Reversing the order of the process—that is, applying i 2 first to r h and then to the 

result — we obtain 

r f = n + dr x =r x + Zj x r, 

= ('*,■ + Z 2 x r,) + zi x (r,- + z 2 x 
= r i + (/2 + Zi) X /*,• + Z| X (z 2 X *■/); (1.10.24c) 

and, therefore, subtracting (1.10.24c) from (1.10.24b) side by side, we obtain 

r f' ~ r f" = Z 2 x (Zi x r,) - zi x (z 2 x r,) = second-order vector in Zi,Z 2 i 

(1.10.24d) 


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CHAPTER 1: BACKGROUND 


that is, to the first order in / l5 ii- 

r/ = r f ", Q.E.D. (1.10.24e) 

Similarly, for an arbitrary number of infinitesimal rotations Zi j 3 C 2 > - - ■ j t0 the first 
order: 

r f = r i + (Zi + li d-) x n- (1.10.24f) 


Angular Velocity 


(i) Angular Velocity from Finite Rotation 

Expanding the rotation tensor (1.10.1 Oe) [with (1.10.9b)] in powers of 7 XY z^ and 
since (with customary calculus notations) 

V = tan(x/ 2 )« = (x/ 2 )« + 0 (x 3 ) = z /2 + 0 (x 3 ), ( 1 . 10 . 25 a) 

we find 


R = 


/ 1 -27 z 27 r \ 

2 7z l -2 lx + 

V 27 y 27 X 1 / 

[Linear rotation tensor = R a ] 


o{j 2 ) 


/I 

0 



/ 

0 

-27 z 

27F \ 

0 

1 

0 

+ 


27z 

0 

-2y x 

\0 

0 

1 ) 


l 

-27 Y 

27z 

0 / 


+ 0 ( 7 2 ) 


[Identity tensor] [Linear rotator tensor = R„' (recall (1.10. lOd, 15d))] 

(1.10.25b) 



/ 0 

-«z 

«y \ 



7 + 

«z 

0 

-«x 

X+0{x 2 ), 

(1.10.25c) 


\-n Y 

«x 

0 




( 0 

-Xz 

Xy n 



7 + 

Xz 

0 

-Xx 

+ o( X 2 y, 

(1.10.25d) 


\-Xr 

Xx 

0 , 




and, with the notations 


r,- = (X, Y,Z) = r, 

r f = r i + Ar i = ( X + AX , Y + A Y, Z + AZ) = r + Ar, (1.10.25e) 
we obtain, to the first order in the rotation angle, 

r+ Ar = R 0 -r= ( l+R 0 ')-r => Ar = Rj - r, (1.10.25fl) 


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§1.10 THE RIGID BODY: GEOMETRY OF ROTATIONAL MOTION 


or, in extenso, 


/AX\ 

( 0 

-Xz 

Xy \ 

( x \ 


H=i 

Xz 

0 

-Xz 

4 

(1.10.25f2) 

\iZ) ' 

V~Xr 

Xx 

0 / 

U/ 



This basic kinematical result states that any orthogonal tensor that differs infinitesi¬ 
mally from the identity tensor, that is, to within linear terms, differs from it by an anti¬ 
symmetric tensor. 

Finally, dividing (1.10.25fl, 2) by At, during which Ar occurs, assuming continuity 
and with the following notations: 

Ytm{AX/At)\ At _^ = dX/dt = v z , etc., Y\m(xx/At)\ At ^ = w x , etc., 


we obtain the earlier found (1.9.1) fundamental kinematical equation of Poisson: 



or, in direct notation, 


v = dr/dt = il ■ r — <>> x r , 


where 

£2 = \xm(R 0 ' / At)\ A angular velocity tensor , 

a> = lim(27 / At)\ Ar „: angular velocity vector 

(axial vector of fl — a genuine vector!). 


(1.10.25g) 


(1.10.25h) 

(1.10.25i) 


(1.10.25j) 


As shown below [(1.10.26f)], (a) the velocities of the points of a rigid body moving with 
one point fixed are, at any instant, the same as they would be if the body were rotating in the 
positive sense about a fixed axis through the fixed point, in the direction and sense of co and 
with an angular speed equal to |o|; and, (b) since both r and v are genuine vectors, so is 
co (a fact that is re-established below). From all existing definitions of the angular velocity, 
this seems to be the most natural; but, in return, requires knowledge of finite rotation. 


(ii) co is a Genuine Vector 

Using the Rodrigues equation (1.10.4b): 

rf — r t = y x (r, + ly), (1.10.26a) 

let us prove directly that the angular velocity m, defined as 

co = lim(2y/zh)|^ r ^ 0 , where y = tan(x/2)«, (1.10.26b) 

is a genuine vector, even though y is not. 

PROOF 

With this in mind, we introduce the following judicious renamings: 

rj = r, Vf = l'/ + Ar = v + Ar. (1.10.26c) 


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CHAPTER 1: BACKGROUND 


Then, eq. (1.10.26a) yields 

Ar = y x [(rx Ar) + r] = y x (2r + Ar ) = (2y) x (r + Ar/2). (1.10.26d) 


Dividing both sides of the above by At, and then letting At —> 0 (while assuming 
existence of a unique limit as Ar —> 0), we obtain 


v = Xim{Ar/At)\ At ^ = lim[(2 y/At) x r]\ At _^ + lim[2y x {Ar/2)\\ At ^ 

= G)xr+0 = tt)Xr (v, r : vectors =>■ m : vector); Q.E.D. (1.10.26e) 


The physical significance of a> is understood by examination of the following case: \ = 
constant, in the direction and sense of the constant unit vector n. Then, with \ —^xAt=> 
y = [tan(xxlt/2)]«, and so (1.10.26b) specializes to: 


to = lim(27 /At) Q = n lim 


2[tan{xAt/2)\ \ 

Jar-* 0 


■ ■ ■= xn ; 


(1.10.26f) 


i.e., here, m has the direction and sense of n (= instantaneous rotation axis), and length 
equal to the angular speed. Hence, Poisson’s formula, (1.10.25h), allows us to draw the 
conclusions following (1.10.25j). 

To complete the proof, let us next show that the line segments <u indeed commute. 
Dividing the composition of rotations equation (1.10.20i) 

73 = 7 1,2 = (7i + 72 + 72 x 7i) /(1 -7i -72) (1.10.26g) 


by At/2, we get 
2 y- i /At= [2yJ At + 2y 2 /At 

+ (At/2)(2y 2 /At) x (2yJAt)] / [l - (At/2) 2 (2y 2 /At) • (2 yi /At )\; 

and then letting At — > 0, while recalling the earlier o-definition (1.10.26b, f), we find 

o) 2 = ^ 1,2 = + tn 2 — o 2 = o 2 \, (1.10.26h) 

that is, simultaneous rn’s obey the parallelogram law for their addition and decom¬ 
position, Q.E.D. 


(iii) to <-> y Differential Equation 

Let us consider a rigid body B with the fixed point O. Its instantaneous angular 
velocity m is related to its Gibbs “vector” y, which carries a typical Zi-particle 

from >•/= r(t) to iy = r(t + At), (1.10.27a) 

by a differential equation. The latter is obtained as follows: in the composition of 
rotations equation (1.10.20i) and in order to create the difference Ay there, we choose 
the rotation sequence 


7i = —7 -> 72 = 7 + ^7, 


(1.10.27b) 


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§1.10 THE RIGID BODY: GEOMETRY OF ROTATIONAL MOTION 


which, clearly, is equivalent to the single rotation y 12 = Ay, and occurs in time At. 
With these identifications in (1.10.20i), the earlier angular velocity definition yields 

m = lim(2dy/d?)|^ 0 = 2{lim(dy/d?) |^ 0 } 

= 21im [(l/dt){[(-y) + (y + Ay) + (y x Ay)\/{\ - (-y) • (y + dy)]}]|^ 0 

= 21im{[(dy/df) + y x (dy/dt)] / [l + y • y + y • dy] }|^ 0 , 

or, finally, 

to = [2/(1 + 7 2 ))[dy/dt + y x (dy/dt)]. (1.10.27c) 

This remarkable formula, due to A. Cayley ( Cambridge and Dublin /., vol. 1, 1846), 
shows that, in general, m and dy/dt are not parallel! 

REMARK 

Equation (1.10.27c) also results if we apply to the formula for the subtraction of 
rotations (1.10.23c), the sequence 

y, = y — Ay — > y 2 = Ay', (1.10.27d) 

which is equivalent to yj 2 = y. Thus, we obtain 

Ay' = [y - (y - Ay) + (y - Ay) x y]/[1 + (y - Ay) • y] 

= (Ay + y x Ay) /(I + 7 2 — y • Ay), (1.10.27e) 

then divide by A t and take the limit as A t —> 0 to obtain 

c« = 2hm(dy7dt)|^ 0 

= 2{[lim(dy/d?) + y x lim(dy/dt)]/(l + 7 2 - y • Ay)} |^ 0 

= [2/(1 + 1 2 )][dy/dt + y x {dy/dt)], (1.10.27f) 

as before. The reader may verify that the sequence yj =y —> y 2 = Ay', which is 
equivalent to y { 2 = y + Ay, also leads to the same formula. 

(iv) Inversion of the Preceding Formula ca = to(y, dy/dt) 

First Derivation. Dotting both sides of that equation, 

(1.10.27c), by y yields 

ym= [2/(1 + 7 2 )] [y • {dy/dt)]\ (1.10.28a) 

while crossing it with y gives 

y x m = [2/(1 + 7 2 )]{y x (dy/dt) + y x [y x (dy/dt)]} 

= [2/(1 + 7 2 )]{y x (dy/dt) + [y • (dy/dt)\y - ^(dy/dt)}. (1.10.28b) 

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CHAPTER 1: BACKGROUND 


Eliminating y-(dy/dt) between (1.10.28a,b) produces 
y x ct) = [2/(1 + 7 2 )]y x (dy/dt) 

- [2 7 2 /(l + 7 2 )](dy/dt) + (y • co)y 

[expressing the first right-side term of the above via (1.10.27c)] 

= { t °- P/(i + 7 2 )](^yM)} ^ p7 2 /(i + 7 2 )K^yM) + (r <o)y 

= to — 2(dy/dt) + (y • to)y, (1.10.28c) 


or, rearranging, finally gives 

2 (dy/dt) = to + (y • a>)y + to x y; (1.10.28d) 

which, for a given to(t), is a vector first-order nonlinear ( second-degree) differential 
equation for y(t) (and can be further reduced to a “Ricatti-type equation”). 

Equations (1.10.27d), and (1.10.27c) clearly demonstrate the one-to-one relation 
between to and dy/dt: if one of them vanishes, so does the other. 


Second Derivation. Applying the earlier rotation sequence 

y x =y^y 2 = Ay', (1.10.28e) 

which is equivalent to y { 2 = y + dy, both occurring in time At, to the composition 
formula (1.10.20i) we obtain 

y + Ay = (y + Ay' + Ay' x y)/(l — Ay' •y ), (1.10.28f) 

from which, subtracting y, we get 

Ay = [Ay' + (y • Ay')y + Ay' xy]/(l- y • Ay'), (1.10.28g) 

and from this, dividing by At and taking the limit as At —*• 0, while recalling that [eq. 
(1.10.27f)] to = 2\mfiAy'/At)\ At ^Q, we re-obtain (1.10.28d). 

For still alternative derivations of the to <-> y equations, via the compatibility of 
the Eulerian kinematic relation v = clr/dt = m / r with the d/dt(.. .)-derivative of 
the finite rotation equation /y = rj (y; r,) [eqs. (1.10.2-4)], see, for example (alpha¬ 
betically): Coe (1938, chap. 5; best elementary/vectorial treatment), Ferrarese (1980, 
pp. 122-137), Hamel (1949, pp. 106-107; pp. 391-393). 


(v) Additional Useful Results 
(a) Starting with 

y = «tan(x/2) 

=> dy/dt = ( dn/dt ) tan(x/2) + n[(dx/dt)/2\ sec 2 (x/2), etc., 
and then using the to <-> y equation, we can show that 

to = ( dx/dt)rt + (sin x){dn/dt) + (1 — cosx)« x (dn/dt). (1.10.29a) 


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§1.10 THE RIGID BODY: GEOMETRY OF ROTATIONAL MOTION 


(What happens if n = constant ?) 

(b) Again, starting with 

y = n tan(x/2) = tan(x/2)(//x) = [tan(x/2)/x]z => dy/dt = ■ • ■ ,etc., 
and then using the to <-> y equation, we can show that 

a = (sin X /x)(dx/dt) + [(1 - cosx)/x 2 ][x x (dx/dt)} 

+ [(1/X) - (sin x/x 2 )\(dx/dt)x 

= dx/dt + [(1 - cos x)/x 2 ][x x (dx/dt)] 

+ [(X - sin x)/x’]{x x [x x (dx/dt)]}. (1.10.29b) 

(c) By inverting (1.10.29b), we can show that 

dx/dt = u> - (x x <»)/2 + (l/x 2 )[l - (x/2) cot(x/2)][x x (x x to)] . (1.10.29c) 
More in our Elementary Mechanics (§13.8 — under production). 

General Rigid-Body Displacement (i.e., no point fixed) 

We have already seen (§1.9) that the most general rigid-body displacement can be 
effected by the translation of an arbitrary base point or pole of it, from its initial to its 
final position, followed by a rotation about an axis through the final position of that 
point (see figs 1.12 and 1.22). Here, we show that the translational part of the above 
total displacement does depend on the base point, but the rotational part—that is, the 
rotation tensor — does not. 

Referring to fig. 1.22, let 

ll' = r p/t» PP"=r f/i , IP = r/\, l'P =r /v , l"P" =l'P" =r n », 

R\ = rotation tensor bringing l'P' to l'P"\ i.e., r^n = R\ -r/y. (1.10.30a) 



Figure 1.22 Most general rigid-body displacement; the rotation tensor is independent 
of the base point (or pole). 

r /1 -»• r /v -► r ir = ^ •'■/id 

fj = r\ + f \ fj, = r v + r /v r f = r v , + r /v , = r v , + R, ■ r jv (r v = r r ). 


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CHAPTER 1: BACKGROUND 


Then, successively, 

PP" =P1+11' + l'P" = -r/i + r l7l + R t ■ r n ,, (1.10.30b) 

or since IP = l'P' (i.e., r/i = r^/), 

rfn = /•,'/! + (/?i -l)-r ,i . (1.10.30c) 

Had we chosen another base point, say 2, then reasoning as above we would have 
found (with some easily understood notations) 

r fn = r 2'a + ( r 2 -1) • r/i ■ (1.10.30d) 

But also, applying (1.10.30c) for P = 2, we have (since r 2 ' = r 2 ") 

r 2 'a = r i'n + ( R i ~ 1 )' r 2 n ■ (1.10.30e) 

Therefore, substituting (1.10.30e) in (1.10.30d) and equating its right side to that of 
(1.10.30c), we obtain 

r yn +(/?!- l)-r 2 /i + ( r 2 - l)-r /2 = r Vn + {R\ -l)-r n , 
from which, rearranging, we get 

(R 1 - 1) • (m - r 2 /i) = {R l -l)-r l2 = (R 2 -l)- r n , (1.10.30f) 

and since this must hold for all body point pairs P and 2 (i.e., it must be an identity 
in them), we finally conclude that 

Ri=R 2 =-=R. (1.10.30g) 

In words: the rotation tensor is independent of the chosen base point; it is a position- 
independent tensor. This fundamental theorem simplifies rigid-body geometry 
enormously and brings out the intrinsic character of rotation. (In kinetics, however, 
as the reader probably knows, such a decoupling between translation and rotation is 
far more selective.) 


1.11 THE RIGID BODY: ACTIVE AND PASSIVE INTERPRETATIONS 

OF A PROPER ORTHOGONAL TENSOR; SUCCESSIVE FINITE ROTATIONS 


A 3 x 3 proper orthogonal tensor may be interpreted in the following consistent 
ways: 

(i) As the matrix of the direction cosines orienting two orthonormal and dextral 
(OND) triads, or bases, and associated axes; say, a body-fixed, or moving, triad t: 


t = (u k ) = 




relative to a space-fixed triad T; 

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(1.11.la) 



§1.11 THE RIGID BODY: INTERPRETATIONS OF A PROPER ORTHOGONAL TENSOR 


T = («*,) = 



(ii) Then, since 


( « x\ 

Uy 

\ u z) 



(1.11.1b) 


I = (/ • i)i + (I -j)j + {I • k)k = A Xx i + A Xy j + A Xz k, etc., cyclically, 
i = (i • /)/ + (* • J)J + (i • A - )A - = + ^y.v J + ^zx A, etc., cyclically, 


the two triads are related by 


T = A• t <^> t = A~‘.T = A t -T, (1.11.1c) 

where 



//•/ 

Ij 

I' k ) 


^ Axx 

A Xy 

A Xz \ 

A = 

Ji 

Jj 

Jk 

= 

''l Yx 

A Yy 

A Yz 


U 

Kj 

K-kj 


\Azx 

A Zy 

A Zz) 


= (A k 'k), A k ' k = cos(x k ',x k ) = u k <-u k [= cos (x k ,x k >) = A kk '}. (l.ll.le) 


The rotation of an OND triad, equation (l.ll.le), T —> t, constitutes the second 
interpretation of a proper orthogonal tensor. 

(iii) The third such interpretation is that of a coordinate transformation from the 
T-axes: 0—x k ' = O—XYZ to the t-axes: 0—x k = O—xyz (of common origin, with 
no loss in generality). In this interpretation, known as passive or alias (meaning 
otherwise known as), the point P is fixed in T-space and the t-axes rotate. Then 
[hg. 1.23(a)], 

OP = r = ^2 x k ,u k' = A/ + YJ + ZK = ^2 x k u k = xi + yj + :k. (1.11.2a) 


(a) PASSIVE (ALIAS) INTERPRETATION (b) ACTIVE (ALIBI) INTERPRETATION 




Figure 1.23 (a) Passive and (b) Active interpretation of a proper orthogonal tensor 
(two dimensions). 


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CHAPTER 1: BACKGROUND 


and so we easily find 

x k > = r • u k , = ■ ■ ■ = A k'kXk, 

x k =r-u k = - = Y^ A kk'X k ' (= A k'k x k '), (1-11 -2b) 

or explicitly, in matrix form, 


(X) 


^ A Xx A Xy A Xz \ 


Y 

= 

A Yx A Yy A Yz 

y 

UJ 


\ A Zx A Zy A Zz ) 

w 


r' = A • r 


Old axes New axes, (1.11.2c) 




^ A Xx A Yx A Zx\ 

( X \ 

y 

= 

A Xy A Yy A Zy 

Y 

\ z ; 


\ A Xz A Yz A Zz ) 

U/ 


r = A T • r' 


New axes Old axes. (1.11.2d) 

For example, in two dimensions [fig. 1.23(a)], the above yield 

( X\ / cos x — sin %\ / x \ f x \ ( cos x sin x \ / X \ 


sin x cos x 
r' = A • r, 




J, 


sin x cos x 

: A t • r' 


(1.11.2e) 


(iv) Under the fourth interpretation, known as active or alibi (meaning elsewhere), 
the axes remain fixed in space, say T = t, and the point P rotates about O, from an 
initial position r, = XI + YJ + ZK to a final one »y = X' 1 + Y'J + Z'K. Then, 
following §1.10, and with A —> R (rotation tensor), 


(X'\ 


( x ) 


(x\ 

Y' 

= R 

Y 

= A 

Y 

\z') 


\z) 


U / 


'/ = R • F 

Final position Initial position, (1.11,2f) 


(X) 

Y 

= R t 

(X'\ 

Y' 

= A J 

(x>\ 

Y' 

\z) 


\Z') 


\ z '! 


F = K T <7 

Initial position Final position. (1.11 -2g) 


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§1.11 THE RIGID BODY: INTERPRETATIONS OF A PROPER ORTHOGONAL TENSOR 


Equations (1.11.2f, g) hold about any common axes; and, clearly, the components of 
R depend on the particular axes used. For example, in two dimensions [fig. 1.23(b)], 
the above yield 

(X'\ = /cosx — sinx\ m (X\ = ( cos X sin X \ (X'\ 

\Y'J Vsinx cos X )\y) U/ l-smx cos X )\Y') 

r f = Rt,-, I", = R T • ry; (1.11.2h) 

and for the new triad (actually a dyad) i, j in terms of the old triad /, J [along the 
same (old) axes], they readily yield 

/cosx\ _ /cosx -sinx\/l\ /-sin x\ = /cosx -sinx\/0\ 

ysinx/ V sin X cosx/V 0 / V C0S X / V sin X cosx/W 

i = R • /, j = R-J. (1.11.2i) 

The passive and active interpretations are based on the fact that: The rigid body 
rotation relative to space-fixed axes ( active interpretation), and the axes rotation 
relative to a fixed body (passive interpretation) are mutually reciprocal motions. 
Hence [fig. 1.24(a, b)]: The coordinates of a rotated body-fixed vector along the 
old axes ( final position, active interpretation), equal the coordinates of the unrotated 
rigid body along the inversely rotated axes ( new axes, passive interpretation). 

It follows that if the body is fixed relative to the new axes and r' = XI + YJ, 
r = xi + yj , then the rotation equations—for example, (1.10.2e)—yields (with 

h + ^new (body-fixed) axes = ** and Vj > Told axes = * ) 

r' = [2/(1 +7 2 )][y X r+ (yc/yj + [(1 - 7 2 )/(l + 7 2 )]r. (1.11.3) 

A correct understanding of the above four interpretations—in particular, the inter¬ 
change of A with A t = A(—x) [and R with R T = R(—x)] in single, and, especially, 
successive rotations (see below)—is crucial to spatial rigid-body kinematics. Lack of 
it, as Synge (1960, p. 16) accurately puts it “can be a source of such petty confusion.” 


(a) ACTIVE INTERPRETATION (b) PASSIVE INTERPRETATION 




Figure 1.24 The final coordinates under x [active interpretation (a)] equal 
the new coordinates under —x [passive interpretation (b)], and vice versa. 

(X ^-rotated vector, old axes = Xjnrotaled vector,-\-rotated axes/ etc.) 


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CHAPTER 1: BACKGROUND 


(i) 

(11) 


Below, we summarize these four interpretations of an orthogonal tensor A or R: 
A = (A k > k ) = («£/ • 14 ): Direction cosine matrix; 


(*) 



J 

= A 

j 

UJ 


w 


or 


T = A • t: Triad rotation; 



(X) 


( x \ 

(hi) 

Y 

= A 

y 


Uv 


v z / 


or rold axes = r ' = A • r new axes = A • r. Passive interpretation 

(Vector fixed; axes rotated); 



{ x '\ 


( x \ 

(iv) 

Y' 

= R 

Y 


\Z') 


U/ 


or ly = R • r,: Active interpretation A = R 

{Vector rotated; axes fixed, and common). 


REMARKS 

(i) In the passive interpretation, we denote the components of A as A k > k , whereas, 
in the active one, we denote them, in an arbitrary but common set of axes, as R k/ (or 

This is an extra advantage of the accented indicial notation, especially in cases 
where both interpretations are needed. 

(ii) The passive interpretation also holds for the components of any other vector; 
for example, angular velocity. 


Successive Rotations 

Let us consider a sequence of rotations compounded according to the following 
scheme: 


T ^ T, ^ T 2 ^ ^ T n _, - T n = t 

A| A 2 A 3 A n _! A n (1.11.4a) 

Then we shall have the following composition formulae, for the various interpreta¬ 
tions. 


(i) Triad Rotation 

T=(A,.A 2 .A n ) • t O t=(A n T .A n _! T . A, t ).T; (1.11.4b) 


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§1.11 THE RIGID BODY: INTERPRETATIONS OF A PROPER ORTHOGONAL TENSOR 


or, in extenso, 


(*) 




J 

(A| • A 2 • • 

■ • A n ) 

./' 

UJ 



U/ 


Initial triad (natural order Final triad, 
of component 
matrices) 





f 1 ] 


= (A n T -A n _ I T .- 

••A, 7 ) 

J 




U / 


Final triad Initial triad. 


(ii) Passive Interpretation 

Here, with some easily understood ad hoc notations, we will have 

T 0 id axes = r' = Aj • r, = Aj • (A, • r 2 ) = ■ ■ ■ = (Aj • A 2 .A n ) • r , 

Tnew axes — T (A| • A2.A n ) • V (A n • A n _[ .A| ) * T , 

or, in extenso, 


( x ) 

Y 

(Ai • A 2 • • 

■ -A n ) 

(*\ 

y 

\z) 



\-J 


Old axes New axes, 


( x ) 



( X \ 

y 

= (A n T • A n _i T • • 

••A, T ) 

Y 




U / 


New axes Old axes. 


(iii) Active Interpretation 

Here, choosing common axes corresponding to T; that is, 

/*,■ = XI + YJ + ZK -> r f = X'l + Y'J + Z'K (= Xi + Yj + Zk), 

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(1.11.4c) 


(1.11.4d) 


(1.11.4e) 
(1.11.4f) 


(111 4g) 

(1.11,4h) 


(1.11.41) 






















CHAPTER 1: BACKGROUND 


we obtain, successively, 

r t = A, T • r n = A T , • (A t 2 • r f2 ) = ■ • • = (A, T • A 2 T .A n T ) • r f 

= (Rl T 'R2 T .RnV/, 

=> jy = (A n • A n _].A!) Vj = (R n • Rn—1.Rl) •*';! 

or, in extenso, 


/x'\ 



( X \ 

Y' 

(Rn • Rn-1 ‘ 

•• -Ri) 

Y 

[z'J 



U / 


Final position 


Initial position, 


( X ) 



(X>\ 

Y 

= (R! T .R 2 T . • 

• -Rn T ) 

Y' 

UJ 



\Z'j 


Initial position 


(1.11.4]) 

(1.11.4k) 


(1.11.41) 


Final position. (1.11,4m) 


Body-Fixed versus Space-Fixed Axes 

The moving triad t and associated axes (O-xyz) may be considered as a rigid body 
going through a sequence of rotations, either about these body-fixed axes themselves, 
or about the space-fixed axes O-XYZ with which it originally coincided. Either of 
these two types of sequences may be used (although the tensors/matrices of rotations 
about body-fixed axes have simpler structure than those about space-fixed axes), and 
their outcomes are related by the following remarkable theorem: The sequence of 
rotations about Ox, Oy, Oz has the same effect as the sequence of rotations of equal 
amounts about OX, OY, OZ, but carried out in the reverse order. Symbolically, 

f R ] 1^2 (^y-fjxed lixcs (R 2 R[ (jpacp-fixedaxes' 


This nontrivial result will be proved in §1.12. 

Thus, for a sequence about space-fixed axes, eq. (1.11.4h) (which expresses the 
passive interpretation for a body-fixed sequence) should be replaced by 


/*\ 

y 

W 


(Si 


= (S n -S 


n—1 


New axes 


S n T ) 


Si ) 1 


n 

Y 

W 

n 

Y 

U/ 

Old axes, 


(1.11.4n) 


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§1.11 THE RIGID BODY: INTERPRETATIONS OF A PROPER ORTHOGONAL TENSOR 


where the S k are the space-fixed axes counterparts (of equal angle of rotation) of the 
R k ; and similarly for the other compounded rotation equations. 

REMARKS 

(i) In algebraic terms, we say that such successive rotations form the Special 
Orthogonal (Unit Determinant) — Three Dimensional group of Real Matrices 
[=SO(3, R)], and are representable by three independent parameters; for example, 
Eulerian angles (§1.12). 

[By group, we mean, briefly, that (a) an identity rotation exists (i.e., one that 
leaves the body unchanged); (b) the product of two successive rotations is also a 
rotation; (c) every rotation has an inverse; and (d) these rotations are associative. See 
books on algebra/group theory.] 

(ii) Some authors call rotation tensor/matrix the transpose of this book’s, while 
others, in addition, fail to mention the distinction between active and passive inter¬ 
pretations. Hence, a certain caution is needed when comparing various references. 
Our choice was based on the fact that when the rotation tensor of the active inter¬ 
pretation is expanded a la Taylor around the identity tensor, and so on (1.10.25a ff), 
it leads to an angular velocity compatible with the definition of the axial vector (cn) of 
an antisymmetric tensor (1.1.16a ff.) f2: fl • r = u> x r; otherwise we would have 
f2 -r = — (o x r. 


Tensorial Derivation of the Finite Rotation Tensor 

Let us consider the following two rectangular Cartesian sets of axes, 0-x k t 
(= O—XYZ , fixed) and 0—x k (= O—xyz, moving), related by the proper orthogonal 
transformation: 

x k' = A k'k x k ^ x k = Yl A kk' x k'y A k'k = A kk' = cos (x k ',X k ). (1.11.5a) 

The corresponding components of the rotation tensor, R k t V and R kh respectively, 
will be related by the well-known transformation rule for second-order tensors 


(1.1.19j ff.): 



R k'l' — A k'k A l'l R kl ^ 

R-kl — ^2 A k'k A l'lRk'l'i 

(1.11.5b) 

or, in matrix form. 



R' = AR- A t 

<£> R = A T R - A, 

(1.11.5c) 

where 



R'= (**'/'), R 

= { R u) > A. = (Aw)- 

(1.11.5d) 


Here, choosing axes 0-x k in which R k/ have the simplest form possible, and then 
applying (1.11.5b, c), we will obtain the rotation tensor components in the general 
axes 0—x k ', R k 'i r , that is, eq. (1.10.10a). To this end, we select Ox k so that x\ = x is 
along the positive sense of the rotation axis «, while x 2 = y, x 3 = z are on the plane 
through O perpendicular to n (fig. 1.25). For such special axes, the finite rotation is a 

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CHAPTER 1: BACKGROUND 



Figure 1.25 Tensor transformation of rotation tensor, between the 
general fixed axes O-XYZ and the special moving axes O-xyz; Ox 
axis of rotation. 


plane rotation of (say, right-hand rule) angle x about Ox, and, hence, there the 
rotation tensor has the following simple planar form: 


10 0 
R = I 0 cos x — sin x 
0 sinx cos% / 

Now, to apply (1.11.5c) we need A. The latter, since 

i = A xX^ T A xY J T A X %K — ft “ nxl T tiyJ T 

becomes 


A = 


(1.11.5e) 


(1.11.5f) 


«X 

A Xy 

A Xz \ 


n Y 

A Yy 

a yz ; 

(1.11 -5g) 

«z 

A Zy 

A Zz / 


...) 

= c(.. 

sin(. 

..)=$(...), (1.11.5c) specia- 


lizes to 


R' = 


"x 

A Xy 


( l 

0 

0 \ 

/ n x 

n Y 

n z 

n Y 

A Yy 

A Yz 

b 

cx 

1 

to 

X 

X 

A Yy 

A Zv 

»z 

A Zy 

A Zz ) 

V° 

sx 

cx / 

\ Axz 

A Yz 

A Zz 


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§1.11 THE RIGID BODY: INTERPRETATIONS OF A PROPER ORTHOGONAL TENSOR 


or, carrying out the matrix multiplications, and recalling that R\'\' = R xx , 
R { n' = R X y, and so on , 

Rxx = n X 2 + (^Xy 2 + A Xz ) COS X, 

R xy = n x n Y + (A Xy A Yy + A Xz A Yz ) cosx — (A Xy A Yz — A Yy A Xz ) sinx, 

Rxz = n x n z + (A Xy A 7) , + A X: A 7= ) cosx + (A- Zy A Xz — A Xy A Zz ) sinx; 

Ryx = n Y n x + (A Yy Axy + A Yz A X z ) cosx + (AxyAzz ~ A Yy A Xz ) sinx, 

R yy = n Y ~ + (A Yy 2 + A y 2 ) cosx, 

Ryz = n Y n z + (AY y A 7y + A Y: A Zz ) cos x — (A Y yA 7z — A Zy A Yz ) sinx; 

Rzx = n z n x + (A Zy A Xy + A Z: A Xz ) cos x — ( A Zy A Xz — A Xy A Zz ) sin x, 

Rzy — n z n Y + (A 7y A Yy + A Z: A Yz ) cosx + (A Yy A Zz — A Zy A Yz ) sinx, 

Rzz = n z + (A Zy 2 + A Zz 2 ) cos x- (1-11.5h) 

However, the nine A k ' k are constrained by the six orthonormality conditions: 

I • J = n x n Y + A Xy A Yy + A Xz A Yz = 0, 

J • K — A Yy-^Zy Yz^-Zz = 0, 

K • I = n z n x + A Zy A Xy + A Zz A Xz = 0; 

I ■ 1 = n x 2 + A x 2 + A Xz ~ = 1, 

J • J = fl Y A y y + A Yz ~ = 1 , 

K-K = n z 2 + A z 2 + A z 2 = 1; (l.ll.Si) 

and also n = u r x u z , or, in components, 

n x = A Y y A Zz — A Zy A Yz . n Y = A Zy A Xz — A Xy A Zz , n z = A Xy A Yz — A Yy A Xz . 

(i.n.sj) 

As a result of the above, it is not hard to verify that the R k 'i<, (1.11.5h), reduce to 

Rxx = n x~ + (1 — n x ~) cos x, 

Rxy = n x n Y + {-n x n Y ) cosx + («z) sinx, 

Rxz = n x n z + (-n x n z ) cosx + Or) sinx; 

Ryx = n Y n x + (~n x n Y ) cos x + (n z ) sinx, 

Ryy = n Y~ + (1 ~ n Y~) cos x, 

Ryz = »r«z + (~n Y n z ) cos x + (-«■y) sin x; 

Rzx = n z n x + (- n z n x ) cos x + (~n Y ) sinx, 

^zr = »z«r + (-«z«y) cosx + («z) sinx, 

^zz = + (1 — n z 2 ) cosx; (1.11.5k) 

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CHAPTER 1: BACKGROUND 


and when put to matrix form is none other than eq. (1.10.10a). We notice that the 
components are independent of the orientation of the O-xyz axes , as expected. 


Angular Velocity via the Passive Interpretation 

Let us consider a generic body point P fixed in the moving frame t: O ijk/O xyz, and 
hence representable by 

*•' = XI + YJ + ZK (space-fixed frame T: O—IJK/ O—XYZ) , (1.11. 6 a) 

r = xi + yj + zk (body-fixed frame; i.e. x,y,z = constant); ( 1 . 11 . 6 b) 

or, in matrix form, 

r' J = (X,Y,Z), r T = (x,y,z). (1.11.6c) 

According to the passive interpretation (1.11.2c) (with A replaced by the rotation 
tensor / matrix R), 


r = R-r, (1.11.6d) 

and, therefore, the inertial velocity of P, resolved along the fixed axes O XYZ equals 

v' = dr'/dt = (dR/dt) - r + R • ( dr/dt ) = (dR/dt) • r + R • 0 

= (dR/dt) • (R t • r') = n'-r ' = o*' X r' , (1.11.6e) 


where [recalling (1.7.30fff.), with A —* R] 

Yl' = (dR/dt) • R t = angular velocity tensor of body frame t relative to the fixed 

frame T, but resolved along the fixed axes O-XYZ, (1.11. 6 f) 

ft* ' = axial vector of $Y; angular velocity vector of t relative to T. along T. (1.11. 6 g) 

The components of the angular velocity along the moving axes can then be found 
easily from the vector transformation (passive interpretation): 

v = inertial velocity of P , but resolved along the moving axes {not to be confused 
with the velocity of P relative to t, which is zero: drjdt = 0) 

= R t -v' = R t - [(dR/dt) -r\ = f l-r=oxr, (1.11.6h) 

where 

Cl = R t • (dR/dt) = angular velocity tensor of body frame t relative to the fixed frame 
T, but resolved along the moving axes O-xyz 

{ = [R T • (dR/dt)] • (R t • R) = R t • [(dR/dt) • R T ] • R 

= R t • ff ■ R; a second-order tensor transformation, as it should be} , (1.11.6i) 

ft* = axial vector of angular velocity of t relative to T, along t [= R T • ft*']. ( 1.11 - 6 j) 

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§1.11 THE RIGID BODY: INTERPRETATIONS OF A PROPER ORTHOGONAL TENSOR 


REMARK 

If R = R( 9l ,< 72 , ft) = R(ftJ, where the q a are system rotational parameters (e.g., the 
three Eulerian angles, § 1 . 12 ), then fY and to 1 can be expressed, respectively as follows: 

Tensor: ft' = ^ ft' a (dq a /dt ), Vector: to' = ^ to' a {dq a /dt ), (1.11.6k) 

where 

ft' a = (dR/dq a ) ■ R t and fl' a • x = (o' a x x , (1.11.61) 

for an arbitrary vector jc; that is, fY can be expressed in terms of the local basis 
{ft' a ', a = 1 , 2 ,3}; and similarly for ft and ft). 

Additional Useful Results 

1. Consider the following two successive (component) rotations: First, from the 
“fixed” frame 0 to the moving frame 1, Ri / 0 = R ls and, next, from 1 to the also 
moving frame 2, R 2 /i = R 2 - Then, by (1.11.4a If.), the resultant rotation from 0 to 2 
will be R = R] • R 2 . Now, let: 

fY/o,o = (dRi/dt) • Ri t : angular velocity tensor of frame 1 relative to frame 0 , 
along O-axes ; 

fY/o,i = R| T • (dRi/dt): angular velocity tensor of frame 1 relative to frame 0 , 
along 1 -axes ; 

ftz/ 1 , i = (dR 2 /dt) • R 2 t : angular velocity tensor of frame 2 relative to frame 1 , 
along 1 -axes ; 

fl 2/u2 = R 2 t • (dR 2 /dt): angular velocity tensor of frame 2 relative to frame 1 , 
along 2-axes ; 

fY/o,o = (dR/dt) -R 1 : angular velocity tensor of frame 2 relative to frame 0 , 
along O-axes ; 

fi 2 /o, 2 = R T • (dR/dt): angular velocity tensor of frame 2 relative to frame 0 , 

along 2-axes ; (1.11.7a) 

(this or some similar intricate notation is a must in matrix territory!) and therefore 

fY/o,o = Ri-tti/o.i- R i T ^ n I /o,i=Ri T -n 1 / o,o-R 1) 
fti/ 1,0 = Rt * fY/i. i • R| T <=> ftz/ 1 , i = Ri • S^ 2 /i,o • R| T ) 
ftz/ 1,1 = R2 • ftz/1,2 • R2 T ^ ftz/ 1,2 = R2 T • ftz/\. 1 ■ R2 ) 

ftz/t>, 1 = Rl T • ^2/0,0 ■ R| = R 2 • f^2/0,2 = R 2 T : 

angular velocity of frame 2 relative to frame 0 , but expressed along 1-axes; etc.; 
i.e., the multiplications Ri^-.^'R ] 7 convert components from 1-frame axes to 
0-frame axes; while R] T (...) -R) convert components from 0-frame axes to 1-frame 
axes; and analogously for R 2 •(...) • R 2 T , R 2 T •(■■■)• R 2 - 
Then, and since R, R t , R 2 are orthogonal tensors, 

(a) fl 2/ 0,0 = (dR/dt) • R t = [d/dt(R, • R 2 )] • (R, • R 2 ) T 

= •■■ = (dRj/dt) -Rj+Ri-KdRj/dt) -Rj^.R , 1 

= /o,o + Ri • ^ 2 / 1.1 ■ Ri T = fii/ 0.0 + ^ 2 / 1,0 

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CHAPTER 1: BACKGROUND 


(theorem of additivity of angular velocities, along O-cuces ); (1.11.7b) 

^2/0,1 = R] T - ^2/0,0 • Rl = R| T • J^l/0,0 ' Rl + ^2/1,1 = ^1/0,1 + ^2/1,1 

(theorem of additivity of angular velocities, along 1-axes ); (1.11.7c) 

^ 2 / 0,2 = R T • (dR/dt) [= R 2 T • rj 2/0 , i ■ R 2 = R T • ^ 2 / 0,0 • R] 

= (Ri-R 2 ) T -[d/dt(Ri-R 2 )] 

= • • ■ = R 2 t • [R! T • (dRi/dt)] • R 2 + R 2 t • (dR 2 /dt) 

= R 2 t • f 2 ]/o, 1 ■ R2 + ^2/1,2 = ^1/0,2 + ^2/1,2 
(theorem of additivity of angular velocities, along 2-axes). (1.11.7d) 


(b) Next, d(.. ,)/dt-differentiating the above, say (1.11.7b), it is not hard to show 
that: 


d^ 2 /o,o/dt = dfl]/o,o/dt + d/dt(Ri • fi 2 /p 1 • Ri T ) 

= dn^o.o/dt + Ri • (dn 2/lj 1 /dt) • R] T + Ri • ! • n 2 /i! — n 2 /i j • • Ri T 

(theorem of non-additivity of angular accelerations, along 0-axes ); (1.11.7e) 

and similarly for dfi 2 /o, i/dt, dfi 2 /o, 2 /dt. The last (third) term of (1.11.7e) shows that 
if the elements of the matrices , fl 1 / 0 , 0 - ^ 2 / 1.1 tire constant, then, in general, the elements 
o/n 2/0 ,o will also be constant if f! 1 /o, 1 and f 2 2 /i,i commute , a well-known result from 
vectorial (undergraduate) kinematics. The extension of the above to three or more 
successive rotations is obvious. 

[As Professor D. T. Greenwood has aptly remarked: “Equations (1.11.7b—e) illus¬ 
trate how the use of matrix notation can make the simple seem obscure.”] 

2. Matrix forms of relative motion of a particle , in two frames with common origin. 
By d/dt(.. ^-differentiating the passive interpretation (1.11.2c), 



Fixed axes Moving axes, (1.11.8a) 

we can show that 


(i) 




( X ) 


f x\ 


d/dt 

Y 

= A- 

d/dt 

y 

+ fi- 

y 



\z) 



K z / 


\ z ) 



{ = A • [relative velocity + transport velocity]}. (1.11.8b) 


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§1.11 THE RIGID BODY: INTERPRETATIONS OF A PROPER ORTHOGONAL TENSOR 



( X ) 



( X ) 



d 2 /dt 2 

Y 

= A- 

d 2 /dt 2 

y 

+ (dfi/dt) • 

y 


UJ 



\ z ) 


\ Z J 


( X ) 




y 

+ 2CI - d/dt 

y 


\ z j 


\ z j 



{ = A- [relative acceleration (d 2 r/dr 1 ) + transport acceleration (a x r + a> x (a> x r)) 

+ Coriolis acceleration (2a> x {dr/dt))]\ ; 

( 1 . 11 . 8 c) 

we point out that, in the matrix notation, the d/dt vs. d/dt difference (§1.7) disappears. 

(iii) If the position of the origin of the moving axes, relative to that of the fixed 
ones, is r 0 = ( X D1 Y 0 ,Z 0 ) J , so that [instead of (1.11.8a)] 


(9 ■ (I) “■(;)■ 

then we simply add d/dt (X 0 ,Y 0 ,Z 0 ) T to the right side of (1.11.8b) and 
d 2 /dt 2 (A OI Y 0 ,Z 0 ) J to the right side of (1.11.8c). 

3. Tensor of Angular Acceleration, and so on. 

(i) By d(.. .)/dt-differentiating (1.7.30i, j): dA/dt = A - Cl = Cl' • A, we can show 
that 


d 2 A/dt 2 = A • E =>- E = A T • (d 2 A/dt 2 ), (1.11.9a) 

where 

E = A+Cl-Cl = A+Cl 2 , (1.11.9b) 

A = Ml /dt : (Matrix of components, along the moving axes, of the) 

tensor of angular acceleration of the moving axes relative 

to the fixed ones (1.11.9c) 

{ = d/dt[A T • (dA/dt)] = (dA T /dt) • (dA/dt) + A T • (d 2 A/dt 2 ) 

= -Cl-Cl+E} . (1.11.9d) 

[In fact, both A and E appear in (1.11.8c). Also, some authors call E the angular 
acceleration tensor, but we think that that term should apply to dfl/dt; that is, 
definition (1.11.9c).] 

(ii) Both E and A are (second-order) tensors; that is, 

E'(=^.' + r2'-0') = A-E-A t E = A t -E'-A, (1.11.9e) 

A'(= dJY/dt) = A • *4.- A t A = A T -.4.'-A; (1.11.9f) 

where, as before, an accent (prime) denotes matrix of components along the fixed 


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CHAPTER 1: BACKGROUND 

(iii) The fixed axes counterpart of (1.11.9a) is: 

d 2 A/dt 2 = E'-A => E' = (d 2 A/dt 2 )-A T , (1.11.9g) 

(iv) It can be verified, independently of (1.11.9a-d) and (1.11.9e-g), that 
Jl 2 = $7 • T2 = —J7 t • 17 = —17 • 17 t 

= • • • = — (dA/dt) T • (dA/dt) = -(dA T /dt) • (dA/dt), (1.11.9h) 

(17') 2 = 17' -17' = —(17 , ) t • 17' = -17' • (17') t 

= • • • = -(dA/dt) • (dA/dt) T = -(dA/dt) • (dA T /dt). (1.11.9i) 

(v) Since d!7/dt is antisymmetric, and 17 • 17 is symmetric (explain this), show that 
the axial vectors of (the nonsymmetric) E and A coincide, and are both equal to none 
other than the vector of angular acceleration a; thus justifying calling A the tensor of 
angular acceleration. 

Finally, if the moving axes are fixed relative to a body B, then 17/17' and AI At are 
respectively, the tensors of angular velocity and acceleration of that body relative to 
the space-fixed axes; and if the earlier particle is frozen (fixed) relative to B (i.e., 
dx/dt = 0, d 2 x/dt 2 = 0, etc.), then (1.11.8b, c) give, respectively, the matrix forms of 
the well-known formulae for the distribution of velocities and acceleration of the 
various points of B (from body-axes components to space-axes components). [For an 
indicial treatment of these tensors, and recursive formulae for their higher rates, see 
Truesdell and Toupin (1960, pp. 439^140).] 


1.12 THE RIGID BODY: EULERIAN ANGLES 

We recommend for concurrent reading with this section: Junkins and Turner (1986, 
chap. 2), Morton (1984). 

As explained already (§1.7, §1.11), the nine elements of the proper orthogonal 
tensor A (or R), in all its four interpretations, depend on only three independent 
parameters. A particularly popular such parametrization is afforded by the three 
(generalized) Eulerian angles. These latter appear naturally as we describe the 
general orientation of an ortho-normal-dextral (OND) body-fixed triad, or local 
frame t = \u k } = (/, j, k) relative to an OND space-fixed frame 
T = {iik 1 } = (/,./, K), with which it originally coincides, via the following sequence 
of three, possibly hypothetical, simple planar rotations (i.e., in each of them, the two 
triads have one axis in common, or parallel, and so the corresponding “partial 
rotation tensor” depends on a single angle): 


(i) Rotation about the (;)th body axis through an angle \(i) = Xi = </; followed by a 

(ii) Rotation about the (y)th body axis (j f i) through an angle Xu) = Xi = S', followed 
by a 

(iii) Rotation about the (k)th body axis (k f j) through an angle X(k) = Xt = 

The angles xi = f (about the original u, = «,-/), xi = S (about the (/-rotated i/ ; —> «■/), 
and X 3 = (about the 0-rotated u k —> u k «) are known as the / —> j —> k Eulerian 
angles. 


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§1.12 THE RIGID BODY: EULERIAN ANGLES 


Of the twelve possible such angle triplets, six form a group for which i fij k = i 
(two-axes group): 

1 —► 2 —> 1, 1 —> 3 —> 1, 2 — > 1 —> 2 , 2 — > 3 —i- 2 , 3 —> 1 —> 3, 3 2 ^ 3; 

and six form a group for which i j k i ( three-axes group): 

1—>2—> 3, 1 —s- 3 —!- 2, 2 —1 —^ 3, 2 —> 3 —> 1, 3 ^ 1 ^ 2, 3 —> 2 —»• 1. 


[Similar results, but with more complicated rotation tensors, would hold for rota¬ 
tions about the space-fixed axes {«*/:/,/, K}. If the partial rotations were about 
arbitrary (body- or space-fixed) axes, then, due to the infinity of their possible 
directions, we would have an infinity of angle triplets. It is the restriction that 
these rotations are about the body-fixed axes {u k } that brings them down to twelve.] 


Eulerian Angles 

The sequence 3 —> 1 —> 3, shown and described in fig. 1.26 [with the customary 
abbreviations: cos(...) = c(...), sin(...) = ,y(...)] is considered to be the classical 
Eulerian angle description, originated and frequently used in astronomy and physics, 
[although “In his original work in 1760, Euler used a combination of right-handed and 
left-handed rotations; a convention unacceptable today” Likins (1973, p. 97)]. 

(1973, p. 97)]. 

Using the passive interpretation and fig. 1.26, we readily find that the correspond¬ 
ing coordinates of the compounded transformation resulting from the above 
sequence of partial rotations about the nonmutually orthogonal axes OZ, Ox' , Oz" 
[i.e., the (originally assumed coinciding) space-fixed O—XYZ and body-fixed O—xyz] 
are related by 


(X\ fct -scp 0\ fx'\ 
Y = scj) ccj) 0 


\Zj 


0 

1/ 









( CCp 

— scj) 

°\ 

/! 

0 

0 ^ 

(*') 




= 

scf > 

ccj) 

0 

0 

c8 

-s9 

// 

y 





u 

0 

1/ 

\0 

s9 

c9 

Wv 





( C(j) 

—scj) 

°\ 

Z 1 

0 

0 \ 

/ ccj) 

—sip 

°\ 

(*\ 

= 

scj) 

ccj) 

0 

0 

cO 

—s9 

scj) 

ap 

0 

y 


u 

0 

1/ 

\o 

sO 

c9 / 

\o 

0 

1 ) 

\ Z J 


R( 

KA) 

R(t' 

,0)- 

R(A" 

III 

73 

73 

O 

• R«|, = 

= R, 



(1.12.1a) 


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CHAPTER 1: BACKGROUND 


cf cip — sf c9 sip 

—ccp sip — scp cOcip 

scps9 

srp cip + ccp cO sip 

—scp sip + ccp c6 cip 

—ccp s9 

sO sip 

s9 cip 

c9 



R or A = (A k ' k ) (=1, if 0,0,0 =0). 


(1.12.1b) 


Classical Eulerian Sequence: ( IJ.K ): 3(0) —► 1 (9) —> 3(0): (/, /, Ac) 
0 < 0 ( precession, or azimuth, angle) < 27r, 

0 < 5 [nutation (i.e., nodding), or pole, angle] < 7T, 

0 < 0 [proper, or intrinsic, rotation angle; or (eigen-) spin] < 2w. 


3(0); Precession 


1(0); Nutation 


3(0); Spin 







In sum: T = R 4 , • [Re • ( R 4 , -t)] = (10, ■ R e ■ 10,) -t = R t 


Figure 1.26 Partial, or elementary, rotations of classical Eulerian sequence: 0 —* 9 —» 0 
(originally: O —xyz = O— x 0 y 0 z 0 = O— XYZ). 


REMARKS 

(i) Equation (1.12.1b) readily shows that if the direction cosines A k , k are known, 
the three Eulerian angles can be calculated from 

0 = tan~*(— A l , 3 /A 2 ' 3 ), 0 = cos _1 (A 3 / 3 ), 0 = tan _1 (A 3 / 1 /A 3 » 2 ). (1.12.1c) 

(ii) If the origin of the body-fixed axes ♦ is moving relative to the space-fixed 
frame O XYZ, then in the above we simply replace X with X — A* and so on, 
cyclically. Then, x,y,z [or x/+,y/+,z/+ (§1.8)] are the particle coordinates relative 
to ♦-xyz. In this case, eq. (1.12.1b) shows clearly that a free (i.e., unconstrained) rigid 
body has six (global) degrees of freedom-. 

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§1.12 THE RIGID BODY: EULERIAN ANGLES 


cp 2 3 = X*, : inertial coordinates of base point (pole) ♦, 

q 4 5 6 = (j),9,’ip: Eulerian angles of body-fixed ♦— xyz relative to ♦— XYZ; 

and the constant x,y,r is the “name” of a generic body particle [more on this in 
chap. 2]. 

Inverting (1.12.1b) — while noting that, since all three component matrices ^ 
are orthogonal, the inverse of each equals its transpose (or using the passive inter¬ 
pretation equations in §1.11) — we readily obtain 



where 

R t = (R(|> • Re • R,) T = R\|/ T • Re T • R(|> T = R-+ • R-e • R-<j> 



( 1 . 12 . 2 ) 


(1.12.2a) 


By adopting the active interpretation, we can show that (along arbitrary but 
common axes) 


(a) 

r f 

= R(k",fi).R(i\9) 

• R(£° = K,<j>)- rj = (Rq, • R e 

•R*) 

(1.12.3a) 

(b) 

n 

= (R\|/ • Re • R*) T • r j 

■ = R(|) T • R 0 T • R\|/ T • ttf = (R_ 

4> • R- 

e • R-+) • r f 



= [R(K : -</>)■ R(i\ 

— 9)-R(k",-fi)] .r <; 


(1.12.3b) 

while, 

by ; 

adopting the rotation of a triad interpretation, we can 

show that 

(a) 


T = 

(R<|) • Re • R\|/) • t, 


(1.12.4a) 

(b) 


t = 

(R—\|/ • R-e • R-4>) • T; 


(1.12.4b) 

where 

T = 

fiI,J,K) T , t = (ij 

,a-) t . 




Next, we prove the following remarkable theorem. 


THEOREM (on Compounded Rotations about Body-fixed versus Space-fixed Axes) 

R + • R e • R* = 1>) • R(i - ', e) • R(*° = k, f) 

= R{K,(/>)-R(I,0)-R(K,il>). (1.12.5a) 

In words: the resultant rotation tensor of the classical Eulerian sequence about 
the body-fixed axes: (f>(k = k° = K) —> 9{i') —> f(k"), equals the resultant rotation 
of the reverse-order sequence about the corresponding space-fixed axes: 

m - e(i) - <kk). 

(i) To this end, we first prove the following auxiliary theorem. 

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CHAPTER 1: BACKGROUND 


Shift of the Axis Theorem 

Let us consider two concurrent axes of rotation described by the unit vectors n and 
//', and related by a rotation through an angle p about a third (also concurrent) axis 
described by the unit vector m; that is, 

n = R(//i, fi) • n = n • R t (/ii, p). (1.12.5b) 

Then, the corresponding rotation tensors about n and but with a common angle 
X, are related by the tensor-like (or, generally, “similarity”) transformation: 

R(«',X) = R (m,fj,)-R(n,x)-R T (m,n). (1.12.5c) 


PROOF 

Applying the rotation formula (1.10.10a) for n —> //' and x, we obtain, successively, 
R(«',X) = R[R(m,p) -fi,x] 

= (cos x) 1 + (sin x) [R(»i, p)-n)x 1 

+ (1 - cos x) [R(m, p) • «] ® [R(m, p) • n] 

[using the fact that, for any vector, v: (R • v) x 1 = R • (v x 1) • R T 
— see proof below] 

= (cosx)l + (sinx)[R(»»,p)-(« x l)-R T (/«,p)] 

+ (1 - cos x) [R(m, m) • ( n ® ») • R T (/m, p)] 

[recalling that R(m,p) *R t (/m, p) = 1] 

= R(m, fj) • [(cosx)l + (sin x)( n x 1) + (1 - cosx)(«® «)] -R T (»»,p) 

= R(m,fi) • [(cosx)l + (sinx)N + (1 - cosx)(n® »)] -R t (»i, p) 

[recalling again (1.10.10a)] 

= R(»i, n) ■ R(«, x) • R T ( n h m), Q-E.D. (1.12.5d) 


[PROOF that (R • v) x 1 = R • (v x 1) • R t 

According to the passive interpretation , v and its corresponding antisymmetric tensor 
V = v x 1 transform as follows: 

R • v = components of v along the old axes =v', 

R • V ■ R t = components of V along the old axes = V'. 

Therefore, 

(R • v) x 1 = v' x 1 = V ' = R ■ V ■ R t = R • (v x 1) • R T , Q.E.D.] 

This theorem allows one to relate the rotation tensors about the initial (») and final 
(i.e., rotated) («') positions of a body-fixed axis. 

(ii) Now, back to the proof of (1.12.5a). Applying the preceding shift of axis 
theorem (1.12.5b, c), we get 

(a) R(k",-ip) = R (/', 9) • R(*\ V>) • R T (*’\ <t>), 

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(1.12.5e) 


§1.12 THE RIGID BODY: EULERIAN ANGLES 


where 

k" = R(i', 9) -k' . (1.12.5f) 

(b) R(i', 9) = R(tf, 0) • R(/, 0) ■ R J (K, 0), (1-12.5g) 

=> R T (i', 0) = R (K, 0) • R t (/, 9) • R t (A', 0), (1.12.5h) 

where 

i' = R(JST, 0) •/; (1.12.51) 

(c) R(^0)=R(tf,0).R(tf,0).R T (JM), (1.12.5j) 

where 

k' = R(K,fi)-K. (1.12.5k) 


Substituting (1.12.5g, h, j) into the right side of (1.12.5e), while recalling that all these 
R’s are orthogonal tensors, yields 

R(*", 0) = [R(tf, 0) • R(/, 0) • R V, 0)] 

• [R(^i 0) • R(^"> 0) • R t (^> 0)] 

• [R(^i 0) • R t (^> 0) • R t (-^g 0)] 

= R(K, 0) • R(/, 9) • R(/f, 0) • R t (/, 9) • R T (tf, 0). (1.12.51) 

In view of (1.12.5g) and (1.12.51), the left side of (1.12.5a) transforms successively to 

R(*",0).R(i',0).R(tf,0) 

= [R (K, 0) • R(/, 9) ■ R(K, 0) • R t (/, 9) ■ R J (K , 0)] 

.[R(tf,0).R(/,0).R T (jr,0)] -R(Jf,0) 

= R(A',0)-R(/,6»)-R(A',0), Q.E.D. (1.12.5m) 

Generally, consider a body-fixed frame O-xyz originally coinciding with the space- 
fixed frame O XYZ. Then the sequence of rotations about Ox (first, xi) —> 
Oy ( second, \i) Oz (third, X 3 ) bas the same final orientational effect as the 
sequence about OZ (first, X 3 ) ~* OY (second, X 2 ) ~> OX (third, xi). [See also 
Pars, 1965, pp. 103-105.] 

Angular Velocity via Eulerian Angle Rates 

Let us calculate the vector of angular velocity of the body frame O-xyz relative to 
the space frame O XYZ, in terms of the Eulerian angles 0,0,0 and their rates 
= df/dt, ijj s = d9/dt, = dtp/df, both along the body- and the space-fixed 
axes. We present several approaches. 

(i) Geometrical Derivation 
By inspection of fig. 1.26 we easily find that 

o) = co^K + ujf) i + iO,p k . 

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(1.12.6a) 


CHAPTER 1: BACKGROUND 


But, again by inspection, along the space basis, 

K = (0)1 + (0)/ + (l)A - , 
i' = (cos </.)/ + (sin <p)J + ( 0 )if, 
k" = (- sin#),/' + (cos 9)k' 

= (— sin 9) [(— sin </>)/ + (cos cp)J] + (cos 9) K 
= (sin # sin 0)/ + (— sin# cos (/>)/ + (cos 9)K\ (1.12.6b) 


and along the body basis, 

K = y "(sin 9) + A: "(cos 9) = (i sin ip + j cos ip) sin # + A: cos #, 

i' = i cos ip —j sin ip, k” = k. ( 1 . 12 . 6 c) 


Inserting (1.12.6b, c) in (1.12.6a) and rearranging, we obtain the representations 

o — u)xl d - coy J ~b = to x i -\- ujyj co z k, (1.12.7a) 


where, in matrix form 


/ 


(o 

C(p 

s4>s9 \ 

/ w 0 \ 

CJy 

= 

0 

S(j> 

—cej) s9 

u e 

\ w z / 


V 

0 

c9 / 

W/ 


Space axes E s(pace )(</>, 9) [no ^-dependence], 


( Ux ) 


( s9sip 

cip 

°\ 

(ujA 

OJy 

= 

s9 cil> 

—sip 

0 

lx) 8 



^ c9 

0 

V 



Body axes 


E b (ody) (6, VO [no (/.-dependence], 


(1.12.7b) 


(1.12.7c) 


Inverting (1.12.7b, c) (noting that, since the axes of are non-orthogonal, the 

transformation matrices E s , E b are nonorthogonal also; that is, their inverses do not 
equal their transposes), we obtain, respectively, 




( —s</> c9 

ccf> c9 

s9\ 

( U ' Y \ 

Ug 

= (1 / sin 9) 

C(ps9 

s(j>s9 

0 

UJy 

\ J 


V A 

— C(j) 

o ) 

\ w z) 


E ,-‘(0,0), 



( A 

cip 

°\ 

( ^ 

(1 / sin 9) 

s9 cip 

—s9 sip 

0 

Uy 


\ —c9sip 

—c9 cip 

s9 / 



E b -‘(0,VO; 


(1.12.7d) 


(1.12.7e) 


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§1.12 THE RIGID BODY: EULERIAN ANGLES 


from which we can also calculate the u>x,y,z o w x y z (orthogonal!) transformation 
matrices. 

REMARKS 

(a) The transformations (1.12.7b-e) readily reveal a serious drawback of the 
3 —1 —*• 3 Eulerian angle description, for 9 = 0 (or ±tt); that is, when Oz coincides 
with OZ (or —OZ), in which case the nodal line ON disappears, sind = 0, and, so, 
assuming </>, ip f 0, eqs. (1.12.7b, c) yield, respectively, 

— { c< P)we, u> Y = (s<p)vg, lo z = => lo x 2 + oj y ~ = ^6> 2 ; (1-12.7f) 

u x = (~Clp)w e , U)y = (~S1p)u) 8l U z = + U)$ => w x 2 +UJ y 2 = UJg 2 ', (1.12.7g) 

which means that knowing ui x Y, 7 /x,y,z(t) [say, after solving the kinetic Eulerian 
equations (§1.17)], we can determine u e uniquely , but not and 

Actually, all twelve generalized Eulerian angle descriptions mentioned earlier, 
Xi Xi -► X 3 > exhibit such singularities for some value(s) of their second rotation 
angle in which case, the planes of the other two angles become indistinguishable] 
From the numerical viewpoint, this means that in the close neighborhood of these 
values of x 2 > il becomes difficult to integrate for the rates < ixk/dt (k = 1,2,3). This is 
the main reason that, in rotational (or “attitude”) rigid-body dynamics, (singularity 
free) four-parameter formalisms are sought, and the reason that the classical Eulerian 
sequence 3 —> 1 —> 3 has been of much use in astronomy (where x,y, z have origin at 
the center of the Earth, and point to three distant stars) and physics; whereas other 
Eulerian sequences, such as 1 —> 2 —>3 or 3—> 2 —> 1 [associated with the names of 
Cardan (1501-1576) (continental European literature), Tait (1869), Bryan (1911) 
(British literature); and examined below] are more preferable in engineering rigid- 
body dynamics; for example, airplanes, ships, railroads, satellites, and so on. 
[Similarly, the position (<p,9,ip) = (0,0,0) represents a singular “gimbal lock”: the 
motions ui^ and uty are indistinguishable since each is about the vertical axis Z; only 
ufy + a;,/, is known. The lu 8 motion is about the X-axis, and so it is impossible to 
represent rotations about the T-axis; it is “locked out”; that is ( 0 , 0 , 0 ) introduces 
artificially a constraint, uj y = 0, to v = 0 that mechanically is not there (then, u> x = cu 9 , 
W>Y — 0, W Z = ^0 + 0 U> x = UJg, W z = Ufy + Ulf).] 

(b) Equations (1.12.7b, c) also show that the components lo x Y,z/x,y,z are quasi or 
nonholonomic velocities', that is, although they are linear and homogeneous combina¬ 
tions of the Eulerian angle rates ui$ = df/dt, to e = dO/dt, u)$ = dip/dt , they do not 
equal the rates of other angles. Indeed, if, for example, uj x = dd x /dt, where 0 X = 
9 X (4>, 9, ip), then we should have 

d9 x /dt = {d9 x /df)[d(p/dt) + ( 39 x /d9)(d9 / dt ) + (d9 x /dip)(dip/dt) 


= (0 ){d(p/dt) + (ccp)(d9 / dt) + (scps9) (dip/dt) 

[by (1.12.7b)], 

(1.12.7h) 

that is, 



d9 x /d(p = 0, 89 x /d9 = c9, d9 x /dip 

>) 

II 

(1.12.7i) 

But, from (1.12.7i), it follows that, in general. 



d/d9(d9 x /d(p) = 0f d/df(d9 x /d9) 

= — scp. 

( 1 - 12 .7j) 


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CHAPTER 1: BACKGROUND 


Hence, no such Ox exists; and similarly for the other ut’s. (An introduction to quasi 
coordinates is given in §1.14; and a detailed treatment is given in chap. 2.) 


(ii) Passive Interpretation Derivation 

(a) Body-fixed axes representation. Since m is a vector, we can express it as the 
sum of its three Eulerian angular velocities: 

ot = 4- u)fl T (1.12.8a) 


where 


ct>0 = ( d<p/dt)K , oj e = (dO/dt)i', a> y, = (dip/dt)k". (1.12.8b) 

Then, using the passive interpretation, (1.11.4h, 7a IT.), we can express (1.12.8a, b) 
along the (new) body axes basis (i,j,k). Since the Eulerian basis ( K,i',k") is non- 
orthogonal, we carry out this transformation, not for the entire o>. but for each of its 
above components o e ,co^, and then, adding the results, we obtain 



( 


( 0 \ 

m _ D T n T _ 

body components - Iv v)/ ^0 

0 

= R_q, • R e • 

0 


W/ 

(UK) 

W/ 


body components *'7 


body components 


sip 

°\ 

p cip 

0 

0 

1/ 



V- 

0 


w< 

/ cip 

sip 

—SlP 

cip 

V 0 

0 

/°v 


0 


W/ 

(ijk) 


/! ° 0\ 

(°) 


( (sO sip) \ 

0 cO sO 

0 

= 

(sO cip)uj, 

1 

o 



V ( C0 H ) 


R 


4 ‘ 


0 

VO/ 


/ocA 

0 

[oj 


{-sip)uj e 

V (o)w fl / 


(1.12.8c) 


(1.12.8d) 


(1.12.8e) 


Adding (1.12.8c-e), we obtain the body axes components, equations (1.12.7c), as 
expected. 


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§1.12 THE RIGID BODY: EULERIAN ANGLES 


(b) Space-fixed axes representation. Proceeding similarly, we find 


(!) = 



C(j) 

S(j> 

0 


—s<j> 

c4> 


0 

R<t> 





( C(j) 

— S(j> 

°\ 

/l 

0 

° \ 

+ 

S(j> 

C(f) 

° 

0 

c9 

-s8 


\o 

0 

1/ 


s9 

c8 ) 



R(i> 



Re 




/o 

0 

V 


( 

+ I S(j) 

V° 


/ s(p sb 

UJg + —C(j)s6 

V c9 




(1.12.8f) 


which is none other than (1.12.7b). 

Let the reader verify that the space -axes representation (1.12.8f) can also be 
rewritten as 


o> = R<i)- 




(° 

+ R(j> • Re • R\|/ • I 0 


while the body-axes representation (1.12.8c-e) can be rewritten as 



(°) 


{ Uf) \ 

(o = • R _ 0 • R_§ • 

0 

+ R-v|/ • R-e • 

0 




\“) 



(1.12.8g) 


(1.12.8h) 


(iii) Tensor (Matrix) Derivation 

We have already seen [(1.7.27e) and (1.7.30i k)] that the space-axes components of 
the angular velocity tensor (vector) are related to its body-axes components 

fl(<n) by the tensor (vector) transformation 

n' = r • fj • r t o n = r t • ft ' • r 

(m' = R-<u <« = R T -c</), (1.12.9a) 

where R, or A, is the matrix of the direction cosines between these axes; and also that 

= (dR/dt) • R t = R • (dR/dt) T [due to d/dt(R • R T ) = dl/dt = 0] 
ft = R t • (dR/dt) = — (dR/dt) T • R. 

dR'/dt = ft'-R [= (R-ft-R T )-R] = R-ft (1.12.9b) 

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CHAPTER 1: BACKGROUND 


(a) Space-fixed axes representation. As we have seen, in the case of the classical 
Eulerian sequence </> —> 9 —> ip: R = R^ • R 0 • R^, and therefore (1.12.9b) yields, 
successively, 

il' = (dR/dt) • R t = d/dt(R + • R e • R v |,) • (R* • R e • R*) 7 

= [(dR<|,/dt) • Re ■ R^ + R^ • (dRe/dt) • R^ + R^ • R 0 • (dR^/dt)] • (R^ 7 ■ Re T • R<|> 7 ) 
= (dR^/dt) • [R e • (R* • R/) ■ R e T ] • R/ 

+ R^ • [(dRe/dt) • (R* • R/) • R e 7 ] • R/ 

+ Rjj, • {R 0 • [(dR^/dt) • R/[ • R e 7 } • R* 7 

= (dR*/dt) -R/ + R* • [(dRg/dt) • R 0 X ] • R,,, 7 + R (t> -R 0 - [(dR + /dt) .R + 7 ] • (R* • R e ) x 
[recalling the definition of tensor transformation (1.12.9a), and (1.12.9b)], (1.12.9c) 

= + R(j) • n'o • R/ + R<t> • Ro • Q\i • (R<t> • Ro) 7 

[£!',(, e,v|) : “partial” rotation tensors, along the space-fixed axes], (1.12.9d) 

from which, after some long but straightforward algebra, we obtain [recalling 
(1.12.1a If.)] 

Qyy = Qxx = 0 , 

Qvt — —Qi'v = Qxy — —Qyx = — n»z = -[d(f)/dt + (c9){dip / dt )], 

Qyy = -Qyy = Qxz = -Qzx = -Wy = (stf>)(d9 / dt) - (c(f) s9)(di///dt ), 

Qyy = Qyy = 0 , 

Qyy — —Qyy = Qyz - —Qzy — ~a>x = -[(ctf>)(d9/dt) + (scps9)(dift / dt )], 

Qyy =Qzz = 0, (1.12.9e) 

which coincide with (1.12.7b), as expected. 

(b) Body-fixed axes representation. Proceeding analogously, we obtain 

£2 = R 7 • (dR/dt) = (R* • R 0 • R + ) 7 • [d/dt(R <t> • R e • R*)] 

= ■ ■ ■ = R/ • Re 7 • [R <t) 7 • (dR^/dt)] • R e • R^ 

+ R + 7 • [R 0 7 • (dRe/dt)] • R + + R/ • (dR^/dt) 

= R vi/ 7 • Rq 7 • £2^ • Ro • R v + R v 7 • £2o • Ry + £2^ 

[£2^,e,vp : “partial” rotation tensors, along the body-fixed axes]. (1.12.9f) 

We leave it to the reader to verify that the above coincides with (1.12.7c). 
Alternatively, one can use the transformation equations (1.12.9a) to calculate £2 /<o 
from £2 '/a>'. (See also Hamel, 1949, pp. 735-739.) 


Cardanian Angles 

This is the Eulerian rotation sequence 3 —> 2 > 1 (fig. 1.27). The angles = 7(3) —> 
X 2 = (3(2) —► x 3 = a(l) are commonly (but not uniformly) referred to as Cardanian 

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§1.12 THE RIGID BODY: EULERIAN ANGLES 



Figure 1.27 Cardanian angles: xt = 7(3) —>■ xi = /3(2) —> X 3 = a(1 )■ 

(i) Rotation (OZ, xi = 7): O-XYZ (space axes) = 0-x 0 y 0 z 0 (initial body axes) —* O-x'y'z'. 

(ii) Rotation ( Oy \2 = PY O-x'y'z' —> 0-x"y"z". 

(iii) Rotation (Ox", y 3 = Q ) : 0-x"y"z" —> O-xyz (final body axes). 


angles. In vehicle and aeronautical dynamics, where such an attitude representation 
is popular, they are called yaw ( 7 ), pitch (0), and roll (a). 

Following the passive interpretation, we readily obtain 


(X) 



f 



Y 

= R- 


- R y • < Rp • 

Rql - 

y 

UJ 


V z 7 

l 


\ Z J 


( H 

—57 

°\ 

/ 

C/9 

0 

5/3 \ 


0 

0 ^ 

(*\ 


C7 

0 


0 

1 

0 

0 

COL 

—SOL 

y 

Vo 

0 

1 / 

V 

—5/3 

0 

c/3/ 

\o 

SOL 

COL J 

W 


R y • Rp R ( 


( 1 . 12 . 10 a) 


/ c/ 3 c 7 

5a 5/3 C7 — ca 57 

ca 5/3 C7 + 5a 57 \ 


c/357 

5a 5/3 57 + ca C7 

ca 5/3 57 — 5a C7 

F 

\ 

5a c /3 

cac /3 / 

W 


( 1 . 12 . 10 b) 


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CHAPTER 1: BACKGROUND 


and, inversely, since R a ,p, Y are proper orthogonal, 


fx 

y 

\- 



(R a T -Rp T .R y T ). 

(R-a - R-P ' R-y) * 



( 1 . 12 . 10 c) 


Angular Velocity Tensors 


Using the basic relations (1.12.9a, b), we can show, after some long and careful but 
straightforward algebra, that (with u 1 = dy/dt, uip = d(3/dt, uj a = da/dt) 



Space axes 


—s~f 

C 7 c/3\ 

(“A 


CJ 

syc/3 

up 


0 

-s/3 ) 

\u a J 

(1.12. lOd) 

i(pace) {if ft) 

[no a-dependence], 




Body axes 


—sf3 

0 

A 

/ CU 7 

c/3 sa 

COL 

0 


cac(3 

—SOL 

0 / 

\U a 


E b (ody) (A a) [no 7 -dependence]. 


(1.12. lOe) 


Inverting (1.12.lOd, e) (noting that, since the axes of w Q ,/ 3, 7 are non-orthogonal , the 
transformation matrices E s ( 7 , 0), E b (/3, a) are nonorthogonal also; that is, their 
inverses do not equal their transposes), we obtain respectively, 


( u l^ 


/ s/3c"f 

s[3sy 

c(3\ 

/u A \ 

Up 

= (1/cos /3) 

—syc/3 

C 7 c/3 

0 

LUy 

\ w «y 


\ n 

S') 

0 ) 

\u Z ) 


E s -‘(/3,7), 



/ 0 

sa 

ca \ 


(1 /cos/3) 

0 

cacfd 

—sa c(3 

Uy 


yc/3 

s/3sa 

5/3 ca ) 

\u z J 


E b 1 (cr, /?); 


(1.12. lOf) 


(1.12. lOg) 


from which it is clear that the Cardanian sequence 3(7) — > 2 (/ 3 ) —> 1(a) has a 
singularity at (3 = ±(7 t/2 ). There, (1.12.lOd, e) become, respectively (for /3 = 7 r/ 2 ), 

u x = {si)up, wy = {ci)ufj, w z = w 7 -w Q => u x 2 + u Y 2 = Up 2 , (1.12.10h) 

u x = —l o 1 + u a , Lo y = ( ca)ojp, u> z = {—sa)u>p => u> 2 + u> y 2 = teg 2 ; (1.12. lOi) 

that is, a unique determination of u> 1 and uj a from u> z , or ui x , is impossible. 


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§1.13 THE RIGID BODY: TRANSFORMATION MATRICES 


Finally, using (1.12. lOf, g), we can obtain the transformation. 

For a complete listing of the transformations between u> x y z = 3 (body-fixed 

axes) and the Eulerian rates d\ 1 , 2 , 3 /^ = v i ,23 (and corresponding singularities), 
for all body-/space-axis Eulerian rotation sequences, see the next section. 


1.13 THE RIGID BODY: TRANSFORMATION MATRICES 

(DIRECTION COSINES) BETWEEN SPACE-FIXED AND BODY FIXED TRIADS; 

AND ANGULAR VELOCITY COMPONENTS ALONG BODY FIXED AXES, 

FOR ALL SEQUENCES OF EULERIAN ANGLES 

Summary of Theory, Notations 

T = («/,/) T = {uy, u 2 ', M 3 ') T = (/,/, K) T : Space-fixed {fixed) triad. 

t = (i/a-) T = {ii 1 ,u 2 ,u 2 ) T = {i,j,k) T : Body-fixed {moving) triad. 

All triads are assumed ortho-normal-dextral (OND), and such that, initially, T = t. 
Eulerian angles (see §1.12): XuXi>X 3 (the earlier </>, 0, or a,/3, 7 ). 


1. Basic Triad Transformation Formula 

T = Rt t= R t T, 

where 

R = {R/ck) = («*' • Uk ) [or {Ak'k)\ '■ Tensor / Matrix of rotation 

= R(«»xi) -R(«/tX 2 ) -R(«fc,X3) = ['(xi) J(x2),Mx3)] 

[Rotation sequence xi —► X 2 X '3 about the body -fixed axes m, —► i/ ; —> « A ] 

= R(«i',X3) •»(«/■',X 2 ) •«(«/',Xi) = [*'(X3) J J , (X2), i'(xi)] 

[Rotation sequence X 3 X 2 — 1 * Xi about the .v/?ace-fixed axes uy —♦ up —> «,/] 
[ 1 , ./,*;= 1,2,3; i',j',k' = l',2',3']; 

and, by the basic theorem on compounded rotations (§1.12), the inverse rotation 

R 1 = R t = R(«*, -X 3 ) • R {uj, -X 2 ) • R(«/, -xi) 

= R(«,-, -xt) • R(«/, -X 2 ) • R(«/c', -X 3 ) 

returns the body-triad t to its original position, i.e. realigns it with the space-triad T. 

How to obtain space-axis rotations; i.e., [k'(xi), j'{x 2 ); ^(xs)]. from a knowl¬ 
edge of body-axis rotations with the same rotation sequence : Xi ~* X 2 ~* X 3 ! he., from 
[/(Xi), j(x 2)1 k{xf)\, and vice versa. An example should suffice; by the above theo¬ 
rem, we will have 

[2(»),3(e),i(»)] = [i'(xj),3'(e),2'(x,)] 

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CHAPTER 1: BACKGROUND 


and, therefore, swapping in the latter X 3 with xi (and vice versa), we obtain 
[1'(Xi), 3 '(x 2 )j2'(x 3 )], which appears in the listing below. Similarly, we have 

[2'(Xi),3'(x2),1'(X3)] = [1(X3),3(X2),2(xi)] 

and swapping in there X 3 with Xi (and vice versa) we obtain [ 1 (xi)i 3 (x 2 )> 2 (x 3 )]- 
Abbreviations: s t (.. .) = sin(x,), c ; (. ..) = cos(x,)- 

2. Angular Velocity Components 
Body-fixed (moving) axes components: 

Cl = R t • (dR/dt) = —(dR/dt) T • R, [due to d/dt(R-R T ) = dl/dt = 0] 

Space-fixed (fixed) axes components: 

O' = (dR/dt) ■ R t = -R • (dR/dt) T ; 
with mutual transformations: 

O' = R • O • R t O = R t • O' • R, u' = R • u uj = R t • J 

where uj = axial vector of O, uj' = axial vector of O' 

[i.e. O • ( vector ) = uj x (vector), etc.]; 

Rotation tensor derivative: 

dR/dt = 0'-R [= (R-0-R T )-R] =R O. 


Listing of Transformation Matrices; and Angular 
Velocity Components 

(Body-fixed vs. Eulerian rates; and corresponding singularities. Notation: 

dX\,2,l/dt = v l,2,3) 


1(a) [1(Xi),2(x2),3(x 3 )] = [3'(X3),2'(x2), l'(Xi)] [Singularity at x 2 = ±(V 2 )]: 

/ C 2 C 3 ~C 2 S 3 Si \ 

S1S2C3 + J3C1 -s 1 ^ 2 5 3 + c 1 c 3 -sic 2 ; 

\-C 1 S 2 C 3 + S Y S 3 C!J 2 5 3 + J!C 3 C,C 2 ) 


W 1 = (c 2 c 3 )vj + (s 3 )v 2 + (0)v 3 
uji = (—c 2 5 3 )vi + (c 3 )v 2 + (0)r 3 
w 3 = ( J 2 ) v l + ( 0) v 2 + (l ) v 3 


V 1 — ( c 2) *[( c 3) w l + 1 £3)^2 + (0 )o2 3 ] 

v 2 = ( c 2) 1 [( c 2‘ s 3) w l + (c 2 c 3 )w 2 + (0)w 3 ] 


v 3 — ( c 2) '[( —J 2 C 3) W 1 + ( s 2 s ’i) UJ 2 + (^ 2 )^ 3 ] ■ 


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§1.13 THE RIGID BODY: TRANSFORMATION MATRICES 


l( b ) [ 1 '(Xi)> 2 '(x 2 ), 3 '(x 3 )] = [ 3 (X 3 ), 2 (x 2 ), 1 (Xi)] [Singularity at x 2 = ±(V 2 )]: 

/ C 2 C 3 SjSjC;, - C X S 3 C x S 2 C 3 + S { S 3 \ 

CiS 3 SiS 2 s 3 + CjC 3 C!S 2 s 3 -SiC 3 ; 

\ $2 s x c 2 C\C 2 ) 


W 1 = (l) v l + (0) v 2 + ( —J 2) v 3 
U 2 = (0)vi + (ci)v 2 + (YiC 2 )v 3 
w 3 = (0) v l + (~‘ v l) v 2 + ( c lC 2 )v 3 


V 1 = { c l) *[( c 2) a; l + (■ sr l' s 2) c * J 2 + ( C 1 J 2) W 3] 
v 2 = ( c 2) * [(0) w l + ( c l c 2) w 2 + (—3 , lC 2 )W3] 
v 3 = ( c 2) * [(0) w l + ( 5 l) w 2 + ( c l) w 3]- 


2(a) [2(xi)i3 (x 2 )j 1(X3)] = [l'(X3),3'(X2)>2'(xt)] [Singularity at X 2 = ±(V 2 )]: 


/ 

c,c 2 

~ c iS 2 c 3 + sty 3 

c l s 2 s 3 + 5 1 c 3 \ 


S 2 

C 2 C 3 

-c 2 s 3 

V 

-S\C 2 

SlS 2 C 3 + Ci S 3 

—S[S 2 S 3 + CiC 3 J 


W 1 — (■ s 2 ) v l + (0)v 2 + (l)v 3 V| — (c 2 ) 1 [(0)^! + (c 3 )w 2 + ( — S 3 )o; 3 ] 

w 2 = ( C 2 C 3) V 1 + ( J 3) v 2 + (0) v 3 v 2 = ( c 2) ' [(O)t^i + {c 2 S 3 )u 2 + (c 2 C 3 )w 3 ] 


W 3 — ( —C 2 5 3 )vi + (c 3 )v 2 + ( 0 )v 3 


v 3 — ( c 2) *[( c 2) w l + ( — S 2 C 3 V 2 + ( J 2- S 3) w 3]- 


2(b) [2 '(xi),3'(x 2 ), l'(X 3 )] = [1(X3),3(x2),2(xi)] [Singularity at X 2 = ±(tt/ 2 )]: 


( c x c 2 

-S2 

S\C 2 \ 

C 1 S 2 C 3 +S,s 3 

C 2 C 3 

S 1 S 2 C 3 - C[S 3 

\C { S 2 S 3 - S!C 3 

C 2 S 3 

s l s 2 s 3 + c l c 3 / 


U\ — ( 0 )V[ + (—Sl)v 2 + (CiC 2 )v 3 
w 2 = (l) v l + (0) v 2 + ( —5 2) v 3 


w 3 — (O)l’l + ( c l) v 2 + (• s 'l c 2 ) v 3 


V 1 = (c 2 ) 1 [ (^ 1 ^*2) F*- 7 1 + ( C 2 )u 2 + (SiS 2 )w 3 ] 
v 2 — (c 2 ) * [(—Sl c 2) w l + (0) w 2 + ( c lC 2 )u; 3 ] 
v 3 = ( c 2) * [( c l) w l + (0)w 2 + (^ 1 )^ 3 ]. 


3(a) [3(xi), 1(X2),2(x 3 )] = [2'(x 3 )> l'(x 2 ),3'(xi)] [Singularity at x 2 = ±(tt/ 2 )]: 


( 

—S\S 2 S 3 + C[C 3 

-■SlC 2 

s x s 2 c 3 + qs 3 \ 


C1S2S3 + .V[C 3 

c x c 2 

C[S 2 C 3 -T S\S 3 

V 

c 2 s 3 

s 2 

c 2 c 3 ) 


Wi — ( — C 2 S 3 )vj + (c 3 )v 2 + (0)v 3 V[ — (c 2 ) 1 [(—J 3 )c* 2 1 + (0)tu 2 + (c 3 )w 3 [ 

w 2 = (* s ’2) v 1 3“ (0)v 2 + (1)^3 v 2 = (c 2 ) [(c 2 C 3 )o2i + (0)o; 2 + (c 2 S 3 )t<2 3 ] 


w 3 — ( C 2 C 3) V 1 + ( J 3) v 2 + (0) v 3 


v 3 — ( c 2) 1 [(‘ s 2‘ s 3) w l + ( c 2) w 2 + ( — • ? 2 C 3) W 3]- 


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CHAPTER 1: BACKGROUND 


3(b) [3'(xi), l'(X 2 ), 2 '(x 3 )] = [2(x 3 ), 1(x 2 ),3(xi)] [Singularity at x 2 = ±(V2)]: 


/ S\S 2 S 3 + C\C 3 C 3 S 2 S 3 - .S' 1 c 3 C 2 S 3 \ 
S\C 2 c \ c i —s 2 ; 


\SlA 2 C 3 - 

Wi = (0)vj + (ci)v 2 + (SiC 2 )v 3 
w 2 = (0)vi + (—i'i)v 2 + (C]C 2 )v 3 

oi 3 = (l)v| + (0)v 2 + (—s 2 )v 3 


C 1 J 3 C 3 + s l s 3 C 2 C 3 ) 

v i = ( c 2 ) 1 [(^i^ 2 )p^i + (cis 2 )u; 2 + (c 2 )oi 3 ] 
v 2 = (c 2 ) 1 [(ci c 2 )u; 1 + (— SiC 2 )u 2 + (0)oi 3 ] 
v 3 = (c 2 ) 1 [(-^ 1 )tDi + (ci)oi 2 + (0)oi 3 ]. 


4(a) [1(xt),3(x 2 ),2(x 3 )[ = [2'(x 3 ),3'(x 2 ), 1 '(xi)] [Singularity at x 2 = ±(tt/2)]: 


/ c 2 c 3 

~s 2 

C 2 S 3 \ 

ClS 2 C3 + SiS 3 

c,c 2 

ClS 2 S 3 S[C 3 

\SlS 2 C 3 - C\S 3 

S\C 2 

+ c l c 3 J 


wi = (c 2 c 3 )vi + (—s 3 )v 2 + ( 0 )v 3 
UJ 2 = (—S 2 )V] + ( 0 )v 2 + (l)v 3 
w 3 = ( C 2 J 3) V 1 + ( c 3) v 2 + (b) v 3 


V 1 = ( c 2 ) *[( c 3 ) a; l + ( 0 )oi 2 + (s 3 )oi 3 ] 
v 2 ~ ( c 2 ) *[( — C 2 J 3) W 1 + (0)oi 2 + ( C 2 C 3) W 3] 
v 3 = (c 2 ) * [(-y 2 C 3 )cu»i + (c 2 )oi 2 + (j 2 S 3 )oi 3 ]. 


4(b) [l'(xi),3'(x 2 ),2'(x 3 )[ = [2(X3),3(x 2 ), 1(xi)] [Singularity at x 2 = ±(V 2 )]: 


/ 

c 2 c 3 

—C 1 J 2 C 3 + ^1^3 

s l s 2 c 3 + C 1 J 3 \ 


s2 

ClC 2 

-SlC 2 

V 

-c 2 s 3 

C]S 2 S 3 + J 1 C 3 

~ s l s 2 s 3 + c l c 3 J 


Wi — (1)V] + (0)v 2 + (s 2 )v 3 


w 2 — (0) v l + ( 5 l) v 2 + ( c l c 2 ) v 3 

01 3 = (o)vj + (ci)v 2 + (—SiC 2 )v 3 


V 1 = ( c 2 ) *[( c 2 ) w l + ( —c l 5 2 ) w 2 + (SiS 2 )oi 3 ] 
v 2 = (c 2 ) *[(0)011 + (SlC 2 )oi 2 + (CiC 2 )oi 3 ] 
v 3 = (c 2 ) *[(0)oii + (Ci)oi 2 + (—Sl)oi 3 ]. 


5(a) [2(xi)> 1 (x 2 ),3(x 3 )] = P'fe), 1 '(x 2 ),2'(xt)[ [Singularity at x 2 = ±(tt/ 2 )]: 


/ S Y S 2 S 3 + CjC 3 S] S 2 C 3 - C\S 3 s,c 2 \ 

c 2 s 3 c 2 c 3 —s 2 ; 


\ CiS 2 S 3 — Sic 3 

oil = (c 2 s 3 )vi + (c 3 )v 2 + (0)v 3 
w 2 = (c 2 c 3) v l + ( — ■ s 3) v 2 + (0)V3 
oi 3 = (—S 2 )vj + (0)v 2 + (l)v 3 


ClA 2 C 3 + SiS 3 C\C 2 J 
V 1 = ( c 2 ) * [(^3)^1 + ( c 3) w 2 + (0)013] 
v 2 = ( c 2 ) *[( C 2 C 3) W 1 + ( — c 2 s l)u 2 + (0)oi 3 ] 
v 3 = (c 2 ) *[(s 2 S 3 )oii + (s 2 C 3 )oi 2 + (c 2 )oi 3 ]. 


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§1.13 THE RIGID BODY: TRANSFORMATION MATRICES 


5(b) [2'(xi), l'(X2),3'(x 3 )] = Pfe), 1(X2),2(xi)] [Singularity at x 2 = ±(tt/2)]: 


-S\S 2 S3 + C1C3 

~c 2 s 3 

C\S 2 S 3 + Y1C3 

■Sl 3 , 2 C 3 + CjS-3 

C 2 C 3 

-CjS 2 C 3 +S!J 3 

-■SlC 2 

s 2 

ClC 2 


UJi = (0)V] + (cj)v 2 + (—^lC 2 )V3 
w 2 = (l) v l + ( 0 )v 2 + (y 2 ) V 3 
W3 = ( 0 )vi + ( 3 'i)v 2 + (CiC 2 )v 3 


V 1 = ( c 2 ) 1 [(•Sl-S , 2 )wi + (c 2 )w 2 + (—CiJ 2 )w 3 ] 
v 2 = (c 2 ) 1 [(ciC 2 )Wi + (0)w 2 + (^lC 2 )cD 3 ] 
v 3 = (c 2 ) 1 [( — ■^l) w l + (0)w 2 + (ci)w 3 ]. 


6(a) [3 (xi),2(x 2 ), Ife)] = [l'(X3),2'(x 2 ), 3'(xi)] [Singularity at x 2 = ±(tt/2)]: 

/C]C 2 C'l .S’ 2 .S' 3 - .S'l C‘3 C 1 Y 2 C 3 +Y 1 Y 3 \ 


^iC 2 ^ 1 3- 2 y 3 + c 1 c 3 JiJ 2 c 3 - ; 


V ^2 

iO\ = (—S 2 )vi + (0)v 2 + (l)v 3 

w 2 = (c 2 ^3) v l + ( c 3)t 2 + ( 0 ) V 3 
w 3 = ( C 2 C 3) V 1 + ( — ‘ S 3 )t 2 + (0) V 3 


C 2 S 1 C 2 C 3 / 

v i = ( c 2 ) * [(0)wi + (53)^2 + (c 3 )w 3 ] 
v 2 = ( c 2 ) *[(0)^1 + (c 2 c 3 )w 2 + (— c 2 ^ 3 )cd 3 ] 

v 3 = ( c 2 ) 1 [( c 2 ) w l + ( s 2 s i)^2 + (j 2 C3)W3]. 


6(b) [3'(xi)>2'(x 2 ), l'to)] = [1(X3),2 (x 2 ),3(xi)] [Singularity at x 2 = ±(tt/2)]: 


/ CjC 2 -Y, C 2 Y 2 \ 

CiS 2 J 3 +SiC 3 -5 1 J 2 3'3 + C 1 C3 -C 2 ^ 3 ; 


\-Cj52C3 + J 1 S 3 
w 1 = (0)v, + (i])v 2 + (CiC 2 )v 3 
w 2 = (0)vi + (ci)v 2 + (—YlC 2 )V 3 
w 3 = (l) v l + (0)v 2 + (s 2 )v3 


S 1 J 2 C 3 + C 1 J 3 C 2 C 3 / 

V 1 = ( c 2 ) * [(— C 1 ‘ y 2 ) < ^’l + (SiJ 2 )w 2 + { c 2 ) ut \ 
v 2 = ( c 2 ) * [(■ s l c 2 ) w l + ( c l c 2 ) w 2 + (0)^] 
v 3 = ( c 2 ) * [( c l) w l + (~Yi)w 2 + (O)^]. 


7(a) [l(xi),2(x 2 ). 1 (x 3 )] = [l'(X3)>2'(x 2 ), l'(Xi)] [Singularities at x 2 = 0 ,±tt]: 
/ c 2 s 2 s 3 s 2 c 3 \ 

sis 2 ~s l c 2 S 2 + qc 3 -Jic 2 c 3 - ; 

\-ClS 2 C 1 C 2 ^3+^ 1 C 3 C l C 2 C 3 ~S l S 3 / 


w i = (c 2 )vi + ( 0 )v 2 + ( 1 ) V 3 
UJ 2 = (*^ 2 *^ 3 ) Tj + (c 3 )v 2 + ( 0 )v 3 

w 3 = (• S 2 C 3) V 1 + ( —J 3) v 2 + (0)V3 


v i = fe) * [(0)^! + (y 3 )w 2 + (c 3 )w 3 ] 
v 2 = (^ 2 ) *[( 0 )wi + (s 2 c 3 )u} 2 + (— 52 ^ 3 ) 023 ] 

v 3 = (^ 2 ) *[(^ 2)^1 + {~ C 2 S 3 )^2 + ( —C 2 C 3) W 3]- 


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CHAPTER 1: BACKGROUND 


7(b) [l'(xi),2'(x 2 )> I'fe)] = [1(X3) ; 2 (x 2 ), Uxi)] [Singularities at x 2 = 0, ±tt]: 

/ c 2 c x s 2 \ 

S 2 Sj —^iC 2 ^3 + CiC 3 — CiC 2 $3 — J 1 C 3 ; 

\-S 2 Ci S\C 2 C 3 + C|.S '3 C\ C 2 C-j — .?| S 3 / 

V 1 = ( 5 2 ) 1 [(‘ y 2 ) aJ l + {~ s \ c 2)^2 + ( c l c 2 ) w 3] 
v 2 = ( 5 2 ) ' [(0) w l + ( c l^ 2 ) w 2 + (—^1^ 2 )w 3 ] 
v 3 = ( J 2 ) ' [(0) w l + (^1)^2 + ( c l) w 3]- 


w i = (l) v i + (0)v 2 + (c 2 )v 3 
L0 2 = (0) V} + (Cj)v 2 + (5i3’ 2 )v 3 

W 3 = (0)vi + (—Aj)v 2 + (Ci3’ 2 )v 3 


8(a) [1 (Xi) i 3(x 2 ), 1(X3)] = [l'(X3),3'(x 2 ), l'(Xi)] [Singularities at x 2 = 0, ±tt]: 

/ c 2 -s 2 c 3 S 2 S 3 \ 

C\S 2 c y c 2 c 3 -S\S 3 -c 1 c 2 5 3 -^ 1 c 3 ; 

\.?|.?2 ^ 1 C 2 C 3 + C 1 J 3 -S', C' 2 ,S ’3 - c, c 3 ) 

V 1 = ( J 2 ) 1 [(0) w l + (~ c 3) w 2 + ( 5 3 ) w 3] 
v 2 = (a 2 ) '[(0)^1 + ( S 2 S 3 )u> 2 + (a 2 C3)o23] 


W 1 = ( c 2 ) v l + (0)v 2 + (l) v 3 
oj 2 = (—■s 2 C 3 )vi + (*y 3 ) v 2 + (0)v 3 
w 3 = (‘ s 2‘ s 3) v l + ( c 3) v 2 + (0) v 3 


v 3 — ( s 2 ) [(• y 2 ) w l + ( C 2 C 3) W 2 + {~ c 2 s 2 ) ljJ 'i}- 


8(b) [l'(xi),3'(x 2 ), l'(X 3 )] = [1(X3),3(x 2 ), Uxi)] [Singularities at x 2 = 0, ±tt]: 

/ C 2 C\S 2 s x s 2 \ 

*y 2 ^3 CiC 2 C 3 -SiS 3 -5 1 c 2 c 3 -c 1 ^3 ; 

\S 2 S 3 C 1 C 2 ^3 + 5 1 C 3 -S x C 2 S 3 + C\C 3 ) 


w i = (l) v i + (0)v 2 + (c 2 )v 3 
w 2 = ( 0 )V[ + (*Sl) v 2 + ( —Ci 3 ’ 2 )V 3 
W 3 = (0) V[ + (Cj)v 2 + (5 1 3’ 2 )v 3 


V 1 = ( J 2 ) '[feVl + ( c l c 2 ) w 2 + ( — ■ s l c 2 ) w 3] 

v 2 = ( 5 2 ) *[(0) w l + (Sl^)^ + (CiS 2 )w3] 
v 3 = ( 5 2 ) '[(0) w l + ( — c l) w 2 + (■S’l)^]- 


9(a) [2(xi), 1 (x 2 ))2(x 3 )] = P'Cto), l'(x 2 ),2'(xi)] [Singularities at x 2 = 0, ±tt]: 

/—S[C 2 S 3 + C{C 3 S’iS't SlC 2 C3 + Ci^ \ 

■^3 c 2 -s 2 c 3 ; 

\ -t'l C 2 S 3 - 3, C 3 C X S 2 CjC 2 C 3 -S\S 3 J 


W 1 — (■y 2 5 3) v l + ( c 3) v 2 + (0) v 3 


V 1 — ( s 2 ) * [feVl + (0) w 2 + (“ c 3) w 3] 


W 2 — (c 2 )vi + (0)v 2 + ( 1 )V 3 V 2 — (s 2 ) [(^ 2 C 3 )cDj + (0)cu 2 + (^ 253 ) 023 ] 

w 3 = ( — a 2 C 3 )v] + (5 3 )v 2 + (0)v 3 V3 = (^ 2 ) 1 [(—c 2 s 3 )cj x + (s 2 )uj 2 + (c z s 3 )uj 3 ]. 


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§1.13 THE RIGID BODY: TRANSFORMATION MATRICES 


9(b) P'(xi), l'(X2),2'(x 3 )] = [2(x 3 ), 1(x 2 ),2(xi)] [Singularities at x 2 = 0 ,±tt]: 


(-S\C 2 Sl + C 1 C 3 s 2 s 3 c l c 2 s 3 +s l c 3 \ 
S\S 2 c 2 —C\S 2 


\-SlC 2 C 3 -Ci5 3 
U>\ = (0)vj + (ci)v 2 + (•S’l‘S'2) v 3 
= (l) v l + (0) v 2 + ( c 2 ) v 3 
w 3 = (0)Vi + (Si)v2 + ( — CiJ 2 )v 3 


^ cic 2 c 3 - s , 3 y 

V 1 = fe) ' [( — ‘ v l c 2) w l + ( 5 2 )w 2 + ( c lC 2 )w 3 ] 
V 2 = (s 2 ) * [(CV 2 V 1 + (0)w 2 + (sis 2 )w 3 ] 

v 3 = fe) * [(^l) w l + (0)^2 + (“ c l) w 3 ]- 


10(a) [2(xi), 3 (x 2 ), 2(x 3 )] = [2'(x 3 ), 3'(x 2 ), 2'(xi)] [Singularities at x 2 = 0, ±tt]: 


/ C X C 2 C 3 ~S X S 3 

-Ci5 2 

c l c 2 5 3 + 5iC 3 \ 

5 2 C 3 

c 2 

5 2 5 3 

\ C 2 C 3 - 

S\S 2 

—5lC 2 Y 3 + CiC 3 / 


= (s 2 C 3 )V! + ( — 5 3 )v 2 + (0)v 3 
w 2 = ( c 2 ) v l + (0)v 2 + (1)V 3 
w 3 = (■ s 2‘ s 3) v l + ( c 3 ) v 2 + (0)v 3 


V 1 = (* 2 ) ' [( c 3 Vl + (0)w 2 + (y 3 )w 3 ] 

v 2 ~ ($ 2 ) 1 [(—■ s '2- s ' 3) Cl; 1 + (0) w 2 + (Y2 c 3 ) w 3] 
v 3 = (^ 2 ) *[( —C 2 C 3 ) W 1 + ( S 2)U 2 + ( —C 2 S 3 )w 3 ]. 


10(b) [2 , (xi),3 , (x 2 ),2'(x 3 )] = [2(x 3 )j 3(x 2 ), 2(xi)[ [Singularities at x 2 = 0, ±tt]: 


/ C X C 2 C 3 ~S X S 3 
Cl ^2 

\"ClC 2 5 3 -SiC 3 
iO\ = (0)V] + (—5j)v 2 + (C[Y 2 ) V 3 
w 2 = (l) v l + (0)v 2 + (c 2 )v 3 

t0 3 = (0)V] + (ci)v 2 + (5i5 2 )v 3 


—5 2 c 3 YiC 2 C 3 + C]Y 3 \ 

c 2 s x s 2 ; 

s 2 s 2 -s { C 2 S 3 + c, c 3 J 
v l = fe) *[( —c l c 2 ) w l + (5 2 )w 2 + (—5!C 2 )w 3 ] 
V 2 = (j 2 ) [(—iV[ 5 2 )(U[ + (0)cu 2 + (C[Y 2 )w 3 ] 

v 3 = { s l) 1 [( c l) w l + (0)(U 2 + (5l)w 3 ]. 


11(a) [3(xi)i l(x 2 )i 3(x 3 )] = [3'(x 3 ) ; l / (x 2 ).3'(xi)] [Singularities at x 2 = 0, ±tt]: 


( ~S\C 2 s 3 + c x c 3 -5!C 2 C 3 - c x s 3 s x s 2 \ 

CiC 2 Y 3 +YiC 3 C\C 2 C 3 5^3 -CyS 2 \ 


\ s 2 s 3 
= (5 2 5 3 )V[ + (c 3 )v 2 + (0)v 3 
w 2 = (52 c 3 ) v 1 + ( — •5 3 ) v 2 + (0)v 3 
w 3 = (c 2 )vi + (0)v 2 + (l)v 3 


S 2 c 3 c 2 ) 

v l = (5 2 ) 1 [(5 3 )W] + (c 3 )w 2 + (0)w 3 ] 
v 2 = (® 2 ) 1 [(5 2 C 3 )wi + (—S 2 S 3 )l0 2 + (0)w 3 ] 
v 3 = (5 2 ) *[( —c 2 53)wi + (—c 2 e 3 )w 2 + (s 2 )u; 3 ]. 


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CHAPTER 1: BACKGROUND 


11(b) [3'Od), l'(x 2 ),3'(x 3 )] = [3(x 3 ), 1 (X2),3 (xt)] [Singularities at x 2 = 0,±tt]: 

/-S l c 2 s 3 + CtC 3 -C l C 2 S 3 -S l C 3 S 2 S 3 \ 

S\C 2 c 3 + c | s 3 c x c 2 c 3 -s x s 3 -S 2 c 3 ; 

V -S’l - 5-2 CiS 2 c 2 ) 


U}\ = ( 0 )l’[ + (c i)v 2 + (^ 1 ^ 2 ) v 3 
w 2 = (0)vi + (—■S , l)v2 + (cn?2) v 3 
Ul 3 = (l)vi + ( 0 )V 2 + (^ 2 ) V 3 


v i = ( s i ) 1 [( ^2)^1 + ( Cj c 2 ) uj 2 + (^2)^3] 

v 2 = (j 2 ) '[(ci^)^! + (— SiS 2 )u2 + ( 0 )cD 3 ] 

v 3 = ( J 2 ) * [( 5 l) w l + ( c l) w 2 + (0)w 3 ]. 


12(a) [3(xi)j2(x 2 )j 3(x 3 )] = [3 '(x 3 ).2'(X2 ) j 3'(xi)] [Singularities at x 2 = 0,±tt]: 


(C\C 2 C 3 -SyS 3 C\C 2 S 3 .4 C 3 Cji 2 \ 

S\C 2 C 3 + Ci5 3 -^ 1 C 2 ^3 + C 1 C 3 S\S 2 ; 


V -*2C 3 

W 1 = (—^2 c 3 ) v 1 + (• s, 3 ) v 2 + (0) v 3 
w 2 = (^ 2‘ S 3 ) v 1 + (c 3 ) v 2 + (0) v 3 

w 3 = (c 2 )vi + ( 0 )v 2 + (l)v 3 


•^3 C 2 J 

v i = fe) *[(— c 3 ) w i + (^3)^2 + ( 0 )w 3 [ 
v 2 = fe) 1 [(• y 2 J 3 ) w l + ( s 2 c l)u 2 + ( 0 )w 3 ] 

v 3 = fe) * [{c 2 C 3 )ui\ + ( — C 2 ^ 3 )W2 + (^2)^3]- 


12(b) [3 '(xi),2'(x 2 ),3 '(x3 )] = [3(x 3 ), 2 (x 2 )> 3(x0] [Singularities at x 2 = 0,±tt]: 

/ C\c 2 c 3 - 5^3 -^ic 2 c 3 - c,5 3 ^ 2 c 3 \ 

C 1 C 2 J 3 + s l c 3 —^iC 2^'3 + C 1 C 3 S 2 S 3 i 

V -CiS 2 SiS 2 C 2 J 


w 1 = ( 0 )Vi + (^l)V 2 + ( —Ci5 2 )V3 
w 2 = (0)V[ + (Cj)V2 + (•S'l^'b 
0J 3 = ( 1 ) Vj + ( 0 ) V 3 + (c 2 ) v 3 


V 1 = (^ 2 ) *[(^<^ 2)^1 + (~ s l c l) u 2 + fe)^] 
v 2 = (^ 2 ) '[(^l^Vl + ( c 1 5 2) w 2 + ( 0 )w 3 ] 
v 3 = (^ 2 ) * [(~ c l) w l + (■S , l)^2 + (0) w 3 ]- 


1.14 THE RIGID BODY: AN INTRODUCTION TO QUASI COORDINATES 

As an introduction to quasi coordinates, and quasi variables in general (a topic to be 
detailed in chap. 2 ), we show in this section that the angular velocity, although a 
vector, does not result by simple d/dt(.. ^-differentiation of an angular displace¬ 
ment; its components along space-/body-fixed axes, say 04 , do not equal the total 
time derivatives of angles or any other genuine (global) rotational coordinates/para¬ 
meters, say 0kt that is, 04 f dO^/dt. This is another complication of rotational 
mechanics, one that is intimately connected with the noncommutativity of finite 
rotations; and it necessitates the hitherto search for connections of the 04 ’s with 
genuine angular coordinates and their rates, like the Eulerian angles <p, 9 , ip. 

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§1.14 THE RIGID BODY: AN INTRODUCTION TO QUASI COORDINATES 


Let us consider, for concreteness, the body-axes components of the angular velo¬ 
city tensor. From 


ft = A T - (dA/dt) = -(dA T /dt) - A, (1.14.1a) 


(1.7.30fff.) we have 

u x = A Xz (dA Xy /dt) + A Yz (dA Yy /dt) + A Zz (dA Zy /dt), etc., cyclically, (1.14.1b) 
or, multiplying through by dt and setting ui x dt = d6 x (just a suggestive shorthand!), 

dO x = Ax-dAxy + A Yz dA Yy + A Zz dA Zy , etc., cyclically. (1.14.1c) 

We shall show that 


8(d9 x ) ~ d(89 x ) ^ 0, etc., cyclically, (1.14.2a) 

where, for our purposes, 6 (...) can be viewed as just a differential of (...), along a 
different direction from d(. . .); that is, with d(. . .) = d j(. . .) and 6(. ..) = d 2 (. ..), 

8{d9 x ) = d 2 (di 9 X ), d{89 x ) = d x ( d 2 9 x ); (1.14.2b) 

and 

S9 X = A Xz SAx y + A Yz 5A Yy + A Zz SA Zy , etc., cyclically. (1.14.2c) 


Now, S (...)-differentiating d6 x and d (...)-differentiating 89 x , and then subtracting 
side by side, while noticing that 


8{dA k , k ) = d(8A k , k ) (k' = X,Y,Z; k= x,y,z), (1.14.3a) 

we get 

8(d9 x ) — d(69 x ) = 8A Xz dAx y — dA Xz bA Xy + 8A Yz dA Yy — dA Yz 6A Yy 

+ bA Zz dA Zy — dA Zz 8A Zy . (1.14.3b) 

Next, in order to express 8A k * k , dA k ' k in terms of 89 k and d6 k , we multiply the 


components of dA/dt = A • L2 (1.7.30i) with dt, thus obtaining 

dA Xz = A Xx d9 y — A Xy d9 x => 8A Xz = A Xx 89 y — A Xy 60 x , (1.14.4a) 

dA Yz = A Yx d9 y - A Yy d9 x => 8A Yz = A Yx 89 y - A Yy 89 x , (1.14.4b) 

dA Zz = A Zx d9 y — A Zy d9 x => 8A Zz = A Zx 89 y — A Zy 89 x , (1.14.4c) 

dA Xy = A Xz d9 x — A Xx d9 z => 8A Xy = A X: 89 x — A Xx 89 : , (1.14.4d) 

dA Yy = A Yz d9 x — A Yx d9 z => 8A Yy = A Yz 89 x — A Yx 89 z , (1.14.4e) 

dA Zy = A Zz d9 x - A Zx d9 z => 8A Zy = A Zz 89 x - A Zx 89 z . (1.14.4f) 


Substituting (1.14.4a f) into the right side of (1.14.3b), and invoking the orthogon¬ 
ality of A = (A k ' k ) [e.g., (1.7.6a, b; 1.7.22d)], we find, after some straightforward 
algebra, the noncommutativity equation 

6(d9 x ) - d(89 x ) = d9 y 89 z - d9 z 89 y . (1.14.5a) 


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CHAPTER 1: BACKGROUND 

Working in complete analogy with the above, we obtain 

6{dO y ) - d(89 y ) = dO z 8Q x - d9 x 80 z , (1.14.5b) 

8{d9 z ) - d{89 z ) = dO x 89 y - d0 y 80 x . (1.14.5c) 

These remarkable transitivity equations [because they allow for a smooth transition 
from Lagrangean mechanics (chap. 2ff.) to Eulerian mechanics (§1.15ff.)] show 
clearly that the 9 X y z are not ordinary (or genuine, or holonomic, or global; or as 
Lagrange puts it “variables finies”) coordinates, like the Eulerian angles </>, 9, ip; 
that is why they are called pseudo- or quasi coordinates. Their general theory, along 
with a simpler derivation of the above, are detailed in chap. 2. 

Similarly, we can show that in terms of space-axes components, the transitivity 
equations are 

6{d9 x ) - d(69 x ) = d9 z 89y - d9 y 89 z , (1.14.6a) 

8{d9y) - d(89y) = d9 x 89 z - d9 z 89 x , (1.14.6b) 

8{dO z ) — d{89 z ) = d9y89 x — d9 x 89 y . (1.14.6c) 


In compact vector form, (1.14.5a, b, c) and (1.14.6a, b, c) read, respectively, 

8 re ,{d0) - d(88) =d0 x 80, (1.14.7a) 

S(d9) - d(S8) =89 xd8, (1.14.7b) 

where dO = d9 x i + d9 y j + dd z k = d6 x l + d9yj + d6 z K => 89 = 89 x i + 80 y j + 89 z k = 
S9 X I + 89 y J + 80 z K => 8 re t(d9) = 6(dO x )i + S(d9 y )j + 8(d9 z )k,d(69) = d(89 x )i + 
d{89 y )j + d(89 z )k; that is, again, 9 is a quasi vector. Here (recall 1.7.30j), fl' = 
(dA/dt) • A t and dA/dt = fl' • A. [More in Examples 2.13.9 and 2.13.11 (pp. 368 ff.).] 

HISTORICAL 

Equations (1.14.5a-c), along with the systematic use of direction cosines to rigid- 
body dynamics, are due to Lagrange. They appeared posthumously in the 2nd edi¬ 
tion of the 2nd volume of his Mecanique Analytique (1815/1816). See also (alphabe¬ 
tically): Funk (1962, pp. 334-335), Kirchhoff (1876, sixth lecture, §2), Mathieu (1878, 
pp. 138-139). 


1.15 THE RIGID BODY: TENSOR OF INERTIA, KINETIC ENERGY 

Introduction, Basic Definitions 

To get motivated, let us begin by calculating the (inertial) kinetic energy T of a rigid 
body B rotating about a fixed point 0\ the extension to the case of general motion 
follows easily. If a> is the inertial angular velocity of B. then, since the inertial velocity 
of a genetic body particle P, of inertial position r, is <u x r = v, we have, successively, 

2 T = ^ dm v ■ v = ^ dm(oj x r) ■ (at x r) 

= £ dm [(co • co) (r • r) — (to • r) (to • #•)] (by simple vector algebra) 

= S dm[uj 2 r 2 — (to-r) 2 ], (1.15.1a) 


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§1.15 THE RIGID BODY: TENSOR OF INERTIA, KINETIC ENERGY 

or, in terms of components along arbitrary (i.e., not necessarily body-fixed) rectan¬ 
gular Cartesian axes O—xyz = 0—x kl in which r = (x,y,z) = (x k ), to = (lu x , u> y , u> z ) = 
M (* = 1,2,3), 

^=SW(S>4(S> 2 )- 


= S’*[(SE 

hi^k^t) (E^ 2 ) ~ (EE^' W/X ^) 


= E E dki Uk u>i 

(Indicial notation) 

(1.15.1b) 

= OJ • I • (0 

(Direct notation) 

(1.15.1c) 

= ft) T • I • to 

(Matrix notation; a>: column vector), 

(1.15.Id) 


where 

l() = I = (-lojd ) = (hi ), 
hi = S dm ( rl Ski ~ x k x,)\ 

Components of tensor of inertia of B, I, at O, along 0-x k , 

r 2 = x k 2 = E XkXk ’ 

or, equivalently, 

hi = Jo hi — Jkh 

where 

Jo =J = ( Jo,kl ) = (Jkl) , 

hi = S XkX > dm 

= Components of Binet’s tensor of B, J, at O, along O-x k , 

Jo = J\\ + J 22 + J33 = $ r 2 dm = Tr J. 

In direct notation, the above read 

/ = ^ [(/■ • r)l — r (g> r\dm , J = ^ (r <8 r) dm. (1.15.2h) 

That I is a (second-order) tensor follows from the fact that, under rotations of the 
axes, T is a scalar invariant while (o is a vector (what, in effect, constitutes a simple 
application of the tensorial ‘‘quotient rule”). This means that the components of I 
along 0—x k ,I k h and along 0-x k /, 4'/', where x k < = JfA k i k x k (proper orthogonal 
transformation), are related by 

h'l' = E E Ak'kAinh-i = ^2 ,E A k'khiA/i' 

= (A -I•A T ) k ' l > [recalling eqs. (1.1.19j ff.)] 

** hi = EE Akk'An'h’i' (Since, indicially, A k i k = A kk i) J . 

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(1.15.2e) 

(1.15.2f) 

(1.15.2g) 


(1.15.2a) 

(1.15.2b) 

(1.15.2c) 

(1.15.2d) 


(E w ***j (E w ^J 


215 


(1.15.21) 






CHAPTER 1: BACKGROUND 


Properties of the Inertia Tensor 

Clearly (and like most mechanics tensors), I is symmetric: fii = /«•; that is, at most 
six, of its nine components, are independent. In extenso, (1.15.2a, b) read 



(lxx Ixy 4z\ 



I = 

Iyx Iyy Iyz 




\ Izx Izy Izz ) 




(S dm O' 2 + 

') — Q dm xy 

—S dm xz \ 

= 

K 

1 

to 

1 

dm (z 2 + x 2 ) 

~ S dm z 


y — Q dm z x 

~ S dm z y 

s dm ( x2 + y 2 ) / 


(1.15.3) 


The diagonal elements of I, l xx , Iyy, I Z z, are called moments of inertia of B about 
O—xyz', and they are nonnegative', that is, Ixx, yy ,zz > 0 - Th e off-diagonal elements of 
/. I xy = I yx , l xz = l 7X , I yz = I ly , are called products of inertia of B about O-xyz, and 
they are sign-indefinite', that is, they may be > 0 , < 0 , or = 0 . 

In view of the above, T can be rewritten as 

2T = l xx U) x " + Iyy U)y~ ~\~ I ZZ LOf 2 ly-y UJy UJy 2 I yz OJ y CJ Z 2 Iy z (jJy LU z . ( 1 . 15 . 4 ) 


Now, evidently, the choice of the axes O-xyz is nonunique. Not only can they be non¬ 
body-fixed (in which case, the I ki are, in general, time-dependent); but even if they are 
taken as body-fixed, (1.15.3, 4) are still fairly complicated. Hence, to simplify matters 
as much as possible, and since the kinetic energy is so central to analytical 
mechanics, we, in general, strive to choose principal axes at O'. O—xyz —> <9-123; 
usually, but not always, body-fixed. Since I is symmetric, such (mutually orthogonal) 
axes exist always; and along them / becomes 


/ 


(h 0 0 \ 
0 I 2 0 

\0 0 ij 


Principal axes representation of inertia tensor, (1.15.5) 


where the principal moments of inertia, at O, I\ 2 3 are the eigenvalues of 


E ^ kl W l = 

that is, they are the roots of its characteristic equation: 

D( A) = — Det(4; - A Skf) = 0 ; ^1,2,3 = A,2,3- 


(1.15.6a) 


(1.15.6b) 


Using basic theorems of the spectral theory of tensors [(1.1.17a ff.)] we can show the 
following: 


(i) At each point of a rigid body B there exists at least one set of principal axes. 

(ii) Since, by (1.15.lb d), the inertia tensor is not only symmetric, but also positive 
definite [i.e., fffflkia k ai> 0 , for all vectors a = (af) f 0 ], all three roots of 
(1.15.6b) are not only real but also strictly positive. 

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§1.15 THE RIGID BODY: TENSOR OF INERTIA, KINETIC ENERGY 


Further 

• If Aj ^ A 2 7 ^ A 3 Aj (all three eigenvalues distinct), then 0-123 is unique. 

• If Aj ^ A 2 = A 3 (two distinct eigenvalues), there exists a single infinity of such sets of 
principal axes; 01 and every line perpendicular to it are principal axes; that is, the 
direction of either 02 or 03 , in the plane perpendicular to 01, can be chosen arbi¬ 
trarily (e.g., a homogeneous right circular cylinder, with O on its axis of symmetry). 

• If \\ = A 2 = A 3 (only one distinct eigenvalue), there exists a double infinity: any three 
mutually perpendicular axes can be chosen arbitrarily as 0-123 (e.g., O being the 
center of a homogeneous sphere). Along principal axes, T, (1.15.4), with 
co = (uq, u> 2 , w 3 ), reduces to 


2 T — T I2^2 T -^ 3^3 . 


(1.15.6c) 


The Generalized Parallel Axis Theorem (“Huygens-Steiner”) 

This explains how I changes from point to point, among parallel sets of axes. 


THEOREM 

Let O-xyz and G-xyz be two sets of mutually parallel axes, and let the coordinates of 
the center of mass of B , G, relative to O , be 


OG = r c = (x G , y G , zg) = (Gr, G 2i G 3 ) = (- a , -b, -c ), (1.15.7a) 


that is, a : b,c = coordinates of O relative to G-xyz. Then, the components of the 
inertia tensor of B at O , l 0 .ku and at G, are related by 

Direct notation : I 0 = 1 G + m{r c 2 l — r G < 8 > r G ) , (1.15.7b) 


Indicial notation: I 0 kl = I G k , + m [ G,.c)j S kl - G k G, ; 

or, in extenso, 


lo = Ig + 


m{b 2 + c 2 ) 
—mba 
—me a 


—mab 
m (c 2 + a 2 ) 
—meb 


—mac 
—mbc 
m(a 2 + b 2 



(1.15.7c) 


(1.15.7d) 


PROOF 

We have, successively, 

(i) io,xx = S dm i(y~ b f + i z - c ) 2 ] 

= S dm(yl + ^ ~ 2b {S dm y) ~ 2b {S dm z'j + ^ dm(b' + c 2 ) 

= I G , XX + 0 + 0 + m(b 2 + c 2 ); etc., cyclically, for I 0iyy , / 0 , zz - (1.15.7e) 

(ii) Io,yz = ~S dm ^ y - b ^ z ~ 

= — ^dm yz + dtu + b (S dm “) — S dm bc 

= Ic.yz + 0 + 0 — mbc; etc., cyclically, for / 0 jcz , I 0iXy - (1.15.7f) 

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CHAPTER 1: BACKGROUND 


More generally, it can be shown that between any two points A , B (with some ad hoc 
but, hopefully, self-explanatory notation), 

Ib = Ia + m(r A / B ~l — r A / B <g> r A / B ) 

+ 2m[(r A / B -r G/A )l -(1/2 )(r A/B ® r G/A +r G/A ® r A/B )] ; (1.15.7g) 

which, when A —> G and r G / A — > 0, reduces to (1.15.7b) (see below). (See, e.g., Lur’e, 
1968, p. 143; also Crandall et al., 1968, pp. 180-182, Magnus, 1974, pp. 200-201.) 

It should be noted that the transfer formulae (1.15.7b, g) are based on definitions 
of moments of inertia about points , like (1.15.2h), not axes, and therefore hold for 
any set of axes through these points; that is, they are independent of the axes 
orientation at A, B. If, however, these axes are parallel, certain simplifications 
occur; indeed, (1.15.7g) then yields the component form, 

IbM = 1A,kl + m \{ x A/B,\ + X A/B,2~ + X A/B, 2 )^kl — x A/B,k x A/B,l\ 

+ 2rn \{x A / B \ X G / A ^\ + X A / B 2 X G / A 2 + X A / B T, X G / A j)S k/ 

— (1/2 )(x A / Bk x G / A j + x G / Ak x A / B/ )], (1.15.7h) 

where r A / B = (x A / BA , x A / B ^, x A / B ^) = coordinates of A relative to B, along axes 
B—xyz = B—x k , and r G / A = (x G / A I , x G / A 2 , x G / A 3 ) = coordinates of G relative to 
A, along axes A—xyz = A—x k (parallel to B—x k )', or, in extenso, with 

X A/B, 1 = X A/B: X A/B,2 = Va/B, x A/B,3 = Z A/Bi X G/A, 1 = X G/A, e tC., 

7/1. \ \ 7.-1 .XX + m{y A / B 2 + Z A/B 2 ) + 2 m(y A / B )’g/a + z A/B z G/A ), etc., cyclically, 

(1.15.71) 

hi.xy = lA,xy ~ ™(yA/B x G/A + X A/Byc/A) ~ ™{ x A/ B yA/B ), etc., cyclically. (1.15.7j) 
If A —> G, then r G / A —> 0, r A / B —> r G / B , and the above reduces to 

^B,kl = Ic,kl + ,n [{ x G/BA~ + X G/B,2~ + X G/B,3~)^kl ~ x G/B,k x G/B,l\ > (1.15.7k) 

from which, in extenso, 

h,xx = Iq,xx + m(yG/B 2 + z g/b~) = Ig,xx + + (“ C )~L etc., cyclically, 

(1.15.71) 

h,xy = hi.xy - ”K x g/b)’g/b ) = hi,xy ~ "»[(-«)(-*)], etc., cyclically; i.e., (1.15.7b—f). 

(1.15.7m) 


Ellipsoid of Inertia 

Let us consider a rectangular Cartesian coordinate system/basis O—xyz/ijk, and an 
axis u through O defined by the unit vector u = (u x) u y , u z ). Then, as the transfor¬ 
mation equations against rotations (1.15.2i) show, the moment of inertia of a body B 
about u, /„„ = /, will be (with k' = /' = u;k,l = x,y,z; A k i k = u k = u l 2 3, 
A k 'i = ui = ni^, etc.) 

I — I XX U X T iyyUy T I -y II - 2 U y U y 21 y-li yll - ~\~ 2 I y-U yll y . 

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(1.15.8a) 


§1.15 THE RIGID BODY: TENSOR OF INERTIA, KINETIC ENERGY 


[For nontensorial derivations of (1.15.8a), see, for example, Lamb (1929, pp. 66-67), 
Spiegel (1967, pp. 263-264)]. We notice that if a> = u>u = {u>u x , u>u v , touf) = 
(uj x , uj y . lo z ), then the kinetic energy of B , moving about the fixed point (9, eq. 
(1.15.4), becomes 


2 T = Ilo 2 . 

(1.15.8b) 

Now, by defining the radius vector 


r = u/I 1/2 =xi + yj + zk [i.e., M = r = (1//) 1/2 ], 

(1.15.8c) 

we can rewrite (1.15.8a) as 


Ixx-^ lyyIzz'z' “I - 2/^xy + 2.1 y Z yz + 2 I zx zx = 1. 

(1.15.8d) 


Since I is positive [except when all the mass lies on «; then one of the principal 
moments of inertia, roots of (1.15.6b), is zero and the other two are equal and 
positive], every radius through O meets the quadric surface represented by 
(1.15.8d), in O-xyz , in real points located a distance r = (l//) 1 ^ 2 from O , and there¬ 
fore (1.15.8d) is an ellipsoid; appropriately called ellipsoid of inertia or momenta! 
ellipsoid. [A term most likely introduced by Cauchy (1827), who also carried out 
similar investigations in the theory of stress in continuous media (“stress quadric”).] 
If the axes are rotated so as to coincide with the principal axes of the ellipsoid — 
that is, 0—xyz—> O— 123 — then (1.15.8d) simplifies to 

Vf + h r 2~ + h r i 2 — 1) 


or 


h/(l//i ) 1/2 ] 2 + h/( 1 // 2 ) 1/2 ] 2 + ['- 3 /(l// 3) 1/2 ] 2 = 1, (1.15.8e) 

where r 123 are the “principal” coordinates of r, and (l//i, 2 , 3)*^ 2 are the semidia¬ 
meters of the ellipsoid. [Some authors (mostly British) define the radius of the 
momental ellipsoid along u (i.e., our r) as 

r = me 4 // 1/2 ~ 7 1/2 , (1.15.8c.l) 

where m = mass of body, and e = any linear magnitude (taken in the fourth power 
for purely dimensional purposes), so that the ellipsoid equations (1.15.8d) and 
(1.15.8e) are replaced, respectively, by 

I xx x 2 + I vy y 2 + I zz z 2 + 2 I xy xy + 2 I yz yz + 2 1 :x zx = me 4 , (1.15.8d. 1 ) 

I x r 2 + I 2 r 2 2 + I 2 r 2 = me 4 . (1.15.8e.l) 

Also, for a discussion of the closely related concept of the ellipsoid of gyration 
(introduced by MacCullagh, 1844), see, for example, Easthope (1964, p. 134ff.), 
Lamb (1929, p. 68 ff.)] However, it should be remarked that not every ellipsoid can 
represent an inertia ellipsoid; in view of the “triangle inequalities” (see below), 
certain restrictions apply on the relative magnitudes of the semidiameters, and 
hence the possible forms of the momental ellipsoid. 

Now, and these constitute a geometrical sequel to the discussion of the roots of 
the characteristic equation (1.15.6b): 

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CHAPTER 1: BACKGROUND 


• If I\ = I 2 = / 3 , the momental ellipsoid is a sphere. All axes through O are principal, 
and all moments of inertia are mutually equal. Such a body is called kinetically 
symmetrical about O. 

• If, say, / 2 = / 3 , the ellipsoid is one of revolution about Ox —all perpendicular 
diameters to Ox are principal axes. Such a body is called kinetically symmetric 
about that axis', or simply uniaxial (Routh). 

The above show that, in general, the ellipsoid of inertia, at a point, is nonunique. 
The ellipsoid of inertia of a body at its mass center G, commonly referred to as its 
central ellipsoid (Poinsot), is of particular importance: As the parallel axis theorem 
shows, if the moment of inertia about an axis through G is known, I G , then the 
moment of inertia about any other axis parallel to it is obtained by adding to I G the 
nonnegative quantity md 2 , where d is the distance between the two axes. 

Finally, the momental ellipsoid interpretation, plus the above parallel axis theo¬ 
rem, allow us to conclude the following extremum (i.e., maximum / minimum) proper¬ 
ties of the principal axes: 

• The principal axes of inertia, at a point O, are those with the larger or smaller moment 
of inertia than those about any other line through that point, /. Quantitatively, if 

h = 4,ax > h > h = 4,in (1.15.8f) 

[ =>■ (l/7i )'/ 2 < (I// 2) 1 / 2 < (I// 3 ) 1 / 2 ], something that can always be achieved by appro¬ 
priate numbering of the principal axes, then 

I mdx > I > Imin- (1.15.8g) 

• The smallest centra! principal moment of inertia of a body, say I G 3 = I Gm in , is smaller 
than or equal to any other possible moment of inertia of the body (i.e., moment of 
inertia about any other space point and direction there); that is, I G m in > I uu . 


Additional Useful Results 

(i) It can be shown that 

+ h <^ 3 + hi h < h + (1.15.8h) 

that is, no principal moment of inertia can exceed the sum of the other two. Equations 
(1.15.8h) are referred to as the triangle inequalities (since similar relations hold for 
the sides of a plane triangle). Actually, this theorem holds for the moments of inertia 
about any mutually orthogonal axes (McKinley, 1981). 

(ii) Let pi 2 3 be the semidiameters (semiaxes) of the ellipsoid of inertia; that is, 
Pi 2,3 = (/ipp) -1 ^ 2 . Then, the third and second of (1.15.8h) lead, respectively, to the 
following lower and upper hounds for p 3 , if pi 2 are given, 

(p 2 -2 +pr 2 r 1/2 < p-s < iP 2 ~ 2 - pr 2 r 1/2 ; (i.i5.8i) 

and, cyclically, for pi 2 ; that is, arbitrary inertia tensors, upon diagonalization, 
may yield (mathematically correct but) physically impossible principal moments of 
inertia! 

As a result of the above, if two axes, say pi and p 2 , are approximately equal, the 
corresponding inertia ellipsoid can be quite prolate (longer in the third direction, 

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§1.15 THE RIGID BODY: TENSOR OF INERTIA, KINETIC ENERGY 


cigar shaped), but not too oblate (shorter in the third direction; flattened at the poles, 
like the Earth). 

(iii) The quantity Tr/ = I xx + f yy + /., (first invariant of I) depends on the origin 
of the coordinates, but not on their orientation. 

(iv) - I xx /2 < I yz < I xx /2, -I yy /2 < I zx < I yy /2, -IJ 2 < I xy < IJ 2. 

(1.15.8 j) 

(v) Consider the following three sets of axes: (a) O-XYZ: arbitrary “background” 
(say, inertial) axes; (b) G—XYZ = G—xyz: translating but nonrotating axes, at center 
of mass G; and (c) G-123: principal axes at G. By combining the transformation 
formulae for hh between parallel axes of differing origins (like O-XYZ and G-xyz) 
and arbitrary oriented axis of common origin (like G-xyz and G-123), we can show 
that 


Ixx = m (Y G ~ + Z G ~) + A x 2 1 { + A X2 2 l2 + A X3 2 I 3 , (1.15.8k) 

I yy = m(Z G - + X G ~) + A Y \~I\ + A Y2 1 2 + A Y3 ~I 3 , (1.15.81) 

Izz = m {X G ~ + Y g ) + A z 2 I\ + A Z 2 2 I 2 + A Z3 2 I 3 ; (1.15.8m) 

Ixy = — mX G Y G + A xx A Yl {I 3 — If) + A X2 A Y2 (I 3 — I 2 ), (1.15.8n) 

Iyz — ~ m Y g Z g + A Y \A Z \(J 3 — I\) + A Y2 A Z2 (I 3 — I 2 ), (1.15.8o) 

Izx = ~ m Z G X G + A zl A X i(I 3 — /J + A Z2 A X2 (I 3 — / 2 ); (1.15.8p) 


where = cos(Odf, Gl) = cos(Gx, Gl), etc., and X Gl Y G , Z G are the coordinates 
of G relative to O-XYZ. The usefulness of (1.15.8k p) lies in the fact that they yield 
the moments/products of inertia about arbitrary axes, once the principal moments of 
inertia at the center of mass are known. 

(vi) 

(a) If a body has a plane of symmetry, then (a) its center of mass and (/ 3 ) two of its 
principal axes of inertia there lie on that plane; while the third principal axis is 
perpendicular to it. 

(b) If a body has an axis of symmetry, then (a) its center of a mass lies there, and (0) 
that axis is one of its principal axes of inertia; while the other two are perpendicular 
to it. 

(c) If two perpendicular axes, through a body point, are axes of symmetry, then they are 
principal axes there. (But principal axes are not necessarily axes of symmetry!) 

(d) If the products of inertia vanish, for three mutually perpendicular axes at a 
point, these axes are principal axes there. [For a general discussion of the relations 
between principal axes and symmetry (via the concept of covering operation), see, for 
example, Synge and Griffith (1959, p. 288 ff.).] 

(e) A principal axis at the center of mass of a body is a principal axis at all points of that 
axis. 

(f) If an axis is principal at any two of its points, then it passes through the center of 
mass of the body, and is a principal axis at all its points. 

(vii) Centrifugal forces: whence the products of inertia originate. Let us consider 
an arbitrary rigid body rotating about a fixed axis OZ with constant angular velocity 
co. Then, since the centripetal acceleration of a generic particle of it P, of mass dm, 
equals v 2 /r = ui 2 r, where r = distance of P from OZ, the associated centrifugal force 

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CHAPTER 1: BACKGROUND 

df c has magnitude df c = dm(uj 2 r), and hence components along a, say, body-fixed set 
of axes 0—xyz{OZ — Oz) will be 

dfcx = dfc(x/r) = dm xui 1 , clf cy = dfi.{y/r) = dm yu? , df c ^ = 0; (1.15.9a) 

where x, y, z are the coordinates of P. Therefore, the components of the moment of 
df c along these axes are 

dM cx = y df cz — z df C y = —dm yzco 2 , 
dM cy = z df cx — x df cz = +dm xzur, 
dM cz = x df C j - y df^ x = 0. (1.15.9b) 

From the above, it follows that these centrifugal forces, when summed over the 
entire body and reduced to the origin O, yield a resultant centrifugal force f c \ 

f c ,x = S =U}2 S dm x = ^ mx G . 

fc,y = S d fc,y =Ujl S dm y = w 2 " 7 -V Gj 

fc, z =S d fc, = 0, (1.15.9c) 

where x G , y G are the coordinates of the mass center of B. G'; and a resultant cen¬ 
trifugal moment My. 

M c x = ^ dM c x = — u) 2 ^ dm yz= +url rz , 

M c y = ^ dM c v = ur ^ dm zx = — ui 2 I xz , 

M ( . z = gdM ( . z = 0. (1.15.9d) 

Equations (1.15.9c, d) show clearly that if G lies on the Z = z axis, then/ f vanishes, 
but M c does not. For the moment to vanish, we must have I yz = 0 and I xz = 0; that 
is, Oz must be a principal axis. In sum: The centrifugal forces on a spinning body tend 
to change the orientation of its instantaneous axis of rotation , unless the latter goes 
through the center of mass of the body and is a principal axis there. Such kinetic 
considerations led to the formulation of the concept of principal axes of inertia, 
at a point of a rigid body [Euler, Segner (1750s)]; and to the alternative term devia¬ 
tion moments, for the products of inertia. We shall return to this important topic 
in §1.17. 


1.16 THE RIGID BODY: LINEAR AND ANGULAR MOMENTUM 

(i) The inertial, or absolute, linear momentum of a rigid body B (or system 5), 
relative to an inertial frame F, represented by the axes I-XYZ (fig. 1.28), is defined as 

p=^dmv = mv G ( G: center of mass of B). (1.16.1a) 

Substituting in the above [recalling (1.7.11a ff.), and with r G / 0 = r G , <n —> f2: angular 
velocity vector of noninertial frame —» axes O-xyz relative to inertial ones I-XYZ] 

Va = Vo + Vc/o = Vo + Vc,rei + ^ x r c (v G , re i = dr c /dt) (1.16.1b) 


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§1.16 THE RIGID BODY: LINEAR AND ANGULAR MOMENTUM 



Figure 1.28 Rigid body (6), or system (5), in general motion relative 
to the noninertial frame O-xyz; S2: inertial angular velocity of O-xyz 
(two-dimensional case). 


readily yields 

P = Ptmm + Preb (1.16.1c) 

/’trans = m ( v o + x r G ): Linear momentum of transport , (1.16. Id) 

p rel = mv G , rei = m(dr G / dt) : Linear momentum of relative motion. (1.16.1 e) 

If B is rigidly attached to the moving frame M, represented by the axes O-xyz (fig. 
1.28), then, clearly, p te] = 0. 

(ii) The inertial and absolute angular momentum of B , relative to the inertial origin 
/, /y /abs = //,. is defined as 

H ^S [rj x dm ( drj/dt)\ = ^ [9? x dm (d^R/dt)\ 

[substituting 9? = r 0 /i + r = r 0 + r =>■ dW/dt = v 0 + v re j + Q x r, v re i = dr/dt] 
= mr 0 x (v 0 + ii x r G ) + mr G x v 0 

+ ^ dm [r x (Q x r)) + ^ dm (r x v rel ), (1.16.2a) 

or, since 

^ dm [r x (Q x r)] = dm [r 2 i2 — (r <g> r) ■ O] 

= ^ dm [r 2 1 — (r ® r)] • fl = Io • , (1.16.2b) 

and calling 

H 0 , rel = S' x (dm r re i): Noninertial and absolute angular momentum of B, about O , 

(1.16.2c) 

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CHAPTER 1: BACKGROUND 

we finally obtain the following general kinematico-inertial result: 

H r = H 0 + r Q x p + mr G x v 0 , Hq = I Q • + H 0?le i. (1.16.2d) 


Special Cases 

(i) If the body B is rigidly attached to the moving frame M , then 

v r ei = o, /> rel = 0, H 0 Te i = 0, Q = o) = inertial angular velocity of B , 

(1.16.3a) 


and, therefore, 

p = m(yo + <*> x r c ) = mva, Hj = I 0 - m + r 0 x p + tnr c x v 0 ■ (1.16.3b) 

(ii) If, farther, O = G, then r G = 0, r 0 = r 0 . and, therefore, 

p = mv G , Hr = I 0 • a) + r 0 x p. (1.16.3c) 


(iii) If 1 = 0 (i.e., rigid-body motion with one point, O, fixed), then r 0 = 0, 
v 0 = 0, and, therefore, 


p = m (a> x r G ) = mv G , H I = H 0 =Io- a> 



r x dm v 


(1.16.3d 1,2) 


It should be pointed out that the above hold for any set of axes, including non-body- 
fixed ones, at the fixed point O', but along such axes the components of Io will, in 
general, not be constant. Equation (1.16.3d2) would then yield, in components 
[omitting the subscript O and with r = (.x A .)], 


H k — ^2, ^ klUJh 

\t = s dm i( r • r ) 7 - ( r ® r )] h=S dm \ (X! XrXr ) Ski - XkX ‘ 


(1.16.4a) 

(1.16.4b) 


If the axes at O are body-fixed , then the I kl are constant; and, further, if they are 
principal, then 


Hk — 4 w k■ 


(1.16.4c) 


Linear Momentum of a Rotating Body 

To dispel possible notions that the linear momentum is associated only with transla¬ 
tion, let us calculate the linear momentum of a rigid body rotating about a fixed point 
♦ . We have, successively, with the usual notations, 


p = mv G = m{m x r G /*) = m(a> x r G ) 

= m(u} x ,u y ,u) z ) x (x G ,y G , z G ) [components along any ♦-axes] 

= m(u y z G - u z y G , u z x G -u x z G , u x v G -u y x G ). (1.16.5a) 


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§1.17 THE RIGID BODY: KINETICS OF TRANSLATION AND ROTATION 


In particular, if the body rotates about a fixed axis through ♦ , say ♦" [recalling 
(1.15.9aff.)], then oj = u z k = (d<j)/dt)k, and so (1.16.5a) reduces to 

p x = — {my G )u> z = — m y u z , p y = (mx G )uj z = m x u z , p : = 0. (1.16.5b) 

These expressions appear in the problem of rotation of a rigid body about a fixed 
axis, treated via body-fixed axes ♦-xyz [see, e.g., Butenin et al. (1985, pp. 266-278) 
and Papastavridis (EM, in preparation)]. 

It is not hard to show that, in this case, the (inertial and absolute) angular 
momentum of the body 


//* = gr x dmv= (H x ,H y ,H : ), (1.16.5c) 

reduces to 

H x = I xz u> z , II v = I yz uj z , H z = I zz uj : = I : lu z . (1.16.5d) 


1.17 THE RIGID BODY: KINETIC ENERGY AND KINETICS OF 
TRANSLATION AND ROTATION 
(EULERIAN "GYRO EQUATIONS") 

We recommend, for this section, the concurrent reading of a good text on rigid-body 
dynamics; for example (alphabetically): Grammel (1950), Gray (1918), Hughes 
(1986), Leimanis (1965), Magnus (1971, 1974), Mavraganis (1987), Stackel (1905, 
pp. 556-563). 

(i) The inertial kinetic energy of a rigid body B in general motion, T, is defined as 
the sum of the (inertial) kinetic energies of its particles: 

2T(B,t) =2T=^dmv-v, (1.17.1) 


or, since 

v = v*+o>xr/* (♦: arbitrary body-fixed point), (1.17.2) 

2T = S dm ( V * +(OX r /«) ' (’’♦ + m X ''/♦) = 2 ( 7 ’transl’n + ^rot’n + ^’g), (1-17.3) 

where 

2T transPn = mv, • v* = mV, 2 : 

(twice of) Translators (or sliding) kinetic energy of B, (1.17.3a) 
2r rot - n = ^dm(a) x ) • (<u x ) =«•/*•<«: (recalling §1.15) 

(twice of) Rotatory kinetic energy of B, (1.17.3b) 

T’cpi’g = x r G/ .) • = mv G/4 • v* = ( mr G/ *)' • v* = (dm Gj +/dt) • v*, 

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CHAPTER 1: BACKGROUND 


or 

T’cpPg = '”»* G/ * • (v* X to) = m Gj + • (v* x to): (1.17.3c) 

Kinetic energy of coupling between r* and to (where »i G /* = mr G /+) 

[= 0; e.g. if G = ♦ , or if v* and to are parallel [♦ on instantaneous screw axis of 
motion (§1.9)]; in which case, T decouples into translatory and rotatory kinetic 
energy]. (1.17.3d) 

• These expressions hold for any axes, either body-fixed or moving in an arbitrary 
manner, or even inertial. But if they are non-body-fixed , the components of r G /* and 
7 * will, in general, not be constant. 

• We also notice that, in there, the mass m appears as a scalar ( nr. T transl > n ), as a 
vector of a first-order moment (/m g /* ■ T’cpPg), and as a second-order tensor 
(I* ■ 7rot' n ). 

• From (1.17.3b), we obtain, successively, 
grad w T mVn = dT voGn /do 

= £ dm v i * • (dv/* /dm) = $ #y* x(tax »y*) dm = £ jy* x (dm v/*) 
= ^♦.relative = [recalling (1.6.5b)]; (1.17.4) 

that is, the angular momentum is normal to the surface T IoVn = constant, in the 
space of the a/s. 

• If v* = 0 — for example, gyro spinning about a fixed point — (1.17.3b) yield 

2 T =X 2T mt i n — I xx lu x ~ T * * * T 2I X y cc x uy T • • • = 77 * • to = /i* * o ^ 0; (1.17.4a) 
that is, since T is positive definite, the angle between 77 * = /j* and to is never obtuse : 

0° < angle ( 77 * , to) < 90°. (1.17.4b) 

• Similarly, we can express T in terms of relative velocities; that is, with 

V = V* + J? X /y* + V/* 5re iative: V/*,relative = dr/Jdt , (1.17.5) 

where Q is the inertial angular velocity of the moving axes. 


Another Useful T-Representation 
We have, successively, 

2 T = ^ dm »’•»’= ^ v • (dm r) = (v* + to x /y*) • (dm v) 

= v* • dm v) + ^ dp • (a) x »y*) (since dm v = dp) 

= V* •/> + ©• (^»y* X <//>) = V* •/) +to - 77 *, absolute- 

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(U7.6) 


§1.17 THE RIGID BODY: KINETICS OF TRANSLATION AND ROTATION 

Kinetic Energy of a Thin Plate of Mass m 
in Plane Motion on its Own Plane (fig. 1.17) 

Using plate-fixed axes ♦— xy, we find [with cos(...) = c(...), sin(...) = s (...)] 

= x G i + y G j = xi + yj, {x,y: constant) (1.17.7a) 

v* = (dX+ / dt)I + ( dY+/dt)J = ( dX/dt)I + ( dY/dt)J 
= (dX / dt){c(j) i — scj) j) + {dY / dt){sf i + ccj) j) 

= [{dX/dt)c4> + {dY/dt)sf\i + [ — {dX/dt)scf> + {dY / dt)c(f]j 
= v x i+v y j, a) = {d(f>/dt)K = {d<f>/dt)k; (1.17.7b,c) 

and so, successively, 

2T tmmVn = m V* • v* = m[{dX/dt) 2 + {dY / dt ) 2 ]; (1.17.7d) 

T cpVg = in r G/4 • (v* x co) = m(x,y, 0) • \{v x , v y , 0) x (0,0, dcj)/dt)\ 

= m{d(j}/dt){VyX — v x y) 

= m{d<f) I dt) {[{dY / dt)x — {dX/dt)y\c<f> — [( dX/dt)x + {dY/dt)y\ s<fi}; 

(1.17.7e) 

2r rofn = w-I.-oj =I. iZZ u z 2 =I{d(j>/df) 2 ; (1.17.7f) 

that is, 

2 T = 2T{dX/dt, dY/dt , d^/dt) 

= m[{dX/dtf + {dY/dt) 2 ] + I{d<j>/dt) 2 

+ 2m{d(j)/dt){[{dY/dt)x — {dX/dt)y\ccf> — [{dX/dt)x + {dY/dt)y\s<f>}. (1.17.7g) 


An Application 

It is shown in chap. 3 that for this three degrees of freedom (DOF) (unconstrained) 
system, defined by the positional coordinates cp = X, q 2 = Y, q 2 = c\>, the 
Lagrangean equations of motion d / dt\dT / d{dq k / dt)] — {dT/dq^) = Qk 
[= system {impressed) force corresponding to q k ]; or, explicitly, angular equation 
(with M = total external moment about ♦): 

I{d 2 <j)/dt 2 ) + 777 { [{d 2 Y/dt 2 )x - {d 2 X/dt 2 )y]ccj) 

- [{d 2 X/dt 2 )xF{d 2 Y/dt 2 )y]s(j)} = M; (1.17.7h) 

which is none other than the (not-so-common form of the) angular momentum 
equation: 


Ua z + {r G /♦ x ma.) z = I{d 2 (j)/dt 2 ) + m[x{a.) y - y{a.) x ] = M, (1.17.7i) 


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CHAPTER 1: BACKGROUND 


where (fig. 1.17), 


*•(?/♦ =x G i + y G j = xi + yj= ••• 

= (. xc(j> —yscj))I + (xsc/) + y cc/))J = XI + YJ, 


(1.17.7J) 


«♦ = ( d 2 X ♦ /dt 2 )I + (d 2 Y. /dt 2 )J = ( d 2 X/dt 2 )I + {d 2 Y/dt 2 )J 
= (d 2 X / dt 2 ) ( ccj)i — s</>./) + (d 2 7/ dr 2 ) (s<j> i + c</) j) 

= [( d 2 X/dt 2 )c(j) + (d 2 Y/dt 2 )scj)\ i + [-{d 2 X/dr)s4> + (d 2 Y/dt 2 )c<t>\ j 


= («*)x 1 + («♦),'./ = 


(1.17.7k) 


x, j-equations (with f x = components of total external force about x, j-axes, 
respectively): 

m[d 2 X/dt 2 — (d 2 (j) / dt 2 ){x s(j> + y c(j>) — (d(j>/dt ) 2 (x c</> — y scj))\ = f x . (1.17.71) 

m[d 2 Y/dt 2 + (d 2 4>/dt 2 )(x c(f> — y scj)) — (dcj)/dt) 2 (xs(j) + y c<j>)\ = f y , (1.17.7m) 

which are none other than 


m{a.) x -m[(d 2 cl)/dt 2 )Y + {d<f)/dt) 2 X} = f x , (1.17.7n) 

m{a.) y + m[(d 2 <t)/dt 2 )X - (dcj)/dt) 2 Y) = f y . (1.17.7o) 


For additional related plane motion problems, see, for example. Wells (1967, pp. 
150-152). 

“British Theorem” 

It can be shown that the (inertial) kinetic energy of a thin homogeneous bar AB of 
mass in equals 

T = {m/6)(v A -v A +v B - v B + v A • v B ) = ( m/6)(v A 2 + v B 2 + v A ■ v B ). (1.17.8) 

(This useful result appears almost exclusively in British texts on dynamics; hence, the 
name; see, for example, Chorlton, 1983, pp. 165-166.) 

Principle of Linear Momentum; Motion of Mass Center 
Since 


v g = •’♦ + <*> x r Gj* ( + : body-fixed point), 


(1.17.9a) 


the principle of linear momentum (§1.6) 


m[dv G jdi) = f (total external force, acting at G ) 


(1.17.9b) 


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§1.17 THE RIGID BODY: KINETICS OF TRANSLATION AND ROTATION 


along body-fixe daxes (i.e., ca = f2) yields, successively, 
m d/dt(v+ + co x r G /f) 

= m(dv+ /dt+ co x v*) + m[(dco/dt) x r G /* + ca x (dr c /+ / dt )] 

[with dco/dt = at, c/r G( /*/c/t = v G y* = ca x r G / 4 ] 

= m(dv+ /dt) + m(co x v*) + a x (mr G ^) + co x [<o x (mr G /*)] = /, 

(1.17.9c) 

or, in terms of the center of mass vector of the mass moment m G/ + = mr G /+ [as in 
(1.17.3c, d)], 

m(dv+/dt) + m(a> x v*) + a x (m c /+ + co x (ca x m G /+) = f. (1.17.9d) 

Along body-fixed axes ♦— xyz, and with m G /+ = (m xyz ), = (v xyz ) there, 
the x-component of (1.17.9d) is 

m[dv x /dt + u) y v z — ut z v y ] + [m z (dco y /dt) — m y (dw z /dt)\ 

+ \u) y (m y w x - m x ut y ) - u z (tn x iu z - m,u x )\ = f x , 

etc., cyclically. (1.17.9e) 


Special Cases 

(i) If ♦ = G, then m G /+ = 0 and, clearly, (1.17.9d) reduces to 

m(dv G /dt+ co x v G ) =/■ (1.17.9f) 

(ii) Along non-body-fixed axes at G, rotating with inertial angular velocity Q, 
(1.17.9b) yields 

m(dv G /dt + fl x v G ) =/; (1.17.9g) 

or, in components, with v G = (v G -. x . y f), 

m(dv G /dt) x = m(dv Gx /dt + 0,, v Gz — Q z v Gy ) = f x . etc., cyclically, (1.17.9h) 

where (. dv G /dt) x = component of a G along an inertial axis that instantaneously coin¬ 
cides with the moving axis Gx, and so on. In general, the v G . xyz are quasi velocities. 

Principle of Angular Momentum; 

Motion (Rotation) about the Mass Center 

Along body-fixed axes ♦— xyz, the principle of angular momentum [§1.6, with • 
(arbitrary spatial point) —> ♦ (arbitrary body point), and //♦.relative = A*], 

dh+/dt + r G /+ x [m(dv+/dt )] = A/ 4 , (1.17.10a) 

becomes 

dli./dt + to x h+ + m G /+ x (dv+/di) + m G /+ x (co x v*) = M* ; (1.17.10b) 

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CHAPTER 1: BACKGROUND 


or, in components, with = {h x ^ y f), m a/+ = ml 'a/* = K, w ), »’♦ = (v x . yz ), and so 
on, 

dh x /dt + u) r h- — oj-h v + m r [dv./dt + to x v y — c o y v x ] 

— m z [dv y /dt + uj z v x — uj x v z \ = M x , etc., cyclically. (1.17.10c) 

[The forms (1.17.9d,e) and (1.17.10b c) seem to be due to Heun (1906, 1914); see 
also Winkelmann and Grammel (1927) for a concise treatment via von Mises’ (not 
very popular) “motor calculus.”] 


Special Cases 

(i) If ♦ = G, then m G = 0, and (1.17.10b) reduces to 

dh G /dt+w xh c =M G . (1.17.10d) 

(ii) Along non-body-fixed axes at G, rotating with inertial angular velocity 12, 
(1.17.10a) yields 

dh G /dt + f2 x h c = M c ; (1.17.10e) 


or, in components, 

( dh G /dt) x = dh Gx /dt + Q v h G - — O z h Gy = M Gx , etc., cyclically. (1.17.lOf) 

(iii) If the axes are body-fixed, then Q = co; and if they are also principal axes, 
then, since (omitting the subscript G throughout) li = I-to.hk = hwk, k = 1,2,3, 
eqs. (1.17.1 Of) assume the famous Eulerian form (1758, publ. 1765): 


I\{du>\/dt) — ( I 2 — — M\, 

I 2 {du 2 /dt) - (/ 3 -Ii)ui 3 ui =M 2 , (1.17.11a) 

Ifiduj^/dt) — (/) — I 2 )u)\ui 2 = My, 

or, alternatively, 

deji/dt — [(/ 2 - I 2 )/Ii\lo 2 oj 2 = Mjl u etc., cyclically. (1.17.11b) 

From the above we readily conclude that: 

• A force-free rigid body in space can rotate permanently [i.e., 
dm/dt = 0 => o> = (a*!, 0,0), or (0, u> 2 , 0), or (0, 0, cufi = constant] only about a central 
principal axis of inertia. Or, if a free rigid body under no external forces begins to 
rotate about one of its central principal axes, it will continue to rotate uniformly 
about that axis; and, if a rigid body with a fixed point, and zero torque about that 
point, begins to rotate about a principal axis through that point, it will continue to do 
so uniformly about that axis. 

• The principle of angular momentum takes the “elementary” form M = d/dt(Iuj) 
only for principal axes of inertia, or if the body rotates about a (body- and space-) 
fixed axis. That is why a central principal axis was called a permanent axis (Ampere, 
1823). 


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§1.17 THE RIGID BODY: KINETICS OF TRANSLATION AND ROTATION 


REMARKS 

(!) Equations (1.17.10a ff.) also hold for any fixed point O. 

(ii) The principles of linear and angular momentum are summarized as follows: 
the vector system of mechanical loads, or inputs [/at G, M G (moments offorces and 
couples )] is equivalent to the kinematico-inertial vector system of the responses, or 
outputs, ( ma G at G, dh G /dt)\ and this equivalence, holding about any other space 
point •, can be expressed via the (hopefully familiar from elementary statics) purely 
geometrical transfer theorem: 

M. = M G + r G/ . x / atG = M g + r G/ . x ma G = dh G /dt + m G/ . x a G . (1.17.12) 

(iii) In general, the direct application of the vectorial forms of the principle of 
angular momentum, either about the mass center G, or a fixed point O, and then 
taking components of all quantities involved about common axes in which the inertia 
tensor components remain constant, is much preferable to trying to match a (any) 
particular problem to the various scalar components forms of the principle. 

(iv) The relative magnitudes of the principal moments of inertia of a rigid body at, 
say its mass center G , I G . 12,3 = Z, 2,3 (he., its mass distribution there) provide an 
important means of classifying such systems. Thus, we have the following classifica¬ 
tion (§1.15: subsection “Ellipsoid of Inertia”): 

• If / = I 2 = / = /, we have a spherical top , or a kineticallv symmetrical body. Then, 


H g = h G = I G • (a = (II) • to = Ioi . 


• If I\ = I 2 f h, the body (or “top”) is symmetric, if / > / 3 , it is elongated , and if f < / 3 , 
it is flattened. 

• If /| f I 2 f f f I\, the body is unsymmetric. 

For further details and insights on these fascinating equations, see Cayley (1863, pp. 
230-231), Dugas (1955, pp. 276-278), Stackel (1905, pp. 581-589). 

Energy Rate, or Power, Theorem for a Rigid Body 

By d/dt(.. ./differentiating the kinetic energy definition 2 T = S dm v • v, and then 
utilizing in there the rigid-body kinetic equation v = v* + a> x r/+, we obtain, suc¬ 
cessively, 

dT/dt = ^ dm v • ( dv/dt ) = ^ dm v • a = ^ dm (v* + m x r/+) • a 

= ^ dm v* • a + ^ dm(co x r /*) • a = dm a^j + <u • dm r/+ x a'j 

= v» • («7 a G ) + a> ■ {S dm [d/dt(r x v) + x v]| 

= v* • ( dp/dt ) -I (O' ( dH./dt+ v* x p), (1.17.13a) 

where (recalling the definitions in §1.6), 

p=^j dm v = m v G \ Linear momentum of body, (1.17.13b) 

dm(r /.* x v): Absolute (and inertial) angular momentum of body, about 
the body-fixed point ♦. (1.17.13c) 

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CHAPTER 1: BACKGROUND 

Invoking the principles of linear and angular momentum (§1.6), we can rewrite 
(1.17.13a) as 


dT/dt = v.-f + a-M ♦. (1.17.13d) 

On the other hand, the power of all forces, d'W/dt = ^ df-v, transforms, succes¬ 
sively, to 


d'W/dt = $df-(v. + ft) x »•/♦) = v* • (jSV/') +<»• ( S r /♦ x d f) 

= v.f+co-M.; ( 1 . 17.1 3e) 


that is, 


dT/dt = d'W/dt, 


(1.17.13f) 


which is the well-known power theorem, proved here for a rigid system. 


Special Case 

If v* = 0 (i.e., rotation about a fixed point), (1.17.13d f) reduce to 

dT/dt = d'W/dt = co -M.. (1.17.13g) 

If, in addition, M » = 0 ( torque-free motion), then d'W/dt = 0 and T = constant 
(energy integral), and M* = dH+/dt = dh+/dt = 0 =>■ //* = /;* = constant (angu¬ 
lar momentum integral). These two integrals of the torque-free and fixed-point 
motion form the basis of an interesting geometrical interpretation of rigid-body 
motion, due to Poinsot (1850s). For details see, for example (alphabetically): 
MacMillan (1936, pp. 204-216), Webster (1912, pp. 252-270), Winkelmann and 
Grammel (1927, pp. 392-398). 


Additional Useful Results 

(i) By multiplying the Eulerian (rotational) equations with w Xi> , jZ , respectively, and 
then adding them, we obtain the following power equation: 

d / dt[(Auj x T BiOy~ + Cuj z ~)/ 2\ — M x ui x T M x ujy T 

i.e., d/dt (Rotational kinetic energy ) = Power of external moments. (1.17.14) 

(ii) Plane motion : Principle of angular momentum for a rigid body B , about its 
instantaneous center of rotation I. We have already seen (1.9.4dff.) that the inertial 
coordinates of the instantaneous center (of zero velocity) /, relative to the center of 
mass G, are 


ri/o = (*>/g, Y,/ g , 0) = {-dYa/dt/w, +dX G /dt/u, 0). (1.17.15a) 

Therefore, application of the principle of angular momentum about /: 

Mi = I G (duj/dt) + ( r G/I x ma G )z, (1.17.15b) 


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§1.17 THE RIGID BODY: KINETICS OF TRANSLATION AND ROTATION 


yields, successively (with I G = mk 2 ), 

Mj = I G (du)/dt) + (m/co)[(dY G / dt, —dX G / dt, 0) x (d 2 X G /dt 2 ,d 2 Y G /dt 2 , 0)] 

= I G (duj/dt) + (m/J)[(dY G /dt)(d 2 Y G /dt 2 ) — {—dX G /dt){d 2 X G /df L )\ 

= I G (du>/dt) + (m/u>)[(dX G /dt)(d 2 X G /dt 2 ) + (dY G /dt)(d 2 Y G /dt 2 )] 

= (m/co) [k 2 u(duj/dt) + ( dX G /dt)(d 2 X G /df 2 ) + (dY G /dt)(d 2 Y G /dt 2 )] 

= (m/cu)(d/dt{(l/2)[k 2 uj 2 + ( dX G /dt ) 2 + {dY G /dtf}}) 

[noting that (dX G /dt) 2 + ( dY G /dtf = v G 2 = rw 2 , r = |r G// |] 

= (1/2 iv){d/dt[m(k 2 + r 2 )u) 2 }}, 

or, hnally, with f = m(k 2 + r 2 ) = mK 2 : moment of inertia of B about I (by the 
parallel axis theorem), 

M, = (l/2w)[^/ht(/ / w 2 )] = Ifdco/dt) + (1/2 )u(dl,/dt) 

= Ifdio/dt) + mr{dr/dt)u>. (1.17.15c) 


Special Cases 

(a) If B is turning about a fixed axis, or if I is at a constant distance from G, then 
dr/dt = 0 and (1.17.15c) reduces to 

M, = Ifidu/dt). (1.17.15d) 

(b) If the axis of rotation is mobile, but the body starts from rest, then, since 
initially u> = 0 and dr/dt = 0, the initial value of its angular acceleration is given 
by (1.17.15d): 

dco/dt= Mfilj. (1.17.15e) 

(c) If the body undergoes small angular oscillations about a position of equili¬ 
brium, then the term df/dt = 2mr(dr/dt) is of the order of the rate dr/dt, and 
therefore (df/dfixi is of the order of the square of a small velocity and so, to the 
first order (linear angular oscillations), it can be neglected; thus reducing (1.17.15c) 
to (1.17.15d), with f given by its equilibrium value. 

In sum, eq. (1.17.15d) holds if the instantaneous axis of rotation is either fixed, or 
remains at a constant distance from the center of mass', or if the problem is one of 
initial motion, or of a small oscillation. In all other cases of moments about I, we 
must use (1.17.15c). For further details and applications, see, for example (alphabe¬ 
tically): Besant (1914, pp. 310-314), Loney (1909, pp. 287, 346-347), Pars (1953, pp. 
403-404), Ramsey (1933, part I, pp. 241-242), Routh (1905(a), pp. 103-104, 171- 
172). Somehow this topic is treated only in older British treatises! 


Rigid-Body Mechanics in Matrix Form 

[Here, following earlier remarks on notation (§1.1), we denote vectors by bold italics, and 
matrices/tensors by bold, roman, upper case (capital) letters; for example, a, A (vectors), 

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CHAPTER 1: BACKGROUND 


A, B (matrices, tensors). This material (notation) is presented here not because we think 
that it adds anything significant to our conceptual understanding of mechanics, but because 
it happens to be fashionable among some contemporary applied dynamicists.] 

By recalling the tensor results of §1.1, and the earlier definitions and notations, 
(1.15.2a ff.), 

I = [(r-r)l — f ®f] dm = — ^ (r-r) dm = (l/2)(TrI)l - J (1.17.16al) 

[ => TrI = 2Tr J], 

J = £j(r®r)dm, (1.17.16a2) 


[/* = axial vector of tensor r and d (...) /dt is inertial rate of change], we can verify the 
following matrix forms of the earlier (§1.15-1.17) basic equations of rigid-body 
mechanics [while assuming that, in a given equation, all moments of inertia and 
moments of forces are taken either about the body’s center of mass, or about a 
body-and-space-fixed point (if one exists), and along body-fixed axes; and suppres¬ 
sing all such point-dependence for notational simplicity, except in eqs. (1.17.16bl 3) 
for obvious reasons]: 


(i) lo = Ig - m re • To = I G + m [(r G ■ r G )\ - r G <g) r G \ 

[r G = r G /o, etc., parallel axis theorem in terms of 1: (1.15.7b)],(1.17.16b!) 


=► Tr I 0 = Tr I G + 2mr G • r G , (1.17.16b2) 

Jo = Jg + m r G ®r G = (Tr I c /2) 1 I G + m r G ® r G 

[Parallel axis theorem in terms of J]; (1.17.16b3) 

(ii) dl/dt = O -1 -I-1 - O t = O -1 — I - r2 

[recalling results of 1.1.20a ff.; to = axial vector of tensor S~2]; (1.17.16c) 

(iii) dl/dt = — (dj/dt) [= -(« • J - J • O)] (1.17.16d) 

(iv) H = I • to = — J • to + (Tr J)to 

(v) H = I • fi T — 0*1+ (Tr I) • $7 = (17 • I) T — IT • I + (Tr I) • Cl (1.17.16el) 

= J- r2 + r2-J = J- r2-(J-^) T =J-f2-^ T -J; (1.17.16e2) 

[H = axial vector of H (angular momentum tensor )] 


(vi) 


(vii) 


M = d/dt (I • to) = (dl/dt) • to + I • ( dto/dt ) [then invoking (1.17.16c)] 

= I • ( dto/dt ) + fl • (I - to) = I • ( dto/dt) + to x (J • to) (1.17.16fl) 

= —[J • (dto/dt) + fi • (J • to)] + (Tr J)(dco/dt); (1.17.16f2) 

M = (E • I) T — E • I + (Tr I) • (dO/dt) (1.17.16gl) 

= E-J-(E-J) t (1.17.16g2) 


[M = axial vector of M ( moment , or torque , tensor ); 

recalling (1.11.9a ff.): E = dJT/dt + f2-fi = «4-l-f2-fi]. 


Additional forms of the above are, of course, possible. 


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§1.17 THE RIGID BODY: KINETICS OF TRANSLATION AND ROTATION 
A Comprehensive Example: The Rolling Disk 

Let us discuss the motion of a thin homogeneous disk D (or coin, or hoop) of mass m 
and radius r, on a fixed, horizontal, and rough plane P (fig. 1.29). 


Kinematics 

Relative to the intermediate axes/basis G—xyz/ijk (defined so that k is perpendicular 
to D, at its center of mass G\ i is continuously horizontal and parallel to the tangent 
to Z>, at its contact point C; and j goes through G, along the steepest diameter of D, 
and is such that ijk form an ortho-normal-dextral triad), whose inertial angular 
velocity Si is 


Si = Si x i + Q y j + Q : k = (u>g)i + (w^ sin 9) j + (w^ cos 0)k, (1.17.17a) 


[where = d<j>/dt, = dO/dt , = dip/dt] the inertial angular velocity of D, co, 

equals 


oj — uj x i -|- uty j -|- c o z k — (u>g)i -|- (c sin $)y T - (cv^ cos 9 to^k 

= S2 + k. 


(1.17.17b) 


In view of the above, the rolling constraint v c = 0 , becomes 


v c = Vg + «> x r c/G = v x i + v y j + v.k + (w x , w y , u z ) x (0, -r, 0) 
= (v x + rw z )i + (■ v y )j + (v z - u x r)k = 0, 


(1.17.17c) 




C/C ■ 


Figure 1.29 Rolling of thin disk/coin D on a fixed, rough, and horizontal plane P. 

O-XYZ: space-fixed (inertial) axes; O-xyz : intermediate axes (of angular velocity o). A B = A, 
C: principal moments of inertia at G. For our disk: A = mr 2 /4, C = mr 2 / 2. 


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235 







CHAPTER 1: BACKGROUND 


from which it follows that 

v x + ru z = 0 => v x = — ruj z = — r (u^cosO + (1.17.17d) 

v y = 0, (1.17.17e) 

v z — u x r =0 => v z = roj x = rui g . (1.17.17f) 

These equations connect the velocity of G with the angular velocity and the rates of 
the Eulerian angles. 

Kinetics 

To eliminate the rolling contact reaction R, we apply the principle of angular 
momentum about C; that is, we take moments of all forces and couples [including 
inertial ones at G; i.e., ~m(dv G /dt) and — dh G /dt] about G (recalling 1.6.6a IT.) to 
give 

M c = dh G /dt + r G / c x \m{dv G /dt)]. (1.17.18a) 

But, with W = weight of disk, and sin(...) = s (...), cos(...) = c(. ..), we have 

(i) M c = r G / c x W = (0, r, 0) x (0, — Ws6, — Wed) = (—rWc8)i; (1.17.18b) 

(ii) dv G jdt = dv G /dt + fl x v G [with the ad hoc notation dv X y iZ /dt = fl VjV ,z] 

= a x i + a y j + a- k + (Q x , Q y , Q z ) x ( v x , v y , v z ) 

= (a x + Q v v z - Q z v y )i 

+ ( a v + Q z v x — C2 x v z )j + ( a z + Q x v y — O v v x )k 
= (a x + v z s8 - v y c0)i + ( a y + v x c8 - v z ui g )j 

+ (a z + VyU e - v x u < j > sd)k\ (1.17.18c) 

(iii) dh G /dt = dh G /dt + f2 x li G [with the ad hoc notation du} Xt y tZ /dt = a Xt y tZ \ 

= + {Aoiy)j + (Ca z )k] + (O x , Q y , Q z ) x (Au x ,Buj y , Cw z ) 

= (Acx x T GC2yLu z — AQ z oj y )i T (Acn y T AQ z ui x — GC2 x co z )j 

T (Go z T AC2 x uj y — Af2 v uj x )k 

= ( Aa x + Cco z u^ s8 — Au> y c8)i + ( Aa v + Au> x c8 — Cco z u> e )j 

T (Col z T Auj y uj(j — A jj x ujy, s())k. 

(1.17.18d) 

and so (1.17.18a) yields the three component equations of angular motion: 

mr(a z + v v — v x s9) + ( Aa x + Cuj z sd — Au y to^ cd) = — Wr c8, (1.17.18e) 

Aa y + Auj x u^cd - Cu : u> 0 = 0, (1.17.18f) 

— mr{a x + v, sd — v v c8) + (C a z + Au> y u e — Au x ui^ sO) =0. (1.17.18g) 


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§1.18 THE RIGID BODY: CONTACT FORCES, FRICTION 


The nine equations (1.17.18e, f, g) + (1.17.17d, e, f) + (1.17.17b, in components) 
constitute a determinate system for the nine functions (of time): qf>, 9, ip; u ^ (quasi 
velocities); v xyz (quasi velocities). We may reduce it further to two steps: 

(i) Using (1.17.17d, e, f) in (1.17.18e, f, g) (i.e., eliminating v xyz ), we obtain 

mr(ra x + ru> z sO) + Aa x + Cu-u^sO — AojyUj^cO = —WrcQ , (1.17.19a) 

Aa y + Au> x u)^ cO — Cco z cu g = 0, (1.17.19b) 

— mr(—ra z + na x uj < j > s9) + Ca z + Aco y ui g — Aco x uj ( j > s9 = 0. (1.17.19c) 


(ii) Using (1.17.17b) in (1.17.19a, b, c) (i.e., eliminating u xy ), we get three equa¬ 
tions of rotational motion in terms of 0, the rates of cp, 9, and the total spin 
U z = LO^, + U),pc9\ 

(A + mr 2 )(d 2 9/dt 2 ) + (C + mr 2 )u z (dcp /dt)s9 — A(df/dt) 2 c9s9 = — Wrc9, (1.17.20a) 
A d/dt [{d<p/di)s9\ + A{d(p/dt){d9/dt)c9 - Cto z (d9/dt) = 0, (1.17.20b) 

(C + mr 2 ){du> z /dt) — mr 2 {dtp / dt)(d9 / dt)s9 = 0; (1.17.20c) 

or, since A = B = mr 2 /A = (1/2) (wr 2 /2) = C/2, 

9: 5r(d 2 9/dt 2 ) + 6 ruj z (d<p/dt) sin0 — r{dcp/dt) 2 sin 9 cos 9 + 4gcos 0 = 0, (1.17.21a) 

<p: 2u> z (d9/dt) — 2(d<p/dt)(d6/dt) cos 9 — (d 2 cp/dt 2 ) sin 0 = 0, (1.17.21b) 

u z . 3(du} z /dt) — 2{dcp/dt){d9/dt) sin0 = 0. (1.17.21c) 


These three nonlinear coupled equations contain an enormous variety of disk 
motions. For simple particular solutions of them, see, for example, MacMillan 
(1936, pp. 276-281); also Fox (1967, pp. 263-267). Once <p(t), 9(t), ip(t) have been 
found, the rolling contact reaction R = {R x . yz ) can be easily obtained from the 
principle of linear momentum: 


mac = m(dvc/dt + fl x va) = W + R=>R = ---=R(t). (1.17.22) 

The details are left to the reader. 


1.18 THE RIGID BODY: CONTACT FORCES, FRICTION 

Recommended for concurrent reading with this section are (alphabetically): Beghin 
(1967, pp. 139-145), Kilmister and Reeve (1966, pp. 81-84, 141-143, 164-177), Peres 
(1953; pp. 62-66); also, our Elementary Mechanics (§20.1,2, under production). 


Introduction and Constitutive Equations 

The forces between two rigid bodies, B and B\, at a mutual contact point C (actually, 
a small area around C that is practically independent of the macroscopic shape of the 
bodies and increases with pressure), say from B to B\ , reduce, in general, to a 
resultant force R and a couple C; frequently, C can be neglected. Decomposing R 

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CHAPTER 1: BACKGROUND 


and C along the common normal to the bounding surfaces of B, B\, say from B towards 
B\ , and along the common tangent plane, at C, we obtain 

R = Rfri + Rj 

= Normal reaction (opposing mutual penetration) 

+ Tangential reaction (opposing relative slipping), (1.18.1) 
C = C N + Cj 

= Pivoting couple (opposing mutual pivoting) 

+ Rolling couple (opposing relative rolling). (1.18.2) 

These components satisfy the following “laws” (better, constitutive equations) of dry 
friction; that is, for a solid rubbing against solid, without lubrication: 

(i) As soon as an existing contact ceases, R = 0. 

(ii) Whenever there is slipping — that is, relative motion of B and B { (v c ^ 0) — 
R n points toward Bp, and R T and v c are collinear and in opposite directions: 

Rj x V (■ = 0, Rj • v'( ■ < 0, (1.18.3) 

and 

Rt = J(Rni v c ); (1.18.4) 

or, approximately (for small relative velocities), 

\R t /R n \ = p: coefficient of friction between B and Bp, a nonnegative constant. 


(1.18.5) 

Frequently, we use the following notation: 

R t = F and R n = N. (1.18.6) 

Then, with jf 7 ! = the above read 

R =Rn + Rj; Rn = Nti, Rj = Ft = —(1.18.7) 

where 

n = common unit normal vector, from B towards B u (1.18.7a) 

t = unit tangent vector, in direction of slipping velocity. (1.18.7b) 

(iii) When v c = 0 (no slipping — relative rest), R N points toward If , while R T can 
have any arbitrary direction and value on the common tangent plane, as long as 

l-^r/^jvl = \F/N\ < /x; or, vectorially, \R x n\ < p\R ■ n|; (1.18.8) 


with the equality sign holding for impending tangential motion. Actually, the p in 
(1.18.8) is called coefficient of static friction, p s ; and the p in (1.18.5) is called 
coefficient of kinetic friction, p K \ and, generally, 

ds > Pk- 

Here, unless specified otherwise, p will mean p K . 

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(1.18.9) 


§1.18 THE RIGID BODY: CONTACT FORCES, FRICTION 


The friction coefficient p is, in general, not a constant but a function of: (a) the 
nature of the contacting surfaces', (b) the conditions of contact (e.g., dry vs. lubricated 
surfaces); (c) the normal forces (pressure) between the surfaces; and (d) the velocity 
of slipping. Further, in the dry friction case (solid/solid, no lubricant), p increases 
with pressure, and decreases with v c ; and this dependence is particularly pronounced 
for small values of v c , so that, if p = p(v c ), then p < p„, where p 0 = p(0). In most 
such applications, we assume that /r is, approximately, a positive constant ( rough 
surface). Then the relation p = tan </> defines the “angle of friction.” If p « 0 ( smooth 
surfaces), then R k, R n = N, Rt ~ 0. If, on the other end, p —> oo (perfect rough¬ 
ness), then v c = 0 throughout the motion, and R can have any direction, as long as 
R N = N points toward . 

(iv) The contact couple C is included in the cases of small p and/or slippingless 
motion as follows: 


(a) If at a given instant and immediately afterwards m N = 0 (i.e., no instantaneous 


pivoting), then 

!CV| < |Cjy ]lnax |, C N<max = f p R N = limiting pivoting moment, (1.18.10) 
f p = pivoting friction/resistance coefficient. (1.18.10a) 

(b) If at a given instant m N f 0, or if it stops being zero at that instant, then 

IC# | = |Cjv, max |; (1.18.11) 

and C N and m N have opposite senses. 

(c) If m T = co rom „g = <» R = 0, then 

\C T \ < |C r , max |, C T max = f.R N = limiting rolling moment, (1.18.12) 
f r = rolling friction/resistance coefficient. (1.18.12a) 


(d) If at a given instant a> T f 0, or if it stops being zero at that instant, then 

\Ct\ = |CV, max |; (1.18.13) 

and C T and m T have opposite senses. 

The coefficients f p and f r have dimensions of length (whereas p is dimensionless!), 
and their values are to be determined experimentally. Theoretically, f p can be related 
to p,ifC N is viewed as resulting from the slipping friction over a small area around 
the contact point C —something requiring use of the theory of elasticity (no such 
relationship can be established for f). It turns out that f p is, generally, five to ten 
times smaller than fp in general, pivoting is produced faster than rolling. 

In closing this very brief summary, we point out that the above “friction laws” sup¬ 
ply only indirect criteria for relative rest or motion (rolling and slipping); that is, if, 
for example, we assume rest and the resulting equations are consistent with it, it means 
that rest is possible, not that it will happen. And if we end up with an inconsistency, 
it means that the particular assumption(s) that led to it is (are) false. Thus, to show 
that two contacting bodies roll (slip) on each other, all we can do is show that the 
assumptions of their slipping (rolling) lead to a contradiction. [For detailed examples 

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CHAPTER 1: BACKGROUND 


illustrating these points, see, for example (alphabetically): Hamel (1949, pp. 543-549, 
629-639), Kilmister and Reeve (1966, pp. 165-177); also Poschl (1927, pp. 484-497).] 


Work of Contact Forces 

Under a kinematically possible infinitesimal displacement of B\ relative to B 
(assumed fixed) that preserves their mutual contact at C, the total elementary 
(first-order) work of the contact actions (of B on Bf is: 


d'W = R-dr c + C-dO 


(1.8.14) 


where 

dr c = elementary translatory displacement of the B , -11 xcd point, at contact, 
relative to B 

(= v c dt , in an actual such displacement), (1.18.14a) 

dO = elementary rotatory displacement of B t relative to B 

(= to dt, in an actual such displacement), (1.18.14b) 

• Since dr c preserves the B/B\ contact, it lies on their common tangent plane at C. 
Then: (a) if R T « 0 (i.e., negligible slipping friction), or (/3) if dr c = 0 (i.e., no 
slipping) and dQ = 0 (i.e., no rotating), then: 


d'W = 0. (1.18.15) 

• If dr c violates contact, but remains compatible with the unilateral constraints, it 
makes an acute angle with the normal toward B\. In this case, if R T « 0 =>■ R ss R N , 
and therefore 


d'W>0; (1.18.16) 

while for elementary displacements incompatible with the constraints, 

d'W<0. (1.18.17) 

• In a real, or actual, displacement d'W becomes 

d'W = (R • v c + C • <n) dt. 

From the earlier constitutive laws, we see that, as long as v c , co N , co T do not vanish , 
the pairs 


(Rti v c)i (Cjv,COv), (C t ,(O t ), 

are collinear and oppositely directed. Hence, frictions do negative work; that is, in 
general, 


d'W < 0. 


(1.18.18) 


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§1.18 THE RIGID BODY: CONTACT FORCES, FRICTION 


If, as commonly assumed, C«0, then 

d'W= (R t • v c ) dt = (F • Vc) dt 

= 0; if F = 0 (frictionless, or smooth , contact) 

= 0; if v c = 0 (slippingless, or rough , contact). (1.18.19) 

It should be stressed that, in all these considerations, the relevant velocities are those 
of material particles , and not those of geometrical points of application of the loads. 


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2 

Kinematics of Constrained Systems 

(i.e., Lagrangean Kinematics) 


I cannot too strongly urge that a kinematical result is a result 
valid forever, no matter how time and fashion may change the 
"laws" of physics. 

(Truesdell, 1954, p. 2) 

It is my belief that students have difficulty with mechanics 
because of an inadequate knowledge of kinematics. 

(Fox, 1967, p. xi) 


2.1 INTRODUCTION 

As complementary reading for this chapter, we recommend the following (alpha¬ 
betically): 


General: Hamel (1904(a), (b)), Heun (1906, 1914), Lur’e (1968), Neimark and Fufaev 
(1972), Novoselov (1979), Papastavridis (1999), Prange (1935). 

Special problems, extensions: Carvallo (1900, 1901), Lobas (1986), Lur’e (1968), 
Stuckler (1955), Synge (1960). 

Research journals (see the references at the end of this book): Acta Mechanica Sinica 
(Chinese), Applied Mathematics and Mechanics (Chinese), Archive of Applied Mechanics 
(former Ingenieur Archiv; German), Journal of Applied Mechanics (ASME; American), 
Applied Mechanics (Soviet —> Ukrainian), Journal of Guidance, Control, and Dynamics 
(AIAA; American), PMM (Soviets Russian), ZAMM (German), ZAMP (Swiss); also 
the various journals on kinematics, mechanisms, machine theory, design, robotics, etc. 


In this chapter we begin the study of analytical mechanics proper with a detailed 
treatment of Lagrangean kinematics, i.e., the theory of position and linear velocity 
constraints (or Pfaffian constraints) in mechanical systems with a finite number of 
degrees of freedom-, that is, a finite number of movable parts; as opposed to contin¬ 
uous systems that have a countably infinite set of such freedoms. All relevant funda¬ 
mental concepts, definitions, equations — such as velocity, acceleration, constraint, 
holonomicity versus nonholonomicity, constraint stationarity (or scleronomicity) 
versus nonstationarity (or rheonomicity) — are detailed in both particle and system 
variables, along with elaborate discussions of quasi coordinates and the associated 
transitivity equations and Hamel coefficients', as well as a direct and readable (and 
very rare) treatment of Frobenius' fundamental necessary and sufficient conditions 
for the holonomicity, or lack thereof, of a system of Pfaffian constraints. 


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§2.2 INTRODUCTION TO CONSTRAINTS AND THEIR CLASSIFICATIONS 


The examples and problems, some at the ends of the paragraphs and some (the 
more comprehensive ones) at the end of the chapter, are an indispensable part of the 
material; several secondary theoretical points and results are presented there. 

This chapter, and the next one on Kinetics, constitute the fundamental essence and 
core of Lagrangean analytical mechanics. 


2.2 INTRODUCTION TO CONSTRAINTS AND THEIR CLASSIFICATIONS 

Positions, Configurations, Motions 

Let us consider a general finite mechanical system 5 consisting of N ( = positive 
integer), free, or unconstrained , material particles. The position r of a generic 
5-particle, P. at the generic time instant, t, relative to an “origin” fixed in a, say 
inertial, frame of reference, F, is defined by the vector function 

r=f(P,t)=r(P,t). (2.2.1) 

The collection of all these particle vectors, at a current instant t, make up a current 
system position , or current configuration of 5, C(f), and its evolution in time consti¬ 
tutes a motion of 5”. The latter, clearly, depends on the frame of reference. Thus, the 
complete description of a motion of 5, if the latter is modeled as a collection of N 
particles, requires (at most) knowledge of 3 N functions of time; for example, the 3 N 
rectangular Cartesian components = coordinates of the N r’s: 

(x l ,y l ,z 1 ;...;x N ,y N ,z N ) = ( x,y,z ) = (£i,... , £ 3 jv) = (2.2.1a) 

These numbers can be viewed as the rectangular Cartesian coordinates of the 
3/V-dimcnsional position vector of a single fictitious, or figurative, particle represent¬ 
ing 5, in a 3/V-dimensional Euclidean space, Et, n , henceforth called the system’s 
unconstrained configuration space', and, therefore, a motion of 5 can be visualized 
as the path traced by the tip of that system position vector in E 3N . Equation (2.2.1) 
can be replaced by 

r =f(r 0 , t', t 0 ) = r(r„, t\ t 0 ), (2.2.2) 

where (fig. 2.1): r 0 = “reference position” of P at the “reference time” t = t„, is used 
to distinguish, or label, the various 5-particles; and the totality of r„’s constitutes the 
reference configuration of 5 at t B , C(t 0 ). For a fixed r a and variable t (i.e., a motion 


Reference configuration: C(t a ) Current configuration: C(t) 



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CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS 


of P), eqs. (2.2.1, 2) give the path of a particle P that was initially at r„. (The same 
equations for fixed t and variable r 0 would give us the transformation of the spatial 
region initially occupied by the system, to its current position at time t.) 

The one-to-one correspondence between r (and /) and r 0 (and t 0 ), of the same 
particle P — that is, the physical fact that “initially distinct particles must remain 
distinct throughout the motion” — requires that (2.2.2) has an inverse : 

r 0 = /~'(r, t\ t 0 ) = g(r, t a ; t) : reference configuration at (variable) time t a . 

(2.2.2a) 

Switching the roles of (r, t) and ( r 0 ,t 0 ), we can view (2.2.2a) as expressing the 
“current” position r 0 in terms of the “reference” position and time (r. t) and 
“current” time t a . From now on, for simplicity, we shall drop, in the above, the 
explicit ( r 0 ,t 0 ) and/or F-dependence [also, replace the rigorous notation//...) with 
r(...), as done frequently in engineering mathematics, except whenever extra clarity 
is needed], and write (2.2.1) simply as 

r = r(i). (2.2.3) 


REMARKS 

(i) For (2.2.2) and (2.2.2a) to be mutually consistent, we must have 

(2.2.2) for t=t 0 ^ (2.2.2a): r = f(r 0 ,t\t 0 ) => r 0 =f(r 0 ,t 0 -,t 0 ) =f~\r,t-,t 0 )- 
(2.2.2a) for t a = t => (2.2.2): r 0 =f~\r,t;t 0 ) => r =f~'(r,t;t ) = f(r 0 ,t\t 0 ); 

(2.2.2b) 

hence, also 

r=f[f{r 0 ,ti\t 0 ),t-,t l ] =f(r 0 ,t;t 0 ), (2.2.2c) 

where t\ is another reference time. 

(ii) In continuum mechanics, (r 0 , t) and (r, t) are called, respectively, material (or 
Lagrangean) and spatial (or Eulerian ) variables; with the former preferred in solid 
mechanics (e.g., nonlinear elasticity), and the latter dominating fluid mechanics (e.g., 
hydrodynamics). (See, e.g., Truesdell and Toupin, 1960, and Truesdell and Noll, 1965.) 

(iii) For systems with a finite number of particles, the dependence on the latter is, 
frequently, expressed by the discrete subscript notation (i.e., r„ —> positive integer 
denoting the “name” of the particle): 

r P = r P (t) = {xp(t),y P (t),z P (t)} (P = 1,... ,N). (2.2.4) 

The simpler continuum mechanics notation, eqs. (2.2.1, 3), dispenses with all un¬ 
necessary particle indices, and allows one to concentrate on the system indices (as 
we begin to show later), which is the essence of the method of analytical mechanics. 
It also allows for a more general exposition; for example, a unified treatment of 
systems containing both rigid (discrete) and flexible (continuous) parts. 


Constraints 

If the N vectors r, and/or corresponding (inertial) velocities v = dr/dt , are functionally 
unrelated and uninfluenced from each other ( internally) or from their environment 
(externally), apart from continuity and consistency requirements, like (2.2.2b,c) — 

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§2.2 INTRODUCTION TO CONSTRAINTS AND THEIR CLASSIFICATIONS 


something we will normally assume — that is, if, and prior to any kinetic considera¬ 
tions, the r’s and v’s are free to vary arbitrarily and independently from each other, 
then S is called (internally and/or externally) free or unconstrained ; if not, 5 is called 
(internally and/or externally) constrained. In the latter case, certain configurations 
and/or (velocities =>) motions are unattainable, or inadmissible; or, alternatively, if 
we know the positions and velocities of some of the particles of the system, we can 
deduce those of the rest, without recourse to kinetics. [Outside of areas like astron¬ 
omy/celestial mechanics, ballistics, etc., almost all other Earthly systems of rele¬ 
vance, and a lot of non-Earthly ones, are constrained — hence, the importance of 
analytical mechanics, especially to engineers.] 

Such restrictions, or constraints, on the positions and/or velocities of S' are 
expressed analytically by one or more (< 3N) scalar functional relations of the form 

f(t,r u ...,r N ;v u ...,v N ) = 0, or, compactly, f(t,r,v) = 0. (2.2.5) 

These equalities are assumed to be; (i) continuous and as many times differentiable in 
their arguments as needed (usually, continuity of the zeroth, first-, and second-order 
partial derivatives will suffice), in some region of the (x,y,z-dx/dt,dy/dt,dz/dt;t); 
(ii) mutually consistent (i.e., kinematically possible, or admissible); (iii) independent 
[i.e., not connected by additional functional relations like F(f ,/ 2 ,...) = 0]; and (iv) 
valid for any forces acting on S, any motions of it, and any temporal boundary! initial 
conditions on these motions (see also semiholonomic systems below). 

Following ordinary differential equation terminology, we call (2.2.5) a first-order 
(nonlinear) constraint, or nonlinear velocity constraint. With few exceptions [as in 
chaps. 5 and 6, where generally nonlinear constraints of the form /(r, r, a, t) =0 
(a: accelerations) are discussed], the velocity constraint (2.2.5) is the most general 
constraint examined here. 

[Other, perhaps more suggestive terms, for constraints are conditions (Victorian 
English: equations of condition', German: bedingungen), and connections or couplings 
(French: liaisons', German: bindungen', Greek: avvbea[ioi', Russian: svyaz’).] 


Special Cases of Equation (2.2.5) 

(i) Constraints like 

</>(?,r) = 0, or [recalling (2.2.1a)], </>(?,£) = 0, (2.2.6) 

are called finite, or geometrical, or positional, or configurational, or holonomic. [From 
the Greek: holos = complete, whole, integral; that is, finite, nondifferential; and 
nomas = law, rule, (here) condition, constraint. After Hertz (early 1890s); also C. 
Neumann (mid-1880s).] 

(ii) Again, with the exception of chapters 5, 6, and 7, all velocity constraints 
treated here have the practically important linear velocity, or Pfajfian, form 

/= S(B.v) + B = 0, (2.2.7) 

where B = B{t,r), B = B(t,r) are known functions of the r’s and t, and Lagrange’s 
symbol S (• • •) signifies summation over all the material particles of S, at a given 
instant, like a Stieltjes’ integral (so it can handle uniformly both continuous and 
discrete situations). Those uncomfortable with it may replace it with the more famil¬ 
iar Leibnizian f (...). 

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CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS 

Multiplying (2.2.7) by dt , which does not interact with S (■ ■ •), we obtain the 
kinematically possible, or kinematically admissible, form of the Pfaffian constraint, 

fdt=^(B-dr) + Bdt = 0. (2.2.7a) 


Degrees of Freedom 

A system of N particles subject to h (independent) positional constraints: 

Mt,r)= 0 (H=l,...,h), (2.2.8) 

and m (independent) Pfaffian constraints: 

f D = S(B d -v) + B d = 0 (D=l,...,m), (2.2.9) 

that is, a total of h + m constraints, is said to have a total of 3 N — (h + m)(> 0) 
degrees of freedom (DOF). This is a fundamental concept whose significance to 
both kinematics and kinetics (of constrained systems) will emerge gradually in 
what follows. 

[Quick preview: DOF = Number of independent components of system vector of 
virtual displacement (§2.3-7) 

= Number of kinetic (i.e., reactionless) equations of 
motion of system (chap. 3).] 


Holonomicity versus Nonholonomicity 

A positional constraint like (2.2.6), since it holds identically during all system motions, 
can always be brought to the velocity form (2.2.7) by d(.. .)/r/t-differentiation: 

d(f>/dt = (df/dr) • v + df/dt = 0; (2.2.10) 

that is, B —> dtp/dr = grad 0 (normal to the E-iN -surface </> = 0) and B —> df/dt. 
However, the converse is not always true: the velocity constraint (2.2.7) may or 
may not be (able to be) brought to the positional form (2.2.6); that is, by integration 
and with no additional knowledge of the motion of the system; namely, without 
recourse to kinetics. If (2.2.7) can be brought to the form (2.2.6), then it is called 
completely integrable , or holonomic (H); if it cannot, it is called nonintegrable, or 
nonholonomic (NH); or, sometimes, anholonomic. This holonomic/nonholonomic 
distinction of velocity constraints is fundamental to analytical mechanics; it is by 
far the most important of all other constraint classifications. [The term anholonomic, 
more consistent than the term nonholonomic seems to be due to Schouten (1954).] 
Schematically, we have 



Holonomic 


Integrable or Holonomic _ 


Nonintegrable or Nonholonomic _ 


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§2.2 INTRODUCTION TO CONSTRAINTS AND THEIR CLASSIFICATIONS 


Hence, a H velocity constraint, like (2.2.10), is actually a positional constraint 
disguised in kinematical form. Before embarking into a detailed study of H/NH con¬ 
straints, we will mention some additional, secondary but useful, constraint classifica¬ 
tions. 


Scleronomicity versus Rheonomicity 

Velocity constraints of the form 

/(r, v) = 0 => df/dt = 0 (2.2.7b) 

are called stationary, otherwise (i.e., if df /dt f 0), they are called nonstationary. If all 
the constraints of a system are stationary, the system is called scleronomic; if not, the 
system is called rheonomic. [From the Greek: scleros = hard, rigid, invariable; 
rheo = to flow; and the earlier nomos = law, rule, decree, (here) condition; that is, 
scleronomic = invariable constraint, rheonomic = variable/fluid constraint. After 
Boltzmann (1897-1904).] For positional constraints and Pfaffian constraints, 
stationarity means, respectively, 

<f>(r) = 0 and $B(r)-v = 0. (2.2.11) 

Catastaticity versus Acatastaticity 
Pfaffian constraints of the form 

$B(t,r)-v + B(t,r) = 0, (2.2.11a) 

are called acatastatic ; while those of the form 

$B(t,r)-v = 0 [i.e, B(t,r) = 0] (2.2.11b) 

are called catastatic. It is this classification [due to Pars (1965, pp. 16, 24) and, 
obviously, having meaning only for Pfaffian constraints], and not the earlier one 
of scleronomicity versus rheonomicity, that is important in the kinetics of systems 
under such constraints. 

REMARKS 

(i) The reason for calling the second of (2.2.11) scleronomic, instead of 

$B(r)-v + B(r) = 0, (2.2.11c) 

that is, for requiring that scleronomic constraints linear in the velocities be also 
homogeneous in them (i.e, have B = 0 =>■ catastaticity), is so that it matches the 
kinematic form generated by d/dt{. . ^-differentiating the scleronomic positional 
constraint (first of 2.2.11): 

<j>{r) = 0 => df/dt = $ (df/dr) • v = 0. (2.2. lid) 

Geometrical interpretation of this requirement: Otherwise, the corresponding con¬ 
straint surface, in “velocity space,” would be a plane with distance from the origin 
proportional to B. That term, representing the (negative of the) velocity of the 

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CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS 


constraint plane normal to itself, is clearly a rheonomic effect. (Remark due to Prof. 
D. T. Greenwood, private communication.) 

(ii) Clearly, every scleronomic Pfaffian constraint is catastatic (B = 0); but cata- 
static Pfaffian constraints may be scleronomic [B = B{r), second of (2.2.11)] or 
rheonomic [B = B(t,r), (2.2.11b)]. 


Bilateral versus Unilateral Constraints 

Equality constraints of the form (2.2.5) are called bilateral, or two-sided , or equality, 
or reversible, or unchecked (after Langhaar, 1962, p. 16); while constraints of the form 

f{t,r,v)> 0 or f(t,r,v) < 0 (2.2.1 le) 

are called unilateral, or one-sided, or inequality, or irreversible. Physically, bilateral 
constraints occur when the bodies in contact cannot separate from each other: for 
example, a rigid sphere moving between two parallel fixed planes, in continuous 
contact with both. In the unilateral case, the bodies in contact can separate: for 
example, a sphere in contact with only one plane, or a system of two particles 
connected by an inextensible string—their distance cannot exceed the string’s length. 
Following Gantmacher (1970, p. 12), we can state that the general motion of a 
unilaterally constrained motion may be divided into segments, such that: (i) in 
certain segments the constraint is “taut” [(2.2.lie) with the =sign; e.g., particle on 
a light, inextensible, and taut string], and motion occurs as if the constraint were 
bilateral', and (ii) in other segments, the constraint is not taut, it is “loose,” and 
motion occurs as if the constraint were absent. Concisely, a unilateral constraint is 
either replaced by a bilateral one, or is eliminated altogether. Hence, in what follows, 
we shall limit ourselves to bilateral constraints. 

REMARKS 

(i) A small number of authors call all constraints of the form (2.2.6) holonomic, as 
well as those reducible to that form; and call all others nonholonomic. According to 
such a definition, bilateral constraints like (2.2.lie) would be nonholonomic! The 
reader should be aware of such historically unorthodox practices. 

(ii) The equations (j){r, t) = 0 and dcj)/dt = S (<9</>/<9r) • v + dcjr/dt = 0 restrict a 

system’s positions and velocities; equation dcj>/dt = 0 is the compatibility of veloci¬ 
ties with (j> = 0. Similarly, the equation 

d 2 (f)/dt 2 = [d/dt(dcj)/dr) ■ v + ( df/dr ) •«] + d / dt(dcj) / dt) = 0 

is the compatibility of accelerations with (j> — 0, d<j>/dt = 0; and likewise for higher 
such derivatives. 

(iii) In the case of unilateral constraints, if at a certain time t: f > 0, then, as 
explained earlier, that constraint plays no role in the system’s motion. But if / = 0, 
then, as a Taylor expansion around t shows, motion that satisfies either of these two 
relations may occur; in the former case df /dt = 0, and in the latter df /dt > 0. Thus, 
the simultaneous conditions / = 0 and df /dt < 0 allow us to detect a possible 
incompatibility between velocities and / > 0. Usually, such conditions occur in 
impact problems (chap. 4; also Kilmister and Reeve, 1966, pp. 67-68). 

(iv) Geometrical!physical remarks: In a system S consisting of several rigid bodies, 
and its environment (i.e., other bodies/foreign obstacles, massless coupling elements: 

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§2.2 INTRODUCTION TO CONSTRAINTS AND THEIR CLASSIFICATIONS 


e.g., springs, cables) the following conditions apply: 


(a) Every condition expressing the direct contact of two rigid bodies of S , or the contact 
of one of its bodies with a foreign obstacle (environment) that is either fixed or has 
known motion (i.e., its position coordinates are known functions of time only), 
results in a holonomic equation of the form (2.2.6); and the corresponding contact 
forces are the reactions of that constraint. 

(b) If, further, at those contact points, friction is high enough to guarantee us (in 
advance of kinetic considerations) slippinglessness, then the positions and velocities 
there satisfy (2.2.7)-like Pfaffian equations (usually, but not always, nonholonomic). 
These conditions express the vanishing of a component of (relative) slipping velocity 
in a certain direction; and, therefore, there are as many as the number of indepen¬ 
dent such nonslipping directions. 

(c) If, in addition, friction there is very high , so that not only slipping but also pivoting 
vanishes, then we have additional (usually nonholonomic) (2.2.7)-like equations; 
that is, linear velocity constraints arise quite naturally and frequently in daily life. 
[Nonslipping and nonpivoting are maintained by constraint forces (and couples), 
just like contact. All these constraint forces are examples of passive reactions; for 
more general, active , constraint reactions, see, for example, §3.17.] 


(v) Holonomic and/or nonholonomic constraints due exclusively to the mutual 
interaction of the system bodies are called internal (or mutual)', while those arising, 
even partially, from the interaction of the system with its environment are called 
external. The associated constraint reactions are called, respectively, internal (or 
mutual) and external. 

(vi) Finally, we repeat that such holonomic and/or nonholonomic constraints 
express restrictions among positions and velocities independently of the equations 
of motion and associated (temporal) initial/boundary conditions, and before the 
complete solution of the problem is carried out. Solving the problem means finding 
r = /•(/): known function of time; then v = clr/dt = v(t): known function of time; and 
these r’s and v’s automatically satisfy the constraints. Under such a viewpoint, 
integrals of the system, like those of linear/angular momentum and energy, assuming 
they exist, do not qualify as constraint equations. 

The (bilateral) constraints, discussed above, are summarized as follows: 


General first-order constraints 


/(') = 0 : 
f{t,r) = 0: 
fir, v) = 0: 
f{t,r,v) = 0: 


Holonomic (integrable) and scleronomic (stationary) 

Holonomic (integrable) and rheonomic (nonstationary) 
Nonholonomic (if nonintegrable) and scleronomic (stationary) 
Nonholonomic (if nonintegrable) and rheonomic (nonstationary) 


Pfaffian velocity constraints 

S B(t,r)-v+ B(t,r) = 0: 
SB(r)-v + B(r) = 0: 
$B(t,r)-v = 0: 

S B W-v = 0: 


Rheonomic and acatastatic 
Rheonomic and acatastatic 
Rheonomic and catastatic 
Scleronomic and catastatic 


(There is no such thing as scleronomic and acatastatic Pfaffian constraint.) 


249 


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CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS 



Figure 2.2 Plane pursuit problem: a dog (D) moving continuously 
toward its master ( M ). 


Example 2.2.1 Plane Pursuit Problem — Catastatic but Rheonomic ( or Nonstation¬ 
ary) Pfaffian constraint. The master ( M) of a dog ( D ) walks along a given plane 
curve: R = R(t) = {X = X(t), Y = T(/)}. Let us find the differential equation of 
the path of D: r = r(t) = {x = x(t),y = v(t)}, if D moves, with instantaneous velo¬ 
city v, to meet M, so that at every instant its velocity is directed toward M 
(fig. 2.2). 

We must have: 


v = parallel to R — r = v[(/? — r)/\R — r|] = ve, 
or, in components, 

dx/dt = v[(X — x)/\R — r\], dv/dt = v[( Y — y)/\R — r|]; (a) 

or, eliminating v between them, 

\Y(t) — y\(dx/dt ) — [df(r) — x](dy/dt) = 0. (b) 

It is not hard to show that this pursuit problem in space leads to the following 
constraints (with some obvious notation): 

[ Y{t ) — y\(dx/dt) — [X{t) — x)(dy/dt) = 0, (c) 

\Z{t) — z](dx/dt) — \X{t) — x\{d:/dt) = 0, (d) 

[Z(0 — z\(dy/dt) — [T(t) — y\(dz/dt) = 0. (e) 

See also Hamel (1949, pp. 770-773). 


Example 2.2.2 Acatastatic Constraints. Let us consider the rolling of a sphere 5 
of radius r and center G on the rough inner surface of a vertical circular cylinder 
A of radius R(> r). Let us introduce the following convenient intermediate axes/ 
basis G 123/ G-ijk (fig. 2.3): Let </> be the azimuth , or precession- like, angle of the 
plane G-13, and z = vertical coordinate of G (positive upward from some fixed 

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§2.2 INTRODUCTION TO CONSTRAINTS AND THEIR CLASSIFICATIONS 



TOP VIEW: 



Figure 2.3 Rolling of a sphere on a vertical circular cylinder. G7: vertically upward; 
G3: horizontally intersects the (vertical) cylinder axis; G2: horizontal, so that G-123 is 
orthogonal-normalized-dextral (OND). 


plane, perpendicular to the cylinder axis). Then, 

v G = inertial velocity of G = (v, = dz/dt , v 2 = (R — r)(d<j>/dt) = (R — r)w^,, v 3 = 0); 

(a) 

or, alternatively, if OG = zK + (R — r)(—k), then (with d(j)/dt = ujf) 

v G = d(OG)/dt = ( dz/dt)K + (R — r)(—dk/dt) = ( dz/dt)i + (R — r){u^j). (b) 

If co = inertial angular velocity of sphere = {to i, cd 2 , cd 3 ), then the inertial velocity of 
the contact point C, v c , is 

v c = V G + to x r c/G = (v 1 ,v 2 ,v 3 ) + (w 1 ,0) 2 ,w 3 ) x (0,0, -r) 

= (v 1 -w 2 r, v 2 +uj { r, v 3 ). (c) 

Therefore: (i) If the cylinder is stationary (i.e., fixed), the rolling constraint is v c = 0, 
or, in components, 

v l —u) 2 r=0 => w 2 = (dz/dt)/r, v 2 + w\r = 0 => u>\ = [1 — (i?/r)]uty, v 3 = 0. (d) 

(ii) If the cylinder is made to rotate about its axis with an (inertial) angular 
velocity Q = (2 (?) = given function of time, the rolling constraint is 

v c = SI x r c/0 = (QK) x (-JM) = (Qi) x (-Rk) = ( QR)j = (0,QR,0), 

or, in components [invoking (c)], 

Vj — uj 2 r = 0 => u 2 = (dz/dt)/r= v z /r, 
v 2 + (jj\r = Q(t)R =$■ u>\ = ojfj, + (i?/r)w r , v 3 = 0, 

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(<=> 
























CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS 


where oj r = Q — = relative angular velocity of cylinder about meridian plane 

G-13. The first of the constraints (e) is nonstationary and acatastatic, even if 
Q = constant. [As explained in §2.5 If., the virtual form of that constraint is 
6p 2 + 68 p- = 0, where dp 2 = v 2 dt and dd\ = co\ dt; and this coincides with the virtual 
form of the catastatic second of the constraints (d). In general, p 2 and 9 X are “quasi 
coordinates” — see §2.9 ff] 


First and second of the constraints (d) in terms of the Eulerian angles of the sphere 
<P, 0, T, relative to the “semiinertial” (translating but nonrotating) axes G XYZ 

We have, successively (recalling §1.12,13), 

V[ = dz/dt = v z , 

co 2 = cos(2, X)cVy T cos(2, T)cuy T cos(2, Z)tu z 
= (— sin 4>)ojx T (cos T (0 )(u z 
= (— sin</>)[cos A>(d0/dt) + sin <P sin &(d'F/dt)\ 

+ (cos 0) [sin <P{d©/dt) — cos <P sin ©(d'F/dt)] 

= ■ ■ • = sin(«P - (j)){d©/dt) - cos($ - (j>) sin ©(d'F/dt), (f) 

that is, the familiar co Y component but with (j> replaced by d> — </>; 
v 2 = {R~ r)(dcj)/dt), 

toi = cos(l, X)cux T cos( 1, Y)lo y T cos( 1, Z)oJz 

= (0)cuy + (0)wy + (l)w z = dd>/dt + cos 0( d'F/dt). (g) 

Therefore, the first and second constraints (d) transform to 

Vi — u) 2 r = dz/dt — r[sin(fp — (/){d©/dt) — cos(<P — (j>) sin 0(d < F/dt)\ = 0, (h) 
v 2 + uqr = (R — r){d(f)/dt) + r[d$/dt + cos ©(d'F/dt)] =0; (i) 

and similarly for the first two of (e). 

Example 2.2.3 Acatastatic Constraints. Let us consider the rolling of a sphere S 
of radius r and center G on a rough surface of revolution with a vertical axis. Let 
us introduce the convenient frame/axes/basis G 123/G ijk shown in fig. 2.4. 
Lurther, let </> be the azimuth, or precession- like, angle of the meridian plane (and 
of plane G-23); and 8 be the nutation -like angle between the positive surface axis 
and the common (outward) normal. Then, with dr//dt = co^, dd/dt = co e , we will 
have 

(i 0 = inertial angular velocity of G —123 = (f2 1 ,f2 2 ,f2 3 ) 

= (-a^sinfl, u^cosfl), (a) 

v G = inertial velocity of G = (v 3 , v 2 ,v 3 ) = (pco s , Rco^ = psindco^, 0), (b) 

where p = radius of curvature of meridian curve of parallel surface at G. 

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§2.2 INTRODUCTION TO CONSTRAINTS AND THEIR CLASSIFICATIONS 




Figure 2.4 Rolling of a sphere on a vertical surface of revolution. C3: along common normal, 
outward; Cl: parallel to tangent to meridian curve, at contact point C; C2: parallel to tangent 
to circular section through C (or, so that G-123 is OND). 


If m = inertial angular velocity of sphere = (uii,ui 2 , w 3 ), then the inertial velocity 
of the contact point C, v c , equals 


v c — v g + 03 x r C/G 

= (vi,v 2 ,v 3 )+ (w 1 ,W 2 ,w 3 ) X (0,0, -r) = (vj -u 2 r, v 2 +uj l r, v 3 ). (c) 

Therefore: (i) If the surface is stationary, the rolling constraint is v c = 0 , or, in 
components, 


Vi — uJ 2 r = 0, v 2 + u\r= 0, v 3 = 0. (d) 

(ii) If the surface is compelled to rotate about its axis with (inertial) angular 
velocity Q = Q(t) = given function of time, the rolling constraint is 

v c = i2 x r c/0 = (0,Q(R - rsind),0), (e) 


or, in components, 


V; — tu 2 r = 0 => u> 2 = V\/r = pug/r, (f) 

v 2 + W|T = Q(R — rsin0) =>■ uji = (R/r)u> r — Qsin9, v 3 = 0, (g) 

where ui r = Q — luj, = relative angular velocity of surface about meridian plane G—13. 
The first constraint (g) is nonstationary and acatastatic, even if Q = constant. 
[As explained in §2.5 ff., the virtual form of that constraint is 8p 2 + 60\r = 0, 
where dp 2 = v 2 dt and d6 x = df, and it coincides with the virtual form of the 
catastatic second constraint (d). In general, p 2 and 6\ are “quasi coordinates”— 
see §2.9 ff.] 


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CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS 


SPECIALIZATIONS 

(i) If the surface of revolution is another sphere with radius p a = p — r = constant , 
since then Vi = (p„ + r)uj s , v 2 = [(p a + r) sin Q\w$, the constraints (f) and the second 
of (g) reduce, respectively, to 


w 2 — [(Po + r )/ r \ u S — [(Po/ r ) + 1]<U0, (h) 

w i = [(Po/ r ) + 1] sin day — 12 sin d. (i) 

(ii) If the surface of revolution is another sphere with radius p 0 = p — r, that is free 
(i.e., unconstrained) to rotate about its fixed center with inertial angular velocity 
(o' = (u\, u/ 2 ,u/ 3 ), then, reasoning as earlier, we obtain the catastatic constraint 
equations 

Vi - uj 2 r = pota'i, v 2 +u>i r=—p 0 u\, v 3 = 0. (j) 

However, if the u>\, uj' 2 , uj' 2 are prescribed functions of time, then the first and second 
of (j) become nonstationary (and acatastatic). 

For additional such rolling examples, including the corresponding Newton-Euler 
(kinetic) equations, and so on, see the older British textbooks: for example, Atkin 
(1959, pp. 253-259), Besant (1914, pp. 353-359), Lamb (1929, pp. 162-170), Milne 
(1948, chaps. 15, 17). 

Example 2.2.4 Problem of Ishlinsky (or Ishlinskii). Let us consider the rolling 
of a circular rough cylinder of radius R on top of two other identical circular and 
rough cylinders, each of radius r, rolling on a rough, fixed, and horizontal plane 
(fig. 2.5). 

Let O-xyz and O-x'y'z' be inertial axes, such that O-xy and O-x'y' are both on 
that plane, while their axes Ox and Ox' are parallel to the lower cylinder generators 





Figure 2.5 Rolling of a cylinder on top of two other rolling cylinders. Transformation equations: 
x = x' cosx - y'sin %, Y = x ' sin X + y' cosx- 


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§2.2 INTRODUCTION TO CONSTRAINTS AND THEIR CLASSIFICATIONS 


and make, with each other, a constant angle x- To describe the (global) system 
motion, let us choose the following six position coordinates: (i) (x,y) = inertial co¬ 
ordinates of mass center of upper cylinder G (as for its third, vertical, coordinate we 
have z = 2r + R): (ii) 9 = angle between +Ox and upper cylinder generator; (iii) ip, 
ip l ,ip 2 = spin angles of the upper and two lower cylinders, respectively. Finally, let r, 
and r 2 be the position vectors of the contact points of the lower cylinders with the 
upper one, relative to G, and v,, v 2 be the corresponding (inertial) velocities. 

The rolling constraints are 

v G + co x r i = Vj and v G + m x r 2 = v 2 . (a) 

Let us express them in terms of components along O-xyz. We have 
v G = (dx/dt,dy/dt, 0) = (v*, v^O) 
co: inertial angular velocity of upper cylinder 

= ((dep/dt) cos 9, ( dcp/dt) sin 9,d9/dt) = (u;^ cos 9, sin 6, uj g ), 

»'i = (-Lv-rcpi) cot 9,-{y-r<p{),-R), 

r 2 = [r<P 2 - 0cosx - x sinx)] cot (9 - x)i' + [r<p 2 ~ (y cos x~x sin x)\j' - Rk' 

= ({ r <t> 2 + x sin x - y cos x) cos 9/ sin(6> - x), 

(rcp 2 + xsinx — jcos x) sin 9/ sin(0 — x), — R), 

= (0,2rwi,0) [where to 1>2 = dcp\ 2 /dt \, 

v 2 = (-2 nv 2 sin x, 2 ru 2 cos x, 0). (b) 

Substituting the above into (a), we obtain the following four constraint components: 

Ft - R^<t, sin 9 - uoircpi - y) = 0, 

v y + Rw^ cos 9 + u) e (rcp l — y) cot 9 — 2no l = 0; 

v x sin(0 — x) — Rco^ sin 9 sin(0 — x) — u e (rcp 2 + x sin x — y cos x) sin 9 

+ 2ruj 2 sin x sin(0 — x) = 0, 

v v sin(0 — x) + R^<p cos 9 sin(0 — x) + w s(t(/> 2 + x sin x — v cos x) cos 9 

— 2rw 2 cosxsin(0 —x) =0. (c) 

For further details, see, for example, Mei (1985, pp. 33-35), Neimark and Fufaev 
(1972, pp. 99-101). It can be shown (§2.11, 12) that these constraints are non- 
holonomic. Therefore, the system has n = 6 globed DOF , and n — m = 6 — 4 = 2 
local DOF (concepts explained in §2.3 ff). 


Example 2.2.5 When is Rolling Holonomic? So as to dispell the possible notion 
that all problems of (slippingless) rolling among rigid bodies lead to nonholo- 
nomic constraints, let us summarize below the cases of rolling that lead to holo¬ 
nomic constraints. It has been shown by Beghin (1967, pp. 436-438) that these are 
the following two kinds: 

(i) The paths of the contact point(s) of the rolling bodies are known cdtead of time ; 
that is, before any dynamical consideration of the system involved and as function of 

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CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS 


its original position, on these bodies. Consider two such bodies whose bounding 
surfaces, 5) and S 2 , are described by the curvilinear surface (Gaussian) coordinates 
(u 1; v t ) and (u 2 , v 2 ), respectively, in contact at a point C. Their relative positions, say 
of Si relative to S 2 , are determined by the values of these coordinates at C and the 
angle </> formed by the tangents to the lines u\ = constant and u 2 = constant (or v 1? 
v 2 = constant) there. Knowledge of the paths of C on both Si and S 2 translates to 
knowledge of the four holonomic functional relations: 

«i = «i(vi), u 2 = u 2 {v 2 ), (j) = (j){u u u 2 ), 5i(m,) = s 2 (u 2 ) ± c; (a) 

where .s, and s 2 are the arc lengths (or curvilinear abscissas) of the contact point 
paths S[ and S 2 , and c is an integration constant. It follows that, out of the five 
surface positional parameters, U\ , vj, u 2 , v 2 , </>, only one is independent; the other four 
can be expressed in terms of that one by finite (holonomic) relations. 

(ii) The bounding surfaces S[ and S 2 are applicable on each other; they touch at 
homologous points and their homologous curves (trajectories of the contact point C 
on them) join together there. This is expressed by the condition of contact, and by 

Ml = U 2 , V! =v 2 , (j) = 0, (b) 

at C; that is, again, a total of four holonomic equations. This condition is guaranteed 
to hold continuously if it holds initially and, afterwards, the pivoting vanishes. Such 
conditions are met in the following examples: 

(a) Rolling of two plane curves (or normal cross sections of cylindrical surfaces Si and 
S 2 ) on each other, and expressed by .sy = ,s 2 ± c. 

(b) Rolling of a body on a fixed surface, which it touches on only two points. For 
example, the rolling of a sphere on a system made up of a fixed circular cylinder and 
a fixed plane perpendicular to it [fig. 2.6(a)], (If the cylinder rotates about its axis in 
a known fashion, the trajectories of the contact points on both plane and cylinder 
are known, but they are unknown on the sphere and, hence, such rolling is non- 
holonomic.) 

(c) Rolling of two equal bodies of revolution whose axes are constrained to meet and, 
initially, are in contact along homologous parallels, or meridians [fig. 2.6(b)], The 
pivoting of such applicable surfaces vanishes. 


(a) 


(b) 




Figure 2.6 Examples of holonomic rolling: (a) rolling of a sphere on a fixed 
circular cylinder and a fixed plane perpendicular to it; (b) rolling of a cone on 
another equal fixed cone. 

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§2.3 QUANTITATIVE INTRODUCTION TO NONHOLONOMICITY 


2.3 QUANTITATIVE INTRODUCTION TO NONHOLONOMICITY 

Let us examine the differences between holonomic and nonholonomic constraints, 
in some mathematical detail, for the simplest possible case: a single particle, with 
(inertial) rectangular Cartesian coordinates x, y, z, moving in space under the 
Pfaffian equation 


adx + bdy + cdz = 0, (2.3.1) 

where a, b, c = continuously differentiable functions of x, y, z. 

[The Pfaffian expression a dx + bdy + cdz is a special differential form of the first 
degree. The total or Pfaffian differential equation (2.3.1) is a specialization of the 
Monge form: 

0 = f (x, y, z; dx, dy, dz) = stationary and homogeneous in the velocity components 

[dx/dt, dy/dt, dz/dt), and hence (since t is absent) only 
path restricting. 

The Monge form is, in turn, a specialization of the general first-order partial differ¬ 
ential equation: 


F[f,x,y,z\ dx/dt, dy/dt, dz/dt) = 0.] 

Now, the constraint (2.3.1) may be nonholonomic or it may be holonomic in 
differential (or velocity) form; specifically, if (2.3.1) can become, through multiplica¬ 
tion with an appropriate integrating factor, p = p[x,y,z), an exact, or perfect, or 
total differential d<j> = df[x,y,z ) of a scalar function <f> = (f>[x,y,z): 

p(a dx + b dy + c dz) = df, (2.3.1a) 

from which, by integration, we may obtain the (rigid and stationary) surface: 

<f>(x,y,z) = constant, or z = z[x,y), (2.3.1b) 

then (2.3.1) is holonomic; if not, it is nonholonomic. 

[Since, as is well known, the two-variable Pfaffian a[x,y) dx + b[x,y) dy has 
always an integrating factor (in fact, an infinity of them), eq. (2.3.1) is the simplest 
possibly nonholonomic constraint. More on this below.] 

In particular, if p = 1 (i.e., dcf) = adx + bdy + cdz), the integrable Pfaffian dcj) is 
exact. Then, 


a = dcj)/dx, b = dcj)/dy, c=d(f/dz, (2.3.2) 

and so the necessary and sufficient conditions for (2.3.1) to be exact are that the 
first partial derivatives of a, b, c, exist and satisfy (by equating the second mixed 
(^-derivatives): 

da/dy = db/dx, da/dz = dc/dx, db/dz = dc/dy. (2.3.3) 

Equations (2.3.3) are sufficient for (2.3.1) to be completely integrable = holonomic; 
but they are not necessary: every exact Pfaffian equation is integrable, but every 
integrable Pfaffian equation need not be exact', in general, a p f 1 may exist, even 
though not all of (2.3.3) hold. In mechanics, we are interested in the holonomicity 
(= complete or unconditional) integrability, or absence thereof, of the constraints. 

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CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS 


Let us now make a brief detour to the general case', the system of m Pfalhan 
constraints in the n{> m) variables x = (xq,..., x n ), 

d'6o = a Dk{x)dxk = 0 [D = 1,..., m(< «)], (2.3.4) 

where rank(a Dk ) = m (i.e., these equations are linearly independent in a certain 
x-region), is called completely (or unconditionally ) integrable, or complete, or 
holonomic, if either (i) it is immediately integrable, or exact', that is, if the 
m d'Oi/s are the exact, or total, or perfect, differentials of m functions 
4>d = <t> d M : 


y a Dki x ) dx k = d<j> D (x)\ (2.3.4a) 

or (ii) each d'dp, although not immediately integrable, nevertheless admits a 
(nonzero) integrating factor d> D (x)', that is, if the 2m (not all zero) functions 
{<P D (x), <f>D{x)\ D = 1,... , m{< n)} and (2.3.4) satisfy 

d>\ d'9\ = <b\ (an dx i + • • • + a\ n dx n ) = df \ (x), 

. (2.3.4b) 

tn d 0 m — (ain i dx\ • • • -f- a mn dx n ) — d(b m (.r). 

or, compactly, <P D d'0 D = d> D fffao k dx] < ) = dcf> D (x), where the {dfo} are (linearly) 
independent. Summing (2.3.4b), over D, we also obtain its following consequence: 

y, d>o d'9 D = y <P D ^ y a Dk dx^J = y df k =d<t> — y (90/ dx k ) dx k 

=> y <b D a Dk = d(p/dx k . 

Clearly, in both cases, (2.3.4a, b), the constraints (2.3.4) are equivalent to the 
holonomic equations 


ffx) = C x ,...,f m {x) = C m , (2.3.4c) 

where the m constants {C D ; D = 1 ,m} are fixed throughout the motion of the 
system. (Elaboration of this leads to the concept of semiholonomic constraints, 
treated later in this section.) If the constraints (2.3.4) are nonintegrable, neither 
immediately nor with integrating factors, they are called nonholonomic; and the 
mechanical system whose motion obeys, in addition to the kinetic equations, such 
nonholonomic constraints, either internally (constitution of its bodies) or externally 
(interaction with its environment, obstacles, etc.), is called a nonholonomic system. 

An alternative definition of complete integrability of the system (2.3.4), equivalent 
to (2.3.4b), is the existence of m independent, that is, distinct, linear, combinations of 
the m d'9o that are exact differentials of the m independent functions/^*): 

/Tn d 9 1 + • • • + [nm d 9 m — dfy ,..., Pmi d 9\ • • * 4- Pmm d 9 m — df m , (2.3.4d) 

where p, DD t = Pdd'( x ), compactly, 

y p DD > d'9 D ' = df D o d'9 D = y M dd > dfjy (D, D 1 = 1,..., m), (2.3.4e) 

[where ( M DD i ) is the inverse matrix of (pdd 1 ), an d both (m x m) matrices are 
nonsingular] and, hence, yield the m independent integrals (hypersurfaces): 

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§2.3 QUANTITATIVE INTRODUCTION TO NON HOLONOMICITY 


f\ = Ci,... ,f m = c m ; that is, the system of eqs. (2.3.4) is completely integrable if 
there exists an m-parameter (n — m) -dimensional manifold that solves them. 
[Frobenius (1877) has shown that if m = n, or n — 1, then the system (2.3.4) is always 
completely integrable — more on this later.] 

Finally, calling the determinant of the coefficients p DD i the multiplicator of (2.3.4) 
[i.e., \p,DD'\ = 0)], and generalizing from the single constraint case (2.3.1), we 

can state that every multiplicator has always the form p F(f ,..., f D ), where F(.. .) 
is an arbitrary differentiable function of the /’s; that is, there exists an infinity of 
multiplicators. 

From the above, it immediately follows that in the case of a single Pfaffian 
equation in the n variables x = (xi,.... x n ) (i.e., for m = 1), complete integrability, 
in a certain x-domain, means that there exists, locally at least, a one-parameter family 
of (n — 1) -dimensional manifolds f(x) = <p(x) — constant = 0, which solves that 
equation. 

[We remark that the solutions of d'6 = a k (x)dxk = 0 are always one-dimen¬ 
sional manifolds, or curves: x k = x k (u), where u = curve parameter. And, generally, 
if the x are functions of the m(< n) new variables (), then 
Xk = x k (u\,..., u m ) is called an m-dimensional solution manifold of d'6 = 0, if, 
upon substitution into it, identical satisfaction results.] 


Problem 2.3.1 Verify that the sufficient (but non-necessary !) conditions for the 
complete integrability of the system of m Pfaffian equations [essentially the dis¬ 
crete version of (2.2.9) for a system of N particles], 

fn dt = Yl ( a Dk dx k + b Dk dy k + c Dk dz k ) + e D dt = 0, (a) 

where D = 1,... ,m(< 3 N), k = 1 and ( a,b,c,e) = continuously differenti¬ 

able functions of (x,y,z, t), are that 


da Dk /dx l 

db Dk /dy l 

dc Dk /dz, 


da D i/dx k , 

db D i/dy k , 
dc m /dz k , 


da Dk /dy , 

db Dk /dz, 
dc Dk /dt = 


= db DI /dx k , 

= dc Dl /dy k , 
de D /dz k ; 


da Dk ldzt = dc m /dx k , 
da Dk /dt = de D /dx k ; (b) 

db Dk /dt = de D /dy k ; (c) 

(d) 


for all k, l = 1,... ,N, for a fixed D. [In fact, the (obvious) choice: a Dk = df D /dx k , 
b D k = df D /dy k , c Dk = d(j) D /dz k , e D = d(j) D /dt; <f> D = x,y,z) satisfies (b-d).] 
Then, (a) simply states that df D = 0; and the latter integrates immediately to the 
holonomic constraints: <p D = (f> D (t-,x,y,z) = (constant) D . 


Introduction to Necessary and Sufficient Conditions for Holonomicity 

Let us, for the time being, postpone the discussion of the general case and return to 
the single Pfaffian equation in three variables, eq. (2.3.1), and find the necessary and 
sufficient conditions for its holonomicity. Assuming that this is indeed the case, then 
from (2.3.1) and the second of (2.3.1b) we readily see that 

dz = (dz/dx) dx + ( dz/dy) dy = (— a/c ) dx + (—b/c ) dy (2.3.5) 

must hold for all dx, dy, dz. Therefore, equating the coefficients of dx and dy of both 

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260 CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS 

sides, we obtain [assuming c f 0, and that z(x,y) is substituted for z in a, b, c] 

dz/dx = — (a/c) and dz/dy =—(b/c), (2.3.5a) 

and since d/dy(dz/dx) = d/dx(dz/dy), we obtain d/dy(a/c) = d/dx(b/c), or, 
explicitly, 

c[da/dy + (da/dz)(dz/dy)\ — a[dc/dy + {dc/dz){dz/dy)\ 

= c[db/dx + (db / dz)(dz / dx)\ — b[dc/dx+ (dc/dz) (dz/dx)], 

and inserting in it the dz/dx- and <9z/<9y-values from (2.3.5a), and simplifying, we 
finally find 

I = a(db/dz — dc/dy) + b(dc/dx — da/dz) + c(da/dy - db/dx) = 0. (2.3.6) 

Equation (2.3.6), being a direct consequence of the earlier mixed partial derivative 
equality, is the necessary and sufficient condition for (2.3.1) to be holonomic. If I = 0 
identically (i.e., for arbitrary x,y,z ), then (2.3.1) is holonomic; if / ^ 0 identically, 
then (2.3.1) is nonholonomic. 

REMARKS 

(i) The form I is symmetric in (x,y,z) and ( a,b,c ); that is, it remains unchanged 
under simultaneous cyclic changes of (x,y,z) and (a,b,c ). 

(ii) Alternative derivation of equation (2.3.6): The mixed partial derivatives rule 
applied to (2.3.1a) readily yields 

d(pb) /dx = d(fia) / dy, d(/j,c) / dx = dffia) / dz, dffic) /dy = d(ffi) /dz. 

Multiplying the above equalities with c, b, a, respectively, and adding them together, 
we obtain (2.3.6); so, clearly, the latter is necessary and sufficient for the existence 
of an integrating factor (for further details, see, e.g., Forsyth, 1885 and 1954, 
pp. 247 ff.). 

(iii) A special case: If a = a(x,y), b = b(x,y), and c = 0, then, clearly, 7 = 0; 
which proves the earlier claim that the two-variable Pfaffian equation 
a(x,y) dx + b(x,y) dy = 0 is always holonomic, that is, for nonholonomicity, we 
need at least three variables. 

(iv) A special form: If (2.3.1) has the equivalent form 

dz = (—a/c) dx + (—b/c) dy = A(x,y,z) dx + B(x,y,z) dy 
= A[x, y, z(x, y)] dx + B[x, y, z(x, y)\ dy 

= A*(x,y) dx + B*(x,y) dy, (2.3.7) 

(or, similarly, dx = • • •, dy = • • •; depending on analytical convenience and/or avoid¬ 
ance of singularities), then the mixed partial derivative rule 

dA*(x,y)/dy = dB*(x,y)/dx, (2.3.7a) 

due to the chain rule (one should be extra careful here): 

dA*/dy = dA/dy+ ( dA/dz)(dzjdy) = dA/dy+ (dA/dz)B, (2.3.7b) 

dB*/dx = dB/dx+ (dB/dz)(dz/dx) = dB/dx + (dB/dz)A, (2.3.7c) 


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§2.3 QUANTITATIVE INTRODUCTION TO NON HOLONOMICITY 


finally yields 


dA/dy+(dA/dz)B=dB/dx+(dB/dz)A; (2.3.7d) 

whose identical satisfaction in .a, y, z, is the necessary and sufficient condition for 
the complete integrability, or holonomicity, of (2.3.7). 

It is not hard to verify that (i) replacing, in (2.3.7d), A with — a/c and B with —b/c, 
we recover (2.3.6); and, conversely, (ii) since (2.3.7) can be written in the (2.3.l)-like 
form: A dx + Bdy + (—1) dz = 0, replacing, in (2.3.6), a, b , c, with A, B, —1, respec¬ 
tively, we recover (2.3.7d). If, in (2.3.7), dA/dz = 0 and dB/dz = 0, then (2.3.7d) 
reduces to dA/dy = dB/dx. Finally, the sole analytical requirement here is the con¬ 
tinuity of all partial derivatives appearing in these conditions (but not those of the 
nonappearing ones, such as dA/dx and dB/dv). 


Example 2.3.1 Let us test, for complete integrability, the following constraints: 

(i) dz = (z) dx + (z 2 + a 2 ) dy; (ii) dz = z( dx + .v dy). 

(i) Here, A = z and B = z 2 + a 2 , and therefore (2.3.7d) yields 

(l)(z 2 + a 2 ) = (2 z)z => z 2 = a 2 , 

that is, no identical satisfaction; or, our constraint is not completely integrable — it is 
nonholonomic. Then, the original equation becomes 

dz = z dx + 2 z 2 dy; 

and so (a) if a = 0, then z = 0 is a constraint integral; but (b) if a ^ 0, then there is no 
integral. For complete integrability, we should have an infinity of integrals depend¬ 
ing on an arbitrary integration constant. 

(ii) Here, the test (2.3.7d) gives xz = z + xz =>■ z = 0; that is, no identical satis¬ 
faction, and therefore no holonomicity. As the original equation shows, this is the 
sole integral. 


Problem 2.3.2 Show that the constraint of the plane pursuit problem (ex. 2.2.1): 

[F(t) — y](dx/dt) — [A(0 — x](dy/dt) = 0, (a) 

or, equivalently, 

[Y(t) — y\dx — \X{t) — x] dy + (0) dt = 0, (b) 

is holonomic if and only if 

[X(t) — x\ / [Y (t) — y\ = (dX / dt) / (dY / dt) [= (dx/dt)/(dy/dt)\. (c) 

Problem 2.3.3 Show that under a general one-to-one (nonsingular) coordinate 
transformation (x,y,z) (u, v, w)\ x = x(u. v, w), y = •••, z = ■■■, 

I = [d(u,v,w)/d(x,y,z)\-l'; 

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(a) 



CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS 


where (with subscripts denoting partial derivatives) 

dO = a dx + bdy + c dz = pdu + qdv + r dw, (b) 

I = I(x,y, z) = a(b, - c y ) + b(c x - a z ) + c(a y - b x ), (c) 

I' = I'{u, v , w) =p(q w - r v ) + q(r u -p w ) + r{p v - q u ), (d) 

and d(u,v,w)/d(x,y,z) = Jacobian of the transformation (f 0); that is, / and I' 
vanish simultaneously; or, the holonomicity of dd = 0, or absence thereof, is co¬ 
ordinate invariant, and hence an intrinsic property of the constraint (a proof of this 
fundamental fact, for a general 1 1 fa 111 an system, will be given later). 

[Incidentally, the transformation law (a) also shows that scalars like / are not 
necessarily invariants (/ f /', in general); in fact, in the more precise language of 
tensor calculus, they are called relative scalars of weight +1, or scalar densities', see, 
e.g., Papastavridis (1999, pp. 46-49).] 


Geometrical Interpretation of the Pfaffian Equation (2.3.1) 

The latter, rewritten with the help of the vectors dr = (dx, dy, dz) and h = (a, b, c) as 

h- dr — 0, (2.3.8) 


means that, at each specified point Q(x,y,z), dr must lie on a local plane perpendi¬ 
cular to the “constraint coefficient vector” h there; or, that the particle P can move 
only along those curves, emanating from Q, whose tangent is perpendicular to h. 
Such curves are called kinematically admissible, or kinematically possible. If (2.3.1,8) 
is holonomic, then all motions lie on the integral surface (2.3.1b); that is, (2.3.6) is the 
necessary and sufficient condition for the existence of an orthogonal surface through 
Q , for the field h = ( a,b,c ) [actually, a family of surfaces <f> = <f>(x,y,z) = constant, 
everywhere normal to h — see below]. We also notice that, with the help of h, the 
condition (2.3.6) takes the memorable (invariant) form: 


/ = h- curl h = 0; or, symbolically, 


a b c 
d/dx d/dy d/dz 
a b c 


0; (2.3.8a) 


that is, at every field point. It is parallel to the plane of its rotation, or perpendicular 
to that rotation and tangent to the surface <f> = constant there [W. Thomson (Lord 
Kelvin) called such fields doubly lamellar ]; while (2.3.7d), with h —> H = (A, B, — l), 
becomes 

H ■ curl H = dA/dy + B(dA/dz) - dB/dx - A(dB/dz ) = (l/c 2 )/z • curl h = 0. 

(2.3.8b) 


Vectorial Derivation of Equation (2.3.8a) 

We recall from vector analysis that a (continuously differentiable) vector is called 
irrotational, or singly lamellar, if (a) its line integral around every closed circuit 

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§2.3 QUANTITATIVE INTRODUCTION TO NONHOLONOMICITY 


vanishes, or, equivalently, (b) if its curl (rotation) vanishes, or (c) if it equals the 
gradient of a scalar. 

Now: (i) If h = ( a,b,c ) is irrotational, then there is a ej> = cj)(x,y,z ) such that 
h = grad <j>, and, therefore, h ■ dr = grad 0 • dr = do = exact differential. 

(ii) If h f irrotational, still an integrating factor (IF) p, = p(x,y,z ) may exist so 
that // h = grad <f>. Then, as before, // h • dr = grad o • dr = def> = exact differential. 

(iii) Conversely, if p = IF, then p h = grad <j> = irrotational; and “curling” both 
sides of this latter, we obtain: 0 = curl(grad 0) = p curl h I- grad p x h, and dotting 
this with h. 0 = p(h • curl It), from which, since pfi 0, we finally get (2.3.8a). In this 
case, since h and grad <j> are parallel: It = (1 / p) grad 0 = v(grad 0), and, therefore, 
curl h = curl (a grad (f>) = grad v x grad 0, so that 

h ■ curl h = v grad <b • ( grad a x grad (f> ) = 0; (2.3.8c) 

that is, the doubly lamellar field h is perpendicular to its rotation curl It. [This condi¬ 
tion is necessary for the existence of an IF. For its sufficiency, see, for example, Brand 
(1947, pp. 200,’230-231), Sneddon (1957, pp. 21-23); also’Coe (1938, pp. 477-478), 
for an integral vector calculus treatment.] These derivations are based on a general 
vector field theorem according to which an arbitrary vector field can be written as the 
sum of a simple and a complex (or doubly) lamellar field: h = grad f + v grad (f). 

Finally, if the Pfaffian constraint is, nonholonomic, then (2.3.1,7) yield one¬ 
dimensional “nonholonomic manifolds”; that is, space curves orthogonal to the 
field h (or H), and constituting a one-parameter family on an arbitrary surface. 


Accessibility 

The restrictions on the motion of the particle P in the two cases 7=0 (holonomic) 
and If 0 (nonholonomic) are of entirely different nature. If I = 0, then P is obliged 
to move on the surface </> = <j)(x,y,z) = 0. If, on the other hand, 7^0, then the 
constraint (2.3.1) does not restrict the (x,y,z), but does restrict the direction 
(velocity) of the curves through a given point (x,y,z). The cumulative effect of 
these local restrictions in the direction of motion (velocity) is that the transition 
between two arbitrary points is not arbitrary, P can move (or be guided through) 
from an arbitrary initial (analytically possible) position, to any other arbitrary fined 
(analytically possible) position, while at every point of its path satisfying (2.3.1, 8); 
that is, the particle can move from “anywhere” to “anywhere,” not via any route we 
want, but along restricted paths. As Langhaar puts it, the particle is “constrained to 
follow routes that coincide with a certain dense network of paths ” (1962, pp. 5-6); like 
kinematically possible tracks guiding the system. 

In sum: (i) Holonomic constraints do reduce the dimension of the space of acces¬ 
sible configurations, but do not restrict motion and paths in there; in Hertz’s words: 
“all conceivable continuous motions [between two arbitrary accessible positions] are 
also possible motions.” 

(ii) Nonholonomic constraints do not affect the dimension of the space of acces¬ 
sible configurations, but do restrict the motions locally (and, cumulatively, also 
globally) in there; not all conceivable continuous motions (between two arbitrary 
accessible positions) are possible motions (Hertz, 1894, p. 78 ff.). 

These geometrical interpretations and associated concepts are extended to general 
systems in §2.7. 

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CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS 


Degrees of Freedom 

The above affect the earlier DOF definition: they force us to distinguish between 
DOF in the large (measure of global accessibility, or global mobility) and DOF in the 
small (measure of local/infinitesimal mobility). We define the former, DOF(L ), as the 
number of independent global positional (or holonomic) “parameters,” or 
Lagrangean coordinates = n (= 3 in our examples, so far); and the latter, 
DOF(S) = /, as n minus the number of additional (possibly nonholonomic) inde¬ 
pendent Pfaffian constraints: f = n — m(> 0). In the absence of the latter, 
DOF(L) = DOF(S): f = n. This fine distinction between DOFs rarely appears in 
the literature, where, as a rule, DOF means DOF in the small. {For enlightening 
exceptions , see, for example, Sommerfeld (1964, pp. 48-51); also Roberson and 
Schwertassek (1988, p. 96), who call these DOFs, respectively, positional(L) and 
motional(S); and the pioneering Korteweg (1899, p. 134), who states that “Die 
anzahl der Freiheitsgrade sei bei ihr eine andere (kleinere) fur unendlich kleine wie 
fur endliche Verruckungen” [Translation: The number of degrees of freedom is 
different (smaller) for infinitesimal displacements than for finite displacements.]} 

As explained later in this chapter (§2.5 flf.), DOF(S) = f equals the number of 
independent virtual displacements of the system', and this, in turn (chap. 3), equals 
the smallest, or minimal, number of kinetic (i.e., reactionless) equations of motion of 
it. In view of this, from now on by DOF we shall understand DOF in the small ; that 
is, DOF = DOF(S) = n — m = /, unless explicitly specified otherwise. The concept 
of DOF in the large is more important in pure kinematics (mechanisms). 

Finally, in the general constraint case, all these results hold intact, but for the 
figurative system "particle” in a higher dimensional space — more on this later. 


Semiholonomic Constraints 

We stated earlier that if 7 = 0, the Pfaffian constraint (2.3.1) is holonomic; that is, it 
can be brought to the form 

df/dt = 0 =>■ <j) = constant = c. (2.3.9) 

Such situations necessitate an additional, albeit minor, classification of holonomic 
constraints into proper holonomic, or simply holonomic, and improper holonomic, 
or semiholonomic ones. In both cases, the constraints are finite (i.e., holonomic), but, 
in the proper case, the constraint constants have a priori fixed values, independent of 
the system’s position/motion; whereas, in the semiholonomic case, those constants 
depend on the arbitrarily specified values of the system coordinates at some “initial” 
instant; that is, semiholonomic constraints are completely integrable velocity 
(Pfaffian) constraints =>■ (generally) initial condition-depending holonomic con¬ 
straints. In the proper holonomic case, the initial values of the coordinates must 
be determined in conjunction with the given constraints and their constants; that is, 
they must be compatible with the latter. However, semiholonomic constraints, being 
essentially holonomic, can be used to reduce the number of independent global/ 
Lagrangean coordinates; and, thus, differ profoundly from the nonholonomic 
ones. Clearly, the proper/semiholonomic distinction applies to rheonomic holonomic 
constraints, like (f>{x, y , z, t) = c. For further details, see (alphabetically): Delassus 
(1913(b), pp. 23-25: earliest extensive discussion of semiholonomicity), Moreau (1971, 
pp. 228-232), and Peres (1953, pp. 60-62, 218-219). 

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§2.3 QUANTITATIVE INTRODUCTION TO NONHOLONOMICITY 


Critical Comments on Nonholonomic Constraints 

The concept of nonholonomicity (in mechanics) has been around since the 1880s, 
and has been thoroughly studied and expounded by some of the greatest mathema¬ 
ticians, physicists, and mechanicians, for example (approximately chronologically): 
Voss, Hertz, Hadamard, Appell, Chaplygin, Voronets, Maggi, Boltzmann, Hamel, 
Heun, Delassus, Caratheodory, Schouten, Struik, Goursat, Cartan, Synge, 
Vranceanu, Vagner, Dobronravov, Lur’e, Neimark, Fufaev, et al. Direct definitions 
of nonholonomicity and analytical tests have been available, on a large and readable 
scale, at least since the 1920s. And yet, on this topic, there exists widespread mis¬ 
understanding and confusion; especially in the engineering literature. For example, 
some authors state that constraints that can be represented by equations like 
</>(r, r) = 0, or <j)(x,y,z,t ) = 0, are called holonomic, and that all others are called 
nonholonomic; for example, Goldstein (1980, p. 12 ff), Kane (1968, p. 14), Kane 
and Levinson (1985, p. 43), Likins (1973, pp. 184, 295), Matzner and Shepley (1991, 
pp. 23-24). Under such an indirect, vague, negative definition, inequality constraints 
like (j> > 0, or (perhaps?!) holonomic ones, but in velocity form, like 

d(j)/dt = (dtj>/dr) • v + d(j>/dt = 0, (2.2.10) 

would be called nonholonomic! Or, we read blatantly contradictory and erroneous 
statements like “With nonholonomic systems the generalized coordinates are not 
independent of each other, and it is not possible to reduce them further by means 
of equations of constraint of the form f(q j,..., q n , t) = 0. Hence it is no longer true 
that the qf s are independent ” (Goldstein (1980, p. 45), emphasis added). Others 
call nonholonomic all velocity constraints that cannot be written in the above 
form <j> = 0, which is correct; but they fail to supply the reader with analytical (or 
geometrical, or even numerical) tools on how to test this; for example, Roberson and 
Schwertassek (1988, p. 96), Shabana (1989, pp. 123, 128). The more careful of this 
last group talk clearly about integrability, exactness, and so on, but restrict them¬ 
selves to only one velocity constraint; for example, Haug (1992, pp. 87-89). Still 
others mix nonholonomic coordinates (quasi coordinates, etc.) with nonholonomic 
constraints, and exactness of Pfaffian forms with (complete) integrability of a system 
of Pfaffian equations, without ever supplying clear and general definitions, let alone 
analytical tests. And this results in defective definitions of the concept of DOF; for 
example, Angeles (1988, pp. 80, 103). Even the (otherwise monumental) treatise of 
Pars (1965, pp. 16-19, 22-24, 35-37, 64-72, 196) is limited to an introduction to the 
subject, albeit a careful and precise one. Finally, there is the recent crop of texts on 
“modern” dynamics, where the problem of nonholonomicity is “solved” by ignoring 
it altogether; for example, Rasband (1983). Only Neimark and Fufaev (1967/1972) 
discuss the nonholonomicity issue clearly, competently, and in sufficient generality 
and completeness to be useful. We hope that our treatment complements and 
extends their beautiful work. 


Extensions/Generalizations of the Integrability Conditions 
(May be omitted in a first reading) 

(i) Single Pfaffian Equation in the n Variables x= (jq,..., x n ): 

d'9 = a^dx% = 0, a^ = a^ix). 

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(2.3.10) 


CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS 


It can be shown that the necessary and sufficient condition for the complete integr- 
ability = holonomicity of (2.3.10) is the identical satisfaction of the following 
“symmetric” equations: 

hip = a k (da,/dx p - da p /dx ,) + a,{da p /dx k - da k /dx p ) 

+ a p (da k /dxi — dai/dx k ) = 0, (2.3.10a) 

for all combinations of the indices k, /, p = 1[For example, one may start 
with the integrability condition of (2.3.1), (2.3.6) (i.e., n = 3) and then use the 
method of induction; or perform similar steps as in the three-dimensional case; 
see, for example, Forsyth (1885 and 1954, pp. 259-260).] Further, it can be shown 
(e.g., again, by induction) that out of a total of n[n — \){n — 2)/6 equations 
(2.3.10a), equal to the number of triangles that can be formed with n given points 
as corners, only rij = ( n — 1)(« — 2)/2 are independent. For n = 3, that number is 
indeed 1: eqs. (2.3.6) or (2.3.8a). Also, if a k / 0, it suffices to apply (2.3.10) only for / 
and p different from k. Finally, with appropriate extension of the curl of a vector to 
^-dimensional spaces, (2.3.10) can be cast into a (2.3.8a)-like form (see, e.g., 
Papastavridis, 1999, chaps. 3, 6). 


Problem 2.3.4 (i) Specialize (2.3.10a) to the acatastatic constraint {n = 4): 

a(t,x,y,z) dx + b(t, x,y,z) dy + c(t,x,y,z) dy + e(t,x,y,z) dt = 0. (a) 

(ii) Show that (a) is holonomic if, and only if, the symbolic matrix 

( a b c e \ 

d/dx d/dy d/dz d/dt , (b) 

a b c e J 

has rank 2 (actually, less than 3); that is, all possible four of its 3 x 3 symbolic 
subdeterminants, each to be developed along its first row, vanish. 

(iii) Further, show that if all such 2x2 subdeterminants of (b) vanish, then (a) is 
exact. 

(iv) Specialize the preceding result to the catastatic case e = 0; verify that, then, 
we obtain (2.3.6). 


Problem 2.3.5 For the Pfaffian equation (2.3.10), define the (« + 1) x n matrix 


/ a. 


P = 


Oil 


\ 

a\ n 


(a) 


\ a n i ... a im J 

where a k / = da k /dxi — dai/dx k (= —a & ); k, l = 1Clearly, a n = • • • = 
a nn = 0. Now, it is shown in differential equations/dilferential geometry that for 
the holonomicity of (2.3.10), it is necessary and sufficient that the rank of P equal 
1 or 2. 

Show that (i) rank P = 1 (i.e., all its 2 x 2 subdeterminants vanish) leads to the 
exactness conditions 


a ki — 0; 


(b) 


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§2.3 QUANTITATIVE INTRODUCTION TO NONHOLONOMICITY 

(ii) rank P = 2 (i.e., all its 3x3 subdeterminants vanish) leads to the earlier 
complete integrability conditions (2.3.10a) 

ctj. Q/p -T- ci i ci pf- -T- a p a k j — 0. (c) 


Problem 2.3.6 Show that for n = 3, equations (b,c) of the preceding problem 
become, respectively, 

a k i = 0 (k, 1= 1,2,3), (a) 


and 


nT#23 + a 2 a 2 \ + a 3 a i2 — 0 (2.3.6)]. 


(b) 


Problem 2.3.7 Consider the Pfaffian equation (2.3.10). Subject its variables x to 
the invertible coordinate transformation (with nonvanishing Jacobian) x —> xin 
extenso: 

Xfc = x k (x k >) o x k > = x k ,(x k ) (k, k' = 1,..., n). (a) 

Show that the requirement that, under that transformation, the Pfaffian form d'O 
remain (form) invariant, that is, 

d'O —> (d'O)' = a k ' dx]j = d'O (= 0), a k > = a k ’(x'), (b) 

leads to the following (covariant vector) transformations for the form coefficients: 

a k < = ( dx k /dx k ')a k & a k = Y (dx k ,/dx k )a k . . (c) 


Problem 2.3.8 Continuing from the previous problem, define the antisymmetric 
quantities 


a M = da k /dx l - dajdx,, (= -a lk ), (a) 

a k <v = da k ,/dx r - da v /dx k , (= -a n ,), (k',l' = 1 ,... ,n). (b) 

Show that under the earlier invariance requirement d'O —> (d'O)' = d'O, the above 
quantities transform as (second-order covariant tensors): 

a k 'i' = EE (dx k /dx k ')(dx,/dx r )a k , & a k , = EE (dx k : /dx k ) (dxf /dxi)a k 'i>. 

(c) 


Problem 2.3.9 Continuing from the preceding problems, assume that the x (and, 
therefore, also the x') depend on two parameters iq and uy. 

x k = x k (ui,u 2 ) and x' k = x' k (u u u 2 ). (a) 

Introducing the simpler notation d'O = dO and (d'O)' = d0', show that 

d 2 (d\0) — d\(d 2 0) = d 2 (d\0') — d\(d 2 0'), (b) 

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267 


CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS 
where 


d\9 = d\Xk = ’y^ ak[(dxk/dui)dui], 
did = Y (ikd 2 x k = y^akKdxk/ duijdui], 

are equivalent to 

EE (i dx k /du x ){dx,ldu 2 )a kl = EE (dx kl /du 1 )(dx r /du 2 )a kr . (c) 


(ii) If the Pfaffian Constraint (2.3.10) has the Equivalent, (2.3.5, 7)-like, Special 
Form: 


dz = y^ b k (x,z) dx k {k = 1,... ,ri), (2.3.10b) 

then, proceeding as in the three-dimensional case, or specializing (2.3.10a), we can 
show that the necessary and sufficient integrability conditions are the n(n — l)/2 
independent identities [replacing n with n + 1 in the earlier «/, following (2.3.10a)]: 

dbk/dxj + ( db k /dz)b, = db,/dx k + (db,/dz)b k (k,l = 1,...,«). (2.3.10c) 

Here, too, only the existence and continuity of the partial derivatives involved is 
needed. 


(iii) General Case of m(< n ) Independent Pfaffian Equations in n Variables 
[In the slightly special total differential equation form, with x = {xd,Xj)\. 

dx D = E b D f(x) dxj or dx D /dxj = b D[ (x) (general form), (2.3.11) 

where ( here and throughout this book) 

D = 1(for Dependent) and / = m+ 1(for Independent), 

b DI = given (continuously differentiable) functions of the m x D = (v 1; ... ,x m ), 
and the (n — m) x k = (x m+ i,... ,x n ). (2.3.11a) 

The system (2.3.11) is called holonomic or completely integrable [i.e., functions 
x D (xi) can be found whose total differentials are given by (2.3.11)], if, for any set 
of initial values x I o , x D o , for which the b DI are analytic, there exists one, and only 
one, set of D functions x D (x r ) satisfying (2.3.11) and taking on the initial values 
x D o at xj 0 . It is shown in the theory of partial (total) differential equations—see 
references below—that: 

For the system (2.3.11) to be holonomic, it is necessary and sufficient that the follow¬ 
ing conditions hold: 

db DI /dx r + y b D ' I f{db DI /dx D f) = db Dr /dx, + y b r yj(db Dr /dx DI ) 

[D,D' = 1,... ,m\ /, I' = m + 1,...,«], (2.3.11b) 


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§2.3 QUANTITATIVE INTRODUCTION TO NON HOLONOMICITY 


identically in the x D , xfs [i.e., not just for some particular motion(s)] and for all 
combinations of the above values of their indices; if they hold for some, but not all, 
m values of D , then the system (2.3.11) is called “partially integrable.” 

Now, and this is very important, as the second (sum) term, on each side of 
(2.3.1 lb), shows, the integrability of the Z>th constraint equation of (2.3.11) depends, 
through the coupling with b D 'p and b D n, on all the other constraint equations of that 
system; that is, each (2.3.11b) tests the integrability of the corresponding constraint 
equation (i.e., same D) against the entire system — in general, holonomicity/non- 
holonomicity are system not individual constraint properties. 

Geometrically, integrability means that the system (2.3.11) yields a field of 
(.n — m)-dimensional surfaces in the n-dimensional space of the x’s; that is, mechani¬ 
cally, the system has (n — m) global positional/Lagrangean coordinates, namely, 
DOF(L) = DOF(S) = n- m. 

Further: 

• With the notation 

bm = b D i(x D ,xf) = boj[xj)(xj),Xi\ = Pdi( x i) = Pdu (2.3.11c) 

and since, by careful application of chain rule to the above, 


d/3 DI /dx r = db DI /dx,i + ^ {db DI /dx D ,){dx D fdx r ) 
= db DI /dx r + ^2 (db D1 /dx D ')b D , v , 


[if x D = x D (xj), then dx D = (dx D /dx r ) dx, = ff b D! (x) dxj] the holonomicity 
conditions (2.3.1 lb) can also be expressed in the following perhaps more intelligible/ 
memorable (“exactness”) form: 

dfoi/dx/' = dpDi'/dx, (/' = m + 1,... ,n); (2.3.lid) 

• It is not hard to verify that the system (2.3.11b, d) stands for a total of 
m(n — 1)(« — 2)/2 identities, out of which, however, only m(n — m)(n — m — l)/2 = 
mf{ f — l)/2 are independent [/ = n — m; as in the general case of the first of 
(2.12.5)]. 

• In the special case where b m = b DI (xj) [Chaplygin systems (§3.8)], (2.3.11b) reduce 
to the conditions: 


dboj/dxji = db DI fdxj [compare with (2.3.1 Id)], (2.3.lie) 

which, being uncoupled, test each constraint equation (2.3.11) independently of the 
others. Last, we point out that all these holonomicity conditions are special cases of 
the general theorem of Frobenius, which is discussed in §2.8-2.11. 

• Equations (2.3.11b, d) also appear as necessary and sufficient conditions for a 
Riemannian (“curved”) space to be Euclidean (“flat”) =4> vanishing of Riemann- 
Christoffel “curvature tensor”; and in the related compatibility conditions in non¬ 
linear theory of strain — see, for example, Sokolnikoff (1951, pp. 96-100), Truesdell 
and Toupin (1960, pp. 271-274). 

• Historical : The fundamental partial differential equations (2.3.11b) are due to the 
German mathematician H. W. F. Deahna [J. fur die reine und angewandte 
Mathematik ( Crelle’s Journal) 20 , 340-349, 1840] and, also, the French mathemati¬ 
cian J. C. Bouquet [Bull. Sci. Math, et Astron., 3(1), 265 ff., 1872], For extensive and 
readable discussions, proofs, and so on, see, for example, De la Valee Poussin (1912, 

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CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS 


pp. 312-336), Levi-Civita (1926, pp. 13-33), and the earlier Forsyth (1885/1954). 
Regrettably, most contemporary treatments of Pfaffian system integrability are 
written in the language of Cartan’s “exterior forms,” and so are virtually inaccessible 
to the average nonmathematician. 


2.4 SYSTEM POSITIONAL COORDINATES AND 

SYSTEM FORMS OF THE HOLONOMIC CONSTRAINTS 

So far, we have examined constraints in terms of particle vectors, and so on. Here, we 
begin to move into the main task of this chapter: to describe constrained systems in 
terms of general system variables. Let us assume that our originally free, or uncon¬ 
strained, mechanical system S, consisting of N particles with inertial position vectors 
[recalling (2.2.4)] 

rp = rp{t) = {x P (t), yp(t),zp(t)} (P=l,...,N), (2-4.1) 

is now subject to h(< 3N) independent positionaljgeometrical/holonomic (internal 
and/or external) constraints 

<t>H{t,r P ) = <j> H {t,r) = <t> H {t\x,y,z) = 0 [H = 1,..., h{< 32V)], (2.4.2) 

or, in extenso, 


0i(*;-*i!j ; i> z i> • • • j ■A'a'iLa'i^aO ~ 0, 


y ti z i) • • •) x Ni Jjvi z n) — 


(2.4.2a) 


where independent means that the (/>!,...,</>/, are not related by a(ny) functional 
equation of the form $(</>!,..., cj) h ) = 0 {In that case we would have, e.g., 
(j) h = F(t;(j>i, 1 ), so that one of the constraints (2.4.2, 2a), i.e., here </)>, = 0, 
would either be a consequence of the rest of them [if F(t; 0,..., 0) =0, while </>/, = 0], 
or it would contradict them [if F(t; 0,..., 0) ^ 0, while (p h = 0]}. 

At this point, to simplify our discussion and improve our understanding, we 
rename the particle coordinates (x,y,z) as follows [recalling ( 2 . 2 . 1 a)]: 

x l — £li y 1 — Z l=^3i-"I X N — £lN-2i Jv — &AT-1! Z N — C 3 V) 

(2.4.3) 


or, compactly, 

x p = &P-2, yp = C 3 T -1 ; z P = iip (P = 1, • ■ • j -^0; (2.4.3a) 

in which case, the constraints (2.4.2a) read simply 

= 0 [H=\,...,h(<3N)-, * = 1,..., 3N\. 

(2.4.3b) 

Therefore, using the h constraints (2.4.2a, 3b), we can express h out of the 3N 
coordinates £= ( x,y,z ), say the first h of them (“dependent”) in terms of the 
remaining n = 3N — h (“independent”), and time: 

= S rf (t;^ + i,...,^ 3 A r) =S d {t\ii) [d=l,...,h ; i = h + 1,..., 3N]; (2.4.4) 

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§2.4 SYSTEM POSITIONAL COORDINATES AND SYSTEM FORMS OF THE HOLONOMIC CONSTRAINTS 


and so it is now clear that our system has n {global) DOF , h down from the previous 
37V of the unconstrained situation. Further, since for h = 37V (i.e., n = 0) the solu¬ 
tions of (2.4.2a) would, in general, be incompatible with the equations of motion 
and/or initial conditions, while for h = 0 (i.e., n = 3N) we are back to the original 
unconstrained system; therefore, we should always assume tacitly that 

0 < h < 37V or 0 < n < 37V. (2.4.5) 

Now, to express this / 2 -parameter freedom of our system, we can use either the last n 
of the £’s [i.e., the earlier = (£ /l+1 ,... , £ 3A /)], or, more generally, any other set of n 
independent (or unconstrained, or minimal), and generally curvilinear , coordinates, 
or holonomic positional parameters 

q = Vh = <h (7),- ■ -,?» = q n {t)\ = {qk = qk{i)\ k = 


or, simply. 


q = (qi,---,q n ), 

related to the via invertible transformations of the type 

6 = 60;?) ?/t = ?*0;6)- 


(2.4.6) 


(2.4.6a) 


[The reader has, no doubt, already noticed that sometimes we use for the totality 
of the independent £’s; i.e., (£ A+1 , • • •, 6w)> and sometimes for a generic one of them; 
and similarly for other variables. We hope the meaning will be clear from the 
context.] In view of (2.4.6a), eq. (2.4.4) can be rewritten as 

6 i = s d 0;6) = s 4*;60;?)] = s At;q), ( 2 . 4 . 6 b) 

that is, in toto, £* = q), * = 1,..., 37V; and so (2.2.4), (2.4.1) can be replaced by 

x P = x P {t, q), y P = y P (t, q), z P = z P (t, q), 


or 


rp = r P (t,q), (2.4.6c) 

or, finally, by the definitive continuum notation, 

r = r(t,q). (2-4.7) 

Let us pause and re-examine our findings. 

(i) The n = 37V — h independent positional parameters q = q(t) are, at every 
instant t, common to all particles of the system (even though not every particle, 
necessarily, depends on all of them); that is, the q s are system coordinates; but 
once known as functions of time they allow us, through (2.4.7), to calculate the 
motion of the individual particles of our system S. The c/s are also called holonomic 
(or true, or genuine, or global), independent (or unconstrained, or minimal) coordi¬ 
nates, although they might be constrained later (!); for short, Lagrangean coordi¬ 
nates', and the problem of analytical mechanics (AM) is to calculate them as functions 
of time. Most authors call them “generalized coordinates” (and, similarly, “general¬ 
ized velocities, accelerations, forces, momenta, etc.”). This pretensorial/Victorian 
terminology, introduced (most likely) by Thomson and Tait [1912, pp. 157-60, 
286 IT.; also 1867 (1st ed.)], though inoffensive, we think is misguiding, because it 

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CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS 


directs attention away from the true role of the cfs: the key word here is not general¬ 
ized but system (coordinates)! The fact that they are, or can be, general—that is, 
curvilinear (nonrectangular Cartesian, nonrectilinear)—which is the meaning 
intended by Thomson and Tait, is, of course, very welcome but secondary to AM, 
whose task is, among others, to express all its concepts, principles, and theorems in 
terms of system variables. Nevertheless, to avoid breaking with such an entrenched 
tradition, we shall be using both terms, generalized and system coordinates, and the 
earlier compact expression, Lagrangean coordinates. 

(ii) The ability to represent by (2.4.7) the most general position (and, through it, 
motion) of every system particle (i.e., in terms of a finite number of parameters), 
before any other kinetic consideration, is absolutely critical (“nonnegotiable”) to 
AM; without it, no further progress toward the derivation of (the smallest possible 
number of) equations of motion could be made. 

(iii) Further, as pointed out by Hamel, as long as the representation (2.4.7) holds, the 
original assumption of discrete mass-points/particles is not really necessary. We could, 
just as well, have modeled our system as a rigid continuum ; for example, a rigid body 
moving about a fixed point, whether assumed discrete or continuum, needs three cf s to 
describe its most general (angular) motion, such as its three Eulerian angles (§1.12). 
In sum, as long as (2.4.7) is valid, AM does not care about the molecular structure/ 
constitution of its systems. [However, as n —> oo (continuum mechanics), the descrip¬ 
tion of motion changes so that the corresponding differential equations of motion 
experience a “qualitative” change from ordinary to partial.] 

(iv) Even though, so far, r has been assumed inertial, nevertheless, the cf s do not 
have to be inertial; they may define the system’s configuration(s) relative to a non- 
inertial body, or frame, of known or unknown motion, and that (on top of the 
possible curvilinearity of the cf s) is an additional advantage of the Lagrangean 
method. (As shown later, the r’s may also be noninertial.) For example, in the double 
pendulum of fig. 2.7, cf> x , cj) 2 , 0 X are inertial angles, whereas 0 2 is not. 

If the constraints are stationary (—> scleronomic system ), then we can choose the 
cf s so that (2.4.7) assumes the stationary form [recalling (2.2.2 ff.)]: 

r P = r P {cf) or r = r(r 0 , q) = r(q); (2.4.7a) 

and, therefore, scleronomicity/rheonomicity (= absence/presence of dr/dt) are 
^-dependent properties, unlike holonomicity/nonholonomicity. 




Figure 2.7 Inertial and noninertial descriptions of a double pendulum: OA, AB. 
Coordinates: <^ 1<2 : inertial; 8 X = fa: inertial; 0 2 = <j> 2 ~ <j>V- noninertial; O, A, C: collinear. 


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§2.4 SYSTEM POSITIONAL COORDINATES AND SYSTEM FORMS OF THE HOLONOMIC CONSTRAINTS 


Analytical Requirements on Equations (2.4.6a-c, 7) 

The n cf s are arbitrary, that is, nonunique, except that when the representations 
(2.4.6c, 7) are inserted back into the constraints (2.4.2, 2a) they must satisfy them 
identically in the q' s, which, analytically, means that 

<Mo£*)=0 => (/> H [t;Ut;q)\= 0 => ^2(d(/) H /dQ(d^dq k ) = d(f) H /dq k = 0, 

where 


H — 1,..., * = 1,..., 37V; k = 1, ...,n(= 3N - //); (2.4.8) 

and where, due to the constraint independence and to (2.4.5), the Jacobians of the 
transformations cj> H £* and £* q k must satisfy 

rank(df H jdt/f) = /;, rank(d^/dq k ) = n (2.4.8a) 

[and since \d^/dq k \ 7 0 =>■ rank(d£,-/dq k ) = n\, in the region of definition of the 
£ and t. In addition, the functions in the transformations (2.4.6a, b) must be of 
class C 2 (i.e., have continuous partial derivatives of zeroth, first, and second order, 
at least, to accommodate accelerations) in the region of definition of the q's, and t. 

Last, conditions (2.4.8a) imply that the representations (2.4.6a, b) have a (non- 
unique) inverse: 

<lk = qtcifO = cj k {t,x,y,z) =q k = q k (t,r). (2.4.8b) 


Additional “regularity” requirements are presented in §2.7. 

Example 2.4.1 Let us express the above analytical requirements in particle vari¬ 
ables. Indeed, substituting into (2.2.8) and (2.2.10): 

v = dr/dt = ^ (dr/dq k )(dq k /dt) + dr/dt (k = 1,..., n), (a) 

we obtain, successively, 

0 = df H /dt = s (d4> H /dr) • ( dr/dq k )(dq k /dt ) + dr/dtj + dcj> H /dt 

= i^ft/dr) ■ (dr/dq k fj ( clq k /dt) 

+ (S • (dr/dt) + df H /dt ) 

= ( d$ H /dq k )(dq k /dt ) + d$ H /dt, (b) 

from which, since the holonomic system velocities dq k /dt are independent, 

d<P H /dq k = 0, (c) 

d<P H /dt = 0. (d) 


Constraint Addition and Constraint Relaxation 

The n cf s (just like the It s) are independent ; that is, we cannot couple them by 
nontrivial functions d>(q) = 0, independent of the problem’s initial conditions, and 
such that upon substitution of the cf s from (2.4.8b) into them they vanish identically 

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274 CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS 

in the <fs and t (i.e., <P[t, q(t, £)] = $(?,£) = 0 is impossible). Thus, as in differential 
calculus, when all the cf s except (any) one of them remain fixed, we are still left with 
a “nonempty” continuous numerical range for the nonfixed cf s; and these latter 
correspond to a “nonempty” continuous kinematically admissible range of system 
configurations (a similar conception of independence will apply to the various q- 
differentials, dq, 6q, .... to be introduced later). However, upon subsequent imposi¬ 
tion of additional holonomic constraints to the system, the n cf s will no longer be 
independent, or minimal. To elaborate: in the "beginning,” the system of particles is 
free, or unconstrained (“brand new”); then, its cf s are the 3 N £’s. Next, it is subjected 
to a mix of constraints; say, h holonomic ones like (2.4.2,2a), and m Pfaffian 
(possibly nonholonomic) ones like (2.2.7,9). Now, the introduction of the 
n = 3N — h cfs, as explained above, allows us to absorb, or build in, or embed, the 
h holonomic constraints into our description; the representations (2.4.6c, 7) guaran¬ 
tee automatically the satisfaction of the holonomic constraints, and thus achieve the 
primary goal of Lagrangean kinematics, which is the expression of the system’s 
configurations, at every constrained stage, by the smallest, or minimal, number of 
positional coordinates needed [which, in turn (chap. 3) results in the smallest number 
of equations of motion. The corresponding embedding of the PfafRan constraints, 
which is the next important objective of Lagrangean kinematics (to be presented 
later, §2.11 flf.), follows a conceptually identical methodology, but requires new “non¬ 
holonomic, or quasi, coordinates”]. Specifically, if at a later stage, h\< n) additional, 
or residual, or non-built-in, independent holonomic constraints, say of the form 

* H <t,q) = 0 (2.4.9) 

are imposed on our already constrained system, then, repeating the earlier proce¬ 
dure, we express the n q' s in terms of n' = n — h' new positional parameters 
q' = {qk'\k' = l,..., n')\ 

qic = qkifqic'), rank(dq/dq') = n', (2.4.10) 

so that, now, (2.4.7) may be replaced by 

r = r (t, q) = r[t, q(t , q')} = r(t, q'); (2.4.11) 

the representation (2.4.7) still holds, no matter how many holonomic and nonholo¬ 
nomic constraints are imposed on the system; but then our cfs will not be indepen¬ 
dent: they have become the earlier £’s. 

This process of adding holonomic constraints to an already constrained system, 
one or more at a time, can be continued until the number of (global) DOF reduces 
to zero: 3N — (h + h' + h" + •••)—> 0. Also, no matter what the actual sequence 
(history) of constraint imposition is, it helps to imagine that they are applied succes¬ 
sively, one or more at a time, in any desired order, until we reach the current, or last, 
state of “constrainedness” of the system. It helps to think of a given constrained 
system as being somewhere “in the middle of the constraint scale”: when we first 
encounter it, it already has some constraints built into it; say, it was not born yester¬ 
day. Then, as part of a problem’s requirements, it is being added new constraints 
that reduce its DOF(L), eventually to zero; and, similarly, proceeding in the opposite 
direction, we may subtract some of its built-in constraints, thus relaxing the system 
and increasing its DOF(L), eventually to 3 N. [Usually, such a (mental) relaxation of 

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§2.4 SYSTEM POSITIONAL COORDINATES AND SYSTEM FORMS OF THE HOLONOMIC CONSTRAINTS 

one or more built-in constraints is needed to calculate the reaction forces caused by 
them (—> principle of “relaxation,” §3.7).] 

In sum: Any given system may be viewed as having evolved from a former 
“relaxed” (younger) one by imposition of constraints; and it is capable of becoming 
a more “rigid” (older) one by imposition of additional constraints. 

For example, let us consider a “newborn” free rigid body. The meaning of rigidity 
is that our system is internally constrained; and the meaning of free(dom) is that, 
when presented to us and unless additionally constrained later, the system is exter¬ 
nally unconstrained; that is, at this point, its built-in constraints are all internal: 
hence, n = 6. If, from there on, we require it to have, say, one of its points fixed 
(or move in a prescribed way), then, essentially, we add to it three external (holo- 
nomic) constraints; that is, n' = n — 3 = 6 — 3 = 3. If, further, we require it to have 
one more point fixed, then we add two more such constraints; that is, 
n" = n' — 2 = 3 — 2= 1. And if, finally, we require that one more of its points 
(noncollinear with its previous two) be fixed, then we add one more such constraint; 
that is, n" = n" — 1 = 1 — 1 = 0. But if, on the other hand, we, mentally or actually, 
separate the original single free rigid body into two free rigid bodies, then we sub¬ 
tract from it six internal built-in constraints (in Flamel’s terminology, we “liberate” 
the system from those constraints) so that this new relaxed system has 
n + 6 = 6 + 6= 12 (global) DOF. 


Equilibrium, or Adapted, Coordinates 

Frequently, we choose, in E 3N , the following “equilibrium,” or “adapted (to the 
constraints)” curvilinear coordinates: 

Xi = 4>i(t;x,y,z) (=0),..., Xh = 4>h(t’,x,y,z) (=0); 

Xh+\ = <t>h+\(t\x,y,z) (^0),..., X'3 N = hN(f,x,y,z) (^0); 

or, compactly, 

Xd = 4>d(t\x,y,z) (=0) (d= 1,...,/*); 

X, = 4>i(t-,x,y,z) #0) (i = h + 1,..., 31V), (2.4.12) 

and X3JV+1 = 4>3N+i = t (/ 0); where <f> d = (0 1; ..., </>/,) are the given constraints, and 
(j) t = ... ,4>3n) are n new and arbitrary functions, but such that when (2.4.12) 

are solved for the 3 N + 1 (t;x,y,z), in terms of (/;%[,... ,X 3 n)> and the results are 
substituted back into the constraints <f> d = 0, they satisfy them identically in these 
variables. In terms of the latter, which are indeed a special case of q s, the constraints 
take the simple equilibrium forms: 

Xd = (Xi = 0,...,Xh = 0), (2.4.12a) 

and so (2.4.7), with q —> \h reduces to 

*■ = r {U X/i+i > • ■ • i X 3 n) =r(t,Xi)- (2.4.12b) 


275 


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CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS 

Clearly, the earlier choice (2.4.4) corresponds to the following special x-case (assum¬ 
ing nonvanishing Jacobian of the transformation): 

Xd = Xd(t,0 =Zd-Bd(t,&) =0 (d= 1 

Xi = xM=ii^ 0 (i = h+\,...,3N). (2.4.12c) 

In practice, the transition from £ to q, Xi is frequently suggested “naturally” by the 
geometry of the particular problem. However, the general method described above 
[but in differential forms; i.e., as d\d = dcj) d (= 0 ) and d\i = dcj), 0 )] will allow us, 
later (§2.11 If.), to build in Pfafhan (possibly nonholonomic) constraints. 

Finally, such equilibrium q, x,’s extend to the case of the earlier described «'(> 0) 
additional constraints. There we may choose the new equilibrium coordinates: 

x'd 1 = d>d' (= 0 ) (d'= 

x'v = ®i' #0) {i' = h'+l,...,n), (2.4.12d) 

so that 

r = r(t,q)^>r(t,x'i')- (2.4.12e) 


Excess Coordinates 

Sometimes, in a system possessing n minimal Lagrangean coordinates, 
q = (< 71 ,..., q n ), we introduce, say for mathematical convenience, e additional 
excess, or surplus, Lagrangean coordinates q E = (q„ + i,..., q„ +e )- Since the n + e 
positional coordinates q and q E are nonminimal—that is, mutually dependent— 
they satisfy e constraints of the type 

F E (t;qi,...,q n ;q n +u---,<ln+e) = F E (t;q,q E ) =0 (E = 1 ,... ,e); (2.4.13) 

and then we may have 

r = r(t,q,q E ). (2.4.13a) 

If we do not need the q E s, we can easily get rid of them: solving the e equations 
(2.4.13) for them, we obtain q E = q E (t;q), and substituting these expressions back 
into (2.4.13a) we recover (2.4.7). For this to be analytically possible we must have 
(see any book on advanced calculus) 

\dF E /dq E ,\^0 (E= E' = n+1,...,n + e). (2.4.13b) 


Example 2.4.2 Let us consider the planar three-bar mechanism shown in fig. 2.8. 
The O-xy coordinates of a generic point on bars OA\ and AiAt, can be expressed 
in terms of the angles <j>i and d> 3 , respectively; similarly, for a generic point P on 
A\A 2 , such that A\P = l, we have 

x = /[ cos^! + /cos^> 2 ) y = h sin^! + /sin^ 2 - (a) 

So, (j>i, 4> 2 , <^ 3 express the configurations of this system; but they are not minimal (i.e., 
independent). Indeed, taking the x, y components of the obvious vector equation 


OAi + A t A 2 + A 2 A 3 + AjO = 0, 


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§2.4 SYSTEM POSITIONAL COORDINATES AND SYSTEM FORMS OF THE HOLONOMIC CONSTRAINTS 



we obtain the two constraints: 

F\ = l\ cos (j)\ T / 2 cos cf)2 T I3 cos (/>2 — L — 0, 

F 2 = l\ sin (j>\ + l 2 sin — h sin ft, = 0. (b) 

Therefore, here, n = 1 and e = 2; knowledge of any one of these three angles deter¬ 
mines the mechanism’s configuration. 

However, it is preferable to work with the representation (a), under (b), because if 
we tried to use the latter to express x and y in terms of either f j, or f 2 , ° r ft, °nly 
(wherever the corresponding Jacobian does not vanish), we would end up with fewer 
but very complicated looking equations of motion. It is preferable to have more but 
simpler equations (of motion and of constraint); that is, requiring minimality of 
coordinates, and thus embedding all holonomic constraints into the equations of 
motion, may be highly impractical. [See books on multibody dynamics; or Alishenas 
(1992). On the other hand, minimal formulations have numerical advantages (com¬ 
putational robustness).] 

Another “excess representation” of this mechanism would be to use the four 
O-xy coordinates of A x and A 2 , (x^yf) and (x 2 ,y 2 ), respectively. Clearly, these 
latter are subject to the three constraints (so that, again, n = 1 but e = 3): 

( x i)~ + (ki ) 2 = (A) - ; ( x 2 — x \Y + (k2 - y\Y = ( 4 ) 2 ; 

(L — x 2 ) 2 + (0 — y 2 ) 2 = (h) 2 - (c) 


Example 2.4.3 Let us consider the planar double pendulum shown in fig. 2.9. 
The four bob coordinates x t , y, and x 2 , y 2 are constrained by the two equations 

( x i ) 2 + (yiY = (A) 2 ) (-T2 — ^i ) 2 + (k 2 ~ ji) - = (^2)“; (a) 

that is, here N = 2 =>■ 2 N = 4, and so the number of holonomic constraints = 
F[ =2 =£- n = 2 N — FI = 2. A convenient minimal representation of the pendulum’s 

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CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS 



configurations is 

x 1 =/ 1 cos 0 1 , >'] = /| sin <p \; 

x 2 = l\ cos^! + / 2 cos<j> 2 , y 2 = A sin + U sin ^ 2 - (b) 


2.5 VELOCITY, ACCELERATION, ADMISSIBLE AND 

VIRTUAL DISPLACEMENTS; IN SYSTEM VARIABLES 

Velocity and Acceleration 

We begin with the fundamental representation of the inertial position of a typical 
system particle P in Lagrangean variables (2.4.7): 

r = r(t,q). (2.5.1) 

[Again, the inertialness of ris not essential, and is stated here just for concreteness. The 
methodology developed below applies to inertial and noninertial position vectors alike; 
and this, along with the possible curvilinearity (nonrectangular Cartesianness) and 
possible noninertialness of the coordinates, are the two key advantages of 
Lagrangean kinematics (and, later, kinetics) over that of Newton-Euler. This will 
become evident in the Lagrangean treatment of relative motion (§3.16).] 

From this, it readily follows that the (inertial) velocity and acceleration of P, in 
these variables, are, respectively, 

v = dr/dt = ^2 ( dr/dq k ){dq k /dt ) + dr/dt = ^ v k e k + e 0 , (2.5.2) 

a - dv/dt = E {dr/dq k )(d 2 q k /dt 2 ) + EE (d 2 r/dq k dq,){dq k /dt)(dq,/dt) 

+ 2^ {9 2 r/dqkdt){dq k /dt) + d 2 r/dt 2 

= E w ^+EE V*V/ e kJ + 2^2 v k e k,o + <?o,o, (2.5.3) 

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§2.5 VELOCITY, ACCELERATION, ADMISSIBLE AND VIRTUAL DISPLACEMENTS 


where 


dq k /dt = v k , 

v= (vi,... 

III 

j* 

ii 

(2.5.2a) 

d 2 q k /dt 2 = dv k /dt = w k , 

W= (Wj,.. 

. ,w„) = (w k ;k = 1,...,«) 


[but, in general, a f ^ 

w k e k + M’ 0 e 0 ; 

see (2.5.4—6) below)], 

(2.5.3a) 


associated with these cf s; and the fundamental (holonomic) basis vectors e k , e 0 , also 
associated with the cf s, are defined by 

e k = e k (t,q) = dr/dq k , e 0 = e 0 (t,q) = dr/dt (or, sometimes, e„ +1 , or e t ); 

(2.5.4) 


and the commas signify partial derivatives with respect to the cf s, t: 

e kj = de k /dq, = de,/dq k = e lik [i.e., d/dq,(dr/dq k ) = d/dq k (dr/dq,)], (2.5.4a) 

e k fi = de k /dt = de 0 /dq k = e ok [i.e., d/dt(dr/dq k ) = d/dq k {dr/dt)}; (2.5.4b) 

we reserve the notation a k for the representation a = Jf a k e k + a o e o- 
Also, note that with the help of the formal (nonrelativistic) notations: 

t = q 0 = q n+ i =>■ dt/dt = dq 0 /dt = dq n+l /dt = v 0 = v„ + \ = 1 7 (2.5.5a) 

and 

d 2 t/dt 2 = dv 0 /dt = dv n+ i/dt = m ’ 0 = w n+ \ = 0, (2.5.5b) 

we can rewrite (2.5.2,3), respectively, in the “stationary” forms: 

v = E v a e m a = w a e a + ^ ^ v a v p e a ^, (2.5.6) 

where, here and throughout the rest of the hook , Greek subscripts range from 1 to 

n+ 1 (=“ 0 ”). 

The v k = dq k /dt are the holonomic (and contravariant, in the sense of tensor 
algebra) components, in the (/-coordinates, of the system velocity or, simply, 
Lagrangean velocities or, briefly, but not quite accurately, “generalized velocities.” 
The key point here is that the velocity and acceleration of each particle, v and a, 
respectively, are expressed in terms of system velocities v = dq/dt and their rates 
w = dv/dt = d 2 q/dt 2 , which are common to all particles, via the (generally, neither 
unit nor orthogonal) “mixed” = particle and system, basis vectors e k , e 0 . The latter, 
since they effect the transition from particle to system quantities, are fundamental to 
Lagrangean mechanics. 

HISTORICAL 

These vectors, most likely introduced by Somoff (1878, p. 155 If.), were brought to 
prominence by Heun (in the early 1900s; e.g., Heun, 1906, p. 67ff, 78 IT.), and were 
called by him Begleitvektoren « accompanying, or attendant, vectors. Perhaps a better 
term would be “H(olonomic) mixed basis vectors” (see also Clifford, 1887, p. 81). 

From the above, we readily deduce the following basic kinematical identities'. 

(i) dr/dq k = dv/dv k = da/dw k = ■ ■ ■ = e k , (2.5.7) 

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CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS 


that is, [with (...)' = d(.. .)/dt\, 

dr/dq k = dr/dq k = df/dq k = • • • = e k (“cancellation of the (over)dots”); 

(2.5.7a) 


and 

(ii) d/dt(dr/dq k ) = d/dt ( dv/dv k ) = de k /dt = d/dq k ( dv/dt ) = dv/dq k , 

or, with the help of the Euler- Lagrange operator in holonomic coordinates: 

E k (...) = d/dt(.. . / dq k ) -d... /dq k = d/dt(.. . / dv k ) - d... /dq k , (2.5.9) 
finally, 

E k (v) = d/dt(dv/dv k ) — dv/dq k = 0. (2.5.10) 

In fact, for any well-behaved function/ = f(t,q), we have 

/ = df/dt = ^2 ( df/dq k )(dq k /dt ) + df/dt, f = d 2 f / dt 2 = • • •, 

=> df/dq k = df/dq k = df/dq k = ■ ■ ■; (2.5.8) 

and 

E k (f ) = d/dt ( df/dq k ) - df/dq k = d/dt ( df/dv k ) - df/dq k = 0 . (2.5.11) 

The integrability conditions (2.5.7, 10) are crucial to Lagrangean kinetics (chap. 3); 
and, just like (2.5.2, 3), have nothing to do with constraints; that is, they hold the 
same, even if holonomic and/or nonholonomic constraints are later imposed on 
the system, as long as the q's are holonomic (genuine) coordinates (i.e., 
q / nonholonomic or quasi coordinates; see §2.6, §2.9). 


Admissible and Virtual Displacements 

Proceeding as with the velocities, (2.5.2), we define the (first-order and inertial) 
kinematically admissible, or possible, and virtual displacements of a generic system 
particle P, respectively, by 

dr = ^2 ( dr/dq k ) dq k + ( dr/dt ) dt = ^ e k dq k + e 0 dt, (2.5.12a) 


6r = ^2 ( dr/dq k ) Sq k = ^ e k 6q k ; (2.5.12b) 


whether the ^-increments, or differentials, dq, 6q, and dt are independent or not (say, 
by imposition of additional holonomic and nonholonomic constraints, later). 

As the above show: 

(i) if dq k = ( dq k /dt ) dt = v k dt, then dr = v dt; 

(ii) if all the dq' s and dt (Sq’s) vanish, then dr = 0 (Sr = 0); and 

(iii) d(dr)/d(dq k ) = d(Sr)/d{Sq k ) = e k . 

These identities (in unorthodox but highly instructive notation) are useful in prepar¬ 
ing the reader to understand, later, the nonholonomic coordinates. 

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§2.5 VELOCITY, ACCELERATION, ADMISSIBLE AND VIRTUAL DISPLACEMENTS 


REMARKS ON THE VIRTUAL DISPLACEMENT 

Let us, now, pause to examine carefully this fundamental concept. First, we notice 
that Sr is the linear (or first-order) and homogeneous, in the Scf s, part of the “total 
virtual displacement” Ar, which is defined by the following Taylor-like r-expansion 
in the first-order increments Sq , from a generic system configuration corresponding 
to the values q, but for a fixed time t: 

Ar = r(t,q + Sq) — r(t,q) = Sr + (1/2) S 2 r + ■ ■ ■. (2.5.13) 

In other words, Sr is a special first position differential, mathematically equivalent 
to dr with t = constant —> dt = 0 (i.e., completely equivalent to it for stationary 
constraints); hence, the special notation S (...): 

dr —> Sr, dq —> Sq, and dt —+ St = 0 (isochrony, always ). (2.5.13a) 

One could have denoted it as d*r, or (dr)*, or z, and so on; but since we do not 
see anything wrong with S(. ..), and to keep with the best traditions of analytical 
mechanics [originated by Lagrange himself and observed by all mechanics masters, 
such as Kirchhoff, Routh, Schell, Thomson and Tait, Gibbs, Appell, Volterra, 
Poincare, Maggi, Webster, Heun, Hamel, Prange, Whittaker, Chetaev, Lur’e, 
Synge, Gantmacher, Pars et al.], we shall stick with it. Readers who feel uncomfor¬ 
table with 6(...) may devise their own suggestive notation; dr and dq won’t do! 

The above definitions also show the following: 

(i) Sr is mathematically equivalent to the difference between two possible/admis¬ 
sible displacements, say d\r and d 2 r, taken along different directions but at the same 
time (and same dt); that is, skipping summation signs and subscripts, for simplicity, 

d 2 r — d x r = \(dr/dq) d 2 q + ( dr/dt) dt] — [( dr/dq ) cfq + (dr/dt) dt] 

= ( dr/dq)(d 2 q — cfq) ~ (dr/dq) Sq = Sr. (2.5.14) 

(ii) For any well-behaved function/ = /(/): Sf = (df/dt) St = 0;butif / = f(t,q ), 
then Sf = (df /dq) Sq f 0 [even though, after the problem is solved, q = <?(/)!]. 

(iii) The virtual displacements of mechanics do not always coincide with those of 
mathematics (i.e., calculus of variations). For example, even though, in general, 
dr Sr, for catastatic systems [i.e., dr = e k (t, q) dq k , Sr = e k (t, q) Sq k ] the 
equality dq k = Sq k ^ dr = Sr is kinematically possible [and in (q, //space dr and 
Sr are “orthogonal” to the /-axis, even though dt 0,St = 0]; whereas, in variational 
calculus we are explicitly warned that dq (parallel to the /-axis) / Sq (perpendicular to 
it). These differences, rarely mentioned in mechanics and/or mathematics books, are 
very consequential, especially in integral variational principles for nonholonomic 
systems (chap. 7). 

As definitions (2.5.12, 13), and so on, show, the (particle and/or system) virtual 
displacement is a simple, direct, and, as explained in chapter 3 and elsewhere, indis¬ 
pensable concept — without it Lagrangean mechanics would be impossible! Yet, 
since its inception (in the early 18th century), this concept has been surrounded 
with mysticism and confusion; and even today it is frequently misunderstood and/ 
or ignorantly maligned. For instance, it has been called “too vague and cumbersome 
to be of practical use” by D. A. Levinson, in discussion in Borri et al. (1992); “ill- 
defined, nebulous, and hence objectionable” by T. R. Kane, in rebuttal to Desloge 
(1986); or, at best, has been given the impression that it has to be defined, or “chosen 
properly” (Kane and Levinson (1983)), in an ad hoc or a posteriori fashion to fit the 

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282 CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS 

facts, that is, to produce the correct equations of motion. For an extensive rebuttal 
of these false and misleading statements, from the viewpoint of kinetics, see chap. 3, 
appendix II. Others object to the arbitrariness of the Sq' s. But it is precisely in this 
arbitrariness that their strength and effectiveness lies: they do the job (e.g., yielding 
of the equations of motion) and then, modestly, retreat to the background leaving 
behind the mixed basis vectors e k . It is these latter [and their nonholonomic counter¬ 
parts (§2.9)] that appear in the final equations of motion (chap. 3), just as in the 
derivation of differential (“field”) equations in other areas of mathematical physics. 
For example, in continuum mechanics, for better visualization, we may employ a small 
spatial element (e.g., a “control volume”), of “infinitesimal” dimensions dx, dy, dz, to 
derive the local field equations of balance of mass, momentum, energy, and so on; but 
the ultimate differential equations never contain lone differentials — only combina¬ 
tions of finite limits of ratios among them; that is, combinations of derivatives. 
Moreover, differentials, actual/admissible/virtual, in addition to being easier to visua¬ 
lize than derivatives, are invariant under coordinate transformations; whereas deriva¬ 
tives are not. [Such invariance ideas led the Italian mathematicians G. Ricci and T. 
Levi-Civita to the development of tensor calculus (late 19th to early 20th century); see, 
for example, Papastavridis (1999).] For example, taking for simplicity, a one (global) 
DOF system, under the transformation q —> q = qft, q), we find, successively, 

Sr = e Sq = ( dr/dq ) Sq = (dr/dq)[(dq/dq') Scf] = [(dr/dq)(dq/dq')\ Sq' = e' Sq' , 
that is, 

e' = dr/dq' = (dq/dq')e e = dr/dq = {dq'/dq)e'. (2.5.15) 

But there is an additional, deeper, reason for the representation (2.5.12b): the 
position vectors r(t,q) and (possible) additional constraints, say ip H i(t,r) = 
0 —> ip H ft,q) = 0, cannot be attached in these finite forms to the general kinetic 
principles of analytical mechanics, which are differential, and lead to the equations 
of motion (§3.2 ff). Only virtual forms ofr and = 0 — special first differentials of 
them, linear and homogeneous in the Sq's [like (2.5.12b)] — can be attached, or 
adjoined, to the Lagrangean variational equation of motion via the well-known 
method of Lagrangean multipliers (§3.5 ff.); and similarly for nonlinear (non- 
Pfaffian) velocity constraints, an area that shows clearly that nonvirtual schemes 
(as well as those based on the calculus of variations) break down (chap. 5)1 
Hence, the older admonition that the virtual displacements must be “small” or 
“infinitesimal.” For example, to incorporate the nonlinear holonomic constraint 
f(x,y) = A ' 2 + y 2 = constant to the kinetic principles, we must attach to them its 
first virtual differential, S<j> = 2(x Sx + y Sy) = 0; which is the linear and homoge¬ 
neous part of the total constraint change, between the system configurations (x,y) 
and (a + Sx, y + Sy): 

Acj) = <f>(x + Sx, y + Sy) - </>(x,y) = [Sf + (1/2) <S 2 (/>] forsman ^ Sf = 0. 

But in the case of the linear holonomic constraint cj> = x + y = constant, that total 
constraint change equals 

Ac/) = Sf = Sx + Sy = 0, no matter what the size of Sx, Sy; 

and both equations, f = 0 and Sf = 0 , have the same coefficients (—> slopes). 

In sum: As long as we take the first virtual differentials of the constraints, the size 
of the Sq' s is inconsequential, whether they are one millimeter or ten million miles! It 

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§2.5 VELOCITY, ACCELERATION, ADMISSIBLE AND VIRTUAL DISPLACEMENTS 


is the holonomic (or “gradient,” or “natural”) basis vectors { e k ; k = 1that 
matter. 

As Coe puts it: “We often speak of displacements, both virtual and real, as being 
arbitrarily small or infinitesimal. This means that we are concerned only with the 
limiting directions of these displacements and the limiting values of the ratios of their 
lengths as they approach zero. Thus any two systems of virtual displacements are for 
our purposes identical if they have the same limiting directions and length ratios as 
they approach zero” (1938, p. 377). Coe’s seems to be the earliest correct and 
vectorial exposition of these concepts in English; most likely, following the exposi¬ 
tion of Burali-Forti and Boggio (1921, pp. 136 ff.). See also Lamb (1928, p. 113). 

The earlier mentioned indispensability of the virtual displacements for kinetics 
will become clearer in chapter 3. Nevertheless, here is a preview: it is the virtual work 
of the forces maintaining the holonomic and/or nonholonomic constraints (i.e., the 
constraint reactions ) that vanishes, and not just any work, admissible or actual; in 
fact, the latter would supply only one equation. This vanishing-of-the-virtual-work- 
of-constraint-reactions (principle of d’Alembert-Lagrange) is a physical postulate 
that generates not just one equation of motion (like the actual work/power equation 
does), but as many as the number of (local) DOFs-, and, additionally, it allows us to 
eliminate/calculate these constraint forces. A more specialized virtual displacement 
—> virtual work-based postulate is used to characterize the more general “servo/ 
control” constraints (§3.17). 


Example 2.5.1 Differences Between Kinematically Admissible I Possible and Virtual 
Displacements. 

(i) Let us assume that we seek to determine the motion of a particle P capable of 
sliding along an ever straight line / rotating on the plane O—xy about O. The config¬ 
urations of / and of P relative to that plane are determined, respectively, by <j) and r, </> 
(fig. 2.10). Since r = r(r, </>): position of P in O—xy, we will have, in the most general 
case, 

dr = (dr/dr) dr + (dr/dcfi) dfi. kinematically admissible displacement of P , 

in O-xy , (a) 

Sr = (dr/dr) Sr + (dr/d(f>) Sf. virtual displacement of P, in O—xy. (b) 



Figure 2.10 On the difference between possible/admissible 
and virtual displacements (ex. 2.5.1: a, b). 


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CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS 


(a) If the rotation of l is influenced by the motion of P relative to it, then <j> is another 
unknown Lagrangean coordinate, like r, waiting to be found from the equations of 
motion of the system P and 1 (i.e., n = 2). Then dr and Sr are given by (a, b), 
respectively, and are mathematically equivalent, (b) If, however, the motion of / is 
known ahead of time (i.e., if it is constrained to rotate in a specified way, unin¬ 
fluenced by the, yet unknown, motion of P), then 

<j> = f(t): given function of time =>■ 

d(j) = df(t) = [df(t)/dt\ dt = u(t) dt ^ 0, but S(j> = Sf(t) = uj(t) St = 0. 

As a result, (a, b) yield 

dr = (dr/dr) dr + (dr/d</>) d<f> = (dr/dr) dr + (dr/d(f>) u(t) dt = dr(t, r; dt, dr), (c) 

Sr = (dr/dr) Sr = Sr(t, r: Sr). (d) 

(ii) Let us consider the motion of a particle P along the inclined side of a wedge W 
that moves with a given horizontal motion: x =f (t) (fig. 2.11). Here, we have 

MMi = M- i M 2 = (dr/dx) dx = (dr/dx)[df (t)/dt\ dt = (dr/dt) dt ~ dt-, (e) 
MM 3 = M { M 2 = Sr= (dr/dq) Sq - Sq\ (f) 

MM 2 ~ dr = (dr/dq) dq + (dr/dt) dt; but Sx = 0. (g) 

(iii) Let us consider the motion of a particle P on the fixed and rigid surface 
<j>(x,y,z) = 0 or z = z(x,y). Then, r = r(x,y,z) = r[x,y,z(x,y)\ = r(x,y), and the 
classes of dr and Sr are equivalent. But, on the moving and possibly deforming 
surface (j>(t;x,y,z) =0 or z = z(x,y; t), r = ■ ■ ■ = r(t;x,y), and so dr ^ Sr: Sr still 
lies on the instantaneous tangential plane of the surface at P, whereas dr does 
not. 



Figure 2.11 On the difference between possible/admissible and virtual displacements 
(ex. 2.5.1: b). 


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§2.5 VELOCITY, ACCELERATION, ADMISSIBLE AND VIRTUAL DISPLACEMENTS 



Example 2.5.2 Lagrangean Coordinates and Virtual Displacements. Let us deter¬ 
mine the Lagrangean description r = r(r 0 -,t,q) and corresponding virtual displace¬ 
ments Sr = ■ ■ ■ for the following systems: 

(i) Two particles, Pi and P 2 , are connected by a massless rod of length l, in plane 
motion (fig. 2.12). For an arbitrary rod point P(X, Y), including P x and P 2 , we have 

X = X l +xcos(j) = X(x-,X 1 ,<f>), Y — Y) + x sin </> = F(x; F b </>), (a) 

or, vectorially, 

r = f*i + xi = r(x; X \, Yi , </>), 0 < x < l. (b) 

Therefore, r 0 = xi', while the (inertial) positions of P x and P 2 are given, respectively, 
by 


r l =r(0-,X u Y l ,(f>) = X l I+Y l J, (c) 

r 2 = r(I ; X h Y), (f>) = (X x + /cos (j))I + (Y { + / sin (j>)J 

= (X { I + Y X J) + /(cos0/ + sin c/rJ) = r t + //. (d) 

Hence, this is a (holonomic) three DOF system: q x = X x , q 2 = Y), q 3 = </>. From (a) 
we obtain, for the virtual displacements, 

SX = 6X\ + x(— sin0) SY = SY\ + x(cos </>) 6<f>] (e) 

or, vectorially, 

Sr = Sri + x hi = Sri + x[(6<f>k) x i] = Sr x + (x 6(f)) j. (f) 


(ii) A rigid body in plane motion (fig. 2.13). For this three DOF system we have 

X = A* + xcos cj) — y sin c/) = X (x, y; A*, </>), 

Y = Y* + xsin (f) + y cos cf) = Y (x, y; Y+,<f>), 

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(g> 





CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS 



(and Z = Z* = 0, say), or, vectorially, 

v = XI+YJ = r(x, y;X*, 3^,0); (h) 

that is, r c = xi + yj and q { = X+, q 2 = Y+, q 2 = cj). Therefore, the virtual displace¬ 
ments are 

8X = 8X. - xsin<t>6<t>-ycosif> 8<f> = 6X. - 64>(Y - 7*), 

8Y = (57* + x cos <j) 8<\> — v sin cf> 8<j> = 8Y+ + 8cj>(X — X»), (i) 

(and 8Z = <5Z* = 0), or, vectorially, 

8r = 5r+ + 8<t> x (r — r 4 ), 6<j) = 84>k. (j) 

The extension to a rigid body in general spatial motion (with the help of the Eulerian 
angles, §1.12, and recalling discussion in § 1 . 8 ) is straightforward. 


2.6 SYSTEM FORMS OF LINEAR VELOCITY (PFAFFIAN) CONSTRAINTS 

Stationarity/Scleronomicity/Catastaticity for 
Positional/Geometrical (=> Holonomic) Constraints in 
System Variables 

We begin by extending the discussion of §2.2 to general system variables, inertial or 
not. Positional constraints of the form cj){q) = 0 (=> dcj)/dt = 0) are called stationary, 
otherwise — that is, if q) = 0 (=> dcj)/dt ^ 0 ) — they are called nonstationary, and 
if all constraints of a system are (or can be reduced to) such stationary (nonstation¬ 
ary) forms, the system is called scleronomic ( rheonomic). Clearly, such a classification 
is nonobjective —that is, it depends on the particular mode and/or frame of reference 
used for the description of position/configuration: for example, substituting r(t, q) 
into the stationary constraint cp(r) = 0 turns it to a nonstationary constraint, 
cj>[r(t, q')\ = tj>(t,q) = 0 (and this is a reason that certain authors prefer to base this 
classification only for constraints expressed in system variables); or, a constraint that 

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§2.6 SYSTEM FORMS OF LINEAR VELOCITY (PFAFFIAN) CONSTRAINTS 


is staLionary when expressed in terms of inertial coordinates ( q ) may very well 
become nonstationary when expressed in terms of noninertial coordinates ( q')\ 
under the frame of reference (i.e., explicitly time-dependent!) transformation 
q —» q'{t,q) <=> q —► q(t,q'), the stationary constraint <j>(q) = 0 transforms to the 
nonstationary one <j>(t,q') =0. Hence, a scleronomic constraint <f>(q) =0 remains 
scleronomic under all coordinate (not frame of reference) transformations 
q —> q'(q) q' —► q(q')\ that is, its scleronomicity under such transformations is 
an objective property. 


Stationarity/Scleronomicity/Catastaticity for Pfaffian Constraints 
in System Variables 

Next, let us assume that the h holonomic constraints have been embedded into 
our system by the n = 3N — h Lagrangean coordinates q. To embed the additional, 
m(< n) mutually independent and possibly nonholonomic Pfaffian constraints 
(2.2.9) into our Lagrangean kinematics and kinetics: first, we express them in system 
variables. Indeed, substituting v from (2.5.2) into (2.2.9), we obtain the Pfaffian 
constraints in system (holonomic) variables: 

Id = § (Bd -v) + B d = ■ ■ ■ = ^ c Dk v k + c D = 0 (D = 1 ,... ,m), ( 2 . 6 . 1 ) 

where 

cat = c Dk (t, q) = $ B d • ( dr/dq k ) = £ B D ■ e k , (2.6.1a) 

c d — c D,n+ 1 — c d,o = c ,d(L q) = B D • (dr/dt) + B D = B D ■ e 0 + B D \ (2.6.1b) 

and rank(c Dk ) = in. Similarly, substituting dr from (2.5.12a) into the differential 
form of (2.2.9), f D dt = 0, we obtain the kinematically admissible, or possible, form 
of these constraints in (holonomic) system variables: 

d'do =f D dt= cat dq k + codt = 0; (2.6.2) 

with d'Oo- not necessarily an exact differential', that is, do may not exist, it may be a 
“quasi variable” (§2.9) and, in view of what has already been said about virtualness, 
namely, dt —» 8t= 0 , the virtual form of these constraints in particle variables is 

6'9 D = ^B D -Sr = 0, (2.6.3) 

and, accordingly, invoking (2.5.12b), in system variables, 

S'0 D = ^ c Dk Sq k = 0 . (2.6.4) 

The above show that, as in the particle variable case, the virtual displacements are 
mathematically equivalent to the difference between two systems of possible 
displacements, d\q and d 2 q, occurring at the same position and for the same time, 
but in different directions: apply ( 2 . 6 . 2 ) at (t, q), for d\q ^ d 2 q, and subtract side by 
side and a (2.6.4)-like equation results. 

And, as in (2.5.12a,b), once the constraints have been brought to these Pfaffian 
forms, the size of the Sq's does not matter; it is the constraint coefficients c Dk that do. 

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CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS 


Now, if in (2.6.1-2), 

(i) dc Dk /dt = 0 => cat = c Dk {q) (2.6.5a) 

and 

(ii) c D = c Dn+ \ = c D q = 0, (2.6.5b) 

=>• constraint: f D = Y c Dk{q)vk = 0, 


for all D = 1, ... ,777 and k = 1, ... , 77 , these constraints are called stationary, other¬ 
wise they are nonstationary, and a system with even one nonstationary constraint 
is called rheonomic; otherwise it is scleronomic. The inclusion of (2.6.5b) in the 
stationarity definition is made so that the velocity form of stationary position con¬ 
straints coincides with that of the stationary velocity constraints: 

<t>D{q) = 0 => df D /dt = Y ( d<j) D /dq k )v k = Y <t>Dk{q)vk = 0. (2.6.5c) 

If only c D = c D n+ 1 = c D q = 0, for allD, but dc Dk /dtf 0 =>■ c Dk = c Dk (t, q) even for 
one value of D and k, the Pfaffian constraints are called catastatic [« calm, orderly 
(Greek)]; otherwise they are called acatastatic. We notice that stationary constraints 
are catastatic, but catastatic constraints may not be stationary; we may still have 
dc D k/dt f 0 for some D and k. As mentioned earlier (2.2.1 la ff.), it is the castastaticl 
acatastatic classification, having meaning only for Pfaffian constraints, that is the 
important one for analytical kinetics, not the stationary/nonstationary one. 

Finally, as (2.6.1b) shows, the acatastatic coefficients c D result from the nonsta¬ 
tionary part of v (i.e., dr/dt ), and the acatastatic part of (2.2.9) (i.e., B D ). From this 
conies the search for frames of reference/Lagrangean coordinates where the Pfaffian 
constraint coefficients take their simplest possible form; a problem that, in turn, leads 
us to the investigation of the following. 


Transformation Properties of c Dk and c D , under a 
General Frame-of-Reference Transformation 

The latter is mathematically equivalent to an explicitly time-dependent coordinate 
transformation: q-^> q = q(t,q ) and t—* t' = t. Then (2.6.1-lb) become 

f D =Y c Dk (X (dq k /dq k ,)v k f + dq k /dt ) + c D 

= ... = Y c Dk' v k' + c ' D (=0) (£,&' = 1,...,7Z; D= 1,..., 777 ); (2.6.6) 

where 

c Dk ' = Y^ ( dq k /dq k ’)c Dk (covariant vector-like transformation in k), (2.6.6a) 

c' D = Y, ( dcjk/dt)c Dk + c D (covariant vector-like transformation in t = 77 + 1, 

with q' n+ \ = t' = t => dt'/dt = 1). (2.6.6b) 

The above readily show that: (i) if dq k /dt = 0 [i.e., q = q(q')] (= coordinate trans¬ 
formation', in the same frame of reference), then c' D = c D ; and (ii) we can choose a 
frame of reference in which c' D = 0; that is, catastaticity/acatastaticity (and statio- 
narity/nonstationarity) are frame-dependent properties. 

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§2.6 SYSTEM FORMS OF LINEAR VELOCITY (PFAFFIAN) CONSTRAINTS 


Holonomicity versus Nonholonomicity 

The m(< n) constraints (2.6.1) are independent if the m x n constraint matrix (c Dk ) 
has maximal rank (i.e., m) at each point in the region of definition of the cf s and t. 

Now, if these constraints are completely integrable = holonomic [i.e., either they 
are exact: c Dk = dh D /dq k and c D = dh D /dt, where h D = h D (t,q) (=0); or they 
possess integrating factors, as explained in §2.2], then there exists a set of n “equili¬ 
brium,” or “adapted (to the constraints)” system coordinates x = (Xti---jXn) i n 
which these constraints take the simple uncoupled form: 

Xi = h{t, q) = 0,..., x,n = h m {t, q) = 0; (2.6.7a) 

Xm +1 = q) + 0,... ,Xn = K(f q) ± 0; (2.6.7b) 

where, as in §2.4, the n — m functions h m+] (t, q),..., h n (t, q) are arbitrary, except that 
when (2.6.7a, b) are solved for the n q’s in terms of the (n — m) xi = (Xm+i, ■ ■ •, Xn) 
and these expressions are inserted back into the m holonomic constraints 
hi(t,q) = 0,... ,h m (t,q) = 0, they satisfy them identically in the xf s and t. The 

X/’s are the new positional system coordinates of this 3 N — (h + m) = 

(3 N — h) +m = n — m = n' (both global and local) DOF : 

q ->■ q' = (xm+i? ■ • ■ ,Xn) = {q\, ■ ■ ■, q'„’)- (2.6.7c) 

This process of adaptation to the constraints via new equilibrium coordinates can be 
repeated if additional holonomic constraints are imposed on the system; and with 
some nontrivial modifications it carries over to the case of additional nonholonomic 
constraints (§2.11: essentially, by expressing this adaptation ... idea in the small; 
i.e., locally, via “equilibrium quasi coordinates”). The importance of this method 
to AM lies in its ability to uncouple constraints, and thus to simplify significantly the 
equations of motion (chap. 3). 

If, on the other hand, the constraints (2.6.1) are noncompletely integrable = non¬ 
holonomic , then the number of independent Lagrangean coordinates (= number of 
global DOF) remains n, but the system has n — m = f DOF (in the small, or local 
case); that is, under the additional m nonholonomic constraints [(2.6.1), (2.6.2)], the n 
q s remain independent (unlike the holonomic case!), but the n v/dq/Sq 's do not —or, if 
the differential increments 6q are arbitrary (if, for example, we let q k become q k + Sq k 
while all the other cf s remain constant ), then they will no longer be virtual ; that is, they 
will not be compatible with the virtual form of the constraints (2.6.4); and similarly 
for the v’s, dcf s. [Of course, if m = 0, then the n cf s are independent and their 
arbitrary increments 6q are virtual; that is, both q’s and bcf s satisfy the existing 
(initial) h holonomic constraints. For example, in the case of a sphere rolling on, 
say, a fixed plane: (a) if the plane is smooth (i.e., m = 0), both the arbitrary <?’s and 
the arbitrary (q + dqf s, are kinematically possible; while (b) if the plane is suffi¬ 
ciently rough so that the sphere rolls on it (i.e., m f 0, and the additional (rolling) 
constraints are nonholonomic), only the q’s are still arbitrary (independent), the 
(q + dqf s are not—or, if they are, the sphere does not roll. For details, see exs. 
2.13.4, 2.13.5, 2.13.6.] 

To find the number of independent Sq’s under the additional m (holonomic or 
nonholonomic) constraints (2.6.1, 2, 4) we must now turn to the examination of the 
following. 


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CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS 


Introduction to Virtual Displacements under Pfaffian Constraints 
(Introduction to Quasi Variables) 

In this case, the particle virtual displacement is still represented by (2.5.12b): 


Sr = J2 ( d, ’/ dc lk) Sq k = Y2 e k Sq k , 

(2.6.8) 

but, due to the virtual constraints (2.6.4), out of the n Sq's only n — m are indepen¬ 
dent; that is, if, now, all n Sq's vary arbitrarily, the resulting Sr, via (2.6.8), will not be 
virtual—denoting a differential increment of a system coordinate by Sq does not 
necessarily make it virtual; it must also be constraint compatible. For example, 
solving (2.6.4) for the first m Sq's, 

Sq D = ( Sq { ,..., Sq m ) = Dependent Sq's, 

(2.6.9a) 

in terms of the last n — m of them. 


Sqj = (Sq m+ 1 ,..., 6q„) = Independent Sq's, 

(2.6.9b) 

we obtain 


Sq D = b oi 6c li ( D = 1. ■ • •, m; I = m+ 1,... 

,n), (2.6.9) 


where b DI = b DI {q, t) = known functions of (generally, all) the q s and t. 
Substituting (2.6.9) into (2.6.8), we obtain, successively, 

Sr = Y^ e k Sq k = e D 6q D + Y^ e i s <h = ^2 e D (5Z bm bq <) + J2 e < hqf ' 

hnally 

either Sr = Sq k , under ^ c Dk Sq k = 0 (6q k , nonarbitrary), (2.6.10a) 
or 6r = ^2f}jSqj (Sqj, arbitrary), (2.6.10b) 

where 

P, = ej + ^2 S>Di e D = dr/dqj + ^ b DI ( dr/dq D ) [see also (2.11.13a ff.)]; 

(2.6.10c) 

that is, the most general particle virtual displacement under (2.6.4) can be 
expressed as a linear and homogeneous combination of the “ narrower" basis {Pj\ 
I = m + 1,...,«}, whose vectors are, in general [and unlike the e k s —recalling 
(2.5.4a If.)], nongradient, or nonholonomic : 

[Ijfdr/dq, ^ dfij/dqr f d[l r /dq, (/,/' = m + 1,...,«); (2.6.11) 

as can be verified directly by using (2.6.10c) in (2.6.11). 

The number of independent Sq's, here n — m = f, equals the earlier defined 
number of local DOFs; and, inversely, we can redefine the number of DOFs in 
the small, henceforth called simply DOF, as the smallest number of independent 
parameters qi = Vj/dqj/Sq / needed to determine v/dr/Sr , for all system particles and 
any admissible, and so on, local motion; that is, the number of DOFs (in the small) 
= minimum number of independent “local positional”, or motional , parameters. Just 
as the number of DOFs in the large, F = n (here), is the minimum number of 

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§2.7 GEOMETRICAL INTERPRETATION OF CONSTRAINTS 


independent positional parameters needed to determine the configurations of all 
system particles in any admissible, and so on, global motion. 

REMARKS 

(i) The / bqi can, in turn, be expressed as linear and homogeneous combinations 
of another set of /-independent motional parameters, say r\p. 6qj = Hir(t,q) Vr 
(I, I' = m + 1,...,«); in which case (2.6.10b) becomes 

Sr = Y l } ! (£ H »' J lr) = E (E H "'fir)vr 

= Y bl ' 7 h' = Y (2.6.10d) 

(ii) As already mentioned, the importance of these considerations lies in kinetics 
(chap. 3), where it is shown that the number of independent kinetic equations of 
motion (= equations not containing forces of constraint) equals the number of inde¬ 
pendent 6q’s. 


Problem 2.6.1 Show that due to the m Pfaffian constraints (2.6.1) (expressed in 
terms of the notation dq k /dt = v k ): 

Y dDfcbt + c D = 0 CD = 1,, m; k = 1,..., n), (a) 

or, equivalently, in the (2.6.9)-like form, in the velocities, 

v D = Y b Di v i + b D (I = m + l,... ,n), (b) 

the additional holonomic constraint q) = 0 satisfies the following (n — m) + 1 
conditions: 

dcb/dq, + ^ t> DI {d(j)/dq D ) = 0 and dd/dt + ^ b D (d(j)/dq D ) = 0; (c) 

which, in terms of the notation eft, q D , q,) = <j>[t, q D (t, qf, cji] = <j> 0 {t,qi) = 0, read 
simply 

d(/) o /dqi = 0 and df 0 /dt= 0 , (d) 

respectively (compare with example 2.4.1.). 


Before embarking into the detailed study of nonholonomic constraints and asso¬ 
ciated “coordinates” (to embed them), and the most general v/r/r/<5r-representations 
in terms of n — m arbitrary motional system parameters, of which the previous 
V/ = q I /dq I /6q I are a special case, let us pause to geometrize our analytical findings; 
and in the process dispel the incorrect impressions, held by many, that analytical 
mechanics is, somehow, only numbers (analysis), no pictures — an impression 
initiated, ironically, by Lagrange himself! 


2.7 GEOMETRICAL INTERPRETATION OF CONSTRAINTS 

Configuration Spaces 

As explained in §2.2, before the imposition of any constraints, the configurations of a 
mechanical system S' are described by the motion of its representative, or figurative, 
particle P(S) = Pin a (clearly, nonunique) 3A-dimensional Euclidean, or noncurved/ 

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CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS 


flat, space, E 3N , called unconstrained, or free, configuration space. [Briefly, Euclidean, 
or noncurved, or flat, means that, in it, the Pythagorean theorem (“distance squared = 
sum of squares of coordinate differences”) holds globally, that is, between any two space 
points, no matter how far apart they may be; see, for example, Lur’e (1968, p. 807 ff.), 
Papastavridis (1999, §2.12.3).] 

The position vector of P, in terms of its rectangular Cartesian coordinates/com¬ 
ponents relative to some orthonormal basis of fixed origin O, in there, is [recall 
(2.4.3 ff.)] 

«=Ki=£i(0>--->6w = 6*(0]- (2-7.1) 

However, as detailed in §2.4, upon imposition on 5 of h holonomic constraints 
and subsequent introduction of n = 3N — h Lagrangean coordinates 
q=[q i = q\(t),... ,q n = q„(t)}, or simply q = (q l ,... ,q n ), the above assumes the 
parametric representation 

£ = §(<,?) = [6 =C\(t;q),...fi 3N = ^ N {t;q)}, (2.7.1a) 

which, in geometrical terms, means that, as a result of these constraints, P can no 
longer roam throughout E iN , but is forced to remain on its time-dependent n-dimen- 
siona! surface defined by (2.7.1a), called reduced, or constrained configuration space of 
the system; actually the portion of that surface corresponding to the mathematically 
and physically allowable range of its curvilinear coordinates q. In differential-geo- 
metric/tensorial terms, that space, described by the surface coordinates q, when 
equipped with a physically motivated metric, becomes, at every instant t, a generally 
non-Euclidean (or curved, or nonflat) metric manifold, M n {t) = M n , usually a 
Riemannian one, embedded in E 3N ; and this explains the importance of 
Riemannian geometry to theoretical dynamics. [Riemannian manifold means one in 
which the square of the infinitesimal distance (“line element”) is quadratic, homo¬ 
geneous, and (usually) positive-definite in the coordinate differentials dq k . In 
dynamics, the manifold metric is built from the system’s kinetic energy (§3.9). 
See, for example, Lur’e (1968, pp. 810 ff.), Papastavridis (1999, §2.12, §5.6 ff.)] 
Schematically, we have 


Configuration spaces 


< Unconstrained: Esn (Euclidean manifold) 
Constrained: M n (Non-Euclidean manifold) 


[N = number of particles, h = number of holonomic constraints, n = 3N - h] 


Now, as S moves in any continuous, or finite, way in the ordinary physical (three- 
dimensional and Euclidean) space, or some portion of it, P moves along a contin¬ 
uous M„-curve, q = q(t). The relevant analytical requirements on such c/’s (§2.4) are 
summarized as follows: 

(i) The correspondence between the q n-tupies and some region of M„ must be one-to- 
one and continuous (additional holonomic constraints would exclude some parts of 
that region from the possible configurations). 

(ii) If As = displacement, in M n , corresponding to the ^-increment Aq, we must have 
\\m{As / Aq k ) f 0, as Aq k —> 0 (k = \,..., n); or dq k /ds (= “direction cosines” of 
unit tangent vector to system path in M„) = finite. The q's are then called regular. 
(See also Langhaar, 1962, p. 16.) 

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§2.7 GEOMETRICAL INTERPRETATION OF CONSTRAINTS 


Event Spaces 

Instead of the “dynamical” spaces Ein and M n , we may use their (formal and 
nonrelativistic) “union” with time t = q 0 = q n+ \, symbolically, 

E 3N+l = £ 3A , x T(ime) and M n+1 = M„ x T(ime). (2.7.2) 

These latter are called (unconstrained and constrained, respectively): manifolds of 
configuration and time (or of extended configuration), or “ geometrical ” space-time 
manifolds, or film spaces', or, simply, event spaces. M„(M„ +1 ) is suitable for the 
study of scleronomic (rheonomic) systems. (One more such “generalized space,” 
the phase space of Lagrangean coordinates and momenta, is examined in chap. 8.) 


Constrained Configuration Spaces and their Tangent 
Planes 

The h stationary and holonomic constraints define, in £ 3Ar , a stationary (nonmoving) 
and rigid (nondeforming) n-dimensional surface M n ; while h nonstationary holonomic 
constraints define, in E m , a nonstationary (moving) and nonrigid (deforming) n- 
dimensional surface M„(t). However, these same nonstationary constraints also 
define, in £ 3A r +1 , a stationary and rigid (n + \)-dimensional surface M n+l ; hence, 
the relativity of these terms! The equations t = constant define 00 1 privileged surfaces 
M„(t) in M„ +1 . Thus, the motion of the system can be viewed either as (i) a stationary 
curve in the geometrical space M n+ p, or (ii) as the motion of the representative system 
point in the deformable, or “breathing,” dynamical space Further, through 

each M„-point q(t) there passes a (n — l)-ple infinity of kinematically possible system 
paths, on each of which the “rate of traverse” dq/dt is arbitrary; and through each 
M n+ \ -point (q, t ) there passes an 77 -ple infinity of such paths, but these latter, since 
there is no motion in M n+l , are not traversed. The kinetic paths of a system in M„ 
and M„ + i are called its trajectories/orbits and world lines, respectively. Additional 
M„/M n+l differences are given below, in connection with nonholonomic constraints. 

Next, and as differential geometry teaches, (i) the set of all (n + l)-ples (dq a ) make 
up the tangent point space (hyperplane) to M n+l at k- T n+ 1 (£); while (ii) the vectors 
{E a = E a (P); a = 1,... , 77 + 1}, defined by dP = dl; = dq = J 2 E„ dq a : vector of 

elementary system displacement determined by P(q) and P(q + dq) (each E a being 
tangent to the coordinate line q a through P) constitute a “natural” basis for the 
tangent vector space associated with, or corresponding to, T „+1 (£); and similarly for 
M„. For simplicity, we shall denote both these point and vector spaces by T n+l (P), 

UP). 

REMARKS 

(i) Without a metric, these tangent spaces are affine. After they become equipped 
with one, they become Euclidean; properly Euclidean if the metric is positive definite, 
and pseudo-Euclidean if the metric is indefinite. As mentioned earlier, in mechanics 
the metric is based on the kinetic energy, and, therefore, it is either positive definite 
or positive semidefinite. 

(ii) It is shown in differential geometry that the condition that dE a =J 2 (" ')apE/3 

be an exact differential [i.e., d/dqry(dE a /dqp) = d/dqp(dE a /dq 1 )\ leads to the 
requirement that be a Riemannian manifold. For details, see, for example, 

Papastavridis (1999, p. 135). 

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CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS 


Pfaffian Constraints 

Let us begin with a system subjected to h holonomic constraints (2.4.2), and, there¬ 
fore, described by the n = 3N — h holonomic coordinates q. Then, a motion of the 
system in the physical space E 3 corresponds to a certain curve in M n (trajectory or 
orbit)/M n+1 (world line ) traced by the figurative system particle P; and, conversely, 
admissible M„/M n+ \ curves represent some system motion. Now, let us impose on it 
the additional m Pfaffian constraints: 

Kinematically admissible form : d'0 D = Cuk dq k + Codt = 0, (2.7.3) 

Virtual form : S'do = Cok Sq k = 0. (2-7.4) 

As a result of the above, we have the following geometrical picture: 

(i) At each admissible M„ + i-point P = ( q,t ), the m constraints (2.7.3) define (or 
order, or map, or form), the \{n + 1) — m] = [(« — m) + 1]-dimensional “element” 
T(n+\)-m{P) = T(n-m)+\(P) = T I+ i(P): tangent space (plane) of kinematically admis¬ 
sible displacements (motions), of the earlier tangent plane T„+1 (P), on which the 
kinematically admissible displacements of the system, dq, and dt lie. Therefore, at 
every P, only world lines with velocities v a = dq a /dt on that plane are possible—the 
system can only move along directions compatible with (2.7.3). 

(ii) At each such point P, the m constraints (2.7.4) define the (n — m) -dimensional 
plane T n _ m (P ): tangent space of virtual displacements (motions), or virtual plane, on 
which the virtual displacements of the system, Sq, lie. Clearly, T n _ m (P ) is the inter¬ 
section of T(„- m ) + ] (P) with the hyperplane dt—+6t = 0 there; symbolically, 
T„- m (P) = T [n _ m)+1 \ St=0 =V n . m (P) ( V for virtual). {And a manifold M n /M„ +l 
whose tangential bundle (i.e., totality of its tangential spaces) is restricted by the m 
nonholonomic equations (2.7.3) [assuming that (2.7.3), (2.7.4) are nonholonomic] is 
called nonholonomic manifold M nn _ m /M n+ln _ m . Some authors call the so-restricted 
bundle, T ln _ m - j+i or T n _ m , nonholonomic space embedded in M„, or M n+i . See also 
Maiffer (1983-1984), Papastavridis (1999, chap. 6), Prange (1935, pp. 557-560), 
Schouten (1954, p. 196).} 

(iii) The given constraint coefficients (c Dk , c D ) define, at P, an (m + 1 (-dimensional 
kinematically admissible constraint plane ( element ) C m+ \(P) perpendicular to 
T m + 1 (P) (with orthogonality dehned in terms of the kinetic energy-based metric); 
while the (c D f) dehne an ;n-dimensional virtual constraint plane (i.e., of the virtual 
form of the constraints) C,„(P) perpendicular to V n _ m (P). Sometimes, C m (P) is 
referred to as the orthogonal complement of V n _ m (P) relative to T„(P). The c D / ( 
can be viewed as the covariant (in the sense of tensor calculus) and holonomic 
components of the m virtual constraint vectors c D = ( c D k ), which, by (2.7.4), are 
orthogonal to the virtual displacements bqk'. c D ■ 6q = CDk^qk — 0- Hence since 
the c D are independent, they constitute a basis (span) for the earlier space C m (P). 
These two local planes are frequently called the nulI[V n _ m (P)\ and range[C„(P )] 
spaces of the m x n constraint matrix (c D „). These geometrical results are shown 
in fig. 2.14 (see also fig. 3.1). 

Let us consolidate our findings: 

(i) Under n initial holonomic constraints, a system can go from any admissible 
initial M„/M n+l - point, P„ to any other final such point, P f , along any chosen 
(-lying path joining P, and P f . 

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§2.7 GEOMETRICAL INTERPRETATION OF CONSTRAINTS 



Figure 2.14 Virtual displacement ( V n _ m ) and constraint (C m ) hyperplanes in 
configuration space (see also fig. 3.1). 


(ii) If the additional m Pfaffian constraints (2.7.3, 4) are holonomic, disguised in 
kinematical form, the local tangent planes become the earlier local tangent planes to 
reduced, or “smaller,” configuration/event manifolds M n _ m /M^ n+l \_ m , inside 
M n /M n+ \. These reduced but finite surfaces contain all possible system motions 
through a given P, —the system can go from any admissible initial 
M n _ m /M( n+l y m - point, P h to any other final such point, Pj, along any chosen 
(M n j M, I+ \)-lying path joining P f and P<\ that is, DOF(local) = 
DOF (global) = n — m. 

(iii) On the other hand, if the additional m Pfaffian constraints are nonholonomic, 
we cannot construct these M n _ m /Mi n+ n_ m . The global configuration/event mani¬ 
folds of the system are still M n /M n+U but these constraints have created, in there, 
a certain path-dependence : any (M„/M„ +1 )-point Pr (in the admissible portions of 
M„/M n+ 1 ) is, again, accessible from any other (M„/M, 1+1 )-point Pj but only along a 
certain kinematical family, or “ network ,” of tracks that is “narrower” than that of 
case (i); that is, the transition Pj —> Pf is no longer arbitrary because of direction-of- 
motion constraints, at every point of those paths. Or, under such constraints, 
all conhgurations/events are still possible, but not all velocities (and, hence, not all 
paths)', only certain M„/M n+ \ -curves correspond to physically realizable motions — 
the system is restricted locally, not globally; that is, n = DOF{global) 
DOF(local) = n — m. We continue this geometrical interpretation of constrained 
systems in §2.11. 


Kinetic Preview, Quasi Coordinates 

The importance of these considerations, and especially of the concept of virtualness, 
to contained system mechanics arises from the fact that most of the constraint 
forces dealt by AM (the so-called “passive,” or contact, ones; i.e., those satisfying 
the d’Alembert-Lagrange principle, chap. 3) are perpendicular to the virtual displa¬ 
cement plane V n _ m , and so lie on the virtual constraint plane C m . And this, as detailed 
in chapter 3, allows us to bring the system equations of motion into their simplest 
form; that is (i) to their smallest possible, or minimal, number ( n ), and (ii) to 
a complete decoupling of them into (n — m) purely kinetic equations—that is, 

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CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS 


equations not containing constraint forces — by projecting them onto the local 
virtual hyperplane, and (m) kinetostatic equations — equations containing constraint 
forces — by projecting them onto the local constraint hyperplane, which is perpendi¬ 
cular to the virtual hyperplane there. This is the raison d’etre of virtualness, and the 
essence of Lagrangean analytical mechanics. In all cases, under given initial/bound¬ 
ary conditions and forces, the system will follow a unique path (a trajectory, or orbit) 
determined, or singled out among the problem’s kinematically admissible paths, by 
solving the full set of its kinetic and kinematic equations. 

Schematically, our strategic plan is as the following: 


Minimal number of 
eqs. of motion (n) 



Kinetic (n — m) 



Uncoupling of eqs. 
of motion: 


Now, if the m Pfaffian constraints are holonomic, their uncoupling (and that of 
the corresponding equations of motion) is easily achieved by “adaptation to the 
constraints,” as explained in §2.4 and §2.6; but, if they are nonholonomic this 
“adaptation” can be achieved only locally, via “equilibrium” nonholonomic co¬ 
ordinates, or quasi coordinates. 

We begin the study of these fundamental kinematical concepts by first examining 
one of their important features: the possible commutativity/noncommutativity of the 
virtual and possible operations, S(.. .) and d(...), respectively, when applied to this 
new breed of “coordinates”; that is, we investigate the relation between 
d[8(quasi coordinate )] and 8[d (quasi coordinate)}. 


2.8 NONCOMMUTATIVITY VERSUS NONHOLONOMICITY; 

INTRODUCTION TO THE THEOREM OF FROBENIUS 

Let us recall the admissible (d) and virtual (6) forms of the Pfaffian constraints 
(2.7.3, 4) (henceforth keeping possible non-exactness accents only when really 
necessary!): 

d9 D = ^c Dk dq k + c D dt = 0 and S9 D = ^2c Dk 5q k = 0, (2.8.1) 

where D = 1,..., m; I = m + 1, ...k (and all other small Latin indices) = 1 
Now, 6(.. .)-varying the first of (2.8.1), and d(. ..)-varying the second, and then 
subtracting them side by side, we find, after some straightforward differentiations 
and dummy index changes, 


d(S9 D ) - 8{d9 D ) = ^ ~2c Dk [d(8q k ) - S(dq k )\ 

+£(£ {dc Dk /dq l - dc Dl /dq k ) dq, 


+{dc Dk /dt - dc D /dq k ) dt^j 6q k 
+ C{)[d(8t) — 8(dt)\, 


(2.8.1a) 


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§2.8 NONCOMMUTATIVITY VERSUS NONHOLONOMICITY 


or, since the last term is zero [St = 0 =>■ d(St) = 0, and, during 8(.. .) time is kept 
constant =>■ S(dt) = 0], and with the earlier notations q 0 = q n+] = t => 8q 0 = Sq„ +l = 
St = 0,c D = c D0 = c Dn+i , and Greek subscripts running from 1 ton + 1 (or from 0 ton): 


d{S0 D ) - S(dO D ) = 22c D k[d(Sq k ) - 5(dq k )\ 

( 2 . 8 . 2 ) 

+ EE ( dc Dk /dq a - dc D Jdq k ) dq a Sq k . 

A final simplification occurs with the useful notations d(S.. .) — 6(d.. .) = D {...), 
and 


C D fja = dc D p/dq a - dc Da /dq, 3 = -C D a0 , (2.8.2a) 


F d = EE C D ka dq a Sq k : Frobenius’ bilinear, or antisymmetric, covariant 

of the Pfaffian forms (2.8.1). (2.8.2b) 


Thus, (2.8.2) transforms to 


DSo = '22 c ° k + F D . 


(2.8.2c) 


Problem 2.8.1 Starting with eqs. (2.5.12a,b): 


dr =22 e k dq k + e 0 dt, 6r = 22 e k Sq k , 


and repeating the above process, show that 

blr = E Dc lk e k- 


(a) 

(b) 


From the above basic kinematical identities, we draw the following conclusions: 

(i) ifc fl to = o, identically in the q’s and t, and for all values of D, k, a then, since 

Dq k = d{Sq k ) — S(dq k ) = 0, (2.8.3a) 

the q k being genuine = holonomic coordinates, it follows that 

D6 d = d{Sd D ) - S{dO D ) = 0; (2.8.3b) 

that is, the do are also holonomic coordinates, the ddo/SOo are exact differentials. 
In this case (2.8.1) may be replaced by m holonomic constraints; which, in turn, may 
be embedded into the system via n’ = n — m new equilibrium coordinates, as 
explained in §2.4. 

(ii) If F d ^ 0, then D0 d ^ 0; or, more generally, we cannot assume that both 
d(dq k ) = S(dq k ) and d(69 D ) = S(d6 D ) hold; it is either the one or the other. (As 
detailed in chap. 7, this realization helps one understand the fundamental differences 
that exist between variational mathematics and variational mechanics. See also 
pr. 2.12.5.) If we assume (2.8.3a) for all holonomic coordinates, constrained or not, 
then D6o ^ 0; that is, the do are nonholonomic coordinates; and, as Frobenius’ 
theorem shows (see below), the constraints (2.8.1) are nonholonomic. 

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CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS 


(iii) If, however, F D = 0, since the dq/bq are not independent, it does not neces¬ 
sarily follow that C D ka = 0. To make further progress — that is, to establish necessary 
and sufficient holonomicity/nonholonomicity conditions in terms of the constraint co¬ 
efficients, c Dk and c D . we need nontrivial help from differential equations/dififerential 
geometry; and this leads us directly to the following fundamental theorem of 
Frobenius (1877). First, let us formulate it in simple and general mathematical 
terms, and then we will tailor it to our kinematical context. 


Theorem of Frobenius 

The necessary and sufficient condition for the complete (or unrestricted) integrability = 
holonomicity of the Pfaffian system: 

X D = Y J ^ DK dx K = Q [D= l,...,m(<F); K,L= 1,...,F], (2.8.4) 

where X nK = X DK (x ]..... x F ) = X nk (x) = given and well-behaved functions of 
their arguments, and rank(X DK ) = m(<F); that is, for it to have m independent 
integrals/ D (x) = C D = constants, is the vanishing of the corresponding m bilinear 
forms: 


( dX DK /dx L - dX DL /dx K )u K v L , (2.8.5) 

identically (in the x’s) and simultaneously (for all D" s), for any/all solutions 
u = (u ] ...., u F ), and v = (v 1; ..., v F ) of the m constraints ^dkVk = 0; that is, 
for any/all rj K —> u K ,v K satisfying 

Y Xdk u k = 0 and ^ Xdk v k = ^ Xdl v l = 0- (2.8.6) 

[Also, recall comments following eqs. (2.3.lie).] 

[If the system (2.8.4) is completely integrable, then, since its finite form depends on 
the integration constants C D (i.e., ultimately, on the initial values of the x’s), then it is 
semiholonomic (§2.3).] 

Adapted to our kinematical problem — that is, with the identifications F —► n + 1, 
u K —> 6q k , v L —> dq a , x — > t, q, and recalling that q n+l = t satisfies the additional 
holonomic constraint Sq n+ 1 = 6t = 0 — Frobenius theorem states that: If 

Fd =d{Sf} D ) -5(d8 D ) = d[y^cpk8q k ) - of ~Fq j 

= E(E ( dc Dk /dq a - dc Da /dq k ) dq a ) 6q k = EE C D ka dq a Sq k = 0, 

(2.8.7) 

for arbitrary dq a = clq k , dq ll+l = dq 0 = dt and 6q k , solutions of the constraints: 

E c Da dq a = °Dk dq k + c D dt = 0 and Y c Dk % = 0, (2.8.1) 


then these constraints are holonomic. 

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§2.8 NONCOMMUTATIVITY VERSUS NONHOLONOMICITY 

The above show that since our dq' s and Sq's are not independent, the vanishing 
of the F D ' s does not necessarily lead to 

C D ka = 0, (2.8.8) 

as holonomicity conditions. For this to be the case, eqs. (2.8.8) are, clearly, sufficient 
but not necessary, they would be if the dq's and Scf s were independent; namely, 
unconstrained. 

This observation leads to the following implementation of Frobenius’ theorem: 
we express each of the («) nonindependent dcf s and Scf s as a linear and homoge¬ 
neous combination of a new set of n — m independent parameters (and dt, for the 
dq' s), insert these representations in F D = 0, and then, in each of the so resulting m 
bilinear covariants (in these new parameters), set its n — m coefficients equal to zero. 
We shall see in §2.12 that, in the general case, this approach leads to a direct and usable 
form of Frobenius’ theorem, due to Hamel. But before proceeding in that direction, we 
need to examine in sufficient detail the necessary tools: nonholonomic coordinates, or 
quasi coordinates (§2.9), and the associated transitivity relations (§2.10). 

REFERENCES ON PFAFFIAN SYSTEMS AND 
FROBENIUS' THEOREM 

(for proofs, and so on, in decreasing order of readability for nonmathematicians): 

Klein (1926(a), pp. 207-214): introductory, quite insightful. 

De la Vallee Poussin (1912, vol. 2, chap. 7): most readable classical exposition. 

Guldberg (1927, pp. 573-576) and Pascal (1927, pp. 579-588): outstanding handbook 
summaries. 

Forsyth (1890/1959, especially chaps. 2 and 11): detailed classical treatment. 

Lovelock and Rund (1975/1989, chap. 5): excellent balance between classical and 
modern approaches. 

Cartan (1922, chaps. 4-10): the foundation of modern treatments. 

Weber [1900(a), (b)]: older encyclopedic treatise (a) and article (b, pp. 317-319). 

Heil and Kitzka (1984, pp. 264-295): relatively readable modern summary. 

Chetaev (1987/1989, pp. 319-326): happens to be in English (not particularly enlighten¬ 
ing). 

Frobenius (1877, pp. 267-287; also, in his Collected Works, pp. 249-334): the original 
exposition; not for beginners. 

Hartman (1964, chap. 6): quite advanced; for ordinary differential equations specialists. 

Outside of Lovelock et al., we are unaware of any contemporary readable exposition of 
these topics in English; i.e., without Cartanian exterior forms, and so on. 


Example 2.8.1 Necessary and Sufficient Condition(s) for the Holonomicity of the 
Single Pfqffian Constraint (2.3.1) via Frobenius’ Theorem: 

dO = a(x,y,z)dx + b(x,y,z)dy + c(x,y,z ) dz = adx + bdy + cdz = 0, (a) 

or, since it is catastatic, 

59 = a 5x + b Sy + c 5z = 0. (b) 


299 


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CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS 

By ^-varying (b), and ^-varying (a), and then subtracting side by side, we find, after 
some straightforward differentiations: 

d{59) — 8{d9) = a[d{Sx ) — h(dx)] + b[d{8y) — 8{dy)} + c[d{8z) — 8{dz)\ 

+ [{da 8x — 8a dx) + {db 8y — 8b dy) + {dc 8z — 8c dz )] 
= {da/dy — db/dx)(dy 8x — 8ydx) + {da/dz — dc/dx){dz8x — Szdx) 

+ {db/dz — dc / dy){dz 8y — 8zdy) 

[substituting into this, dz = {—a/c) dx + {—b/c) dy and 8z = 
(— a/c) 8x + (— b/c) 8y, solutions of the constraints (a, b), respec¬ 
tively; since here n — m = 3 — 1 = 2 = number of independent differ¬ 
entials (for each form of the constraint); we could, just as well, 

substitute dx = • • • dy + • • • dz and 8x = ■■ ■ 8y H- 8z, or dy = ■ ■ ■ 

and 8y = ■ ■ ■ ] 

= ■ ■ ■ = ( da/dy — db/dx)(dy 8x - 8ydx) 

+ {da/dz — dc/dx){—b/c){dy8x — 8ydx) 

+ {db/dz — dc/dy){—a/c){dx8y — 8xdy) 

= \{da/dy - db/dx) + {b/c){dc/dx - da/dz) 

+ {a/c){db/dz — dc / dy)]{dy 8x — 8ydx). 

Setting d{89) — 8{d0) = 0, and since now the bilinear terms dySx and Sydx are 
independent, we recover the earlier holonomicity condition (2.3.6). 


Vectorial Considerations 

Equations (a)/(b), in terms of the vector notation 

h={a,b,c ), dr = {dx,dy,dz), and 8r = {8x, 8y, 8z), (d) 

state that 

h ■ dr = 0 and h ■ 8r = 0; (e) 


that is. It is perpendicular to the plane defined by the two (generally independent) 
directions dr and 8r, through r = (. x,y,z ). On the other hand, the second of (c) states 
that 

d{59) — S{d9) = curl h ■ {dr x Sr) = 0, (f) 


that is, curl h is perpendicular to the normal to that plane; and, hence, excluding the 
trivial case dr x 8r = 0, curl h lies on that plane. Accordingly, b and curl h are 
perpendicular to each other: 

h curl It = 0. i.e., (2.3.8a). (g) 


Example 2.8.2 The Two Independent and Catastatic Pfaffian Constraints: 

d9 = a{x,y,z)dx + b{x,y,z) dy + c{x,y,z)dz = adx + b dy + cdz = 0, (a) 

dO = A{x,y, z) dx + B{x , y, z) dy + C{x,y,z) dz = Adx + Bdy + Cdz = 0, (b) 


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§2.9 QUASI COORDINATES, AND THEIR CALCULUS 


when taken together (i.e., n = 3, m = 2) will always make up a holonomic system; 
even if each one of them separately (i.e., n = 3, m = 1) may be nonholonomic! 

Solving (a) and (b) for any two of the dx, dy, dz in terms of the third, say dx and 
dy in terms of dz , we obtain 


dx = e(x, y, z) dz and 

and, similarly, since (a) and (b) are catastatic, 

dy =f(x,y,z)dz; 

(c) 

6x = e(x,y,z ) & 

Therefore, we find, successively, 

d{50) - 5{d6) = 

and 

Sy=f{x,y,z)8z. 

(d) 


= ••• = (■■ ■ )(dy6x — 6ydx ) + (• • -)(dz8x — Sz dx) + (• • -)(dz8y — 8zdy ) 
= [using (c) and (d)] = ■■■ = (•• -)(dz8z — 8zdz) = (• • -)0 = 0; (e) 


and, similarly, 

d{5&) - 6(dO) = •■■ = (•••)(*&- fe<fe) = (•■■)() = 0; Q.E.D. (f) 

Proceeding in a similar fashion, we can show that: a system of n — 1 (or n) indepen¬ 
dent Pfaffian equations, in n variables [like (2.8.1) with m = n — 1 or n] is always 
holonomic. This theorem illustrates the interesting kinematical fact that additional 
constraints may turn an originally (individually) nonholonomic constraint into a 
holonomic one (as part of a system of constraints); see also §2.12. 


2.9 QUASI COORDINATES, AND THEIR CALCULUS 

Let us, again, consider a holonomic system S described by the hitherto minimal, or 
independent, n Lagrangean coordinates q = (q x ,... ,q n ), and hence having kinema¬ 
tically admissible I possible system displacements {dq, dt) = (, dq i,..., dq n ; dt). Now, at 
a generic admissible point of S”s configuration or event space {q, t ), we can describe 
these local displacements via a new set of general differential positional and time 
parameters ( d9,dt) = ( d0 l: ..., d0 n ; d6 n+ \ = dd 0 ), defined by the n + 1 linear, homo¬ 
geneous, and invertible transformations: 

d0 k = ^ a k i dq/ + a k dt , dd n+x = dd 0 = dq n+l = dq Q = dt , (2-9.1) 

rcmk(a k {) = n => Det[a k [) f 0, (k, l = 1,... ,n), (2.9.1a) 

where the coefficients a k / and a k = a k n+x = a k0 are given functions of the q's and t 
(and as well-behaved as needed; say, continuous and once piecewise continuously 
differentiable, in some region of interest of their variables). Inverting (2.9.1), we 
obtain 


dq t = A[ k d0 k + Aj dt , dq„+i = d6 n+l = dd 0 = dt, (2.9.2) 
rank(Aj k ) = n => Det(Aj k ) f 0, (k, l = 1,..., n), (2.9.2a) 

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CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS 

where the “inverted coefficients” A /k and A\ = A/„ +1 = A m become known functions 
of the q’s and t, and are also well-behaved. Clearly, since the transformations (2.9.1) 
and (2.9.2) are mutually inverse, their coefficients must satisfy certain consistency, or 
compatibility, conditions; so that, given the as, one can determine the A’s and vice 
versa. Indeed, substituting dq t from (2.9.2) into (2.9.1), and d6 k from (2.9.1) into 
(2.9.2), and with 8 k/ = Kronecker delta (= 1 or 0, according as k = l, or k ^ /), we 
obtain the inverseness relations: 


^ ^ ^ ^ A r i G-kr &kl i 

^ ^ &kr A /• — ^ ^ A r Clk r Clki 

(2.9.3a) 

^ ^ Alr&rk = ^ ^ ClykA-lr &kh 

^ ^ Alr&r = y ^ Clr-A-iy = Aj. 

(2.9.3b) 


Further, with the help of the unifying notations a k = a kn+] and A t = A /n+l , the 
definitions a n+xk = 6 n+]k (= 0) and A ll+l l = 8 n+1 j (= 0), and recalling that Greek 
subscripts have been agreed to run from 1 to n + 1, the transformation coefficient 
matrices in (2.9.1) and (2.9.2) take the (« + 1) x (n+ 1) “Spatio-Temporal” forms: 


a = 


/«ii • 

&\n 






( a k , 

a k 




a n i ' 

^nn 

a n,r 7+1 

V o 

1 

\ o ■ 

■■ 0 

1 ) 




»s 


0 

1 




(2.9.4a) 


( A \ 


A = 


V 0 


M n 


0 


l l,n+l 


1 n,n +1 


A k i 

A k 

0 

1 


As 

Ay 

0 

1 


= {A h \ (2.9.4b) 


1 / 


Then (2.9.1, 2) assume the simpler (homogeneous) forms: 

= ^ dqp = ^2 Afa d6 1 , (2.9.5) 

while the consistency relations (2.9.3a) read simply 

a A 1 or ^ ' aps A$y — ^ ~ A^apg (2.9.6a) 

and from this we obtain the “spatio-temporally partitioned” matrix multiplications: 


a S 

3y 

0 

1 


As 

Ay 

0 

l 


a sAs 

a s A x ay 

0 

1 


1 


0 


that is, 


a§A§ — 1 and a^Ay -I- a-j- — 0; 


(2.9.7a) 


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§2.9 QUASI COORDINATES, AND THEIR CALCULUS 


and, similarly, the consistency relations (2.9.3b) read 


(aA) T = A T a T = 1 or 


from which 


that is, 



As^s — 1 


^ ' Ay; Ufip ^ @S(3 ^7 S ^ 


A-S^S 


, A x as + a x 


and A x a s + a x — 0. 


Let us recapitulate the notations used here: 


(2.9.6b) 


(2.9.7b) 


(i) Matrices are shown in roman and bold; vectors in italic and bold; 

(ii) (.. .) T = transpose of square matrix (...); 

(iii) a s ,A s = (nx n) spatial , or catastatic, submatrices of a and A, respectively; and 
a x ,A T = (n x 1) temporal, or acatastatic, submatrices of a and A, respectively; 

(iv) 1 = square unit, or identity, matrix (of appropriate dimensions); 

(v) 0 = zero matrix (column or row vector of appropriate dimension); and 

(vi) Here, commas in subscripts — for example, a^+i ,A/ = A /n+1 — are used only to sepa¬ 
rate the spatial from the temporary of these subscripts, for better visualization; that is, 
no partial differentiations are implied, unless explicitly specified to that effect. 


Thus, for example, for (3 —> k and 7 —> / eqs. (2.9.6a) yield 
^ ^ ttkr^rt T i 6/tf > 'y ' ctkfAfi 6 £/, i.e., first of (2.9.3a), 

for {3 —* n + 1 and 7 —»/ they yield 

'y \ tin+\,r A r t T a t 24-1 5 1 An+iy 4+i,o i.e., 0 + 0 0, 

while for (3 —* k and 7 —> n + 1 they yield 

^ j/ ^r,n+ 1 T dk,n+\ 4 ,«+1 b, i.e., second of (2.9.3a), 

and similarly with (2.9.6b). 


Specializations, Remarks 

(i) If (a/a) is an orthogonal matrix — that is, if 

a ki = A i k and Det(a /f/ ) = ±1, (2.9.8a) 

then the spatial parts of (2.9.3a, 3b) are replaced, respectively, by 

Y a kr a lr = Y a h a kr = 4 / and Y a ri a rk = Y c ‘ rk a >' 1 = Skl ’ ( 2 - 9 - 8b ) 

and, similarly, for the full (« + 1) x (« + 1) a and A matrices. 

(ii) As shown in chap. 3, and foreshadowed below, it is the spatial/catastatic 
submatrices as and A s that enter the equations of motion; not the temporary/acata¬ 
static submatrices a x and A x . The latter, however, enter the rate of energy, or power, 
equations (§3.9). In what follows, we shall have the opportunity to use all these, 

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CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS 


mutually equivalent and complementary notations; primarily the indidal and secon¬ 
darily the matrix ones. All have relative advantages/drawbacks, depending on the 
task at hand. 


Velocities and Virtual Displacements 

Just as we defined new general kinematically admissible/possible system displace¬ 
ments via (2.9.1, 2), etc., we next define the following: 

(i) The corresponding general system velocities (dO —> oj dt); 

u k = Y a ki{dqi/dt) +a k = Y a kidi + a k = Y ak ' v ‘ + ° k ’ 


u) n+ \ = u) 0 = dq n+i /dt = dt/dt = 1 (isochrony ), (2.9.9) 

or, compactly, 

U P~Y a h( d q y /dt) = Y a h v v (2.9.9a) 

and, inversely, 

d q,/<lt = qi = vi = Y A ikU k + A h dq n+l /dt = bj n+l =dt/dt=\, (2.9.10) 

or, compactly, 

dq~ 1 ldt = q 1 = v 1 = Y^A 1 Qtjjp\ (2.9.10a) 

and 

(ii) The corresponding general system virtual displacements ( d6 —> 60, 
d0 n+ 1 50, 1+]l =6t = 0): 

69 k = Y a k' 6 dt, S0 n+X = Sq n+ i = St = 0; (2.9.11) 

and, inversely, 

Sqi = Y A ’ k S9k ’ Sq "+ l ~ Sq ° ~ S9 «+ l ~ S9 ° = St=0. (2.9.12) 

If the dO and dt describe an actual motion, then d0 k = to k dt. But it would be incor¬ 
rect to set 60 k = ui k 6t, because of the ever present (better, ever assumed) virtual time 
constraint 5t = 0; whereas, in general, 56 k f 0! 

Next, let us examine the integrability of these Pfalfian forms (not constraints!) 
(2.9.1, 11), of our hitherto n DOF system. 


Bilinear Covariants, Integrability, Quasi Coordinates 

Indeed, proceeding as in §2.8, and assuming that d(6q k ) = 6(dq k ), constraints or not, 
we find that the Frobenius bilinear covariants of (2.9.1, 11), d(60 k ) — 6(dO k ), equal 

d{60 k ) - 5{dO k ) = EE {da kl /dq s - da ks /dq,) dq s 6q, 

+ Y ( da k i/dt - da k /dq,) dt6q, 

= Y dipkiSqi■ 

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(2.9.13) 


§2.9 QUASI COORDINATES, AND THEIR CALCULUS 


Now, with the help of these expressions, and since the dq s and 8q's are (as yet) 
unconstrained (i.e., m = 0), like the q s, we can enunciate the following “obvious” 
theorems, in increasing order of specificity: 

• The necessary and sufficient conditions for th e particular Pfaffian/onn (not constraint!) 

d0 k = ^ a k / dqi + a k dt, or in virtual form 89 k = ^ a kl 8q h (2.9.14) 

to be an exact differential — that is, for the hitherto shorthand symbols d9 k and 89 k to 
be the genuine (first and total) differentials of a bona fide function 9 k = 9 k (q,t) 

(—» holonomic coordinate) —is that its bilinear covariant (2.9.13), vanish. 

• The necessary and sufficient condition for a Pfaffian form (2.9.14) to be the exact 
differential of 9 k [since its n 8q' s in (2.9.13) are arbitrary] is that its associated n 
Pfaffian forms 


di>kl = ^2 ( 9a «/ dc ls ~ da ks /dqi) dq s + ( da kl /dt - da k / dq,) dt, (2.9.15) 


all vanish; that is, dipu = 0 for all / (= 1,...,«). 

• The necessary and sufficient condition for a Pfaffian form (2.9.14) to be the exact 
differential of 9 k [since the n dq’s and dt in (2.9.15) are arbitrary] is that the following 
n(n+ l)/2 integrability (or exactness) conditions hold: 

da k //dq s — da ks /dqi = 0 and da k i/dt — da k /dqi = 0, (2.9.16) 

identically in the q's and t, and for all values of /, s (= 1,...,«). [For additional 
insights and details, see, for example, Hagihara (1970, pp. 42—46), Whittaker (1937, 
p. 296 ff.).] 

Hence, if (2.9.16) hold for all k= 1,..., n, the n 9's are just another minimal set of 
Lagrangean coordinates, like the q's: 6 k = 9 k (q u ..., q,,-, t); and u k = d6 k /dt are the 
corresponding holonomic Lagrangean (generalized) velocities. But if, and this is the 
case of interest to AM, 

da k //dq s — da ks /dqi ^ 0 or da k i/dt — da k /dcji ^ 0, (2.9.17) 

even for one I, s, then ut k is not a total time derivative, and d0 k is not a genuine 
differential of a holonomic coordinate 9 k , only the dO k /56 k /u) k are defined through 
(2.9.1, 9, 11). Such undefined quantities, 9 k , are called pseudo- or quasi coordinates [a 
term, most likely, due to Whittaker (1904)], or nonholonomic coordinates’, and the ut k , 
depicted by some authors by symbolic (.. ^'-derivatives, like 

uj k = d'6 k /dt = 6 k = 9 k , etc., (2.9.18) 

instead of d9 k /dt, are called quasi velocities. From now on we shall assume, with no 
loss in generality, that all (2.9.17) hold, and therefore all 9 k are quasi coordinates. 
{We notice that, the isochrony choice d9 n+l = dq n+x = dt, resulting in [recalling 
(2.9.4a, b)] 

— d n -\-\,k 0; ^ 7 t+l,H+l ^n+l,n+\ 1) (2.9.19a) 

and 

^ n-\-l,k — ^n+\,k ^5 (2.9.19b) 

guarantees that 9 n+ \ remains holonomic.} 

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CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS 


REMARKS 

(i) Let us consider, for simplicity, the catastatic version of (2.9.9), 

u k = '^2a kl (t 1 q)v, = u k (t,q,q= v). (2.9.20) 


If the q’s are known/specified functions of time t, then integrating (2.9.20) between an 
initial instant t 0 and a current one t we obtain the line integral 


Qk{t) - e k {t 0 ) = 


Uk[r,q(T), v(r)] dr = 




a ki[r,q{r)]vi(T)\ dr, (2.9.20a) 


similar to the work integral of general mechanics and thermodynamics. Since this is 
the integral of an inexact differential, as calculus/vector field theory teach, 0 k [t) 
depends on both t (current configuration) and the particular path of integration! 
history followed from t 0 to t; it is point- and path-dependent. If it was a genuine 
global coordinate, it would be point-dependent , but path-independent. 6 k (t ; tf) is a 
functional of the particular curves/motion {g(r), t a < t < t}\ 

(ii) As will be explained in §2.11, the satisfaction of (2.9.16) guarantees that 9 k , as 
defined by (2.9.14), is a holonomic coordinate; and that property will hold even if, at 
a later stage, the dq k /6q k /v k become holonomically and/or nonholonomically con¬ 
strained. One the other hand, if 9 k is originally [i.e., as defined by (2.9.14)] nonholo- 
nomic, then upon imposition on the latter’s right side of a sufficient number of 
additional holonomic and/or nonholonomic constraints, later, it will become holo¬ 
nomic; but that would be a different Pfaffian form. 

In sum: once a holonomic coordinate, always a holonomic coordinate; but once a 
nonholonomic coordinate, not always a nonholonomic coordinate. 

(iii) The local transformations a v = Ap v Ep Ep = ^2 a v pa v , where [recalling 
discussion in (§2.7)] each Eg is tangent to the coordinate line dqp at (q, t) and all 
together they constitute a holonomic basis for the local tangent space T„+1 > and the 
coefficients satisfy the earlier (2.9.3a, 3b), define a new but, generally nonholonomic 
basis there: that is, ff a I; dO r .\ nonexact differential =>■ daJdOp f dap/dO n [where the 
nonholonomic gradients, d/dOp, are defined in (2.9.27 ff.)]. And, in view of 

55 ^ E 0 = 51 v 0 E 0 = 55 ^(55 a ^ a v) = 55 (55 a v3 v a) a v = 55 ^ 

= 55 w /3 a /3> 


the up are simply the nonholonomic components of the system velocity vector, while 
the vp are its holonomic components. [The system basis {a, ( } plays a key role in the 
geometrical interpretation of Pfaffian constraints (§2.11.19a ff.)] 

(iv) The precise term for the 6 k s is “nonholonomic (local) system coordinates,” 
and for the u k s “nonholonomic system velocity parameters,” or “( contravariant ) 
nonholonomic components of the system velocity ” (Schouten, 1954/1989, pp. 194— 
197). We shall call them collectively quasi variables’, and their symbolic calculus, if 
proper precautions are taken, is quite useful. As Synge puts it: “In the theory of 
quasi-coordinates in dynamics, however, it pays to live dangerously and to use the 
notation d6 k [in our notation]. Otherwise we shall be depriving ourselves of a very 
neat formal expression of the equations of motion” (1936, p. 29). On the symbolic 
calculus of quasi variables, see also Johnsen (1939). 

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§2.9 QUASI COORDINATES, AND THEIR CALCULUS 


Example 2.9.1 The most common example of quasi velocities in mechanics is the 
components of the (inertial) angular velocity of a rigid body moving, with no loss 
in generality here, about a fixed point O, resolved along either space-fixed 
(inertial) axes O XYZ, w Y ,a)y,w z ; or body-fixed (moving) axes O xyz.uj x ,uj y ,Lj z . 
If cp —> 9 —> ip are the three Eulerian angles 3 —> 1 —> 3, then for body-axes, and 
with the convenient notations s(. ..) = sin(...) and c(.. .) = cos(...), and 
d<j>/dt = cd6/dt = u>g, dip/dt = we have (§ 1 . 12 ) 

uj x = ( sipsh+ (cip)uig + (0 )uty, (a) 

iOy = (cips8)uj <l> + {-sip)u e + (0)w* (b) 


LO z — (c9)u)0 + (0)ujg + (c) 

that is, with k = x — > 1, y —>2, z —>3; and / = </>—» 1, 6 — >2, ip — >3, the nonvan¬ 
ishing elements of ( a ki ) are 

a n = sips9, a l2 = cip-, a 2 \ = cipsO, a 22 = —sip-, a 2 \ = cO, u 33 = 1. (d) 
Clearly, not all (2.9.16) hold identically here. For example, 

da i2 /dq 2 f da\ 2 /dq 2 : d(cip)/dip f 9(0)/ 88: —sipfO; (e) 

except in the special ( nonidentical!) case: ip = 0, 2-7T. If we set cu x = d9 x /dt, then 


= 


UJ X \8(t),iP(t) ; cu^(t), oj s {t)\ dr + 9 X ( initial): path dependent ; (f) 


that is, 9 X is an (angular) quasi coordinate, and co x an (angular) quasi velocity; and 
similarly for 9 y ,9 z -,lu v ,uj z -, that is, they are quasi variables (if the (p, 9, ip are uncon¬ 
strained). However, if we impose additional constraints, for example, (p = constant , 
9 = constant (fixed-axis rotation), then (a-c) reduce to 

u x = 0 , uj v = 0 , u z = dip/dt =>■ u x ,L 0 y ,u z -. holonomic velocities-, (g) 


9 z (t) — 9 Z ( t 0 : initial) 


[dip(r)/dr\ dr = ip(t) — ip{t 0 \ initial ): path independent .(h) 


Problem 2.9.1 Let the reader verify that the corresponding space-fixed 
components 9 X ,9 Y ,9 Z and lo x ,u> y ,lu z (such that u x =d0 x /dt, etc.) are also, 
respectively, quasi coordinates and quasi velocities; and that under additional 
constraints they too may become holonomic variables. 


Particle Kinematics in Quasi Variables 

Due to the 9 <-> q transformation relations (2.9.1, 2, 9, 10, 11, 12), the (inertial) 
velocity, acceleration, kinematically admissible/possible displacement, and virtual 
displacement, of a typical system particle, obtained in §2.5 in holonomic variables, 
assume the following quasi-variable representations, respectively: 

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CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS 


(i) Velocity: 



(ii) Acceleration: 


a = ■■■ = Y^ s k (du> k /dt) + terms not containing (dw/dt)' s; 
= Y^ e k d> k + terms not containing cb’s, 


(2.9.22) 


(iii) Kinematically possible/admissible displacement: 


dr = E C/t (E Afci dO/ A.fc dt'j €q dt — • • • — ^ ^ dQfc £q dt 5 (2.9.23) 


(iv) Virtual displacement: 


Sr = E e k (E A kl 69 ‘) = ■ ■ ■ = E Ek S0k\ 


(2.9.24) 


where the fundamental, generally nongradient, n + 1 particle and system vectors s k 
and s„ +1 = £q, corresponding to the 0’s, nonholonomic counterparts of the gradient 
vectors e k and e n+l = e 0 , which correspond to the q’s [recalling (2.5.4-4b)], and 
defined naturally by (2.9.21-24), obey the following basic (covariant vector-like) 
transformation equations: 


= E ( dv l/ djJ k)ei = Y A lk e h 


(2.9.25a) 


e k = Y, {9u)i/9v k )^i = E a,kEl [comparing with (2.9.11, 12)]; (2.9.25b) 



(2.9.26a) 


e 0 = Y^ a k^k + «o = — E AkCk e ° [recalling (2.9.3a, 3b)]. (2.9.26b) 

Clearly, if the e vectors are linearly independent (and \a k/ \, \A k i\ 0), so are the e 
vectors; even if the q’s and/or dq/dt= v’s get constrained later. And, as with the 
(^-representation (2.5.12b), so with (2.9.24): the size of the 60’s is unimportant; it is 
the e’s that matter, because they are the ones entering the equations of motion (chap. 3)! 

Quasi Chain Rule, Symbolic Notations 

The above, especially (2.9.24), suggest the adoption of the following very useful 
symbolic quasi-chain rule for quasi variables: 


dr/dO k = Y (9r/dq,)(dvi/duj k ) = Y ( dr / dt li) [d{dqi)/d{dO k )} 


= Y ( dr / d di) [d(6q,)/d(60 k )\, 


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§2.9 QUASI COORDINATES, AND THEIR CALCULUS 


or, simply, 

dr/d9 k = Y A^dv/dq ,): i.e, (2.9.25a); (2.9.27) 


and, inversely, 

dr/dq k = Y (dr/d9,)(duj,/dv k ) = Y a ik(dr/d9,): i.e., (2.9.25b). (2.9.28) 

Similarly, for a general well-behaved function / =f(q, t), and recalling (2.9.12), we 
obtain, successively, (i) for its virtual variation 5f : 

Sf = Y fa = E ( 5 //%t) (E ^k/duti) 60 ,) = Y (Qf/Wi) 

(2.9.29) 

that is, 

df/dO, = Y C Of/ 8 *' ) (fok/&>t) = Y Mdf/dq k ) , (2.9.30a) 

and, inversely, 

df/dq k = Y (9f/de,) (a j,/dv k ) = Y Mdf/dd,) ; (2.9.30b) 


and (ii) for its total differential df [recalling (2.9.2)]: 

df = Y ( d f/ d( la) dc l3 = Y ( d f/ dc lk ) dq k + ( df/dt) dt 

= Y ( d f/ d dk) (E Akt + Ak dt ) + ( d f / dt ) dt 

= E(E A kiW/dq k )) do, + (£ A k {df/dq k ) + 3//0f) dt 

= Y W/Wi) do, + (d//d0 o ) dt, (2.9.31) 


where we have introduced the additional symbolic notation [recalling that 

0 0 = 0„+l — U n+ 1 = 1 ] ; 

d.../dd n+l = Y( d ---/ d( la)(dvp/duj n+l ) 

= Y ( d ■ ■ ■ /dq k )(dv k /duj n+ i) + {d... /dt)(dv n+l /du) n+l ) 

= Y A k(d.../dq k ) + d.../df, (2.9.32) 


instead of the formal extension of (2.9.30a) for 9 : —* 9 n+l . This latter we shall denote 
by d.../d(t)\ 

d.../d(t) = Y( d ■ ■ ■ / d< lk){dv k /duj n+l ) = Y A k{d.../dq k ); (2.9.32a) 


so that (2.9.32) assumes the final symbolic form 

<9... /d6„ + i = d... /89 0 = d... /d(t) + d... /dt. 


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(2.9.32b) 


CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS 


Inversely, we have 

d.../dt= ^ ~2(du a /dv n+x )(d.../d6 a ) = d... /d9„ +l + Y a k (d... /d6 k ), 

(2.9.32c) 


and, comparing this with (2.9.30a, b), we readily conclude that 

8... /d{i) = Y A k (d... /dq k ) =-22 a k (d... /d0 k ). (2.9.32d) 

Such (by no means uniform) symbolic notations are useful in energy rate/power 
theorems in nonholonomic variables (§3.9). 


Some Fundamental Kinematical Identities 

From the above (2.9.21 ff.), we readily obtain the following fundamental kinematical 
identities, nonholonomic counterparts of (2.5.7-10), and like them, holding indepen¬ 
dently of any subsequent holonomic and/or nonholonomic constraints. 

(i) dr/d9 k = dr/d9 k = dr/dO k = dr/duj k = ■ ■ ■ = e k , 
or 

dr/dO k = dv/doj k = da/duj k = ■■■ =e k ; (2.9.33) 

(ii) dq k /d9, = dq k /d9, = dq k /d0, = dq k /du, = ■■■ = A kh 
or 


dq k /ddi = dv k /du>i = dw k /du>i = ■••= A k p (where dv k /dt = w k ) (2.9.34) 

(iii) dOk/dqi = duj k /dv, = duj k /dw, = ■■■ = a kh (2.9.35) 

with formal extensions for 9 n+ \ = q n+i = t. The du k /dt = d 1 9 k /dt 2 are called (not 
quite correctly) quasi accelerations', while the 9/lo/lj/ ... are referred to, collectively, 
as (system) quasi variables. 

(iv) We have, successively, 

d(dr/d9 k )/dt = d(dv/dto k )/dt = ds k /dt 

= ^ik e /) / dt = Y^ [( dA lk /dt)ei + A Ik (dei/dt )] 

= Y ( dAi k /dt)ei + Y A lk (dv/dq,) [recalling (2.5.7,10)]. (2.9.36) 

But by partial (^/-differentiation of v(q, v, t) = v[q, v(q, u >, t),t\ = v*(q, u, t ), we hnd 
dv*/dq, = dv/dq, + Y (dv/dv,.)(dv r /dq,) = dv/dq, + Y (9v r /dq,)e n 

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§2.9 QUASI COORDINATES, AND THEIR CALCULUS 


and so 


E A ik{dv/dcj!) = Y A ik{dv*/dq,) ~YY A ik( dv r/dqi)e r 


= dv*/d6 k — YY a ik(dv,./dqi)e r . 


Therefore, returning to (2.9.36), we see that it yields 


d&k/dt — dv*/dQfc — ^ ^ ^dA.jfcjdt — E A r k{dvi/dq r )\ e, ± 0, (2.9.36a) 


that is, unlike the H coordinate ease (2.5.10), 

E k *(v*) = (i dv*/dd k y - dv*/dd k = d/dt(dv*/du k ) - dv*/dO k ^ 0. (2.9.37) 

This nonintegrability relation is a first proof that, in general, the s k basis vectors are 
nongradient , or nonholonomic. More comprehensible and useful forms of E k *(v*) are 
presented in the next section. 

[Some authors call the s k vectors “partial velocities.” However, in view of (2.9.33), 
they could just as well have been called partied positions , or partial accelerations , or 
even partial jerks (recall that da/dt =j= jerk vector , and therefore dj/du k = s k ), etc. 
Perhaps a better term would be nonholonomic mixed basis vectors (i.e., nonholonomic 
counterpart of Heun’s Begleitvektoren ).] 


A Useful Nonholonomic-Variable Notation 

Frequently, for extra clarity, we will be using the following “(.. .)*-notation”: 



With its help: 

(i) Equations (2.9.21), (2.9.22), (2.9.33) become, respectively, 
v(t,4,v) = Y e k(t,q)v k + e 0 (t,q ) = Y E k{Uq)Uk + So(t,q) = v*(t,q,uj); (2.9.39) 


a(t, q , v, w) = Y^ e k(U q) w k + no other q = tv-terms 

= Y^ £ /c(h q)u k + no other w-terms = a*(t, q , u, u); 


(2.9.40) 


dr/dO k = dv*/duj k = da*/du> k = ■■■ = s k ; 


(2.9.41) 


(ii) The quasi-chain rule (2.9.30a) and its inverse (2.9.30b) generalize , respectively, 
to 


dj*/de, = Y (dT/dq k )(dv k /dui) = Y Mdf*/dq k ), (2.9.42a) 


and 


dj*/dq k = Y (df*/ae,)(di 0 ,/dvk) = Y «ft(0/W); 

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(2.9.42b) 



CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS 


also, we easily obtain the related chain rules [recall derivation of (2.9.37), and 
(2.9.42a)] 

df*/dq k = df/dq k + Y {df/dvi){dv,/dq k ), (2.9.43a) 


df*/dOi = Y A ki(df*/dq k ) = Y A ki df/dq k + Y ( df/dv,){dv r /dq k ) 


(2.9.43b) 


(iii) The following genuine (i.e., ordinary calculus) chain rule , and its inverse, hold: 
dfo/cko, = Y (df/dv k )(dv k /a^,) = Y A ki(9f/dv k ), (2.9.44a) 


df/dv k = Y (dr/^di^/dvk) = Y a, k (df*/dio,). (2.9.44b) 

We notice the difference between (2.9.42a, b) and (2.9.44a, b); the former are non- 
vectoricil transformations, just symbolic definitions; while (for those familiar with 
tensors) the latter are genuine covariant vector transformations. 

(iv) Finally, invoking (2.9.11, 12, 42a, b), it is not hard to see that 

Y (dff/m k ) 80 k = Y (5/7 dq k ) 6q k . (2.9.45) 


Some Closing Comments on Quasi Coordinates 

The theory of nonholonomic coordinates and constraints is, by now, a well estab¬ 
lished and well understood part of differential geometry/tensor calculus and 
mechanics, with many fertile applications in those areas. Its long and successful 
history has been created by several famous mathematicians, such as (chronologi¬ 
cally): Gibbs, Volterra, Poincare, Fleun, Hamel, Synge, Schouten, Struik, 
Vranceanu, Vagner, Kron, Kondo, Dobronravov et al. And yet, we encounter con¬ 
temporary statements of appalling ignorance and confusion, like the following from 
an advanced “Tract in Natural Philosophy” devoted to rigid kinematics: “It appears 
that the reason why many a book on classical dynamics follows Kirchhoff’s 
approach is a lack of understanding of the kinematics of rigid bodies. Thus, one 
finds extensive discussions on ill-defined — or, sometimes, totally undefined — esoteric 
quantities such as quasi-coordinates and virtual displacements ,” (Angeles, 1988, p. 2, 
the italics are that author’s). 


2.10 TRANSITIVITY, OR TRANSPOSITIONAL, RELATIONS; 

HAMEL COEFFICIENTS 

So far, our system remains a holonomic (H) one, with n = 3 N — h DOF. Now, to be 
able to either (i) embed to it additional Pfaffian (possibly nonholonomic) constraints in 
their “simplest possible form” or, even if no such additional constraints are imposed, 
(ii) express the equations of the problem in quasi variables, or (iii) do both, we need 
to represent the right sides of the Frobenius bilinear covariants of the Pfaffian forms 
of its quasi variables, (...) dqSq [recall (2.9.13)], in terms of the latter’s differentials, 
(...) d6 69. [By simplest possible form we mean uncoupled from each other; and, as 

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§2.10 TRANSITIVITY, OR TRANSPOSITIONAL, RELATIONS; HAMEL COEFFICIENTS 


detailed in chap. 3, this leads to the simplest possible form of the equations of 
motion.] To this end, we insert expressions (2.9.2 and 12) into the right side of 
(2.9.13), and group the terms appropriately. The result is the following generalized 
transitivity, or transpositional, equations (Hamel’s Ubergangs-, or Transitivitdts- 
gleichungen ): 

d{S0 k ) — S(dO k ) = y a k i[d(Sq/) — 6(dq /)] + EE l k a(i d0 p S0 a 

= E a kll d ( S( ll) ~ S (dqt)\ + E E E d6 0 S6 r [ since b °n+ 1 = St = 0] 

= E a k‘ i d + E E E de s 66 r + Y 7 k r dt SO,., 

( 2 . 10 . 1 ) 

(again, we recall that all Latin (Greek) indices run from 1 to n (1 to n + 1)) where the 
so-defined 7 ’s, known as Hamel ( three-index ) coefficients, are explicitly given (and 
sometimes also defined) by 

E = EE {da kp /dq e - da ke /dq p )Ap r A es 
= EE 0 da kb /dq c - da kc /dq b )A hr A cs 

+ E ( da kb/dt - da k ,n+\ldqb)A hr A n+Xs 

+ E ( da k, n +i/dq c - da kc /dt)A n+X/ A cs 
+ (da Kn+l /dt - da kin+l /dt)A n+hr A n+l s , (2.10.1a) 

or, due to A n+Xr = 6 n+Xr = 0 which leads to the vanishing of the last three groups/ 
sums of terms, finally, 

l k rs = E E ( da kb/dq c - da kc /dq h )A br A cs -, ( 2 . 10 . 2 ) 

and 

E,n +1 = -7^+1/ - 7*1 = E E ( da kp/ dc le - da k Jdqp)Ap r A En+l , (2.10.3) 

or, with a kn+ \ = a k , A kn+] = A k , and since A„ +l: „ +1 = 6 n+ M+1 = 1, finally, 

e-EE (da kb /dq c da kc /dq b )A br A c -\- ^ i()a kb /<)t da k /dq b )A br . (2.10.4) 

[The 7 ’s are a significant generalization of coefficients introduced by Ricci (mid- 
1890s), Volterra (1898), Boltzmann (1902) et al.; and, hence, they are also referred 
as “Ricci/Boltzmann/Hamel (rotation) coefficients.” See, for example, Papastavridis 
(1999, chaps. 3, 6 ).] 

It is not hard to show [with the help of (2.9.3a, b)] that (2.10.1) inverts to 
d{Sq k ) - S(dq k ) = Y A kl [[d{S0,) - S/d0,)\ 

-EE 7 ^ d0 s 60 r — 7 ^ dt <5d r |. (2.10.5) 

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CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS 


For an actual motion, dividing both sides of (2.10.1) and (2.10.5) with dt [which does 
not interact with 8 (...)], we obtain, respectively, the (system) velocity transitivity 
equation and its inverse: 

(S6 k ) - 8uj k = 55 «jy[(<5<7/) - 8v,\ + 55 7 » Mr + 55 7*r (2-10.6) 

(%)■ -«v t = 55 ^,{[w- m-55 E -55 7 U^}- ( 2 - 10 - 7 ) 


Properties of the Hamel Coefficients 

(i) Clearly, these coefficients depend, through the transformation coefficients ag £ 
and Ag e , on the particular v <-> u> choice; that is, they do not depend on any particular 
system motion. 

(ii) The 7 * contain the contributions of (a) the acatastatic terms a k and A k , and of 
(b) the explicit time-dependence of the homogeneous coefficients of the v <+> u> trans¬ 
formation. Hence, for scleronomic such transformations (i.e., a k = 0 => A k = 0, and 
da k i/dt = 0 => dA kl /dt = 0) they vanish; but for catastcitic ones, in general, they do 
not. In fact then, as (2.10.4) shows, they reduce to 

7 * = 55 ( da k b/di)A br (for catastatic Pfaffian transformations). (2.10.4a) 

(iii) The matrix y k = ( 7 fc „) is, obviously, antisymmetric', that is, 

7 fe „ = — 7 k sr => y k rr : diagonal elements = 0 (k,r,s = 1 ,...also n+ 1 ). 

( 2 . 10 . 8 ) 

To stress this antisymmetry in r and s, we chose to raise /c; that is, we wrote j k rs 
instead of y rks , or 7 krs , or y rsk , and so on. [Nothing tensorial is implied here, although 
this happens to be the tensorially correct index positioning; see, for example, 
Papastavridis (1999, chaps. 3, 6 ).] Hence, each matrix y k can have at most 
n(n — l )/2 nonzero (nondiagonal) elements. 

(iv) From the above, we readily conclude that 

Yl- f-1 A \ Yl~\~ 1 A H~\~ 1 I - 1 A H~\~ 1 A 

T e/t H =" 7 kl 0? T k,n -fl 7 n+ljc 0? T n+l,n+l 0 

[k, l = 1,... e,P = 1— ,n-n + 1], (2.10.9) 

and from this (recalling that a n+i k = 8 n+ik = 0 ), that 

{86 „ + 1 )' — 8u> n+l = d/dt(8q n+l ) — 8(dq„ +l /dt ) = d/dt(8t) — 8(dt/dt ) 

= 55 E 7" +1 » w, 89,. + 55 7” +1 r , n+1 89,. = 0 + 0 = 0 , ( 2 . 10 . 10 ) 

which, essentially, states that 

d{89 n+l ) - 8(d9„ +l ) = d(8t) - 8(dt) = rf(0) - 6{dt) = 0-0 = 0, (2.10.10a) 

as it should, and also shows that ( 2 . 10 . 1 ) and ( 2 . 10 . 2 ) also hold for k = n + 1 . 

(v) In concrete problems, the analytical calculation of the nonvanishing 7 ’s is 
best done, as Hamel et al. have pointed out, not by applying (2.10.1a^4), which 

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§2.10 TRANSITIVITY, OR TRANSPOSITIONAL, RELATIONS; HAMEL COEFFICIENTS 


are admittedly laborious and error prone, but by reading them off as coefficients 
of the bilinear covariant (2.10.1,6), in terms of the general subindices: 
°, * = 1 ■•..,«;«+ 1 : 

d(89 t ) — 8{d6f) = -h (7*0*) dQ+ 80 o -\ -. (2.10.11) 

Also, this task is independent of any particular assumptions about d(6q) — 8(dq); 
and, hence, assuming that for cdl holonomic coordinates d(8q k ) = 8{dq k ), or equiva¬ 
lently ( 8q k )' = 8(q k ) = 8v k (Hamel viewpoint — see also pr. 2.12.5), even if they (or 
their differentials) become constrained later , we may safely and conveniently calculate 
all the nonvanishing 7’s from the simplified, and henceforth definitive, transitivity 
equation: 

d(69 k ) - 8{d9 k ) = EE j k rs de s 8e r +^^ k r dtse r . (2.10.12) 

Finally, dividing the above with dt, and so on, we obtain its velocity form: 

W-4 = EE 60 r + 7*r 60,-, (2.10.13) 

a representation useful in Hamilton’s time integral “principle” in quasi variables 
(chap. 7). Unfortunately, the transitivity equations, and their relations with the 
7’s, are nowhere to be found in the English language literature (with the exception 
of Neimark and Fufaev, 1967 and 1972, p. 126. IT.); although the definition of the 7’s 
via (2.10.1a, 2) appears in a number of places. This unnatural situation produces an 
incomplete understanding of these basic quantities. 

REMARK (A PREVIEW) 

As will become clear in chapter 3, the expression for the system kinetic energy (and 
the Appellian “acceleration energy”) are simpler in terms of quasi variables, such as 
the w’s and dui/df s, than in terms of holonomic variables like the v’s and dv/dt’s. 
And this leads to formally simpler equations of motion in the former variables than 
in the latter; for example, the well-known Eulerian rotational rigid-body equations 
(§1.17) are simpler in terms of such quasi variables than, say, in terms of Eulerian 
angles and their (...)'-derivatives. But there is a catch: to obtain such simpler-look- 
ing Lagrange-type equations of motion — that is, equations based on the kinetic 
energy and its various gradients — we must calculate the corresponding 7’s; some¬ 
thing that, even with utilization of (2.10.11-13) and other practice-based short cuts, 
requires some labor and skill. On the positive side, however, the 7’s supply an 
important “amount” of understanding into the kinematical structure of the parti¬ 
cular problem; and Appellian-type equations in quasi variables may not contain the 
7’s, but they have other calculational difficulties. In sum, there is no painless way to 
obtain simple-looking equations of motion in quasi variables. 


Problem 2.10.1 Verify that the transitivity equations, say (2.10.12), can be 
rewritten as 


d(60 k ) - 8{dO k ) = EE 7 \ s (d9 s S9 r — 89 s d9 r ) + 7 k r dt89 n (a) 

where X] X] ’ means that the summation extends over r and s only once ; say, for s<r. 
[We point out the following interesting geometrical interpretation of (a): each of its 

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CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS 


double summation terms is proportional to a 2 x 2 determinant, which, in turn, 
equals the area of the infinitesimal parallelogram with sides two vectors on the 
local “9 S 9, .-plane,” at its origin ( q , t) in configuration/event space, of respective 
rectangular Cartesian components (dQ s , dd r ) and (66 s , 69 r ) there; with the factor 
of proportionality being y k rs . That parallelogram is the projection of the generalized 
parallelogram with sides d6 = (d0 { ,..., d6 n ) and 69 = (69 1 ,..., 69„), at (q, t ), on the 
“6 s 6 r -plane” (see, e.g., Boltzmann, 1904, pp. 104-107; Webster, 1912, pp. 84-87, 
381-383; also Papastavridis, 1999, §3.14).] 


Other Expressions for the 7's 

By ^-differentiating (2.9.3a) and then rearranging so as to go from the (da/dq )'s to 
the (dA/dq)’s, we obtain 

E ( da kb/dq c )A br = ~Y a kh(dA hr /dq c ), (2.10.14a) 

E ( da kc/dq h )A cs = ~Y a kc(dA cs /dq b )-, (2.10.14b) 

then, substituting the above into (2.10.2), and renaming some dummy indices, we 
obtain the equivalent 7-expression: 

7 k rs = YY a kb[Acr(dA hs /dq c ) - A cs (dA br /dq c )]. (2.10.15) 

For j —> h + 1, the above yields an alternative to the (2.10.3), (2.10.4) expression for 

k _ k 
7 r,«+1 = 7r- 


Problem 2.10.2 Show that yet another 7-expression is 

A = EE (A hr A cs A cr A hs ) (dei kh /dq c ). (a) 

and similarly for 7* v , + i = 7* (see also Stiickler, 1955; Lobas, 1986, pp. 34-36). 


Some Transformation Properties of the 7's 
(i) With the help of the following useful notation: 

a be = da k b/dq c - da kc /dq b = -a k cb , (2.10.16a) 

a k b,n+ 1 = a \ = da kh /dt - da k /dq b (2.10.16b) 

[recalling (2.9.16); also similar notation in (2.8.2a)], the 7-definitions (2.10.2) (2.10.4) 
are rewritten, respectively, as 

7 \s = E E ak bcA br A c „ r y l \. = Y, E ak bcAb r A c + E ak bA br ■ (2.10.17a, b) 

With the help of the inverseness conditions (2.9.3a, 3b) and a number of dummy 
index changes, it is not too hard to show that (2.10.17a,b) invert, respectively, to 

^ he ^ \ ^ 7 rs®rb®sc ; ^ h ^ ^ ^ ^ 7 rs®rb®s 7 ^ ^ 7 r@rb‘ 


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(2.10.18a,b) 





§2.10 TRANSITIVITY, OR TRANSPOSITIONAL, RELATIONS; HAMEL COEFFICIENTS 


The above transformation equations show that if the a\ c and a\ vanish [recall con¬ 
ditions (2.9.16)], so do the ~/ k rs and 7 *; and vice versa; that is, the vanishing of 7 * ... 
constitutes the necessary and sufficient condition for d0 k /69 k to be an exact differential, 
and hence, for 9 k to be a holonomic coordinate. If the dq/8q/v are unconstrained, as is 
the case so far (i.e., m = 0 ), this new set of exactness conditions in terms of the 7 ’s 
does not offer any advantages over (2.9.16); the a k kc and a k k are easier to calculate 
than y k rs and 7 *. As shown in the next section, the real value of the 7 ’s, in questions 
of holonomicity, appears whenever the dq/6q/v are constrained (m f 0 ). 


REMARK 

For those familiar with tensors, the transformation equations (2.10.17a, b) show that 
the 7 * and a k transform as covariant tensors in their two subscripts; that is, both 
are components of the same geometrical entity, the a' s, its holonomic components in 
the local “coordinates” dq/8q, and the 7 ’s, its nonholonomic components in the local 
“coordinates” d9/89, at (q, t). In precise tensor notation, using, for example, 
accented ( unaccented) indices for nonholonomic ( holonomic ) components, summation 
convention over pairs of diagonal indices of the same kind (i.e., both holonomic, or 


both 

cs 

k 

7 « - 


nonholonomic), and 


the 


k' 

7 rV 


notational changes: A hr —> A f —> Af, 
a k rs —> 7 *„ (= holonomic components), 
(= nonholonomic components), the transformation equations 


and 


FF be 


with 

k ' 

->■ a be 


(2.10.17a) read 


k' 


= AfA 


k' 

■'7 rsi 


(2.10.17c) 


and similarly for (2.10.17b) (2.10.18b). Such elaborate notation is a must in 
advanced differential-geometric investigations of nonholonomic systems. 
Fortunately, it will not be needed here. 

(ii) The invariant definition of the 7 ’s via the transitivity equations (2.10.1) and 
( 2 . 10 . 12 ) readily shows that, contrary to what one might conclude by casually 
inspecting their derivative definition via (2.10.2-4), these nontensorial coefficients, 
known in tensor calculus as geometrical objects of nonholonomicity (or 
anholonomicity), are independent of the original holonomic coordinates q, and 
thus express geometric properties of the local/differential basis dO/89/ui. In 
particular, it follows that if the 7 ’s do (not) vanish, when based on some ( q,t ) 
frame of reference, they will (not) vanish in any other frame ( q',t ), obtainable from 
the original frame by an admissible transformation. 

(iii) However, under a local transformation d9 k d9 k ', that is, at the same (q, t)- 
point, the 7 ’s, do change, in the earlier mentioned nontensorial fashion. 

[(a) For further details on tensorial nonholonomic dynamics see, for example, 
Dobronravov (1948, 1970, 1976), Kil’chevskii (1972, 1977), MaiBer (1981, 1982, 
1983-1984, 1991(b), 1997), Papastavridis (1999), Schouten (1954), Synge (1936), 
Vranceanu (1936); and references cited there, (b) For transitivity equation-based 
proofs of these statements, see, for (ii): ex. 2 . 12 . 2 , and for (iii): ex. 2 . 10 . 1 ; and for 
a derivative definition-based proof, see, for example, Golab (1974, pp. 140-141).] 


Noncommutativity of Mixed Partial Quasi Derivatives 

Below we show that the second mixed partial symbolic quasi derivatives of an 
arbitrary well-behaved function / =f(q,t,...), in general, do not commute: 

d/d9 k (6f/d9,) f d/d9,(df/d9 k ). (2.10.19) 

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CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS 


Invoking the basic quasi-derivative definition (2.9.30a, b), we obtain, successively, 

d 2 f/de k do, = d/de k {df/dei) = J^a rk {d/dq r {^2 A sl {df/dq s ))} 

= EE {8 2 f/dq r dq s ) + A rk {dA sl /dq r )(df/dq s )] 

= EE A rk A sl {d 2 f/dq r dq s ) + a bsA rk {dA sl /dq r )) (df/d6 b ), 

and, analogously (with k —> / and / —> k in the above). 


5 2 //90 ,&) k = d/d0i{df/d0 k ) =■■■ 

= EE A r iA sk (d 2 f/dq r dq s ) + a b s A,i{dA sk /dq r )) (8f/89 h ), 

and therefore subtracting these two side by side, and recalling the 7 -dehnition 
(2.10.15), we obtain the following alternative transitivity /noncommutativity relation: 

d 2 f/d9 k 89 l -d 2 f/d9 l d9 k = d/do k {df/dOi) - d/d9,(df/do k ) 

= E{ EE“4T, (dA sl /dq r ) - ^ r/ (5^/%)] | ( df/d6 h ) 

= E^E9//^ a ); ( 2 . 10 . 20 ) 

which expresses noncommutativity in terms of (9... /50)-derivatives, rather than 
(d.../6.. ^-differentials, as ( 2 . 10 . 1 ) and ( 2 . 10 . 12 ) do. 

REMARK 

In the theory of continuous (or Lie) groups, it is customary to write X k f for our 
df/89 k , (2.9.30a); that is, 

d.../dd k = X k - •• = E {d.../dq,){dv,/duj k ) = A lk {8... /dq,). (2.10.21) 

The differential operators X k are called the generators of that group. In this notation, 
equation ( 2 . 10 . 20 ) is rewritten as 

[*it, X,]f = £ 7 b k ,(X h f), (2.10.22) 

where [X k ,X,] = X k X, — X,X k = Y^ 1 h k i(X b ): commutator of group. For further 
details, see texts on Lie groups, and so on; also Hamel (1904(a), (b)), Hagihara 
(1970), McCauley (1997). 


Problem 2.10.3 Extend (2.10.20) to the case where one or both of 9 k , 9, are the 
(0„+i)th “coordinate”, that is, 9 — > t. 

Problem 2.10.4 The choice / —> q r in (2.10.20), and then use of (2.9.34), yields 
the symbolic identity 

d 2 q r /d9 k 89, - d 1 q,./d9 l 89 k = ^ 7 b kl {dq r /89 b ) = ^ A rb f b kl . (a) 

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§2.10 TRANSITIVITY, OR TRANSPOSITIONAL, RELATIONS; HAMEL COEFFICIENTS 


Solving (a) for the 7’s, derive the following alternative symbolic expression/definition 
for 7: 

7 \i = Y a br(d 1 <lr/dQk dOi ~ d 2 q,./de l d6 k ) . (b) 


HINT 

Multiply (a) with a sr and sum over r, and so on. 


Nonintegrability Conditions for a Nonholonomic Basis 

Since (2.10.20) holds for an arbitrary /, let us apply it for f —> r = r(t,q). In this 
case, df/d6 h — > dv/d6 h = z h , and thus we obtain the basic nonintegrability conditions 
for the nonholonomic basis {s k ; k = 1 

d&,/dd k - dz k /d0 1 = Y £h, (2.10.23) 

or, compactly, 

[e^, £/] = J h ki E h — commutator of basis {e^}. (2.10.23a) 

In differential geometry, such bases are called nonholonomic, or noncoordinate, or 
nongradient; that is, they are not parts of a global coordinate system; like the 
{e k = dr/dq k } for which, clearly [recalling (2.5.4a)], 


de,/dq k - de k /dq, = [e k , e,] = 0. (2.10.23b) 

In sum: the vanishing of the 7’s is the necessary and sufficient condition for the 
corresponding basis to be holonomic; or gradient, or coordinate. 

We leave it to the reader to show that (2.10.23) also hold for k, l = n + 1; that is, 
e^t. 


A Fundamental Kinematical Identity 

Here, with the help of (2.10.23), we will complete the derivation of the basic identity 
(2.9.37). Indeed, since z k = dv/du k = dv*/du> k = z k (t, q), and [recalling (2.9.21)] 
v = v*(t, q,uS) = e k Lo k + £„ +1 = E k UJ k + £ (h we obtain, successively, 

(i) d / dt(dv / du> k ) = d/dt(dv*/dui k ) = dz k /dt 

= Y ( d£ k/dqi)vi + ds k /dt 

[recalling the inverse quasi chain rule (2.9.30b)] 

= Y (5Z a ri(9 £ k/90r)) v/ + de k /dt [recalling (2.9.9)] 

= Y ( ds k /d6 r )(u) r - a r ) + ds k /dt 
= E (wk - E ( ds k /d9 r )a r + ds k /dt. (2.10.24a) 


(ii) dv/dQ k = dv*/d0 k = Y^ (' dz r /d6 k )uj r + dz 0 /dd k . 

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(2.10.24b) 


CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS 


Therefore, subtracting the above side by side, and recalling (2.9.32a, d), we obtain 

de k /dt - dv/dd k = Y (■ de k /dO r - ds r /d9 k )uj r + (ds k /dt - ds 0 /dd k ) ~53 {ds k /d9,.)a r 

= Y ( d£ kl de r ~ ds r /d9 k )u r 

+ (de k /d9 0 - 53 A s(9e k /dq s fj 

- (ds 0 /d9 k - Y ( ds k /d9 r )a 

= E ( ds k/ dS p ~ 9sp/d9 k )u 0 

-zMz a rs (ds k /d9 r )^ - Y a r (ds k /d9 r ) 

{for the first sum we use (2.10.23), with / —> k, k —► (3, b —> r [recalling (2.10.9)]; and 
by the second of (2.9.3a) the last two sums add up to zero} 

= E (E (2.10.24c) 


and so, finally, 

E k *(v) = E k *(v*)\ Hamel vector of nonholonomic deviation of a particle 

= d/dt(dv/du> k ) - dv/d9 k = d/dt(dv* / dui k ) — dv*/d9 k = ds k /dt — dv/d9 k 

= EE 7V £ '- W /3 — EE YikUfir +53 7 n+i,* w «+i 8 r [swapping k and /] 

= - E E 7 « ujiEr ~ E tY,- = - E (E 7« w /+ Yk )+ 

= ~Yh T k *r\ (2.10.25) 

where 

h' k = 53 Yu + + Yk = E 7 C3 +9 • Two-index Hamel symbols. (2.10.25a) 

This fundamental kinematical identity, in its various equivalent forms, like the tran¬ 
sitivity equations ( 2 . 10 . 1 , etc.), shows clearly the difference between holonomic and 
nonholonomic coordinates (not constraints): for the former, E k (v) = 0; while for the 
latter, E k *(v) = E k *(v*) f 0. It is indispensable in the derivation of equations of 
motion in quasi variables (§3.3). 


Problem 2.10.5 Transitivity Relations for System Velocities. 

(i) Show that for the general nonstationary transformation (with <37 = v/) 

W>k = 53 a kl V l + a k v l = E A ' kUJ k + A h ( a ) 

the following transitivity identities hold: 

Ei{u k ) = d/dt(duj k /dv,) - duj k /dqi = Y (E E/ 3 +?)+■/ 

= E (E T k rs U s + 7 A () a H EE 53 hk r a rl- ( b ) 

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§2.10 TRANSITIVITY, OR TRANSPOSITIONAL, RELATIONS; HAMEL COEFFICIENTS 


(ii) Then show that, in the stationary case, (b) specializes to 

E,(uj k ) = d/dt(du k /dv,) - duj k /dq, = EEAow; (c) 

that is, the first line of (b) with ft —> s. 

(iii) Show that, as a result of the above, the transitivity equations (2.10.13), 
become 


(69 k y - Sco k = EE 7 « W, 60, + E 7 k r Mr - E l,k >■ 66 >- 

= EE E l {aJ k )A lr 66 r = EE (dvi/du> r )Ei(uj k )S9 r , (d) 

where the (9vi/duj r )Ei(uj k ) can be viewed as the nonlinear generalization of the h k r 
(§5.2). 

Problem 2.10.6 By direct (//^-differentiations of Sr = E k-M k and 
dr = ^ e k d0 k , respectively (assume stationary systems, for algebraic simplicity but 
no loss in generality), and then use of 

de k = d (E Aik e ?j = E e t + Ai k def), (a) 

and 

dei = E (de,/dq r ) dq r = EE (■ de t /dq r )A rs d9 s , 
dA/ k = E ( dAik/dq ,.) dq r = EE {8A lk /dq r )A rs d9 s , (b) 

and similarly for hr/, = <5( ^4 /a^/) = ..., and then recalling the 7-definitions, obtain 

the following basic particle/vectorial transitivity equation : 

<*(&■) - W = E { [<W - <W] + E E 7« d °r Ms } £ A3 ( C ) 

or, dividing by dt, its equivalent velocity form 

(Sr)' -Sv = E {[(%)' - Su) k\ + EE 7 k rsU r S9sj s k . (d) 

Replacing in the above r with ft = 11, extends it to the nonstationary / 
rheonomic case. 

[Note that (c) and (d) are independent of any (/(<5</) — <5(d<7) assumptions. 
Therefore, since 

d(Sr) - S(dr) = E [^(^/) - S(dqi)]ei = EE [d(&y,) - <5(^/)]a«£ k , (e) 

if we assume d(Sq/) — 6(dq t ) = 0 (Hamel viewpoint), then d(<5r) — 6(dr) = 0, and this 
leads us back to the transitivity equations (2.10.12) and (2.10.13).] 

Example 2.10.1 Local Transformation Properties of the Hamel Coefficients. Let 
us find how the 7’s transform under the admissible (and, for simplicity, but with 
no loss in generality) stationary quasi-variable transformation 9 —> 8': 

d9 k ' = E a k'k d9 k <s> d9 k = E A kk ' dd k >, 

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(a) 


CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS 


where a k ' k = o-k'k( c j), A kk' = A k k'{q), and all Latin indices run from 1 to n. We find, 
successively, 

d(60 k/ ) - 8(dO k i) = d(^2 <*k'kMk) ~ $(^2 a k'k d °k) 

^ ^ [ dci^'k 80k 4 - cik'k d(80fc) ficL^'k dO^ Q-k'k 8(^d0k)\ 

= E{(E {da k , k /dq p ) dq^j 89 k + a k , k d{86 k ) 

~ (E ( da k'k/dq P ) H,) de k - a k ’ k 8{dO k ) | 

[recalling that dq p = ^ A pr d9 r , etc.] 

= E a k , k [d(S0 k ) - 8(d9 k )} 

+ EE E [( da k'k/dq p )A pr d6 r 86 k - (i da k . k /dq p )A pr dO k 89,] 

= E a k'k (E E l\c d9 c 89^j 
+ EEE \{da k ' k /dq p )A pr - (, da k , r /dq p )A pk ] d9,.89 k 
= E E E a k'kl k hc A cc' dO c ) (E A bb , 89 h ^j 
+ E E E [( da k'k/dqp)A pr -(da k , r /dq p )A pk \ (E A rr , </0 r ')(E 
=EEEEE ( [ a k'k A rr' A U'l k lr ) ^r' 

+ EEEE C da k ' k /d9 r - da k , r /d9 k )A rr ,A kl , d9 r , 89,, ; (b) 

and since, by definition, 

</(<%,) - «$(<%,) = EE 1 k ' l , r ,d6 r ,89 l ,, (c) 

we conclude that 

l k l'r' = E E E a k'k A U'Arr'l k lr + E E ( da k'k/dOr ~ dd k , r /89 k )A k ,, A,,.,. (d) 

In tensor calculus language, the transformation equation (d) shows that the j k , r do 
not constitute a tensor, if 7 k lr = 0 (i.e., if the 9 k are holonomic coordinates), it does 
not necessarily follow that 7 k ,, r , = 0; and that is why these quantities are called, 
instead, components of a geometrical object. However, if the second group of terms 
(double sum) in (d), which looks (symbolically) like a Hamel coefficient between the d9 k 
and (19 k ,, vanishes, the 7^,. transform tensorially. In such a case, we call the d9 k and d9 k , 
relatively holonomic ; that happens, for example, if the coefficients a k , k are constant. 

For futher details, and the relation of the 7’s to the Christoffell symbols (§3.10) 
and the Ricci rotation coefficients, both of which are also geometrical objects, see, for 
example (alphabetically): Papastavridis (1999), Schouten (1954), Synge (1936), 
Vranceanu (1936); also, for an alternative derivation of (d), see Golab (1974, pp. 
141-142), Lynn (1963, pp. 201-203). 


a kl' 


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§2.11 PFAFFIAN (VELOCITY) CONSTRAINTS VIA QUASI VARIABLES 


We have developed all the necessary analytical tools of Lagrangean kinematics. In 
the following sections, we will show how to apply them to the handling of additional 
Pfaffian (possibly nonholonomic) constraints. 

For quick comparison, when working with other references, we present below the 
following, admittedly incomplete, but hopefully helpful, list of common 7 -notations 
in the literature: 

(i) Our notation (also in Papastavridis, 1999): 7 a b c = y h ac (sometimes, for extra 
clarity, a subscript dot is added between a and c, directly below b). 

(ii) Authors whose notation coincides with ours: Dobronravov (1948, 1970, 1976), 
Golomb and Marx (1961), Gutowski (1971), KiFchevskii (1972, 1977), Koiller 
(1992): la \. 

(iii) Authors whose notation differs from ours: Butenin (1971), Fischer and Stephan 
(1972), Neimark and Fufaev (1967/1972), Whittaker (1937; but his a kt is our <%): 
7 abc ; Corben and Stehle (1960): j acb ; Nordheim (1927): y cba ', Rose (1938): j hac ', Pasler 
(1968): — 7 bac ; Djukic (1976), Funk (1962), Lur’e (1961/1968), Mei (1985), Prange 
(1935): 'y c h a ; Kilmister (1964, 1967): 7 “/; MaiBer (1981): A c b a ; Desloge (1982): a abc ; 
Stiickler (1955): /3 abc ; Heun (1906): (3 ach ; Winkelmann and Grammel (1927): (3 cab ; 
Morgenstern and Szabo (1961): f3 bac ; Hamel (1904(a), (b)): /3 acb ; Hamel (1949): 
(3 b a ' c ; Schaefer (1951): 0 c b a ; Vranceanu (1936): w b c \ Wang (1979): K A B C ; Schouten 
(1954): 2Q c h a ; Levi-Civita and Amaldi (1927): rj b \ ca . 


2.11 PFAFFIAN (VELOCITY) CONSTRAINTS VIA QUASI VARIABLES, 

AND THEIR GEOMETRICAL INTERPRETATION 

Let us, now, assume that our hitherto holonomic n (= 3N — h )-DOF system is sub¬ 
jected to the additional m independent Pfaffian constraints [recalling (2.7.3 and 
2.7.4)]: 

Kinematically admissible!possible form: 


X! °Dk dq k + c D dt = 0 , 

( 2 . 11 . 1 a) 

Virtual form: 


c ° k Sqk = °’ 

( 2 . 11 . 1 b) 

Velocity form (with dq k /dt = v k ): 


c Dk v k + CD = 0 ; 

( 2 . 11 . 1 c) 


where D = 1,..., m (< n ), k = 1and the constraint independence is expressed 
by the algebraic requirement rank(c Dk ) = m. Since additional holonomic constraints 
(in any form) can always be embedded, or built in, with a new set of fewer cf s, we 
can, with no loss of generality, assume that all constraints ( 2 . 11 . 1 ) are nonholo¬ 
nomic. 

Now, and in what constitutes a direct and natural extension of the method 
of holonomic equilibrium coordinates (§2.4) to the embedding Pfaffian constraints, 
we introduce the following equilibrium quasi variables (Hamel’s choice): 

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CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS 


Kinematically admissible!possible form: 



dO D = a Dk dc lk + aD dt (= 0) , 

(2.11.2a) 


dd, = Y^ a Ik dq k + a, dt {f 0), 

(2.11.2b) 


d() n+] = d0 o = dq n+x = dq 0 = dt (^ 0); 

(2.11.2c) 

Virtual form: 

S6 d = ^2 a Dk Sq k (=0), 

(2.11.2d) 


M I = '52a lk 6q k (^0), 

(2.11.2e) 


© 

x 

III 

+ 

III 

+ 

(2.11.2f) 

Velocity form: 

OtD = ^ a Dk v k + a D (— 0); 

(2.11.2g) 


aJi = a lk v k +aj (^0), 

(2.11.2h) 


O 

II 

III 

O 

> 

III 

+ 

Ill 

o 

3 

III 

+ 

K 

3 

(2.11.2i) 

where (here and throughout the rest of the book): D = 1 ,...,m (< n) = Dependent , 
/ = m+ 1 = Independent [additional dependent ( independent ) indices will be 

denoted by D'. D", .. . ( l',I ", ...)]; and the coefficients a k i, a k are chosen as follows: 

(i) a Dk = c Dk and a D = c D [i.e., 0 D = Xd, recall (2.6.2—4; 

2.8.1)], (2.11.3) 

(ii) The a Ik and a r 
for the dq/6q/v in 

are arbitrary , except that when eqs. (2.11.2) are solved (inverted) 
terms of the independent d0/86/u>, respectively; that is, 

Kinematically admissible!possible form: 



dq k = ^ A k t d6[ + A/ dt (yf 0), 

(2.11.4a) 


© 

2b 

o 

^3 

III 

+ 

^3 

III 

o 

^3 

III 

+ 

(2.11.4b) 

Virtual form: 

8q k = Y2 A ki Mt 0) , 

(2.11.4c) 


© 

II 

III 

o 

<3> 

III 

+ 

<3? 

III 

o 

'-O 

III 

+ 

<-0 

(2.11.4d) 

Velocity form: 

v k = ^2 AkjUJ I Al > 

(2.11.4e) 


o' 

dk 

II 

ill 

o 

3 

+ 

3 = 

III 

O 

> 

III 

+ 

(2.11.4f) 


and then these results are substituted back into (2.11.1a-c) and (2.11.3), they satisfy 
them identically. Other choices of 0’s and a’s are, of course, possible (see special 
forms/choices, below), but Hamel’s choice (2.11.2) is the simplest and most natural, 
because then our Pfaffian constraints assume the simple and uncoupled form: 

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§2.11 PFAFFIAN (VELOCITY) CONSTRAINTS VIA QUASI VARIABLES 


Kinematically admissible!possible form: 


Virtual form: 


Velocity form: 


dO j) — 0, 

(2.11.5a) 

b 

II 

(2.11.5b) 

lo d — 0; 

(2.11.5c) 


and, as a result (already described in §2.7 and detailed in ch. 3), the equations of 
motion decouple into n — m kinetic equations (no constraint forces) and m kineto- 
static equations (constraint forces). 


Constrained Particle Kinematics 

In view of the constraints (2.11.5), the particle kinematical quantities (2.9.23-26) 
reduce to the following: 

Kinematically admissible!possible displacement: 



dr = ^ £[ d9j + e„ +1 dt = ^ £/ d9j + £ 0 dt; 

(2.11.6a) 

Virtual displacement: 



6r = £ £ i 66 1 ; 

(2.11.6b) 

Velocity: 

v = ^ £ I U] + £„ + i = ^ £/ U>1 + £ 0 ; 

(2.11.6c) 

Acceleration: 

a = ^ £/ 0/ + terms not containing w. 

(2.11.6d) 


Special Forms/Choices of Quasi Variables 

1. Once we have chosen the equilibrium quasi variables dd/69/ui, we can move to 
any other such set dO' / SO' /a/, defined via linear (invertible) transformations of 
the following type: 

dO k ' = £ cik'k dOk + &k' dt — ^ ^ cifcti dOj cifc' dt 0), (2.11.73.) 

d0 {n+l y = dO n+l = dq n+l =dt 0); (2.11.7b) 

and, inversely \{a k ' k ), {A kk i)\ nonsingular matrices], 

d9 k = ^ A kk f d6 k ' + A k dt —> d0 D = 0 and d9 k f 0; 
and similarly for 69 k ', u> k >. 

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(2.11.7c) 


CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS 


2. If the Pfaffian nonholonomic constraints are given in the quasi-variable forms: 

X a D'k db k + a D ' dt = 0, or X a D'k S9 k = 0, or X c ‘ D ' k Uk + a °' = °> 

(2.11.8a) 

then, proceeding a la Hamel again, we may introduce new quasi variables by 
dO ^ ~ a pi/- d0 k a d' dt = 0, dO j' = ^ ) ci,> k dO k -T- a,> dt / 0; (2.11.8b) 

or 

fiQp' = X a D'k = 0; 80 1 ' = X a I'k 7^ Oj (2.11.8c) 

or 

UJD 1 = X a D'k u} k + = 0, W/' = X a l'k^>k + a V 7^ 0> (2.11.8d) 

where, again, the coefficients ap k , a // are arbitrary; but when (2.11.8b-d) are 
solved for the dO/80/ui in terms of the dO'/80'/to', and the results are substituted 
back into (2.11.8a), they satisfy them identically (see also their specialization in 
item 4, below). 

3. Frequently, the Pfaffian constraints (2.11.1) are given, or can be easily brought to, 
the special form [recalling (2.6.9-11), and, using the notation dq k /dt = v k ]\ 

dq D = X b oi dc li + b D d t, or 8q D = X b Di or v D = X b Di^i + b D, 

(2.11.9) 

where the coefficients b DI , b D are known functions of q and t\ that is, the first m 
(or dependent) dqa/8q[)/v[) are expressed in terms of the last n — m ( independent ) 
dq,/8q,/v /. [In terms of the elements of the original mxn constraint matrix 
( c Dk ) = (%), we, clearly, have ( b DI ) = — (ct DD 'fi l (a DI ), and so on. See also pr. 
2.11.2.] 

Now, the transformations (2.11.9) can be viewed as the following special choice 
of d0/80/cu: 

cl0 D = dq D — X b Di dc li ~ b D dt = 0, dO, = dq, / 0, d0„ +l = dq n+l = dt / 0; 

(2.11.10a) 

80 D = 8q n - X b DI 8qj = 0, 80, = 8q, / 0, 80 n+x = 8q n+l =8t= 0; 

(2.11.10b) 

u D = Vd ~ X b Di y i - b D = 0, ujj = v/ / 0, ui n+] = v„ +1 = dt/dt =1/0. 

(2.11.10c) 


The above invert easily to 

dq D = d6 d T ^ ^ b di d6j T dt = ^ ^ ^/)/ dOj T bp dt : 

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§2.11 PFAFFIAN (VELOCITY) CONSTRAINTS VIA QUASI VARIABLES 


dq I = dOj, dq n+l = dd n+l = dt; (2.11.11a) 

Sq D = 69 d + "22 6>di 6qi = ^ b DI 6q ,, 

6qi=60j, 6q„ + i = 69 n+ i = 6t = 0; (2.11.11b) 

v d = + 22 b DI uJi + b D = 22 b DI u}j + b D , 

v/ = w / , v„ +1 = v 0 = w„ +1 = w 0 = dt/dt= 1. (2.11.11c) 


Comparing (2.11.10, 11) with (2.11.2, 4) we readily conclude that, in this case, the 
(mutually inverse) transformation matrices a and A [recalling (2.9.4a ff.)] have the 
following special forms: 


a = 

(i 

0 1 

-bn+1 N 

0 

A = 

1 \ b 

0 1 

b n+ l^ 

0 


v 0 0 

1 J 


v 0 0 

1 J 


that is, 



( 2 . 11 . 12 ) 


(2.11.12a) 


(2.11.12b) 


where b = ( b DI ), b n+1 = {b D ^ n+x = b D ); and, of course, satisfy the consistency rela¬ 
tions (2.9.3a, b). For a slight generalization of the choice (2.11.10c), see pr. 2.11.2. 


Particle Kinematics 

In this case, the particle kinematical quantities [recalling (2.5.2 ff.) and (2.11.6a If.), 
and that e n+ \ = e 0 = dr/dt] specialize to 

dr = 22 e k dq k + e n+l dt = 22 e n dq D + 22 e, dq, + e n+l dt 

= 22 e ° (5Z ^ DI ^ c li + b D dt'j + "22 e i d< h + e n+x dt 

= *22 Pi d< h +Pn+i dt = ^2 Pidqi + P 0 dt, (2.11.13a) 

*■ = ■■■ = £/»/«?/, (2.11.13b) 

v = £ P/Vi + P„+i = v(t,q,vj) = v 0 , (2.11.13c) 

a = 22 Pi^’i + terms not containing V/ = a(t , q , vj, v k ) = a 0 , 

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(2.11.13d) 















CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS 


where (s/ —► Pi): 

Pi = e i + ^ b DI e D , P n+l = p 0 = e„ + i + Y^ b D e D = e 0 + ^ b D e D . 

(2.11.13e) 


REMARK 

It should be pointed out that under the quasi-variable choice (2.11.9), and, according 
to an unorthodox yet internally consistent interpretation [advanced, mainly, by 
Ukrainian/Soviet/Russian authors, like Suslov, Voronets, Rumiantsev; and at 
odds with the earlier statement (§2.9) that the q s are always holonomic coordinates], 
the < 7 /, and hence also the q D , are no longer genuine = holonomic coordinates, but 
have instead become quasi-, or nonholonomic coordinates; even though one could 
not tell that very well from their notation. To avoid errors in this slippery terrain, 
some authors have introduced the particular notation (q) (Johnsen, 1939); we shall 
use it occasionally, for extra clarity. Thus, specializing (2.9.27), while recalling the 
first of (2.11.12b), we can write 

dr/d{q I ) = Y (dr/dq k )(dv k /dv,) 

= Y ( dr / d( bD)(dvD/dv 1 ) + Y {dr/dq v ){dv v /dvj) 

= dr/dq , + Y h Di(dr/dq D ) = Y A diC D + Y A i'/ e i' 

= Y^ b DI e D + Y^ bri e ]' = Y1 bDi e D + e i = Pi': (2.11.14a) 

and analogously for P n+l = P Q . Similarly, with the helpful notation [(2.11.13c)]: 
v = v(t, q, v) = • • • = v a (t, q, Vj) = v 0 , chain rule, and recalling (2.11.9), we obtain 

dv 0 /dv, = dv/dv, + Y {dv/dv D ){dv D /dv ,) = e, + Y e obDi = Pi', (2.11.14b) 

that is, the fundamental identities (2.9.33) specialize to 

dr/diq,) = dv a /dv I = dajdvj = ■■■ = Pj = P^t.q) (2.11.14c) 

[not to be confused with the analogous holonomic identities (2.5.7, 7a)]. 

Equation (2.11.14a) gives rise to the special symbolic quasi chain rule (see also 
chap. 5): 


d.../d[q,) = Y( d ---/ dl lk){dv k /dv I ) 

= Y( d "' / dc JD)(dv D /dv,) + Y( d --- / dqi')(dv r /dvi) 

= d.-./dqj + Y b D i(d.../dq D ); (2.11.15a) 

which, when applied to v D , yields 

dv D /d(q /) = Y (dv D /dq D '){dv D '/dv,) + Y (dv D /dq r )(dv r /dv,) 

= dv D /dq, + Y b D 'i(dv D /dq D -). (2.11.15b) 

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§2.11 PFAFFIAN (VELOCITY) CONSTRAINTS VIA QUASI VARIABLES 


Generally, applying chain rule to 

f=f(t,q,v) =f[t,q,v D (t,q,v I ),v I ] = f 0 (t,q,v,) =f 0 , (2.11.15c) 

we obtain the useful formulae 

dfo/dv, = df/dv, + Y (df/dv D )(dv D /dv,) = df/dv, + Y b D i{df/dv D )\ 

(2.11.15d) 

and 

dfo/dq i = df/dq, + Y ( df/dv D ){dv D /dq, ); (2.11.15e) 

while (2.11.15a,b) are seen as specializations of 
df 0 /d{q,) = dfo/dq, + Y ( df 0 /dq D )(dv D /dv,) 

= dfo/dq, + E bmidfo/dqf] [notation, not chain rule!]. (2.11.15f) 


Problem 2.11.1 With the help of the above symbolic identities [recall (2.11.12 ff.)] 
show that: 

(i) dq k /dO, = dv k /duj, -> dq k /d{q,): 

dqD/dfjD') = A D d' = boD'i dq D /d{q,) = A D , = b D ,, dq,/d{q D ) = A, D = 0, 
dq,/d{q,') = A, r = 6„f. (a) 

(ii) dd k /dq, = dw k /dv, -> d{q k )/dq,: 

d(q D )/dq D ' = a DD ’ = <W, d{q D )/dq, = a DI = -b DI , d(q,)/dq D = a, D = 0, 
d[q,)/dq r = a IV = 8„:. (b) 

[Notice that dq D /d{q,) = b D , f d{q D )/dqi = -b D ,.] 

(iii) d... /d0 11+ 1 —> d... /d{q n+i ) [recall (2.9.32 ff.), and since A kn+l = A k \ 

= Y A ^ d ■ ■ ■ ! dq ^+ 9 ■ ■ ■ / dt 
= Y A °( d ■ ■ ■ +E A ^ d ■ • • / dqi )+ 9 ■ ■ • / dt 

= Y b D {d.../dq D ) + 0 + d.../dt = d.../d{t) + d.../df, (c) 

which for r yields the earlier (2.11.13e). 

(iv) dr/dfio ) = Y ( dr/dq k )(dv k /dv D ) = Y e k A kD = ■■■ = dr/dq Dl 

i.e., p D = e D . (d) 

(v) dp,/{q ,0 f dp,,/d{q,y, (e) 

which is a specialization of (2.10.23), and shows clearly that the basis {p,} is non¬ 
gradient. 


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CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS 


4. Occasionally, the constraints appear in the (2.11.9)-like form, but in the quasi 
variables dO/89/uj [special case of (2.11.8a)]: 

dOf) = B di dOj + B d dt ; or 89 D = B DI 89 /; or ui D = B di loi + B Dl 

(2.11.16) 


where the coefficients B DI , B D are known functions of q and t. 

To uncouple them, proceeding as before, we introduce the following new equi¬ 
librium quasi variables dO' / 80 ' /uj' (to avoid accented indices, we accent the quasi 
variables themselves): 

d9 'd = d9 0 — 'y ' BDi d9j — B dt = 0, dO 1 j = d9j ^ 0, dO’ nJr \ = d0 n+ j = dt ^ 0; 

(2.11.17a) 

89' D = 89 d — B di 80j = 0, 89'j = 80j 0, 80' n+ \ = 89 n+i = 8t = 0; 

(2.11.17b) 

u 'd = u D — ^ B di loj — B d = 0, lo'j = ujj ^ 0, u/ n+1 = w„ +1 = dt/dt = 1 ^ 0; 

(2.11.17c) 

which invert easily to 


d9 j) — d9 i) T y ^ B i) j d9 / T Bd dt — y ( Bdi dO / T Bd dt , 


d0[ = dO' j, 

d@n+l — ^0 — dt , 

(2.11.18a) 

89 d = 89' D 

+ ^2 Bdi SO'i = ^ B D i 89'j, 


"hs 

II 

qS" 

o 

II 

III 

+ 

III 

+ 

(2.11.18b) 

U>D — u'd + ^ B D j Lo'i + B d — ^ B D 1 Oj'j + B d , 


UJi = u'j, 

w „+l — u'n+l — dt/dt — 1. 

(2.11.18c) 


Clearly, (2.11.16) (2.11.18) bear the same formal relation to (2.11.8a) that 
(2.11.9) (2.11.11) bear to (2.11.2) (2.11.4). 

In sum, the possibilities are endless and, in practice, they are dictated by the 
specific features and needs of the problem at hand. The essential point in all these 
descriptions is that, ultimately, they express the n dq/8q/v in terms of n — m 
independent parameters d9 I /89 I /ijj I ', and if the nonholonomic constraints are in 
coupled form, either among the dq/8q/v or among another set of n quasi variables 
dO/89/uj then, following Hamel, we introduce new equilibrium quasi variables 
dO'/80 '/lu' such that d0' D /80 ' D /uj' d = 0 and dOj/89'jju'j ^ 0. And, as already stated, 
this uncoupling of the Pfaffian constraints is the main advantage of the method. 


Problem 2.11.2 Consider the homogeneous Pfaffian constraints, 

U> D = ^ <-<Dk v k — ^ a DD' v D' + ^ a DI’ v I’ (= 0), 
u>i = V/ = ^ 8 ID 'V D ' + ^ 8/j'Vj' (= ^ fifth1 7 ^ o) ; 

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(a) 

(b) 



§2.11 PFAFFIAN (VELOCITY) CONSTRAINTS VIA QUASI VARIABLES 


where D, D' = 1,..., m,I, I 1 = m + l,... ,n; that is, (with some easily understood 
notations) 


a =>- a s = ( a k ,) = 


( a DD') ( a DI')\ ( ( a DD') ( a DI ') 

(a ID i) (aw) ) \ (0/ d ') (<)//') 

(i) Verify that its inverse (assuming that a is nonsingular) equals 

, i^DD') (Ajj/')\ ( (flDZ)') ~( a DD') ( a D'I ‘) 

A => A s = ( A kt ) = 


(c) 


(d) 


V (A/d') (A//') / y (0 /d') (<5//0 / 

(ii) Extend the above to the nonhomogeneous case; that is, wd=X) a Dk v k+ a D (=0), 

uji = V/ (^ 0). 

(iii) Verify that the earlier particular choice (2.11.9 If.) is a specialization of the 
above. 


Geometrical Interpretation of Constraints 

(May be omitted in a first reading.) We begin by partitioning the mutually inverse 
n x n matrices of the virtual transformation between bq <-> 66, a s = (a k i) and 
A s = (A kl ), into their dependent and independent parts: 


a s = 



(2.11.19a) 


A s = (A d | = (A kD | A kI ). (2.11.19b) 

Clearly, 

/ 3 dAd 3 dAi\ /I 0\ 

a s A s = = . (2.11.19c) 

V a iAo a iA! ) \0 1 / 


Next, we partition these submatrices in terms of their dependent and independent 
(column) vectors as follows [with (.. .) T = transpose of (...), and using strict matrix 
notation for vectors and their dot products, instead of the customary vector notation 
used before and after this subsection]: 


/V\ 


a D = 


a I 


W) 

(a m+ ?\ 

V ) 


a D T — («i, • ■ ■, «,„), a D T — (a D \,... a Dn ), (2.11.20a) 


a i T = (« m +u---'«»)' «/= («/u •••«/«). (2.11.20b) 


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CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS 


Ad — (Ai, ■ • • ,A m ), 


( A ' T ) 

U,v 


A| ( A m +\ :••••> A/;) , 



— {A\d, ■ ■ ■, A„ D ), (2.11.20c) 


A, t = {A u ,...,A nI ), (2.11.20d) 


Also, since (a D • a D T ) 1 • (a D • a D T ) = 1 


and A d t • a D T = 1, it follows that 


Ad T — ( a D -a D T ) ' -a D- 


(2.11.20e) 


Now, in linear algebra terms, the virtual form of the constraint equations 

X] a DkSq k = 0 [rank(a Dk ) =m{<n)\, (2.11.21a) 

[we note, in passing, that rank(a Dk ) mxn = rank(a Dk \a D )^ mx ^ n+l ^] or, in the above- 
introduced matrix notation, 

a D • Sq = 0 (one matrix eq.), a D T • Sq = 0 [m vector (dot product) eqs.], 

(2.11.21b) 

state that every virtual displacement ( column ) vector Sq 1 = (Sq l ,..., Sq„), at the point 
(q, t), lies on the local (n — m)-dimensional tangent/null/virtual plane of the (virtual 
form of the) constraint matrix a D = (%.), T„_ m (P) = V n _ m (P) = V„_ m (§2.7, 
suppressing the point dependence); or, equivalently, that Sq is always orthogonal 
to the local m-dimensional range space/constraint plane ofa D T ,C m (P) = C,„ (which 
is orthogonally complementary to V n _ m ). 

Next, in view of our quasi-variable choice, that is, Sq = Aj • S6j, where 
S0j T = (S6 m+U ..., S9 n ), the (n — m) vectors (A m+1 ,..., A n ) = {A f } constitute a 
basis for while the m constraint vectors (« 1; ... ,a m ) = {a D } constitute a 

basis for C,„. Or, all ( Sq k ) satisfying (2.11.21a, b), at (q 7 t), form a local vector 
space V n _ m , which is orthogonal to the local vector space C m built (spanned) by 
the m constraint vectors a D T = (fl^i, • ■ ■, flz>„)- 

More precisely, expressing (2.9.3a, b) in the above matrix/vector notation, we 
have 


(i) 

yi a ik^ki’ — a t T • Ap — S n ’ (I, /' — m + 1,... 

,«), 

(2.11.22a) 


or a, • A, = 1, or A I T -a ] T = l; 


(2.11.22b) 

that is, the columns of aj T and A t , or the rows of aj and Aj T , namely, the vectors {«/} 
and { Aj }, are mutually dual , or reciprocal , bases of V n _ m ; and 

(ii) 

a Dk^kD' — a D T • A/)' — S DD ' (D 7 D' — 1 ,.. 


(2.11.22c) 


or a D -A D = l, or A D T -a D T = l 

5 

(2.11.22d) 


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§2.11 PFAFFIAN (VELOCITY) CONSTRAINTS VIA QUASI VARIABLES 


that is, the columns of a D T and A D , or the rows of a D and A D T , namely, the vectors 
{a D } and {A D }, are mutually dual bases of C m . Clearly, if the { a D } are orthonormal , 
so are the {A D }, and the two bases coincide; and similarly for the bases {«/}, {Aj}. 
Likewise, from (2.9.3a, b) we obtain 

(iii) ^ a Dk A kI = a D T -A, = S D1 = 0 (D = 1,..., m; I = m + 1, 

or a D • Aj = 0, or A t T • a D T = 0; 

that is, the vectors {a D } and {Aj} are mutually orthogonal. 

(iv) J2 a/kA kD = a, T -A D = 6 W = 0 (/ = «?+ 1,..., n; D = 1,..., m ), 

(2.11.22g) 

or a I -A D = 0, or A D T -a I T = 0. (2.11.22h) 

that is, the vectors {«/} and { A D } are mutually orthogonal. Equations (2.11.22f) and 
(2.11.22h) state, in linear algebra terms, that the “virtual displacement matrix” A t is 
the orthogonal complement of the “constraint matrix” a D . 

[Hence, the projections of an arbitrary system vector M = (M x ,..., M n ) on the 
local mutually orthogonal (complementary) subspaces V n _ m and C m , are, respec¬ 
tively, 

Null/ Virtual space projection P ...(•• •): 

Y A kI M k = (A[ T ■M) i = A, t -M = M i = ^V(Null)(-^0 = P V (Virtual) (M ); 

(2.11.23a) 


• ,«), 

(2.11.22e) 
(2.11.22f) 


Range/constraint space projection P ...(•• •): 

Y A kD M k = (A D T • M) d = A d 7 • M = M d = f 3 /?(Range)(^) = ^C(Constraint) (^)-] 

(2.11.23b) 

The above hold, locally at least, for any velocity constraints, be they holonomic or 
nonholonomic. However: (a) If the constraints are nonholonomic, the corresponding 
null and range spaces are only local; at each admissible point of the system’s con¬ 
strained configuration (or event) space; but (b) If they are holonomic, then these 
spaces become global; that is, the hitherto /(-dimensional configuration space is 
replaced by a new “smaller” such space described by n — m Lagrangean coordinates, 
as detailed in §2.4 and §2.7. 

Tensorial Hors d'Oeuvre 

These projection ideas, originated by G. A. Maggi (1890s) and elaborated, via 
tensors, by J. L. Synge, G. Vranceanu, V. V. Vagner, G. Prange, G. Ferrarese, 
P. MaiBer, N. N. Poliahov et al. (1920s 1980s), are very useful in interpreting the 
general problem of AM [i.e., of decoupling its equations of constrained motion into 
those containing the forces resulting from these constraints and those not containing 
these forces], in terms of simple geometrical pictures of the motion of a single 
“particle” in a generalized system space. They have become quite popular among 
multibody dynamicists, in recent decades; but, predominantly as exercises in linear 

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CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS 


algebra/matrix manipulations, that is, without the geometrical understanding and 
insight resulting from the full use of general tensors. 

To show the advantages of the powerful tensorial indicial notation, over the 
noncommutative straightjacket of matrices, we summarize below some of the 
above results. With the help of the summation convention [over pairs of indices, 
one up and one down, from 1 to n; and where, here, capital indices (accented and/ 
or unaccented), signify nonholonomic components], we have the following: 


(a) Equations (2.11.21a, b), and their inverses: 

89° = a D k 8q k = 0, 8q k = A k , 6(t. (2.11,24a) 

(b) Equations (2.11.22a, b): 

a I k A k I ' = <5/<) (2.11.24b) 

(c) Equations (2.11.22c, d): 

44 = 4 , (2.11.24c) 

(d) Equations (2.11.23a): 

P v {M) = A k I M k = M I , (2.11.24d) 

(e) Equations (2.11.23b): 

P c (M) = A k D M k = M D . (2.11.24e) 

The summation convention explains why, in order to project the (covariant) M k , 
above, we dot them with At, and A k D , instead of a\, a D k , respectively. [Briefly, the 
a 1 (A,) build a nonholonomic contravariant (covariant) basis in V n _ m , while the 
a D (A D ) build a nonholonomic contravariant (covariant) basis in C m .] 

Last, a higher level of tensorial formalism may be achieved, if, as described briefly 
in (2.10.17c), we use accented (unaccented) indices to denote nonholonomic (holo- 
nomic) components; for example, successively: a k , —> ct) —> A k k , A k , —> A k , —> A k k >; 
so that a-A = 1 reads A k k A k ,i = 8 k and similarly for the other equations. For 
further details on tensorial nonholonomic dynamics, see, for example, Papastavridis 
(1999) and references cited therein. 


2.12 CONSTRAINED TRANSITIVITY EQUATIONS, AND 
HAMEL'S FORM OF FROBENIUS' THEOREM 

Constrained Transitivity Equations 

Let us begin by examining the transitivity relations (2.10.1) under the Pfaffian con¬ 
straints (2.11, 2a flf.), dd D = 0, 86 d = 0, and their implications for the latter’s holo- 
nomicity/nonholonomicity. Indeed, assuming d(6q k ) = 8(dq k ) for all k=l 
whether the dq/8q are constrained or not (what is known as the Hamel viewpoint , 

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§2.12 CONSTRAINED TRANSITIVITY EQUATIONS 


see pr. 2.12.5), the general transitivity equations (2.10.1) reduce to 

d(66 D ) - 6(d6 D ) = EE 7 V df) r 89, + I n i dtS0 h (2.12.1a) 

d(89,) - 8(d9 k ) = EE 'l I ri "dO I »8e r + ^2'y I I 'dt66it. (2.12.1b) 

From the above we conclude that, even though uj D (t) = 0 (or a constant), or 
d9 D (t) = 0, or 69 D (t) = 0, from which it follows that (66 D )' = 0 or d(89 D ) = 0, yet, 
in general, d(89 D ) — 8(d9 D ) f 0 => —8(d9 D ) f 0! Specifically, as (2.12.1a) shows, 

-8(d9 D ) = EE ^ D n' d9[' 89j + ^ 7 ^ dt89; f 0 (in general); (2.12.1c) 

that is, we cannot assume that both d(Sq k ) = 8(dq k ) and d(89o) = 8(d9o)(= 0)! This 
is a delicate point that has important consequences in time-integral variational prin¬ 
ciples for nonholonomic systems (see Hamel, 1949, pp. 476-477; and this book, 
chapter 7; also pr. 2.12.5). 

The Frobenius Theorem Revisited (and Made Easier to Implement) 

We have already stated (§2.8) that the necessary and sufficient condition for the 
holonomicity of the system of m Pfaffian constraints 

d9 D = E a Dk dq k + a D dt = 0, 69 D = E a Dk 8q k = 0 (D = 1,... ,m), 

(2.12.2) 

that is, for the existence of m linear combinations of the d9 D = 0, or 89 D = 0, that 
equal m independent exact differential equations df\ = 0 ,..., df m = 0 => f\ = 
constant ,..., f m = constant , is the identical vanishing of their Frobenius bilinear 
covariants [recall (2.9.13)] 

d(89 D ) - 8(d9 D ) 

= EE (da Dk /dq l - da Dl /dq k ) dq, 8q k + ^2(da Dk /dt - da D /dq k ) dt 8q k , 

(2.12.3) 

for all dq k , dt, 8q k solutions of (2.12.2). From this fundamental theorem we draw 
the following conclusions: 

(i) If the dq k , dt, 8q k are unconstrained, that is, if m = 0, then the identical satisfaction 
of the conditions 

a°ki = da Dk /dq, - da D ,/dq k = 0, a D k = da Dk /dt - da D /dq k = 0, (2.12.3a) 

for all k, / = is both necessary and sufficient for the holonomicity of 9 D 

(§2.9). 

(ii) But, if the dq k , dt, and 8q k are constrained by (2.12.2), then the vanishing of 
d(89 D ) — 8(d9 D ) does not necessarily lead to (2.12.3a). 

To obtain necessary conditions for the holonomicity of the system (2.12.2), we must 
express the dq k , dt, 8q k , on the right side of (2.12.3), as linear and homogeneous 
combinations of n — m independent parameters (Maggi’s idea); that is, we must take 
the constraints (2.12.2) themselves into account. Indeed, substituting into (2.12.3) the 

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CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS 


general solutions of (2.12.2) [recalling (2.11.4)]: 

dq k = E A kI ddj + A k dt, 6q k = E A kI 60 j, (2.12.4) 

we obtain (2.12.1a). From this, it follows that (and this is the crux of this argument), 
since the 2 (n — m) differentials dO k , 80j are independent/unconstrained, the conditions 

7 V = 0 and 'y°i, n +i = l°i = 0 , (2.12.5) 

for all D = 1,... ,m; I, I 1 = m + 1,... ,n {i.e. maximum total number of distinct/ 
independent such j D ’s is [m(n — m){n — m—\)/2]+m(n — m) = m(n — m)(n — m+l)/2 = 
mf(f+ l)/2 ,/ = n — m}, are both sufficient and necessary for the holonomicity of the 
Pfaffian system (2.12.2). 

• Since [recalling the 7 -definition (2.10.2 ff.)] (2.12.5) can be rewritten as 

A- = EE (da nh /dq c - da Dc jdq h )A hl A cV = EE aD bcA hI A cI , = 0, (2.12.5a) 

^, = EE ( da Db /dq c - da Dc /dq h )A h[ A c + E ( da Dh /dt - da D /dq h )A hI 

= EE a ^hc^blA c + E aD b^bl = 0) (2.12.5b) 

we readily recognize that the (identical) vanishing of (all) the 7 fl .’s does not neces¬ 
sarily lead to the vanishing of (all) the a D kc , a n k , while the vanishing of all the latter 
leads to the vanishing of all the 7 z> . ’s; that is, (2.12.3a) lead to (2.12.5, 5a, b) but not 
the other way around. Hence, (2.12.3a) are sufficient for holonomicity but not 
necessary, whereas (2.12.5, 5a, b) are both necessary and sufficient. 

• Since, as (2.12.5a, b) make clear, each y D ( 7 "°) depends, in general, on all the 
coefficients a Dk ,A kI (a Dk ,A kI ;a D ,A k ), the holonomicity/nonholonomicity of a(ny) 
particular constraint, of the given system ( 2 . 12 . 2 ), depends on all the others; that 
is, on the entire system of constraints. In other words: eqs. (2.12.5) check the holo¬ 
nomicity, or absence thereof, of each equation d0 D , S0 D = 0 against the entire sys¬ 
tem; that is, there is no such thing as testing an individual Pfaffian constraint, of a 
given system of such constraints, for holonomicity; doing that would be testing the 
new system consisting of that Pfaffian equation alone (i.e., m = 1) for holonomicity. 
In short, holonomicity/nonholonomicity is a system property. 

As Neimark and Fufaev put it “the existence of a single nonintegrable constraint 
(in a system of constraints) does not necessarily mean a system is nonholonomic, 
since this constraint may prove to be integrable by virtue of the remaining constraint 
equations” (1972, p. 6 , italics added). However [and recalling (2.10.16a-18b)], we can 
see that the identical vanishing of all coefficients y k rs and 7 * n+l = 7 * (for all 
r,s = 1 ,...,«) in 

d{86 k ) - 6{dd k ) = ■■■ + ( 7 *.) d6 69+ ( 7 *.) dtSO, (2.12.5c) 

independently of the constraints d0 D ,60 D = 0 (or, as if no constraints had been 
applied; and which is equivalent to a k rs = 0 , a k r = 0 , identically), is the necessary 
and sufficient condition for that particular 9 k to be a genuine/Lagrangean coordinate ; 
that is, a kr = d6 k /dq r , a k = d0 k /dt. 

Let us recapitulate/summarize our findings: 

(i) Pfaffian forms (not equations), like 

d6 k = a k /(q) dqi {k = 1 ,..., n'\ l=\,...,n\ n and n unrelated) ( 2 . 12 . 6 ) 

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§2.12 CONSTRAINED TRANSITIVITY EQUATIONS 


(for algebraic simplicity, but no loss in generality, we consider the stationary case), 
are either exact differentials, or inexact differentials. 

If their dq s are unconstrained , then the necessary and sufficient conditions for d6 k 
to be exact, and hence for 9 k to be a holonomic coordinate , are 

a k rs = 0 (r,s= (2.12.7) 

In this case, each of the n' forms d9 k is tested for exactness independently of the 
others; k, in (2.12.7), is a free index, uncoupled to both r and s. If n = n', then, as 
already stated, conditions (2.12.7) can be replaced by 

l\s = 0 (r,j= (2.12.8) 

but since calculating the 7 ’s requires inverting ( 2 . 12 . 6 ) for the n dq's in terms of the 
n dO' s, eqs. (2.12.8) offer no advantage over eqs. (2.12.7). 

If eqs. (2.12.7) hold, then dd k remains exact no matter how many additional con¬ 
straints may be imposed on its dq’s later. For, then, we have 

d(69 k ) - 8(d9 k ) = EE (■ da kr /dq s - da k Jdq r ) dq s 8q r = 0; (2.12.9) 

that is, if 9 k is a holonomic coordinate, it remains holonomic if additional constraints 
be imposed among its dq, 8q's, later. This is the meaning of Hamel’s rule: 
d(8q k ) = 8(dq k ), for all cf s, constrained or not. 

If eqs. (2.12.7) do not hold, d9 k is inexact; but it can be made exact by additional 
constraints among its dq, 8q' s; that is, if 9 k is a quasi coordinate, it may become a 
holonomic coordinate by imposition of additional appropriate dq, 8q constraints. 
For example, let us consider the Pfaffian form (not constraint) 

d9 = a(x, y, z ) dx + b(x, y, z) dy + c(x, y, z) dz. 

Under the additional constraints y = constant =>■ dy = 0 and z = constant =>■ 
dz = 0, it becomes dd = a{x, y, z) dx = f(x)dx = exact differential, even if, origin¬ 
ally, x was a quasi coordinate. 

(ii) Pfaffian systems of constraints 

d0 D = ^2 a Dk dq k = 0 , 89 D = ^ a Dk 8q k = 0, (2.12.10a) 

are either holonomic or they are nonholonomic. The necessary and sufficient condi¬ 
tions for holonomicity are 

d(89 D ) - 8(d9 D ) = EE y n , v d9 r 80f = 0, (2.12.10b) 

or, since the d9i, 89j are independent, 

r y D ir = 0 {D = 1,... ,m\ I,I 1 = m + 1,... ,n); (2.12.10c) 

that is, the “dependent” q’s relative to their “independent” indices (subscripts) should 
vanish; or, the components of the dependent (constrained) Hamel coefficients along the 
independent (unconstrained) directions vanish. 

{This, more easily implementable, form of Frobenius’ theorem seems to be due to 
Hamel (1904(a), 1935); also Cartan (1922, p. 105), Synge [1936, p. 19, eq. (4.16)], and 
Vranceanu [1929, p. 17, eq. (9'); 1936, p. 13].} 

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CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS 


In closing this section, we repeat that Frobenius’ theorem is about the integra- 
bility of systems of Pfaffian equations, like (2.12.2), not about the exactness of 
individual Pfaffian forms, like (2.12.6). 


Example 2.12.1 Special Case of the Hamel Coefficients, via Frobenius’ Theorem. 
Let us calculate the Hamel coefficients corresponding to the special constraint form 


dq D = £ bm d<h ' Sq ° = Y bm 6qi ’ 


(a) 


where b DI = h DI {q), and formulate the necessary/sufficient conditions for their 
holonomicity. We begin by viewing (a) as the following special Hamel choice [sta¬ 
tionary version of (2.11.10a—12b)]: 


dq D - Y b Dl dq I = °> 

d9, = 

- dq, f 0, 

+ 

III 

g- 

+ 

III 

g. 

ik 

© 

(b) 

8q D — Y, b DI bc lt = 

89, = 

8q, f 0, 

fi@n +1 — fitfn +1 — fit 0, 

(c) 

Md + £ b Di d9j = Y^ 

b D , d9,, 

^3 

II 

^3 

III 

+ 

^3 

III 

+ 

^3 

(d) 

89d + £ b DI 8q, = Y^ 

8di fiqi, 

8q, = 89,, 

© 

II 

III 

+ 

III 

+ 

(e) 


also, since here dq, = d9,, we can rewrite the system (a) as 


where 


dqk — £ BkI dqi = £ ^kt dbfy 


(B a ) 


^ u\,m+l • • • b\rfi 


b m ,m +1 

* • fifWl 

1 

■ 0 

0 

•• 1 / 


(f) 


Since 9, = q,, we shall have 'y I a g = 0; while (2.12.9), with k^> D and (f), becomes, 
successively, 


d(89 D ) - 8{d9 D ) = YY a D rs dq s 8q r = YY a ^{Y dqi) (£ B it' 6c h') 

==££££ (a D rs b s , b rV ) dq, 8q r 
= - = ££ 7 D ridq, 6q,< 

= £ £ 7 D iv dqr 8q, = Y, £ 7 °//' d8 r 89, , 


(g) 


where (expanding the sums in r and s, with D, D ', D" = 1, ..., m\ /, I' = m + 1, .. ., n) 
l D i'i = £ £ t^D'D" t>D"i t>D'r + £ cPd'i b d'i' + £ ti D rn' b,y, + a D i'i\ (h) 
or, since cP D i D « = a DD ' D « — a DD " D ', where commas denote partial differentiations 


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§2.12 CONSTRAINED TRANSITIVITY EQUATIONS 

relative to the indicated cf s and [by (2.11.12—12b)] a DD ' —* S DD ', a DI —> —b DI , 
ci id —> 0, a n t —> S rr : 

(0)b D "ib D 'i' + [0 — {—db DI /dq D ')\ b D 'i' 

+ El [( -,% £/7 %d') ~ 0]^ 7 + [{-db D i</dqi) - (-db DI /dq r )\, 

or finally, 

7 D /7 = \dbpi/dq,' + ( db DI /dfopfep'/'l - [db Dr fdq r + ^ (db Dr >/dq D >)b r y r ] 

= —w D jh = = Voronets (or Woronetz) coefficients ; (i) 

clearly, a specialization of 7 Z /'/. Thus, (g) becomes 

d(se D ) - = EE 7 D I'I dqi Sqr = EEA ' dqj bqj', (j) 

and, since the dqj and <5^/ are independent, by Frobenius’ theorem, the necessary 
and sufficient conditions for the holonomicity of the system (a) are 

H’V = 0, (k) 

which are none other than the earlier Deahna-Bouquet conditions (2.3.11b ff.). 
REMARKS 

(i) With the help of the symbolic notation (2.11.15a), we can rewrite (k) in the 
more memorable form, 

7 D i'i = w D W = db DI /d(q r ) - db Dr /d(q I ). (1) 

(ii) In the special “Chaplygin (or Tchapligine) case” (§3.8), where b DI = 
b D Mm+ 1 ,- • -,qn) = b DI (q D ), the above reduce to 

l D ri = db DI /dq r - db Dr /dq, = t D lv . (m) 

Problem 2.12.1 Continuing from the previous example, show that eqs. (i) for the 
Voronets coefficients also result by direct application of the definition (2.10.2) 

7 V = EE (da Dk /dq r - da Dr /dq k )A kI A rI , (a) 

to the constraints (ex. 2.12.1 :a) in the equilibrium forms (ex. 2.12.1 :b e). 

HINT 

Flere [recalling again (2.11.12—12b)]: a DD ' = b DD ', a DI = ~b DI , a ID = 0, a ir = 6 n >; 
Add 1 = bf) r y, A di = bph A[ D = 0, Ajy = 6 n >. 


Problem 2.12.2 Continuing from the above, show that in the general 
nonstationary case 

dq D = ^2 b oi dc h + b D dt , dq, = ^ Sir dq r = dq u (a) 

bq D = E bDI Sqi = E b]V 6qi ' = bqi ~ b ° l = bDl( d’ ^)> h ° = b ° q )> ( b ) 

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CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS 


the 7 D ii' remain unchanged, but we have, the additional nonstationary Voronets 
coefficients: 


1 D I,n +1 = 7 D I = ~W D I,n+\ = 


= db D /dq, + Y (db D /dq D ')b D '! 
= db D /d(q,) - db DI /d(q n+ i ) 


db DI /dt + Y ( db DI /dq D ')b D - 


(c) 


[recalling the symbolic (2.9.32ffi), (2.11.15): A k —> b D , and ]C (d ■ ■ ■ /dq D )b D = 

a.../5(t)]. 


REMARK 

In concrete problems, use of the above definitions to calculate the iv-coefficients is 
not recommended. Instead, the safest way to do this is to read them off directly as 
coefficients of the following bilinear difference/covariant: 

d(S0 D ) - 6(dO D ) = • • • + 7 D IV dO r <50/ + • • • + 7 D j dttoj + • • • 

== • • • — w jj! defat + * ■ * — w / dt + • • •. (d) 


Problem 2.12.3 Continuing from the preceding problem, verify that: 

(i) in the catastatic Voronets case, the w D n i remain unchanged, while w D I = db DI /dt, 
and 

(ii) in the stationary Voronets case, the w D r[ ' remain unchanged, while it’ 23 / = 0. 
Problem 2.12.4 Continuing from the above problems, verify that 


(i) 

O 

II 

C"~ 

(/ = m + 1,..., n; (3,e= 1-- n; n + 1); 

(a) 

(ii) 

7 V = 0 

(D,D' = 1,... , 777 ; e = 1- n + 1); 

(b) 


(recall that 8 / = qj is 

a holonomic coordinate). 



Problem 2.12.5 Continuing from the above example and problems, consider 
again the nonstationary constraints in the special form (2.11.10a ff.): 

d c lD — E ^di dqi + bo dt, btlD = E b Dl v ° = E b D/ vi + b D , (a) 

where b DI = b DI {t,q), b D = b D (t,q), and, as usual, v k = dq k /dt. 

Show by direct (//^-differentiations of the above, and assuming that d(6qj) — 
6(dqj) = 0, that 

d(Sq D ) - b{dq D ) = EE w L //< dqi* bqi + / ' w°i dt bqj, (b) 

or, dividing both sides by dt, 

{& c Id) — ^(,9d) = d) ~ 8 v d = E (E 11 ' Vr = E vJ?/ ( c ) 


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§2.12 CONSTRAINED TRANSITIVITY EQUATIONS 


that is, in general, and contrary to the hitherto adopted Hamel viewpoint (§2.12), 
d(8q D ) ^ 6 ( dq D ), as if the q D are no longer holonomic coordinates! 

REMARKS 

The alternative (and, as shown below, internally consistent) viewpoint exhibited by 
(c) [originally advanced by Suslov (1901-1902), (1946, pp. 596-600), and continued 
by Levi-Civita (and Amaldi), Neimark and Fufaev, Rumiantsev, and others], is 
based on the following assumptions: 

(i) If the n differentials/velocities dq/hq/v are unconstrained, then we assume that the 
Hamel viewpoint holds for all of them; that is, d(6q k ) = S ( dq k ) (k = 1 ,...,»). 

(ii) But, if these differentials/velocities are subject to m (a)-like constraints, then we 
assume that the Hamel viewpoint holds only for the independent of them, say the 
last n — m, but not for the dependent of them, that is for the remaining (first) nr. 

Suslov viewpoint: d(8q,) — 8(dq,) = 0, but d(8q D ) — S(dq D ) ^ 0. (d) 

Let us examine this quantitatively, from the earlier generalized transitivity equations 
(2.10.1, 5): 

d(89 k -) — 8(d9, k ) = Y, a k i[d(8qi) — 8(dqj)\ + EE 7 \ s dd s 89,. + E 7 \dt86 n (e) 
d{8q k ) - 8{dq k ) = ^zl«{[J(^) -8{d9i)\ -EE 7 l rs d9 s 89,. -E 7 Uhd,.}. (f) 

(a) Hamel viewpoint'. d(8q k ) = 8{dq k ), always. Then, since d9 D , 89 D = 0, (e) yields 

d{89 D ) - 8{d9 D ) = EE 1 D IV dOp 89, + 7 D I dt 89/ [by (pr. 2.12.4: b)[ (g) 

= -EE w D ji' dq,< 8q, —^ w D I dt 8q, [by (ex. 2.12.1: g, j)[; (h) 

d(89,) - 8(d9,) = EE 7 I VV ! d9,» 89r + Y y v dt89 r = 0 

[by (pr. 2.12.4: a)]. (i) 

(b) Suslov viewpoint [for the Voronets-type constraints (a)]. Since here, A dd i = 8 DD >, 
A id = 0, Ajp = 8,p, eq. (f) yields, successively, 

(1) 0 = d(8q,) - 8(dq,) 

= 4{[« - S(d9 r )] ~ E E d’ri" d9j» 89, v - Y 7 V dt69 v ) 

= [d{89j) - 8(d9j)\ - EE 7 I i'i" dOp' 89p -E 7 1 p dt 89ji 

= d{89[) - 8(d9j) [by (pr. 2.12.4: a)]; (j) 

(2) d{8q D ) - 8{dq D ) 

= E 'W{i<*(«M - - EL 7 D ' IV d9p 89, - Y l D 'i dt89 ,} 

= \d{89 D ) - 8{d9 D )\ - EE 7 D ,pd9p89, -E 7 D ,dt89j 

= [d(89 D ) - 8(d9 D )] + EE w D jp d9p 89, + Y w °i dt Mi 

[by (ex. 2.12.1: i), (pr. 2.12.2: c)[ 

-EE w D ,p dcjp 8q, + E w°,dt8q, [by (b)[, 

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CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS 


and comparing the last two expressions of d(8q D ) — 8(dq D ), we immediately con¬ 
clude that 

d(S0 D ) - 8{d9 D ) = 0 . (k) 

Hence: In the Suslov viewpoint we must assume that d(89 k ) = 8 (d9 k ) (k = 1,..., n). 

Both viewpoints are internally consistent; but, if applied improperly, they may give 
rise to contradictions/paradoxes. Hamel’s viewpoint, however, has the advantage of 
being in agreement with variational calculus (more on this in §7.8). 


Problem 2.12.6 Consider the special stationary dO dq transformation: 

dd D =y^ a DD ' dq D i (= 0 ) and d9j =dq k 0 ; the 9j are holonomic coordinates), (a) 

where a DD t = a DD fq x ,..., q m ) = a DD fq D ). Show that, in this case, the Hamel co¬ 
efficients are 

(i) 7 ( da Dd '/dq d « - da Dd «/dq dl )A d , D ,A d « D «, (b) 

where D, D', D", d' , d" = 1,... ,m and dq D t A D s D d9 D ; and 

(ii) 7 k rs = 0 , for any one of k,r,s greater than m. (c) 


Example 2.12.2 Transformation of the Hamel Coefficients under Frame of 
Reference Transformations. Let us again consider, for algebraic simplicity but no 
loss in generality, the stationary Pfaffian constraint system: 

d9 n = a Dk dq k = 0, 89 D = Y a Dk % = 0. (a) 

Further, let us assume that (a) is nonholonomic; that is, ^ D U ’ f 0. Now we ask the 
question: Is it possible, by a frame of reference transformation q —> q'(t, q), to make 
the constraints (a) holonomic? In other words, is it possible to find new Lagrangean 
coordinates q k > = q k '(t,q k ), in which the corresponding (dq k > d9 k ) Hamel co¬ 
efficients "fiq 1 ) 0 ]]' = 7 ,D n' vanish? Below we show that the answer to this is no; that 
is, if a system of constraints is nonholonomic in one frame of reference, it remains 
nonholonomic in all other frames of reference obtainable from the original via 
admissible frame of reference transformations. 

Indeed, we find, successively [with 7 {q) D w = 7 D w], 

0 f d(89 D ) - 8(d9 D ) = EE 1 n r,d9,b9 r = EE a D rs dq s 8q r 

[dq s /dq s ') dq s i + (dq s /dt) dt'j ' ( dcj r / dq r f 8q r '^j 

= EEEE [(dq s /dq s ’)(dq r /dq,.')a D rs \dq s > 8q r - 
+ EEE [( dq s /dt)(dq,./dq l j)a D rs \ dt 8q r , 

-EE a D r ' s i dq s ’ 8q r t + E a D r ’ dt8q r > 

[where a D r > s > = da Dr fdq s i — da Ds '/dq r ', etc.] 

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§2.12 CONSTRAINED TRANSITIVITY EQUATIONS 

= E E aD ''' s ' (E J ^ s ' 1 + A s’ dtj A,.: i' 80 ['^ ^ a D r i A r iji 80j^j dt 

= EEEE {a n r i s ’A s ’iA r ’ji) dOj 80i‘ +E(EE + E aD r'A,'i^j dt 80 j* 
= EE r )' D Vi dO, 80 r + ^ A(b) 

from which, comparing with the first line of this equation, we readily conclude that 

7 (?) Z /'/ = l{q') D i'i and ffiq') D i> = 0 ; (c) 

that is, the Hamel coefficients remain invariant under frame of reference transforma¬ 
tions', or, these coefficients depend on the nonholonomic “coordinates” 0 k but they 
are independent of the particular holonomic coordinates frame used for their 
derivation. 

Incidentally, this derivation also demonstrates that the 7 -definition (2.10.1) is 
both practically and theoretically superior to the more common (2.10.2-4). 

REMARKS 

(i) We are reminded that the transformation properties of the 7 ’s under local 
transformations: dO k <=> dO k ', at (q,t), have already been given in ex. 2.10.1. 

(ii) The reader can easily verify that if, instead of (a), we had chosen a general 
nonstationary dO <^> dq transformation, we would have found 7 {q') D r = l{q) D i', 
instead of the second of (c). Also, then, 

c10,. = a rs dq s + a r dt = E a «(E (dq s /dq s ') dq s f + ( dqjdt ) dt^j + a r dt 

= ^ a n .t dq,.' + a' r dt, 


from which we can readily deduce the transformation relations among the coeffi¬ 
cients a(q), a(q') [recall ( 2 . 6.6 ff.)]. 


Problem 2.12.7 (see Forsyth, 1890, p. 54.) Verify that a system of n independent 
Pfaffian constraints in the n (or even n + 1) variables; that is, 

d0 k = E a ki dq, = 0 (k, l = 1 ,... ,n), (a) 


is always holonomic. 


Problem 2.12.8 Alternative Formulation of Frobenius’ Theorem. It has been 
shown, by Frobenius and others (see, e.g., Pascal, 1927, p. 584), that the Pfaffian 
system: 


d0 D = E a Dk (q) dq k = ° {D=l,...,m;k=\,...,n), (a) 

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CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS 


is holonomic if, and only if, each of its m (n + m) x (n + m) antisymmetric 
“Frobenius matrices”: 


0 

D D 

a 12 a \3 ' 

' a D \n 

an ■ 

d m \ 

D 

a 21 

0 a D a • 

• a °2n 

a n ■ 

d m 2 

r, D 
a n\ 

a D n 2 

■ 0 

a ln ■ 

&mn 

a n 

a \2 

d\n 

0 ■ 

■ 0 

d m 1 

dm2 ■■■ ' 

&mn 

0 • 

• 0 


where a D kl = da Dk /dq l - da Dl /dq k = a DkJ - a D lk = -a D lk (e.g., a D u = ~cP 2 1 , 
a D n = —a\ | =>■ a D \\ = 0, etc.), has rank 2m. 

Apply this theorem for various simple cases: for example, m = 0 (i.e., dq k uncon¬ 
strained), m = 1 (one constraint), and m = 2 (two constraints). 


Example 2.12.3 Geometrical Interpretations of the Frobenius Conditions (May be 
omitted in a first reading.) In terms of the earlier (2.11.20a ff.) m constraint vectors 
a D = (a D i, ■ ■ ■ ,a D „) and n — m virtual vectors A k = (A n ,... ,A ln ) (in ordinary 
vector, nonmatrix notation), Frobenius’ conditions first of (2.12.5) assume the 
following forms: 

(i) First interpretation: From the a D we build the antisymmetric tensor: 

0 °ki) = (- a\) = ( da Dk /dq, - da D ,/dq k ). (a) 

These can be viewed as the holonomic (covariant) components of the “ rotation or 
curlling) of a D "\ a D kt = —(curl a D ) kl . Also, we recall that A k/ = dv k /du>i. As a result 
of the above, (2.12.5): 

{da Dk /dq, - da m /dq k )A kI A u , -EE aD kiA k iAn' — 0, (b) 

assumes the (covariant) tensor transformation form, in k, I: 

7V = EE (dv k /du,)(dvi/duj r )a D k i 

= A r ■ curla D ■ A, = ^ ^[A v )\curla D ) lk (Aj) k = 0; (c) 

that is, the (covariant) components of the curl of the dependent!constraint vectors a D 
along the independent nonholonomic directions Aj should vanish. 

(ii) Second interpretation: The Frobenius conditions (first of 2.12.5), rewritten 
with the help of the alternative expression (2.10.15) and the quasi chain rule 

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§2.13 GENERAL EXAMPLES AND PROBLEMS 


(2.9.30a) as 


^ = ££ {a Dh [A c i(dA hr /dq c ) - A cI ,{dA bI /dq c )}} 

— £ a Db(dA h i>/ddj - dA hI /d6ji) = 0, (d) 

state that the constraint vectors a D should be perpendicular to the 
(n — m) [n — — l)/2 vectors: 

A w = {^2 A ci( dA bi'/dq c ) ~ ^rA cI ,(dA hI /dq c )^ = {dA^jdQ, - dA hI /dd r ) = -A VI , 
that is, 

1°iv — a D ' A iv = 0 - ( e ) 

Similarly for the nonstationary/rheonomic Frobenius conditions (second of 2.12.5): 

d A 
7 / = 0. 

For further details, including the precise positioning of indices, as practiced in 
general tensor analysis (and not observed in the above discussion!), see, for example, 
Papastavridis (1999, §6.9). 


2.13 GENERAL EXAMPLES AND PROBLEMS 


Example 2.13.1 Introduction to the Simplest Nonholonomic Problem: Knife, Sled, 
Scissors, and so on. Let us consider the motion of a knife S, whose rigid blade 
remains perpendicular to the fixed plane O-xy, and in contact with it at the point 
C(x,y), and whose mass center G lies a distance b 0) from C along the blade 
(fig. 2.15). The instantaneous angular orientation of 5 is given by its blade’s angle 
with the +Ox axis <j>. 

Let us choose as Lagrangean coordinates: q\ = x, q 2 = y, q 2 = <j>. If v = ( dx/dt , 
dy/dt, dz/dt = 0) = (y Y , v y , 0) = (inertial) velocity of C, and u = (cos </>, sin (j>, 0): 
unit vector along the blade, then the velocity constraint is 

v x u = 0 =>■ (sin 4>)v x + (— cos <j})v y = 0, or dy/dx = tan (/). (a) 



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CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS 


Since this is a stationary (and, of course, catastatic) constraint, we will also have, for 
its kinematically admissible/possible and virtual forms, respectively, 

(sin f) dx + (— cos <j>) dy = 0 and (sin</>) 8x + (— cosf) 6y = 0. (b) 

Other physical problems leading to such a constraint are: 

(i) A racing boat with thin, deep, and wide keel, sailing on a still sea. Since the water 
resistance to the boat’s longitudinal motion is much larger than the resistance to its 
transverse motion, the direction of the boat’s instantaneous velocity must be always 
parallel to its keel’s instantaneous heading; 

(ii) a lamina moving on its plane, with a short and very stiff razor blade (or some 
similar rigid and very thin object: e.g., a small knife) embedded on its underside. 
Again, the lamina can move only along the instantaneous direction of its guiding 
blade; 

(iii) a sled; 

(iv) a pair of scissors cutting through a piece of paper; 

(v) a pizza cutter, etc. 

Application of the holonomicity criterion (2.3.6) or (2.3.8a) to (a), (b), with 
h = (sin <j>, — cos cj), 0) and dr = (dx. dy, df) [as if x,y,<j> were rectangular Cartesian 
right-handed coordinates] yields 

I = h- curl h = It - [(<9/cbc, d/dy, d/df) x (sin </>, — cos <j), 0)] 

= (sin</>, — cos </>, 0) • (— sin </>, cos (/>,0) = — 1 f 0; (c) 

that is, the constraint (a), (b) is nonholonomic. This means that, although the general 
(global) configuration of 5 is specified completely by the three independent coordi¬ 
nates x,y,cj>, not all three of them can be given, simultaneously, small arbitrary 
variations; that is, although there is no functional restriction of the type 
f(x,y,4 ») = 0, there is one of the type g(dx,dy,dcf)-,x,y,(j)) = 0, namely the Pfaffian 
constraint (a), (b). Put geometrically: the blade has three global freedoms (x,y,(f>), 
but only two local freedoms (any two of dx.dy.df). Since n = 3 and m = 1, this is 
the simplest nonholonomic problem; and, accordingly, it has been studied extensively 
(by Bahar, Caratheodory, Chaplygin, et al.). 

The independence of x,y, </> can be demonstrated as follows: we keep any two of 
them constant, and then show that varying the third results in a nontrivial (or 
nonempty) range of kinematically admissible positions: 


(i) keep x and y fixed and vary <j) continuously; the constraint (a), (b) is not violated 
[fig. 2.16(a)]; 

(ii) keep y and (f> fixed. Varying x we can achieve other admissible configurations with 
different x’s but the same y and <j>\ 

but to go from one of them to another we have to vary all three coordinates [fig. 
2.16(b)]; 

(iii) similarly when x and <f> are fixed and y varies [fig. 2.16(c)], 

The precise kinetic path followed in each case, among the kinematically possible/ 
admissible ones, depends on the system’s equations of (constrained) motion and on 
its initial conditions. 

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§2.13 GENERAL EXAMPLES AND PROBLEMS 


(a) (c) 

(2) 





Figure 2.1 6 Global motions of a knife showing the independence of its three positional 
coordinates. We can always, through a suitable finite motion, bring the knife to a position and 
orientation as close as we want to any specified original position and orientation; that is, the 
relation among x, y, <j> is nonunique. 


An Ad Hoc Proof of the 

Impossibility of Obtaining a Relation f(x,y,f) = 0 

Let us assume that such a constraint exists. Then c/-varying it, and with subscripts 
for partial derivatives, yields 

df = f x dx +f y dy+f 0 df = 0, (d) 

or, taking into account the constraint in the form: dy = (tan f ) dx, 

df = {f x +f y tan <j>) dx + (/ 0 ) df = 0, (e) 

where now dx and df are independent. Equation (e) leads immediately to 

U = 0 => / = / {x, v) and f x +f y tan f= 0. (f) 

By {d/df) -differentiating the second of (f), while observing the first of (f), we obtain 

f y {\/cos 2 f) = 0, (g) 

from which, since in general 1/cos 2 f f 0, it follows that f y = 0 => / =/(x). But 
then the second of (f) leads to f x = 0 =>- / = constant {independent of x,y,f), and as 
such it cannot enforce the constraint f{x, y, f) = 0. Hence, no such/ exists (with or 
without integrating factors). 

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CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS 

However, if the knife was constrained to move along a prescribed path, on the 
O-xy plane, the system would be holonomic! In that case, we would have in advance 
the path’s equations, say in the parametric form: 

x = x{s) and y = y(s) (s = arc length), (h) 

from which <j> could be uniquely determined for every s [i.e., 0 = 0(s)], via 

dy/dx = dy/ds j dx/ds = y'(s)/x'(s) = tan 0(F). (i) 

This is somewhat analogous to the basic variables of Lagrangean mechanics q k , 
dq k /dt = v k , which, before the problem is solved, are considered as independent , 
and then, after the problem is completely solved, become dependent through time. 


Example 2.13.2 The Knife Problem: Hamel Coefficients. Continuing from the 
preceding example: in view of the constraint (a), (b) there, and following Hamel’s 
methodology (“equilibrium quasi velocities,” §2.11), let us introduce the following 
three quasi velocities: 

w, = (- sin 4>)v x + (cos f)v y + (O)v^ (= 0), 

u 2 = (cos 0) v Y + (sin (j))v y + (O)v 0 = v (^ 0), 

w 3 = (0)v x + (0)v > , + (l)v^ = 0), (a) 

where v = velocity component of the knife’s contact point C; and hence v x = vcos 0, 
v v = vsin0, and the constraint is simply uq = 0. Clearly, since 

9(cos cfj/dff 9(0 )/dx and 9(sin cjf/dff 9(0) /dy, (b) 

uj 2 = v is a quasi velocity, that is, v f total time derivative of a genuine position 
coordinate, or of any function of x,y, 0. Inverting (a), we obtain 

v x = (—sin0)w! + (cos (j))u) 2 + (0)w 3 , 

v y = (cos + (sin0)w 2 + (0)w 3 , 

v <l> = (0) w i + (0) w 2 + (l) w 3- ( c ) 

If a and A are the matrices of the transformations (a) and (c), respectively, then we 
easily verify that a = A, and Deta = Det A = — sin 2 0 — cos 2 0 = — 1 (i.e., nonsin¬ 
gular transformations). Further, we notice that (a), (c) hold with u ; 12 ,3 and v x , v y , v$ 
replaced, respectively, with dd l 23 = iv l23 dt and ( dx,dy,df ) = {v x , v y , v^) dt: and, 
since they are stationary, also for 69 x 2 3 and bx, by, <50. 

Next, by direct ^/^-differentiations of 69 u d9 x , and then subtraction, we find, 
successively, 

d[69f) ~ 6{d9 y ) = d[{— sin0) 6x + (cos 0) by + (0) 6f\ 

~ <5[(— sin 0) dx + (cos 0) dy + (0) df\ 

= (— sin <p)(dbx — bdx) + (cos </>)(dby — bdy) 

— cos <f>d(j)bx — sin (f>d<j>by + cos <f> dx b<f> + sin 0 dy be/) 

= 0 + 0 — cos<£(1) d9 2 [{— sin0) 69 x + (cos0) b9 2 + (0) <5d 3 ] — • • • 


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§2.13 GENERAL EXAMPLES AND PROBLEMS 


[i.e., expressing dx, 6x, dy, 6y, df, Sc/) from (c), with replaced with 


dO l2 3 , S9 X23 ] and so we, finally, obtain the differential transitivity equation: 

d(69i) - S(dO x ) = dd 2 S9 3 - d9 3 Sd 2 , (d) 

[i.e., d(S9 1 ) — 8{dO x ) f 0, even though S9 X = 0 and d6 x = 0]; and, also, dividing this 
by dt, which does not couple with 6 (...), we obtain its (equivalent) velocity transi¬ 
tivity equation: 

(59 1 ) --6u) 1 = (0)69 l + (-l)u> 3 S9 2 + (l)u) 2 69 3 (fO). (e) 

Similarly, after some straightforward differentiations, we find 

(59 2 ) '-Su) 2 = (1)u) 3 S9 1 + (0)S9 2 + (-1)uj 1 S9 3 (= 0), (f) 

(69 3 )-- 6u) 3 = (0)69 1 + (0)69 2 + (0)69 3 (= 0). (g) 

From (e, f, g) we readily read off the nonvanishing Hamel’s coefficients: 

7V(Z)=1;/,J' = 2,3): 7 23 = ~l\i = -1; (h) 

(I = 2- k,l = 1,3): 7 2 i3 = -7n = 1 ■ « 


REMARKS 

(i) Since not all y D n ’ —> (k, l = 2,3) vanish, we conclude, by Frobenius’ the¬ 

orem (§2.12), that our constraint, in any one of the following three forms: 

Velocity: 

= (-sin0)v x + (cos0)vj, + (O)v0 (=0), (j) 

Kinematically admissible: 

d6 x = (— sin c/>) dx + (cos 0) dy + (0) dc/> (= 0), (k) 

Virtual: 

S6 X = (—sin </>) Sx + (cos</>) Sy + (0) 8</> (= 0), (1) 


is nonholonomic. 

(ii) The fact that upon imposition of the constraints S6 x = 0, uq = 0, the transitivity 
equation (f) yields (69 2 )' — 6lo 2 = 0 does not mean that 

dd 2 = (cos</>) dx + (sin^) dy + ( 0 ) dc/) = vdt (^ 0 ), (m) 

or 

S9 2 = (cos c/>) Sx + (sin </>) Sy + (0) Scj) 0), (n) 

are exact; it does not mean that 0 2 is a genuine (Lagrangean) coordinate. For 
exactness, we should have 7 2 W = 0 (k, I = 1,2,3) => ( S9 2 )' — Su> 2 = 0, independently 
of the constraints ui l /d9 l /S9 l = 0. [We recall (§2.12) that Frobenius’ theorem tests 
the holonomicity, or absence thereof, of a system of Pfaffian equations of constraint ; 
whereas the exactness, or inexactness, of a particular Pfaffian form , like d0 2 and 
d9 3 {f 0) is a property of that form; that is, it is ascertained by examination of 
that form alone, independently of other constraint equations. In sum: constraint 
holonomicity is a system (coupled) property; while coordinate holonomicity is an 
individual (uncoupled) property .] 

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CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS 

(iii) Since cc 3 = dtp/dt = v r3 is a genuine velocity, 7 3 w = 0 (k,l= 1,2,3); as 
expected. 


Hamel Viewpoint versus Suslov Viewpoint 

So far, we have assumed Hamel’s viewpoint; that is, 

d(Sx) = 8(dx), d(8y ) = 8(dy), d(8cj)) = 8(d(f>)\ (o) 

and d{80i) f 8{dO x ), in spite of the constraint 80\ = 0 and dQ\ = 0 [and that even if 
di^86\) = 0, still -6(d0i) f 0!]. 

Let us now examine the Suslov viewpoint: with the analytically convenient choice, 
q D = y and q, = x,(j), we can rewrite the constraint as 

d9 x = dy — (tan </>) dx = 0 and 89 x = 8y — (tan </>) 8x = 0 [instead of (a)], 
or 

dy = (tan </>) dx + (0) dcf) and 8y = (tan f) 8x + (0) <5</>; (p) 

and, therefore, the corresponding transitivity equations become [instead of (d)-(g)] 

Dependent: d(8y ) — 8{dv) = d^xtan^) — <5(fi(xtan</>) = • • • 

= \d{8x) — 8{dx)\ tan tf>+ (1/cos 2 (f>)(d(t>8x — dx8(j)) 

= (1/cos 1 (j))(d4> 8x — dx8(j>) f 0, (q) 

Independent: d(8x) — 8(dx) = 0, d(8(j>) — 8(d(f) = 0; (r) 

from which we readily read olf the sole nonvanishing Voronets symbol: 

= -vrV = 1/ cos 2 ^>. (s) 

Under Hamel’s viewpoint, using the same variables, front <5y = (tan^>) <5x (i.e., 
89 x =0) it follows that d(8y) = d(8x) tan <(>+(1/cos 2 <(>) df8x [i.e., d(80 x ) =0]; but 
from dy = (tan f) dx (i.e., dd x = 0) it does not follow that 8(dy) = 
8{dx) tan </> — (1/cos 2 ^) 8cf>dx [i.e., 8(d9 x ) f 0]. 


Problem 2.13.1 Consider a knife (or sled, or scissors, etc.) moving on a 
uniformly rotating turntable T (fig. 2.17). In T-fixed (moving) coordinates O-xyf, 
its constraint is 

(sin^)^ + (— cos cj))v v = 0 [v x = dx/dt , v y = dy/dt]. (a) 

Show that in inertial (fixed) coordinates 0-XY<P, where 

X = (cos 6)x + (— sin 9)y + (0)0, 

Y = (sin 9)x + (cos 9)y + (0)0, 

<f> = (0)x+(0)y + (!)</> + 9, (b) 


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§2.13 GENERAL EXAMPLES AND PROBLEMS 



and 6 = wt, to: constant angular velocity of O-xy relative to O-XY [i.e., say, 
X = X(x,y,t), etc.], the constraint takes the (acatastatic) form (with v v = dX/dt, 
Vy = dY/dt), 

(sin <P)v x + (— cos <P)v Y + w[(cos <P)X + (sin <P)Y] =0. (c) 


Example 2.IB.3 Rolling Disk—Vertical Case. Let us consider a circular thin disk 
D , of center G and radius r, rolling while remaining vertical on a fixed, rough, 
and horizontal plane P (fig. 2 . 18 ). (The general nonvertical case is presented later 
in ex. 2 . 13 . 7 .) This system has four Lagrangean coordinates (or global DOF ): the 
(x,y,z = r) coordinates of G, and the Eulerian angles cj) (precession) and ijj (spin). 
The constraints z = r (contact) and 6 = 7r/2 are, clearly, holonomic (H). The 
velocity constraint is v c = 0 (where C is the contact point); or, since along the 
fixed axes O-XYZ [with the notation dx/dt = v x , dy/dt=v y \ dcj)/dt = uu, 
d'f/dt = i tty]: 

v G = (v x ,v y , 0), CJ = (-Uty sin <j>, u^cos(/>, w 0 ), and r c/G = (0,0, -r), 


=> v c = v G + O) x r c / G = ■ ■ ■ = (v x - rui^ cos cj>, v y - rw^ sin <j>, 0) = 0, (a) 

or, in components, in the following equivalent forms: 

Velocity: v x = r uy, cos </> and v y = ruj^ sin 0, (b) 

Kinematically admissible: dx = (r cos <fr) dip and dy = (r sin <p) dip, (c) 
Virtual: 6x = (rcosqi) Sip and 6y = (rsin^) Sip. (d) 


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CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS 




X - % cosp^ r\pcosp. y = V 0 xinj-ry simp; u «r (cosp, simp), n = (sirtp. cosp) 


Figure 2.18 Rolling of vertical disk on a fixed plane. 


As shown below, these constraints are nonholonomic (NH). Hence, the disk is a 
scleronomic NH system with / = n — m = 4 — 2 = 2 DOF in the small. 

It is not hard to see that imposition, on (b-d), of the additional H constraint 
dcp = 0 =>■ <j) = constant, say tp = 0 , would reduce them to the well-known H case 
of plane rolling : dx = r d'tp =>■ x = r ip + constant, and dy = 0 =>■ y = constant. 
{Also, the problem would become H if the disk was forced to roll along a prescribed 
O-XY path. For, then, the rolling condition would be [with s: arc-length along (c)] 
ds = rdip =>■ s = rip + constant, and (c) would yield the parametric equations 
x = x(s) and y = y(.s); that is, for each s there would correspond a unique x,y,ip, 
and cp [from (b-d)], and that would make the disk a 1 ( global) DOFH system.} 


Ad Hoc Proof of the Nonholonomicity of the Constraints (b-d) 

Let us assume that we could find a finite relation f(x,y,<p,ip ) = 0 , compatible with 
(b-d). Then (with subscripts denoting partial derivatives), we would have 

df = f x dx +f y dy +4 dtp +f^dip= 0. (e) 

Substituting dx and dy from (c) into (e) — that is, embedding the constraints into it — 
yields 

{rf x cos tp + rf y sin <p +4) dip + (/ 0 ) dtp = 0, (f) 

which, since now dip and dtp are independent, gives 

4 = 0 f =f(x,y,ip) and rf x cos <p + rf y sin 0+4 = 0. (g) 

Next, (9/<9</>)-difTerentiating the second of (g) once, while taking into account the first 
of (g), yields 

— rf x sin tp + rf y cos tp = 0, (h) 

and repeating this procedure on (h), while again observing the first of (g), produces 

—rf x cos tp — rf. sin tp = 0. (i) 


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§2.13 GENERAL EXAMPLES AND PROBLEMS 


[Further (d/<90)-differentiations would not produce anything new.] The system (h), 
(i) has the unique solution, 

fx = 0 and f y = °, (j) 

due to which the second of (g) reduces to = 0. It is clear that the above result in 
/ = constant, and such a functional relation, obviously, cannot produce the con¬ 
straints (b d) — no f{x,y,(j>,tp) exists. Geometrically, this nonholonomicity has the 
following consequences: Starting from a certain initial configuration, we can roll the 
disk along two different paths to two final configurations with the same contact 
point—namely, same final (x,y), but rotated relative to each other, that is, with 
different final (0,0). If the constraints were H, then 0 and 0 would be functions 
of (x,y) and the two final positions of the disk would coincide completely. 


Proof that the Constraints (b-d) are NH via Frobenius’ 

Theorem 

Let us rewrite the two constraints (c, d) in the equilibrium forms: 

Kinematically admissible : 

dd l = dx — (rcos0) dfi = 0, dd 2 = dy — (rsin0) dif = 0, (k) 

Virtual: 89\ = 8x — (rcos0) Sift = 0, 89 2 = 8y — (rsin0) Sip = 0. (1) 

It follows that the corresponding bilinear covariants (2.8.2 ff.) are 

d{89 ]) — 8{d9f) = • • • = (rsin0)(d0<50 — dip8cf>), (m) 

d(89 2 ) — 8{d9 2 ) = ■ ■ ■ = (—r cos <f>)(d(p8ij.’ — dfidfi), (n) 

and, clearly, these vanish for arbitrary values of the independent differentials 
dc\>,8c\),dfi,8fi, if sin0 = O and cos0 = 0. But then the constraints (c) reduce to 
dx = 0 => a: = constant and dy = 0 => y — constant, which is, in general, impossible. 
Hence, the constraints are NH [one can arrive at the same conclusion with the help 
of the 7 ’s (§2.12), but that is more laborious]. 


Problem 2.IB.2 Continuing from the previous problem (vertically rolling disk), 
show that its velocity constraints can be expressed in the equivalent form: 

v G • u = v x cos 0 + v y sin 0 = ru>^, (a) 

v G -n = — y Y sin0+ v_ F cos0 = 0, (b) 

where u and n are unit vectors on the disk plane (parallel to O-XY) and perpendi¬ 
cular to it, respectively (fig. 2.18). [Notice that (b) coincides, formally, with the knife 
problem constraint.] 


Example 2.13.4 Rolling Sphere — Introduction. Let us consider a sphere of center G 
and radius r, rolling without slipping on a fixed, rough and, say, horizontal plane P 
(fig. 2.19). The complete specification of a generic sphere configuration requires five 
independent (minimal) Lagrangean coordinates. As such, we could take the (inertial) 

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CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS 



coordinates of G ( X , Y), and the three Eulerian angles (</>, 9, ip) of body-fixed axes 
G-xyz relative to translating (nonrotating) axes G-XYZ. The contact constraint is 
expressed by the holonomic (H) equation, Z = vertical coordinate of G = r. The 
rolling constraint is found by equating the (inertial) velocity of the contact point of the 
sphere C with that of its (instantaneously) adjacent plane point, which here is zero; 
us = inertial angular velocity of sphere. Using components along O-XYZ axes throughout, 
we find 

v c = v g + m x r c/G = dPc/dt + (*> x (— rK) 

= ( v Xi v Yi 0) + ( w Xi w z) x (0,0, —r) = • • • = (v x — rujy, v Y + ruj x , 0) = 0. 

Hence, the rolling conditions are 

Vj - ruy = 0, v Y + rujx = 0; (a) 

or, expressing the space-fixed components u> x Y in terms of their Eulerian angle rates 

(§ 1 - 12 ), 

Vx _ r(sin(j)u)g — sin 9 cost/) uifi) = 0, v Y + r(cos()iw e + sindsin(/uu^,) = 0; (b) 
or, further, in kinematically admissible form, 
dX — r(sm(j)d9 — sin 9 cos (f>dip) = 0, dY + r(cos cj)d9 + sin 9 sin (f>dip) = 0; (c) 
or, finally, since these constraints are catastatic, in virtual form, 

8X — r(sin $ 69 — sin 9 cos <j> 6ip) = 0, <5T+ r(coscj)89+ sin 9 sin (f> 8i/j) = 0. (d) 

[Absence of pivoting would have meant the following additional constraint: 

( <u ) normal to sphere atC ^ij> T COS 0 UJ t p 0, 

or 

d4> + cos 9 dip = 0 =>■ d(f>/dip =—cos 9 = h(9)\. 

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(e) 








§2.13 GENERAL EXAMPLES AND PROBLEMS 


As shown later, the constraints (a-d) are nonholonomic (NH). [We already notice 
that (c), for example, do not involve d<p, and yet the constraints feature sm<j> and 
cos (/>.] Mathematically, this means that it is impossible to obtain them by differen¬ 
tiating two finite constraint equations of the form F(X, Y, <p, 9, ip) =0 and 
E(X, Y, <p, 9, ip) = 0; that is, the coordinates X,Y,(p,9,ip are independent. But 
their differentials dX, dY, dtp , d6, dip, in view of (a-d), are not independent; that is, 
in general, only three of them can be varied simultaneously and arbitrarily. We say 
that the sphere has five DOF in the large, hut only three DOF in the small: 
f = n — m = 5 — 2=3. (Had we added pivoting, we would have / = 2.) 

Kinematically, the above mean that the sphere may roll from an initial config¬ 
uration, along two different routes, to two final configurations, which have both the 
same contact point and center location (i.e., same X, Y), but different angular 
orientations relative to each other (i.e., different cp,9,ip). If the constraints (a-d) 
were holonomic—for example, if the plane was smooth —it would be possible to 
vary all X , Y, (p, 9 , ip independently and arbitrarily without violating the (then) con¬ 
straints; namely, the sphere’s rigidity and the constancy of distance between G and 
C. Further, the sphere can roll from any initial configuration, with the sphere point 
Cj in contact with the plane point P h to any other final configuration, with the sphere 
point Cf in contact with the plane point Pf. To see this property, known as acces¬ 
sibility (§2.3), we draw on the plane a curve ( 7 ) joining C, and Pf, and another curve 
on the sphere (6), of equal length to ( 7 ), joining C, and Cf. Now, a pivoting of the 
sphere can make the two arcs ( 7 ) and (6) tangent, at C, = P h Then, we bring Cj to Pf 
by rolling ( 6 ) on ( 7 ). A final pivoting of the sphere brings it to its final configuration 
(see also Rutherford, 1960, pp. 161-162). 


A Special Case 

Assume, next, that the sphere rolls without pivoting, and also moves so that 
9 = constant = 9 0 . Let us find the path of G. With 9 = constant =>• dQ = 0, the roll¬ 
ing constraints (c) reduce to 

dX + r(sin0 o ) cos <p dip = 0, dY + r(sin9 0 ) sirup dip = 0; (f) 

and the no-pivoting constraint (e) to 

dcp/dip = — cos 9 a = constant. (g) 

This leaves only n — m = 5 — 4 = 1 DOF in the small. Taking <p as the independent 
coordinate and eliminating dip between (f), with the help of (g), yields 

dX = r(tan 9 0 ) cos <p dcp, dY = r(tan 9 0 ) sin (p d(p, (h) 

which integrates readily to the curve (with X 0 and Y„ as integration constants): 

X — X 0 = r(tan0 o ) sin/ Y — Y 0 = — r(tan 9 0 ) cos cp; (i) 

that is, G describes, on the plane Z = r, a circle of radius r tan0 o . 

[These considerations also show how imposition of a sufficient number of 

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CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS 


additional holonomic and/or nonholonomic constraints turns an originally non- 
holonomic system into a holonomic one.] 


Example 2.IB.5 Rolling Sphere on a Spinning Table — Introduction. Let us extend 
the previous example to the case where the plane P is not fixed, but rotates about 
a fixed axis OZ perpendicular to it with, say, constant (inertial) angular velocity 
12. In this case, the rolling condition expresses the fact that the contact points of 
the sphere and the plane, C, have equal inertial velocities: 

( ^c) sphere = Oc)pl an e : V G + « x r c/G = Qx V c/0 (= Q X p)' (a) 

or, in terms of their components along inertial (background) axes O-XYZ / O-IJK: 

(vx,vy,0) + (u}x,to Y ,toz) x (0,0, —r) = (0,0,0) x ( X , Y, 0), (b) 

from which we easily obtain the two rolling conditions: 

v y — rujy = — Of, Vy T rujx = QX. (c) 

Next, expressing wj, u> Y in terms of their Eulerian angles (between translating/ 
nonrotating axes G-XYZ and sphere-fixed axes G-xyz ) and their time rates, as in 
the preceding example, we transform (c) to 

v x — r(sin (puj e — sin 9 cos </> ojfi) + QY = 0, 

v Y + r(cos (picg + sin 9 sin <p ujfi) — QX = 0. (d) 

The O-proportional terms in (d) are the acatastatic parts of these constraints, and 
arise out of our use of inertial coordinates to describe the kinematics in a noninertial 
frame; had we used plane-fixed (noninertial) coordinates, the constraints would have 
been catastatic in them. It is not hard to see that the kinematically admissible (possible 
and virtual forms of these constraints are, respectively (note differences between them 
resulting from constraint 6t = 0), 

iIX — r(sin <f>d9— sin 9 cos <j) dip) + (Q Y) dt = 0, 

dY + r(cos 4>d9 + sin 9 sin <j) dip) — (QX) dt = 0; (e) 

6X — /-(sin tp69 — sin 9 cos </> Sip) = 0, 

6Y + r(cos cp 89 + sin 9 sirup 6ip) = 0. (f) 


Example 2.13.6 Rolling Sphere on Spinning Table — the Transitivity Equations. 
Continuing from the preceding example, let us show that its rolling constraints 
(c-f); as well as those of its previous, stationary table case) are nonholonomic; that 
is, the system has n = 5 DOF in the large , and / = /; — /// = 5 — 2 = 3 DOF in the 
small. 

In view of the structure of these constraints, we choose the following equilibrium 
quasi velocities (with the usual notations: dX/dt = v x , ■ ■ ■, dcp/dt = ...): 

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§2.13 GENERAL EXAMPLES AND PROBLEMS 


Dependent: 

Ui = v x — rujy + QY = v x — r(sin<j)ujg — cos</>sin 0uty) + QY = v x — rcj 4 + QY (=0), 

(a) 

oj 2 = Vy+ ru> x — C2X = v Y + r(cos (j>uig + sin (f> sin 9 uX) — QX = v Y + rui 3 — QX (= 0), 

(b) 

Independent: 

w 3 = u> x = (cos <j>)u) e + (sin cj) sin 9)u ^ (^ 0), (c) 

UJ 4 = u> Y = (sin (j))u s + (—cost/) sin (^ 0 ), (d) 

w 5 = w z = ( 1 )^V + (cos 9)u>^ ( 7 ^ 0 ), (e) 

u) 6 = dt/dt= 1 ( isochrony ). (f) 

Recalling results from §1.12, we readily see that these partially decoupled equations 
invert to 


'’1 

= Vx = 

uj\ H- y CJ4 — £2 Y 

(without enforcement of constraints u> x 2 = 

= 0), 

(gl) 


= Vy = 

UJ2 — ? ^3 + QX 

(without enforcement of constraints uq 2 = 

= 0), 

(g2) 

v 3 

= ^ = 

(— cot# sin 0)u; 3 + i 

(cot 9 cos (j))uj 4 + u 5 , 


(g 3 ) 

v 4 

= UJg = 

(cos </>)w 3 + (sin 4 >)ui 4 , 


(g 4 ) 

V 5 


(sin (j)/ sin0)u; 3 + (- 

-cos (j>/ sin 9)lo 4 , 


(g 5 ) 

v 6 

= dt/dt 

= U) 6 =l. 



(g6) 


The virtual forms of (a-g 6 ) are as follows [note absence of acatastatic terms in 
(hi, 2 )]: 

Dependent: 

69 { = 6X— r69 Y = 6X+ (—r sine/)) 69+(r cos ft sin 9) 6ip = 6X — r69 4 (=0), (hi) 

69 2 = 6Y + r69 x = 6Y + (rcosc/)) 69 + (rsin <?!> sin0) 6ip = 6Y + r69 3 (=0), (h2) 


Independent: 

69 2 = 69x = (cos cj)) 69 + (sin cj)sin 9) 6 i/j (=/= 0), (h3) 

69 4 = 69 Y = (sin cjs) 69+ (—cos cj) sin 9) 6tp (^ 0), (h4) 

69 5 = 69 z = (1) 6<j) + (cos 9) 6 r/> (^ 0), (h5) 

69 6 = 6 q 6 = 6 t = 0 (isochrony)-, (h 6 ) 

6 q\ = 6 X = 69\ + r69 4 , (il) 

6 q 2 = 6 Y = 69 2 -r69 3 , (i2) 

6 q 3 = 6 (j> = (— cot 9 sin (js) 69 3 + (cot 9 cos 0 ) 69 4 + 69 5 , (i3) 


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CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS 


8q 4 = 80 = (cos (j>) 80 3 + (sin c/>) 89 4l (i4) 

8q 5 = 8ip = (sin^/ sin#) 89 3 + (— cos 0/sin#) 89 4 , (i5) 

8q 6 = 8t = 89 6 = 0. (i6) 

Now we are ready to calculate Hamel’s coefficients from the transitivity equations 
(§2.10): 

(86 k ) ' - 6u) k = ^ Y ^ r P “P S0 r = YY 6d >■ + Y ^ ' S9 " 0) 

where k,r,s = /? = 1, - - -, 6; J k r = 1 k r ,„+i = 7^6- 

By direct differentiations, use of the above, and the indicated shortcuts [and 
noting that, even if Q = Q(t) = given function of time, still 8Q = 0], we obtain, 
successively, 

(89 1 )' — Scoi = (8X — r89 Y ) — 8(v x — raj Y + QY) 

= [(<Wf)' - 8v x ] - r[(89 Y y - Su Y \ -Q8Y 
= 0 — r[(89 4 y — 8 uj 4 ] - Q8Y 

[invoking the rotational transitivity equations (§1.14 and ex. 2.13.9), and (i2)] 

= —r(c Oz 80x — ^x 89 z ) — Q(892 — y 89 3 ) 

= —r(uj 5 80 3 — u> 3 89 5 ) — Q(89 2 — r89 3 ), 

or, finally, 

(89 1 )’ — 8uj\ = (—r)uj 5 86 3 + (r)u> 3 89 5 + (-Q) 89 2 + (r(2) 89 3 \ (kl) 

(89 Y)' — 8lu 2 = (8Y + r89 x )' — <5(vy + ruj x — QX) 

= [(h7)- - 8vy} + r[(89 x y - 8u x ] + Q8X 
= 0 + r[(89 3 y - 8 cj 3 ] + Q 8X 

[invoking again the rotational transitivity equations and (il)] 

= v(iOy89/ — (jJ/ 80 y ) 4 - C2(80\ 4 - y89 4 ) 

= y(co 4 89 3 — cu 3 89 4 ) + Q(89\ 4 ~ y 89 4 ) , 

or, finally, 


(<5#2)" — 8 co 2 = (— r)ui 3 89 4 + (y)lu 4 89 5 + (Q) 89\ 

+ (r£2) 664 ] 

(k2) 

and, again, the rotational transitivity equations (with X — 

>3, Y -► 4, Z - 

-> 5) give 

( 86 3 )' — 8 cj 3 = w 4 89 5 — qj 5 89 4i 


(k3) 

(89 4 )' — 8 ui 4 = uj 5 89 3 — u > 3 89 5 , 


(k4) 

(89 Y) — 8 uj 3 = lo 3 89 4 — cu 4 89 3 . 


(k5) 


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§2.13 GENERAL EXAMPLES AND PROBLEMS 


Comparing (j) with (kl-5) we readily find that the nonvanishing 7 ’s are 

1 1 1 1 _ 1 r, 1 1 _ 1 r, / n \ 

7 35 = -7 53 = -C 7 26 = “7 62 = 7 2 = ““, 7 36 = ~7 63 = 7 3 = ' ^ l 11 ) 

7 2 45 = - 7 2 54 = -r, 7 2 16 = - 7 2 61 = 7 2 1 = Si, 7 2 46 = - 7 2 64 = 7 2 4 = r Q; (12) 

7 3 45 = - 7 3 54 = 7 4 53 = - 7 4 35 = 7 ? 34 = - 7 ? 43 = ~ 1 [= -1 {permutation symbol )]. (13) 
Here, Dependent) = 1,2 and I, I'independent)) = 3,4,5. Therefore, 

l D w ; 7 1 35 = —^ 0,7 2 45 = —^0; 7°/ : 7 ' 3 = 7 2 4 = ^ ^ 0; (m) 

and so, according to Frobenius’ theorem (§2.12), the system of Pfaffian constraints 
u>i = 0 and uj 2 = 0 is nonholonomic, in both the catastatic (rolling on fixed plane) and 
acatastatic (rolling on rotating plane) cases; that is, for any given Q = Q(t). 


Example 2.13.7 Rolling Disk on Fixed Plane. Let us consider a thin circular disk 
(or coin, or ring, or hoop), of radius r and center G, rolling on a fixed horizontal 
and rough plane (fig. 2.20). A generic configuration of the disk is determined by 
the following six Lagrangean coordinates: 

X, Y,Z: inertial coordinates of G; 

Eulerian angles of body-fixed axes G-xyz relative to the cotranslating but 
nonrotating axes G-XYZ (similar to the rolling sphere case). 



/' = costj) I + siiuj) J. j' = cos8 m a , + sin8 K. k' = -sin8 u N + cos8 K; 
u N = -snuj) I + cost/) J, u n = i' 


Figure 2.20 Geometry and kinematics of circular disk rolling on fixed rough 
plane. Axes: C-nNZ = C-x’NZ: semifixed; C-nn'z = G-x'y'z 1 : semimobile; 
G-xyz: body axes (not shown, but easily pictured); C-XYZ-. space axes. 

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CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS 


[In view of the complicated geometry, we avoid all ad hoc, possibly shorter, treat¬ 
ments, in favor of a fairly general and uniform approach. An alternative description 
is shown later.] 

The vertical coordinate of G, clearly, satisfies the holonomic constraint 

Z=rsind, (a) 

and this brings the number of independent Lagrangean coordinates down to five'. 
X , Y, <j>, 9, ?/>; that is, n = 5. 

The rolling constraint becomes, successively, 

0 = v c = v G + co x r c / G (to: inertial angular velocity of disk) 

= V G - !'(,) X ) 

= v G — r(u> x ’i' + ujy'j 1 + u z 'k') x j' (semimobile o-decomposition) 

= v G — ru> x ’{i' x j') — ruj z i(k' x j') 

= v G -rUx'{k')-ru) z '(-i), (b) 

from which we obtain the constraint components along the two “natural" (semi¬ 
fixed) directions n(i') and N(u n ): 

(i) 0 = v c -i' = v G •/' + ru> z i or v G] „ + rw,< = 0 ; (cl) 

(ii) 0 = v c • u N = v G ■ u N - ru> x '{k' • u N ) + ru z >(i' ■ u N ) = v G ■ u N - ru x '{k' • u N ), 
or 

v GN — rw x i cos( 7 r /2 + 9) = v GtN + ruj x t sind = 0. (c2) 

The third semifixed direction component gives the earlier constraint (a): 

0 = v c -K = v G -K-ru x '(k' •K) + ru z '{i'-K) = v G • K - rw x >(k' ■ K), 
or, since w x i = ui e , 

0 = t g ,z — rtog cos 9 =>■ dZ — r cos 9 d9 =0 => Z — r sin 9 = constant —> 0. 

Equations (cl, 2) contain nonholonomic velocities. Let us express them in terms of holo¬ 
nomic velocities exclusively. It is not hard to see that, with v G = (X, Y,Z ) = (vx, Vy. vz), 

(i) u> x i = Log, u y ' = (sind)a; 0 , uy = (cos 0 )w^ + uy; (dl) 

(ii) v Gi „ = v G -i' = v G -u„ 

= (y x l + v Y J + v z K) • (cos(j)I + sin </>/) = (cos^)vx + (sin</>)v y ; (d2) 

(iii) Vg,n = v g -u n 

= (v x I + v Y J + v z K) • (— sind>7 + cos 4>J) = (— sind>)vx + (cos^)vy. (d3) 

With the help of (d 13), the constraints (cl, 2) take, respectively, the holonomic 
velocities form: 

(i) v c -i = v c> „ = (cos^)vx + (sin 0 )v y + r(oty + cos 0 c^) = 0 , (el) 

(ii) v c ~u N = v CjA r = (— sin 0 )vx + (coscjfivy + rsinOug = 0 . (e 2 ) 

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§2.13 GENERAL EXAMPLES AND PROBLEMS 


In view of (el, 2), we introduce the following equilibrium quasi velocities: 
Dependent: 

w i = v c,n — v G,n + ruJ z' = (cos</>)vy + (sin<^)vy + (rCOS 0 )cU 0 + (r)u>^ (= 0 ), (fl) 

u> 2 = v c,n = v g,n + rsinduv = (— sin</>)vy + (cos</>)vy + (rsm9)ajg (= 0 ); (f 2 ) 


Independent (semimobile components of o>): 

= W„ = W x ' = Ug (^0), (f3) 

UJ 4 = w„' = Uy> = (sin 6)u}^ 0), (f4) 

W 5 = U) z = LJ Z ' = (cos 9)u>q 1 + Ufy (^ 0). (f5) 

These catastatic, and partially uncoupled, equations invert easily to 

V] = v x = (cos^)wi + (— sin0)(u 2 + (r sin 0 sin </>)w 3 + (— rcos<fi)u) 5 , (gl) 

v 2 = v Y = (sin<^)u;i + (cos^>)w 2 + (—rsin0cos^>)w 3 + (—rsin0)w5, (g2) 

v 3 = (l/sin0)u; 4 , (g3) 

v 4 =uj e = u} 3 , (g4) 

V5 = kty = u 5 - (cot 6)u} 4 . (g5) 


Below, we show that the constraints u>i = 0 and w 2 = 0 are nonholonomic; that is, 
n = 5 global DOF, m = 2 —> / = n — m = 3 local DOF. 

Indeed, by direct d- and ^-operations on (fl g5), and their virtual forms (which 
can be obtained from the above velocity forms in, by now, obvious ways), and 
combination of simple shortcuts with some straightforward algebra, we find, succes¬ 
sively, 

(MiY - < 5^1 = [(Sp Gi „y - <5v G] „] + r[( 66 5 Y - &u 5 ] [where dp Gn = v Gj „<*] 

= ■ • • = {[(cos(/>) + (sin ^>) 6 Y\ — 6[(cos0)vy + (sin0)v y ]} 

T r( uj 4 69 3 — (u 3 69 4 ) 

= • • • = {w 0 [(— sin^) 6 X + (cos 0) 6 Y] — 6 cj)[(— sin0)vy + (cos^)vy]} 

T /' ( UJ 4 (50 3 — CU 3 b0 4 ) 

= (^ Sp G jf - v Gi y <5<)>) + r(uj 4 69 3 - w 3 <50 4 ) [where dp GN = v G A , <fr] 
= [(cu 4 / sin 0) 6p G2 ^ — Vg,n(^® 4 / sin 0)] -t- r(c o 4 69 3 — co 3 <50 4 ) 

= (cu 4 / sin 9)(6p GN + rsin9 69 3 ) — (69 4 / sin0)(v GA r + /-sin0w 3 ) 

= (1 / sin 9) (uj 4 6 p CN - v CjN 69 4 ) [where dp c , N = v CjA r dt\ 

= (1 / sin 9) ( 0 J 4 69 2 — tu 2 69 4 ) = 0 

(after enforcing the constraints 60 2 ,uj 2 = 0), (hi) 

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CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS 

(69 2 ) — 6 u >2 = [{ 8 p Gi n) — ^ v g,n\ + [(rsin#60 3 ) — <S(r sin0(u 3 )] 

[since w 3 sin 9 = uj s sin 9 is integrable, the second bracket term vanishes] 

= [(— sin (j>) 8 X + (cos (j>) 8 Y\ — 6 [(— sin^)v z + (cos</>)vy] 

= — lj^cos cj) 6 X + sin (j> 8 Y) + 6 tj>[( cos(f>)v x + (sin^)vy] 

= 8 p G „ + 8 <j> v Gt „ 

= — (u> 4 / sin0)(^0! — rS9 5 ) + (<50 4 / sin#)(u 7 — ruj 5 ) 

= — ( 1 / sin 0 )(u ; 4 89 x — 07 ^a ) + ( r / sinff)(w 4 89 5 — u 5 89 4 ) 

= (r/ sin 6 )(u >4 89 5 — u > 5 89 4 ) ^ 0 

(even after enforcing the constraints £ 0 1 ,u 7 =O); (h 2 ) 

(80 3 y — Slot, = 0 (independently of constraints) =>- 6> 3 = holonomic coordinate, (h3) 

( 89 4 )' — 8 u >4 = (sin 989)' — 6(sin9ut^) = (cos 9) 8 cj) — 89) 

= (cot 9) (w 3 89 4 - 0 J 4 89 2 ), (h4) 

( 89 $)’ — 8 cj 5 = ( 8 ip + cos 98(f))' — 8 (w^ + cos = (sin 9){u>^ 89 — ujg 8 c/)) 

= 0 J 4 89 2 — cn 3 894 ', (h5) 

and since Z = r sin 9, with uj 6 = v z — r cos 9 ujg => 89 6 = 8 Z — r cos 9 89, we get 
(89 6 y- 8 uj 6 = ( 8 Z - rcos989)'~ 8 (v z — rcos9aj e ) 

= { 8 Zy- r(cos9)'89 — r cos 6(89)' — 8 v z + r(— sin0) 89tog + r cos 9 8 ujg = 0, 

(independently of the other constraints) as expected. 

From the above, we immediately read off the nonvanishing 7 ’s: 

7*24 = 7^ 42 = 1/sin 9- 

7 2 4 i = - 7 2 i 4 = 1/ sin 6 », 7 2 54 = ~ 7 2 45 = r/ sin 6 >; 

7 4 43 = - 7 4 34 = cot0; 

7 5 34 = -7 5 43 = 1- (i) 

Flere, Dependent) = 1,2 and /, I'(independent) = 3,4,5. Therefore, 

7V : 7 2 54 = ''/sin 6 >/ 0 ; (j) 

and so, according to Frobenius’ theorem (§2.12), the system of Pfaffian constraints 
07 = 0 and u > 2 = 0 is nonholonomic. We also notice that to calculate all nonvanishing 
7 ’s, we must refrain from enforcing the constraints u 7 , 89 j = 0 and uj 2 , 89 2 = 0, in 
the earlier bilinear covariants. 

Rolling Constraints via Components along Space Axes 
With reference to tig. 2.20, we have, successively, 

r c / G = — (rcos9)u N — ( rsin9)K = (— rcos9)(— sin (/> / + cos </>/) + (— rsin9)K 

= (r cos 9 sin <j>)I + (—r cos 9 cos tj>)J + (—r sin 9)K, (kl) 


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§2.13 GENERAL EXAMPLES AND PROBLEMS 


o) = oj x I + u Y J + w z lf [recalling formulae in §1.12] 

= [(cos (j))ug + (sin cj) sin 9)to^\l + [(sin (j))u e + (— cos 4> sin d)co^]J 

+[w, i> + (cos 9)u)^,\K, (k 2 ) 

and, of course, 

V'c; = V X I + V Y J + V Z K. (k3) 

Substituting these fixed-axes representations into the constraint (b): 
0 = v c = v G + a) x r c / G , and setting its components along I,J,K, equal to zero, 
we obtain the scalar conditions: 

v x + r( cos (j> cos 9u) z — sin 9 to Y ) 

= v x + r[(cos cj) cos d)co >0 — (sin 0 sin 9) lu b + (cos 4>) u>^\ = 0, 

v y + r(sindo;x + sin</>cosdu; z ) 

= v Y + r[(sin c/)COS 9)u) ( j > + (cos</)sind)u ; 9 + (sin^)u;^] = 0, 

v z — r cos 9 (cos (j)Lu x + sin (j)uj Y ) 

= v z — rcos9ui s = 0 => Z = rsia9 (i.e., holonomic). (k4, 5,6) 

We leave it to the reader to verify that (k4, 5) are equivalent to the earlier (el f2); 
and, also, that they can be brought to the (perhaps simpler) form, 

[{X/r) + sin 0 cos d]' + (cos = 0, [(Y/r) + cos ^ cos 0]' + (sin^)ufy = 0. 

(k7) 


Constraints and Transitivity Equations in Terms of the 
(Inertial) Coordinates of the Contact Point of the Disk (X c , Y c ) 

[This is a popular choice among mechanics authors (e.g., Hamel, 1949, pp. 470 ff., 
478-479; Rosenberg, 1977, pp. 265 ff.) but our choice — that is, in terms of the 
coordinates of the disk center, G — shows more clearly the connection with the 
Eulerian angles.] 

Taking the fixed-axes components of the obvious relation r G = r c + v G / c , and 
then d/dt(. . ^-differentiating them, we obtain (consulting again fig. 2.20, and with 
v c,x — dX c /dt , v C Y = dY c /dt) 

(i) X = X c — (rcos9) sin</> => v x = v CjX — (rcost/)cos9 )u>j, + (rsm(f>sm9)u}g, (11) 

(ii) Y= Y c + (r cos 9) cos (/) => v Y = v c Y - (rsin^cos#)^ — (rcos^sind)o; e ; (12) 

and substituting these v x , v Y expressions into (k4, 5), respectively, we eventually 
obtain the simpler forms 

v cx + (r cos(/>)uty = 0 and v CY + (r sin0)w^ = 0. (13) 

(The above can, also, be obtained by ad hoc knife problem-type considerations.) 

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CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS 


In view of (13), we introduce the following new equilibrium quasi velocities: 


Dependent: 

= T CjX + (rcos^Wy, (=0), (ml) 

W 2 = Vc,Y + ( r sin 4>)u)^ (= 0); (m2) 

Independent: 

w 3 = 0) => q 3 .. = 0, (m3) 

w 4 = w e (^0) => 7 4 .. = 0, (m4) 

w 5 = (cos0)v CiZ +(sin0)v Cj y [= by (13)]; (m5) 


or, instead, the equivalent but simpler, knife-type, quasi velocities: 
Dependent: 


Qi = {- sin4>)v c ,x +{cos(j))v CiY [= 0, by (13)], (nl) 

Q 2 = r + (cos <f>) Vc,x + (sin d 1 ) Vc, r = t + W 5 (= 0 ); (n 2 ) 

Independent: 

0 3 = ( 7 ^ 0 ), (n3) 

Q A = ui A = uj e (y^O), (n4) 

^5 =^ 5 = (cosd)v c ,x +(sind)Tc,r [=-r + Q 2 =-r u$). (n5) 

Inverting the above yields 

vi = v c ,x = (— sin 4>)Q\ + (cos <j))Q 5 , (ol) 

v 2 = v CY = (cos^)^ + (sin</))i 2 5 , (o 2 ) 

v 3 = W 0 = f2 3 , (o3) 

V 4 = UJg = Q 4 , (o4) 

v 5 = = ( l / r )(^2 — ^ 5 )- (° 5 ) 


By direct d/d-dififerentiations of (nl-5), use of (ol-5), and the obvious notation 
d0 k = Q k dt; k= 1,...,5, we obtain the corresponding transitivity equations as 
follows: 

{80 { )' — 8 Q 1 = [(— sind) 8 X C + (cosd>) 8 Y C \ - <5[(— sind>)v cz + ( cos ( t ) ) v c,Y} 

= ■ ■ ■ = cos 4>{ v c,x 8<j> — 8X C ) + sin 4>{vcj 8<f> — w^SYc) 

= cos </>[(— sind>f2[ + cos <j)Q 5 ) 80 3 — f2 3 (— sin c/)80\ + cos cj)80 5 )] 

+ smd>[(cosd>f 2 [ + sind>f2 5 ) 80 3 — ^(cosd’iS©! + sind>h@ 5 )] 

= Q 5 80 3 -Q 3 80 5 , (pi) 


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§2.13 GENERAL EXAMPLES AND PROBLEMS 


(<50 2 ) — <5f2 2 — [r fi'ip T (cos <j>) 8X G -\- (sin (j>) (iF^j — 8\rui^ T (cos (jf^c.x T (sin </>) Vc,r] 
= • • • = sin f(v G x 8f — 8X G ) -t- cos <j>(ca^, 8Y g — v G y 8(f) 

= sin <)>[(— sin (/>£?! + cos </> 0 5 ) <50 3 — f2 3 (— sin</>5@i + cos <j><50 5 )] 

+ cosc/)[Qt,(cos(/)6&\ + sin f8Qf) — (cos^Oi + sin^0 5 ) 60 3 ] 


— f2 3 60\ — C2\ <50 3 

(= 0, upon imposition of the constraints <50,, Q x = 0), (p2) 

(<50 3 )‘ — 6Q 3 = (8(f)' — 8ui^ = 0 (0 3 = holonomic), (p3) 

(<50 4 )' — 8Q 4 = (66)' — 8 uig = 0 (0 4 = holonomic) , (p4) 

(<50 5 )’ — <5f2 5 = [(cos</>) <LF C + (sin0) <5F C ]' - 5[(cos^)v C x + (sin<j>)v C y] 

= (-r&tpy - «(-r^) + (Se 2 y-SQ 2 = O+Q 3 80 X -Q x 80 3 

— Qj 80 x — Q\ <50 3 

(= 0, upon imposition of the constraints ^0 I ,f2 1 =O); (p5) 


that is, just like the knife problem (ex. 2.13.2), all the 7 ’s are either ±1 or 0. 
Finally, here, Dependent ) = 1,2 and /, I'independent) = 3,4,5. Therefore, 

l D n' ■ 735 = -753 = 1 ^0; (j) 

and so, by Frobenius’ theorem (§2.12), the constraint system Q x = 0 and f2 2 = 0 is 
nonholonomic. 

Problem 2.13.3 Rolling Disk in Accelerating Plane. Continuing from the 
preceding example, show that if the plane translates (i.e., no rotation), relative to 
inertial space, with a given velocity ( v x (t ), Vy(t), v z (t)), and the new inertial axes 
O-XYZ are chosen so that OZ is always perpendicular to the translating plane, 
and X, Y, Z are the new inertial coordinates of the center of the disk G, then the 
rolling constraints take the rheonomic form 

(cos<£)[Fy - v x (t)\ + (sin <j>)[V Y - v Y (t)\ + (rcos0)w 0 + (r)u^ = 0, (a) 

(— sin 4>)[V x - v x (t)\ + (cos<^)[F y - v Y (t)\ + (/• sin 6 ») cd 0 = 0 , (b) 

where V x = dX/dt, V Y = dY/dt; = d<j>/dt, and so on; that is, they are the same as 
in the fixed plane case, but with v G replaced with v G — v c (where v c is the inertial 
velocity of contact point of disk with plane). 

Example 2.13.8 Pair of Rolling Wheels on an Axle. Let us discuss the kinematics 
of a pair of two thin identical wheels, each of radius r, connected by a light axle 
and able to turn freely about its ends (tig. 2 . 21 ), rolling on a fixed, horizontal, and 
rough plane. For its description, we choose the following (six —>) five Lagrangean 
coordinates: 

(X, Y,Z = r): inertial coordinates of midpoint of axle, G; 

f: angle between the O-XY projection of the axle (say, from G" toward G') and 
+OX; 

ifi', ill"', spin angles of the two wheels. 

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CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS 



Figure 2.21 Rolling of two wheels on an axle, on fixed plane. 


Here, the constraints are v c > = 0 and v c „ = 0, where C' and C" are the contact 
points of the two wheels. However, due to the constancy of G"G' (and 
C"C' = 2b) and the continuous perpendicularity of the wheels to the axle, these 
conditions translate to three independent component equations, not four; say, the 
vanishing of v c » and v c » along and perpendicularly to the axle (the “natural” 
directions of the problem). Let us express this analytically: since 


V C' — V G' + °V x r C'/G' — ( V G + 0>A x r G'/G ) + °V x r C'/G' 

\(D W ' and m A : inertial angular velocities of first wheel and axle, respectively] 

= (vx, vy, 0) + (0,0, ty,) x ( b cos 4>, b sin </>, 0) + (ay cos <£, ay sin <j>, w 0 ) x (0,0, -r) 
= (v x — b a^ sin (j) — r ay sin 0, v Y + b cos (j> + r ui y cos <j>, 0); (a) 

and similarly, for the second wheel [whose inertial angular velocity is co w » = 
(ty,» cos </>, uy< sin u^)], 

v C " = (vy + buifiSmcj) — ray/ sin </>, v Y — bto^ cos(/) + ray/ cos (j), 0); (b) 

the constraints are (with u ... for unit vector): 

0 = v c > = v c / • u„ = v c , • (cos 4 >, sin <j>, 0) = v c> = v c » • u n = v c „ • (cos <j>, sin qb, 0), 

or 

v C ',n = v C",n = V_yCOS(/) + v y sin<() = 0; (cl) 

and 

v C ',t = v c' • u t = v c' • (— sin 0, cos 4 >, 0) = — v x sin< j>+ v Y cos(/) + bco ( / ) + ray = 0, 

(c2) 

v C ",t = v C " ■ u, = v c » • (— sin qb, cos cf), 0) = — v x sin 4> + v Y cos <f> — bui^ + ry» = 0. 

(C3) 


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§2.13 GENERAL EXAMPLES AND PROBLEMS 


[The above can also be obtained by simple geometrical considerations based on fig. 

2 . 21 .] 

By inspection, we see that (c2, 3) yield the integrable combination 

2buj^ + r(uj^ - = 0 =>■ 2b 4> = c - r(ip' - ip"): 

[c = integration constant, depending on the initial values of ft"]. (d) 

Hence, we may take X, Y, ip', ip", as the minimal Lagrangean coordinates of our 
system, subject to the two knife-like nonholonomic (to be shown below) constraints 

v C',n = v C",n = v x cos </> + v y sin </> = 0; (el) 

Vc',t = -Vysin^-I- vy cos (j) T (r/2)(uy + uy<) = 0; (e2) 

that is, n = 4, m = 2 =>■ f = n — m = 4 — 2 = 2 DOF in the small , and 4 DOF in the 
large. 

In view of (el, 2), we introduce the following equilibrium quasi velocities: 


uq = (cos(j>)vx + (sin^)vy (=0); (fl) 

w 2 = (-sin</>)v A - + (cos^)vy (^ 0); (f2) 

(^0); (f3) 

uq = 2(—Vjy sin cf> + Vy cos </>) + r{ut^i + uy,») (= 0); (f4) 

w 5 = 2 bto^ + r (— uty») (= 0; tu 5 = holonomic velocity); (f5) 

which invert easily to 

v x = (cos (f))iOx + (— sin <p)ui 2 , (f6) 

Vy = (sin0)uq + (cos</>)w 2 , (f7) 

w 4 , = (0)uq + ( 0 )(jJ 2 + ( 1 )^ 3 , (f8) 

= (l/2r) (—2cu 2 — 2r u> 2 + uq + uq), (f9) 

= (1 /2r)(—2 w 2 -f- 2ruq -T uq. — uq). (flO) 


Comparing the above with the quasi velocities of the knife problem (ex. 2.13.2), to 
be denoted in this example by lo k , we readily see that we have the following corre¬ 
spondences: 

K K K / y, 1 \ 

CDi —S- CD 2-> w 2 —> 1, Uq —> U> 3. (ill) 

Hence, and recalling the transitivity equations of that example, we find 

{86 — <Suq = {89 k 2 )' — 8uj k 2 = lo k 2 86 k x - u> K x 89 K 3 = uq 86 2 — uq 89 2 (^ 0), (gl) 

(86 2 )' — 8uj 2 = {89 k [)' — 8uj K \ = lo k 2 89 k 2 — u/q 89 k 2 = uq 89 2 — uq 89\ {=/= 0), 

(= 0, after enforcement of the constraints 89 \, uq = 0), (g2) 

{80^)' — 8oj 2 = 0 ( 0 3 = (j) = holonomic coordinate), (g3) 

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CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS 


( 504 ) - 5w 4 — 2[(502) — boj 2 \ + r\(8i/j + 6 i/j ) — 5(afy' T ^ 5 //')] 

= 2(uq50 3 — w 3 50[) + 0 (= 0, after enforcing 50! ,04 = 0), (g4) 

(50 5 ) ' - 5w 5 = 2b[{8cj)y - 5w 0 ] + r[(5V>' - 5w")' - 5(uy - uy,)] 

[=0 =>- 05 = 2 bcf> + r(ip' — ip") = holonomic coordinate], (g5) 

The above immediately show that the nonvanishing 7 ’s equal ±1, as in the knife 
problem; and since here D = 1,4; /, I' = 2,3 and 

IV '■ 7*23 = — 7*32 = 1 7^ Oj (h) 

the system of Pfaffian constraints uj x = 0 and w 4 = 0 is nonholonomic. 

For additional wheeled vehicle applications, see also Lobas (1986), Lur’e (1968, 
pp. 27-31), Mei (1985, pp. 35-36, 168-175, 437^139), Stuckler (1955—excellent 
treatment). 


Example 2.13.9 Transitivity Equations for a Rigid Body in General (Uncon- 
trained) Motion. As explained in §1.8 IT., to describe the general spatial motion of 
a rigid body B we employ, among others, the following two sets of rectangular 
Cartesian axes [and associated orthogonal—normalized-dextral (OND) bases]: (i) a 
body-fixed set ♦ xyzj* ijk (noninertial), where ♦ is a generic body point (pole); 
and (ii) a space-fixed one O-XYZ/O IJK (inertial), where O is a generic fixed 
origin. Frequently (recalling §1.17, “A Comprehensive Example: The Rolling 
Coin”), we also use other “intermediate” axes/bases that are neither space- nor 
body-fixed: +-x'y'z'/+-i'j'k'\ for example, axes + XYZ translating, or 
comoving, with B but nonrotating (i.e., ever parallel to O-XYZ). 

Let us examine the transitivity equations associated with the translation of B with 
pole ♦, and its rotation about ♦ (earliest systematic treatment in Kirchhofif, 1883, 
pp. 56-59). 

(i) Rotation. As shown in §1.12, the transformation relations among the spatial 
and body components of the inertial angular velocity of B , a>, and the Eulerian 
angles (and their rates) between ♦ -xyz and O-XYZ (or ♦ XYZ) are [with 
s... = sin..., c... = cos...] as follows: 

Body axes components (assuming sin0 0): 

lo x = (s6 sip)^ + (cip)u e , uj y = (sOcip)^ + (~sip)uj e , u z = (c9)^ + oty; (al) 

=> u<t, = {sip/s9)u x + (ctp/s0)u)y, ivg = (cip)u x + (~sip)u y , 

uty = (— cot 9s(j>)io x + (— cot 0 np)u> y + u> z . (a 2 ) 

Space axes components (assuming sin 0^0): 

U X = {cf)uj e + (sc/)s9)u)^ Wy = WH + (-c 0 j 0 )w^ UJ Z = w# + (c 0 )u^; (bl) 

= r > 1 jj 0 — ( — cot 0 S(p)ui z T (cot 0 cc/djcoy T tog — (ccp)cu % T (sfijcoy , 

uty = ( s(p/s9)u x + {—c(j)/s9)io Y . (b 2 ) 

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§2.13 GENERAL EXAMPLES AND PROBLEMS 


Since these transformations are stationary, they also hold with the oj xy ,, z -x,y,z 
replaced by the d9 xyrXY .7 or 69 xyz . X j^z and the replaced by or 

6(j >,..., respectively. 


Transitivity Equations 

(a) Body axes: Differentiating/varying the first of (al), while invoking the 
“d<5 = 6d ” rule for cf>, 9, ip, we obtain, successively, 

(69 x )' — 6u> x = [(s9sip) 6(f> + (cip) 66\ — 6[(s9si/))+ (cip)u>g\ 

= c9 sip(u>g 6<j> — 69) + s9 cip(uj^ 6cf> — 6tp) + sip(uig 6ip — 69 ), 

and substituting u>^,... /6(j ),... in terms of u x ,... /69 x ,..., from (a 2 ), we eventually 
hnd 

(69 x )' — 6 uj x = ui z 69 r — ui y 69 z , (cl) 

and similarly, for the other two, 

(69 v )' — 6u> y = oj x 69 _ — uj, 69 x , (c2) 

(69 z )' — 6 lo z = ui y 69 x — uj x 69y. (c3) 

Hence, the nonvanishing 7 ’s are 

l X yz = -7 \y = 1, I y zx = “7 V xz = 1, l\y = “7 %x = lj ( d l) 

or, compactly [with k,r,s —> x,y,z : 1 , 2 ,3], 

7 \s = £ krs =(k-r)(r- s)(s - k)/2 

= ±1, according as k,r,s/x,y,z are an even or odd permutation of 1,2,3; 

and = 0 in all other cases: Levi-Civitapermutation symbol (1.1.6 ff.). (d2) 

(b) Space axes: Applying similar steps to (bl, 2), we eventually obtain 


(69 x ) — 6 ujx — coy 69 x — uiz 69y , (el) 

and similarly, for the other two, 

(69 Y )' — 6 lo y = ui z 69 x — ui x 69 z , (e 2 ) 

(69 z ) — 6 coz = 69 y — cuy 69 x ■ (c3) 

Hence, the nonvanishing 7 ’s are 

X X , Y Y , Z Z , / ri N 

7 yz — ~7 zy — ~ 1) 1 zx — 7 xz — 1) 7 xy — ~7 yx — ~ b (11) 

or, compactly (with k , r, s —> X, Y, Z: 1,2,3), 

7 K rs = ~ £ krs = £ rks ■ ( f 2 ) 


The above show clearly that the orthogonal components of m, aj xyz and u x ,y,z are 
nonholonomic; while the nonorthogonal components are holonomic (and, 

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CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS 


again, this has nothing to do with constraints, but is a mathematical consequence of 
the noncommutativity of rigid rotations). 

REMARK 

These transitivity relations and 7 -values, (cl-f2), are independent of the particular 
oJ x _ y ^x,Y,z ^ relationships (al-b 2 ); they express, in component form, in¬ 

variant noncommutativity properties between the differentials of the vectors of 
infinitesimal rotation and angular velocity. A direct vectorial proof of these proper¬ 
ties is presented in the next example. 

(c) Intermediate axes: Such sets are the following axes of ex. 2.13.7: (i) G-nn'z = 
G-x'y'z' , with OND basis G i'j'k' = G u n ii n k; and (ii) G-nNZ , with OND basis 
G-u n u N K (. u„ = i' = unit vector along + nodal line); and they are called by some 
authors semimobile ( SM ) and semifixed ( SF ), respectively. 

Below, we collect some kinematical data pertinent to them. Since their inertial 


angular velocities are (consult fig. 2.20) 

vi^M = vi K T luq i — u)q i T (sin O j T cos 0 k ) 

= (u e )i' + sin 9)j' + (w 0 cos0)*' 

= UJ,pK + UJg U„ = V) SF + U)g U n = Vi" + Vig H„, (gl) 

0 ) SF = Vi" = K, (g2) 

we will have the following relations for the rates of change of their bases: 

du n /dt = a' x u„ = u)<pu N = uj^cosdu,/ — sin6 k')-, (g3) 

du n t/dt = a' x «„/ = (—u^cosd)w„ + ojgk'-, (g4) 

dk'/dt = a' x k' = (a^sin 6 )u„ — cogu n r, (g5) 

dii„/dl = a" x */„ = u} 0 ti N ; (g6) 

du N /dt = a" x u N = — (g7) 

dK/dt = ca" xK = 0 . (g8) 


Finally, the body angular velocity along the SM axes, thanks to the second line of 
(gl), equals 


VI = Vi' + u^k' = ( ujg)i' + (cu^sind)/ + (u>^ + o^cos 0)k' 

= w x < 1 + Uyj' + ay k (g9) 

and since this is a scleronomic system, (g9) holds with uyy z ', replaced with d0 x ' y < z < 
and 66 x 'y z i ; and replaced with d</>, hd, t/i/' and 6</),69,6iIj, respectively. 

From the above, by straightforward differentiations, we obtain the rotational 
transitivity equations in terms of semimobile components'. 

( 69 x ')' — 6u> x i = 0 (9 x i = 9 = holonomic coordinate => 7* = 0), 

(69 y t)' — 6u) v i = cot 9(u) x ' 69 v ' — ay 69 x i), 

( 69 : i )' — 6u > z 1 = (ay 69 x i — uj x i 69 r '); 


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(hi) 

(h 2 ) 

(h3) 


§2.13 GENERAL EXAMPLES AND PROBLEMS 


and hence the nonvanishing 7 ’s are (assuming cot 0 = finite) 


= cot 9, 


7 x'y' = 


-7 'y'x' = 1 - 


(h4) 


Note that (hl^t) are none other than (h3-5) and (i) of ex. 2.13.7. 

(ii) Translation of pole (or basepoint) ♦. Let us assume that ♦ has inertial 
position: 

OP = p = Px I + Py J + Pz^i (h5) 


and, therefore, inertial velocity: 


v = dp/dt 

= ( dpx/dt)I + ( dp Y /dt)J + ( dpz/dt)K = v x I + v Y J + v zK 
[along space- axes: v x = dp x /dt = v • 7, etc.; 

Px,Y,ziyx,Y,z)'- holonomic coordinates (velocities) of ♦.] 

= v x i + v y j + v z k = ( dp x /dt)i + ( dp y /dt)j + ( dp z /dt)k 
[along body- axes: v x = dp x /dt = v • i, etc.; 

Px, y ,z( v x,j,z): nonholonomic coordinates (velocities) of ♦.] (h 6 ) 

Clearly, the above velocity components are related by the following vector transfor¬ 
mations: 

V x = COS(x, X) V X + COS(x, Y) Vy + cos(x, Z)v z , etc. 

v x = cos(X, x)y,. + cos(3f,y)v > , + cos(X,z)v,, etc., (h7) 

and, since this is a scleronomic system, their differentials are related by 


dp x = v x dt = (v • /) dt = dp • i = cos(x, X) dp x + cos(x, Y) dp Y + cos(x, Z) dp z , etc, 
Sp x = bp • i = cos(x, X) Sp x + cos(x, Y) bp Y + cos(x, Z) 6 p z , etc. (h8) 

Next, since 

di = dQ x i, Si = 56 x i etc., (il) 

where dO = a> dt = d9 x i + dO y j + dO z k = dfK + dO u n + dfik = elementary ( inertial) 

kinematically admissible/possible rotation vector , hence SO = 59 x i + 59 v j + 59- k = 
5(j)K + 59 u„ + 5ipk = ( inertial) virtual rotation vector, we find by direct calculation [with 
(...)' = inertial rate of change, for vectors] 

{SPxY = ( Sp-i)' = {Spy-i + 6p- (/)■ = {Spy -i + 5p-{(0 x 1 ), 

5{dp x /dt) = 5v x = 5{v • i) = 5v ■ i + v • Si = 5v • 1 + v • {50 x i), (i2) 

and subtracting the above side by side, while noting that {Sp)' — Sv = 

5{dp/dt) — Sv = Sv — Sv = 0, we obtain the x-component of the pole velocity transi¬ 
tivity equation : 

(5p x )' — 5v x = 5p • {a x i) — v • {50 x i) = (5p x 0 ) • i — (v x 50) ■ i 
= (5p x a) — v x 50) • i; 


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( 13 ) 


CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS 


and similarly for its y and z components. Hence, our pole transitivity equation can be 
written in the following vector form [with d (...) and S re i (...) denoting differentials of 
vectors, and so on, relative to the moving, here body- fixed, axes]: 

d(8p)/dt — 5 re i(dp/dt) = 5p x to v x 80, (i4a) 

or 

Sreiv = d(8p)/dt + co x Sp+v x 80. (i4b) 


In component form, along ♦-xyz, (i4a) reads 




(5p x y - 

8v x = (u z 5p y 

~u y 

8Pz) - (Vy 

86 z 

- v z 86y), 

(i5) 



( 5p y y - 

8v v = (u} x 8p z 

-L0 : 

8p x ) - (V- 

80 x 

— Tv 56 z ), 

(i 6 ) 



(8p z y - 

II 

~u x 

8p y ) - (v„ 

56 y 

- v y 8o x y, 

(i7) 

and, therefore. 

the nonvanishing 7 ’s are [with accented (unaccented) indices for the 

components 8p, 


)] 








7 X y'z = - 

-7 V = 1 

and 

7 V 

= - 

-*/ , = 1 

1 z y A 5 

(18) 



7 * z'x = - 

'7 V = 1 

and 

I y zx' 

= - 

- 7 ^ = 1 , 

(i9) 



7 Z 'x'y = - 

-7 Z yx' = 1 

and 

t 

1 xy' 

= - 

Z 1 

-7 y'z = 1- 

(ilO) 

Semifixed 

axes 

+-u„u n K. Here, we have 





v« = 

iPnY 

= *’ • = 

■S . 

II 

cos^ 

’ + (p Y )'sm4, 


(Jl) 

v N = 

E (Pn) 

)' = V-U N 

= ~{px)' sin 

4 +(p Y y cos 4 , 



(J2) 

v z = 

E iPz) 

= v-K 

= (PzY => 7 

Z 

0 (i.e., v z = 

holonomic velocity). 

(j3) 


We leave it to the reader to show that (recalling the earlier semimobile axes kine¬ 
matics) 

(6p„Y - 6v„ = — [aty 8p x - {p x )'S(f>\ sin 4 + 8p Y - (prY 84} cos 4 

= ^ 6p N - v N 84 = (1 / sin 4) (u y Sp N - v N 86 ,,), (j4) 

(Spn) ' - Sv n = -[^0 8p x - (px)' 84) cos 4+ [(p Y )' 84 - Sp Y ) sin 4 

= -Up8p n + (p n y 54= (1/sin 4)[{p„y 86 y - Wy 8p„\; (j5) 

and hence that the nonvanishing 7’s are (with some, easily understood, ad hoc 
notation; and assuming that sin 8 ^ 0) 

7 n Ny = -7 n yN = V sin 6, j N yn = -7 N ny = 1/ sin 6. (j6) 

[Recalling ex. 2.13.7 (rolling disk problem), eqs. (d2, 3), (hi, 2), (i), etc.] 

For related discussions of the rigid-body transitivity equations, see also Bremer 
(1988(b)) and Moiseyev and Rumyantsev (1968, pp. 7-8). 

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§2.13 GENERAL EXAMPLES AND PROBLEMS 


Example 2.13.10 Cardanian Suspension of a Gyroscope. Let us consider a 
gyroscope suspended a le Cardan (fig. 2.22). The rotation sequence 

cp = (^[precession ) —> q 2 = 9(nutation ) —> cp = f(spin) 

(i.e., 3 —> 2 —> 1, in the Eulerian angle sense of §1.12) brings the original axes 
G-XYZ, through the intermediate position G-x'y'z' (outer gimbal), to the also 
intermediate position G-xyz (inner gimbal). 

Now: (i) The inertial angular velocity of the outer gimbal co 0 , along outer gimbal- 
fixed axes , is 

U)o,x' = o, tx>o,y' = 0, <x> 0 ,z' = (a) 

(ii) the inertial angular velocity of the inner gimbal coj, along inner gimbal-fixed axes, 
is 

Vi,x = ~^0 sin 6 , U) hy = uj Si w /iZ = cos 6 ; (b) 

and (iii) the inertial angular velocity of the gyroscope co, along inner gimbal-fixed 
axes, is 

lo x = — (jOj, sin 6 , uiy = ug, co z = u )0 cos 9. (c) 

Let us find the transitivity equations corresponding to these quasi velocities. 
Equations (c) can be rewritten as 

pfi = p-h = ( — sin 9 ) 10 ^ + (0)wg + (l)u )0 [f 0), (d) 

uj 2 = oj y = (O)lo 0 + (\)uj s + (O)u >0 [f 0), (e) 

P>3 = P4 = ( cos 9)^0 + (0)Wfl + (0)aty (^0); (f) 




Figure 2.22 Kinematics of Cardanian suspension of a gyroscope. 

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CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS 


and their inverses are readily found to be 

vi = ufy = (0)04 + (0)^ + (l/cos0)w z , (g) 

v 2 = io g = ( 0 )w Y + (1 )u> y + ( 0 )c j z , (h) 

v 3 = = (1 )uj x + (0)w v + (sin 9/ cos 8 )u> z . (i) 

From these stationary relations, and assuming d(8cp < ) = S(dq k ) (k = x,y, r), we 

obtain, successively, 

(i) d(S8 x ) - S(d8 x ) = d[(- sin 9) 8<j> + Sip] - <5[(- sin 8 ) dcp + dip] 

= • • • = (cos 9)(dcp89 — ddScp ) = • • • = d8 z 68 y — dd y 68 : ; (j) 

(ii) d(89 y ) — 8(d9 y ) = 0 (=>■ 9 y = holonomic coordinate); (k) 

(iii) d(S9 z ) — S(dd z ) = h[(cos 9) Sep] — <5[(cos 9) dcp] 

= ■ ■ ■ = (sin 9) {dcp 89 - d9 Sep) = ■ ■ ■ = (tan 9) ( d8,88 y - d9 y 68 ,); (1) 

and so the nonvanishing 7 ’s are (assuming 8 ^ ± ir/2) 

l x yz = -1% = 7 23 = -7 ‘32 = +1, (m) 

7 % == 7 3 23 = -7 3 32 = tan 6 ». (n) 

Example 2.IB. 11 An Elementary ad hoc Vectorial Derivation of the Rotational 
Rigid-Body Transitivity Equations. Let us consider, with no loss of generality, a 
free rigid body B rotating with (inertial) angular velocity m about a fixed point O. 
Then, as is well known (§1.9 ff.), and since this is an internally scleronomic system, 
the (inertial) velocity/kinematically admissible displacements/virtual displacements 
of a typical 5-particle of (inertial) position vector r, are, respectively, 

v = uj x r => dr = dO x r, 5r = SO x r, (a) 

where dO = ut dt, and d(...)/5{...) are kinematically admissible/virtual ( inertial ) var¬ 
iation operators. Now, d(.. .)-varying the last of (a), S(.. .)-varying the second, and 
then subtracting the results side by side, while invoking (a) and the rule 
d(8r) — 8(dr) = 0, we obtain, successively, 

0 = d[8r) — 5(dr) = [d[8Q) x r + SO x dr] — ]5(d0) x r + dO x Sr] 

= [d(S0) x r+ SO x (dO x /■)] — [S(dO) x r + dO x (SO x /•)] 

= [d(50) — S(dO)] x r + [SO x (dO x r) — dO x (SO x r)] 

[and applying to the second bracket (last two triple cross-products) the cyclic vector 
identity, holding for any three vectors a,b,c: ax(bxc) + bx(cxa) + 
c x (a x b) = 0, with the identifications: a —» SO, b —> dO , c —> r] 

= ]d(50) — S(d0)] x r + (SO x dO) x r, 

from which, since r is arbitrary, we finally get the fundamental and general inertial 
rotational transitivity equation: 

d(50) — S(d0) = dO x SO. (b) 

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§2.13 GENERAL EXAMPLES AND PROBLEMS 


Dividing the above with dt, which does no interact with these differentials (and 
noting that, by Newtonian relativity, dt = dt), we also obtain the equivalent tran¬ 
sitivity equation in terms of the angular velocities: 

d(50)/dt — S(o = (oxSO. (c) 

Next, let us find the counterparts of (b, c) in terms of relative differentials/variations, 
i.e. relative to moving (here body-fixed) axes, to be denoted by d(... )/S re i( .. .). Applying 
the well-known kinematical operator identities [(§1.7 ff.)] 

d(...) = d(...)+ddx (...), 5(...) = 6 rel (...) + 60 x (...), (d) 

(which immediately yield dr = 0 and S re i r = 0, as expected) to 56 and d0, respec¬ 
tively, and then substracting side by side, we find, successively, 

d(56) — 5(dd ) = 8(58 ) — 5 re i(dd) + ( d0 x 56 — 56 x dd ) 

= 8(56) — 5 re i{dd) + 2(dd x 56), (e) 

or, invoking (b) for its left side and rearranging slightly, we get, finally, 

8(56) — 5 re i(dd) = 56 x d6; (f) 

and dividing by dt, we also obtain its velocity equivalent, 

d(58)/dt — 5 re ico = 58 x a). (g) 

The kinematical identities (f, g) are the noninertial counterparts of (b, c). 

The difference between (b, c) and (f, g) often goes unnoticed in the literature. 
To understand it better, let us write them down in component form, along space- 
fixed axes ♦ - A YZ and body-fixed axes ♦-xvz. Only the first equations are shown 
(i.e., X, x); the rest follow cyclically: 

Space-fixed (inertial) axes: 

d(56 x ) — 6(dO x ) = dOy 50/ — d0/ 50y . or ( 50 x ) — 5lo x = ccy 56/ — io/ 50y : 

(hi) 

Body-fixed (noninertial) axes: 

d(56 x ) — 6(dd x ) = d0 : 59 v — dd v 66 z , or (58 x )' — 5ui x = w z 56 — ai y 69 z ; (h2) 

which, naturally, coincide with equations (cl-f2) of ex. 2.13.9, and §1.14. 

[When dealing with derivatives/differentials of components, we may safely use the 
same notation (...)' /d(.. .)/6 (...) for both space and body such changes; here, the 
intended meaning is conveyed unambiguously]. 


Additional Special Results 

(i) Applying the second of (d) for to, and then equating the resulting fou-expres- 
sion with that obtained from (c), we get d(56)/dt — a> x 56 = <5 re /tu +56 x a), or, 
simplifying, d(58)/dt = 5 re im', or, equivalently (multiplying with dt), 

d(50) = 5 re i(dd). (i) 

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CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS 

(ii) Starting from (c), and then invoking the first of (d), we obtain, successively, 

8a> = d{8Q)/dt — a) x 56 = [d(56)/dt+co x 56} — oj x 56 
= d(86)/dt [= 5 re i(o + 89 x <u, by (g)]; 

that is, 

5(o = d(58)/dt, or, equivalently, 5(dd) = d(56)\ (j) 

which is “symmetrical" to (i). 

(iii ) Applying the first of (d) for <u yields 

dm=d(o, or, equivalently, d(dd) = d(dO); (k) 

but the second of (d) shows that 

5w ^ 5rei(o, or, equivalently, 5{d6) ^ 5 re i(dd). (1) 

Problem 2.13.4 Rigid-body Transitivity Equations. Using the results of the 
preceding example and its notations, show that, for a rigid body rotating about a 
fixed point, 

d(8r) — 5 re i(dr) = (56 x dO) xr / 0, (a) 

or, equivalently (dividing by dt = dt), 

d(5r)/dt — 8 re iv = (86 x m) x /• ^ 0; (b) 

even though d(8r) — 8(dr) = 0; that is, the rule d(6.. .) = 6(d.. .) is not frame- 
invariant] 


Example 2.13.12 A Special Rigid-Body Transitivity Equation — Holonomic Coordi¬ 
nates. Continuing from the above examples, we show below that, for a rigid body 
rotating about a fixed point, the following transitivity/nonintegrability identity 
holds: 


E k ((o ) = d/dt(d(o/dv k ) — dco/dq k = m x (d(o/dv k ). (a) 

For such a system (with k = 1,2,3; and q k = angular Lagrangean coordinates', e.g., 
Eulerian angles <j>, 9, ip) we will have 

(o = m(q k , dq k /dt = v k ) = c o(q , v) 

= linear and (for our system, also) homogeneous function of the v^-’s 
= 'y^ (da>/dv k )v k (by Euler’s homogeneous function theorem) = E CkVk, 

(b) 

[definition of the c k s; also, recalling (1.7.9a, b)] from which it follows that 

dd = oj dt = c k dq k . 


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§2.1 3 GENERAL EXAMPLES AND PROBLEMS 


and since this is a scleronomic system 

se-'EckSq* ( c ) 

and so the basis (quasi) vectors c k = dco/dv k (independent of the v k s) can also be 
defined symbolically by 

Ck = 90/dq k = d{dO)/d(dq k ) = d{50)/d(6q k ). (d) 

Now, let us substitute the above representations into the earlier (inertial) transitivity 
equation 

d(86)/dt — 8ct) = oj x 86. (e) 

We find, successively, 

(i) Left side [we assume that (8q)‘ = 5(dq/dt) = <5v]: 

d{86)/dt — 8(o = d/dt(^^{d(o/dv k )8q^j — ^ [{dm/dq k )8q k + (dm /dv k )8v k ] 
= • - = [d/dt{d(o/dv k ) - dco/dq k \8q k = ^ E k {m) 8q k . (f) 


(ii) Right side: 

ai x 86 = tt) x ( y^(dco/dv k )8q k ) = ^ [to x (dco/dv k )\8q k , (g) 

and therefore (since the 8q k are independent—but even if they were constrained that 
would only affect the equations of motion) equating the right sides of (f) and (g), the 
identity (a) follows. 

In terms of the earlier c k vectors, (a) reads 

dc k /dt = to x c k + dco/dq k = ^ (c/ x + dc,/dq k )v,. (h) 

Finally, applying the first of (d) of ex. 2.13.11 to dto/dv k , and inserting the result 
into (a, h) produces the following interesting result: 

E k ,rei{(o) = d/dt{dm/dv k ) — doj/dq k = 0 or dc k /dt = dco/dq k . (i) 


Problem 2.13.5 Using the well-known kinematical result 


du k /dt = a) x u k , 


(a) 


where {u k = u k {q)} is, say, a body-fixed basis rotating with inertial angular velocity <u 
(like the earlier i. j, A'), with the co-representation (b) of the preceding example: 

(D = co{q k , v*) = co(ry, v) = ^ c k v k , (b) 

show that 


du k /dqi = Ci x u k [note subscript order], 


(c) 


i.e., (1.7.9c). Clearly, such a result holds for any vector h = h{q) rotating with angu¬ 
lar velocity co: db/dqi = c/ x h = {da>/dv k ) x h. Also: (i) d / dt{db / dq{) = 
d/dqfdb/dt ), and (ii) db/dvi = 0. 


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CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS 


Problem 2.IB.6 By direct substitution of the representations 

(16 = mdt = Y^ Ck dq k and 89 = Y^ c k 5q k (a) 

into the earlier inertial rotational transitivity equation [ex. 2.13.11: eq. (b)]. 

d{59) - 5{d9) =d0 x 59, (b) 

and some simple differentiations, show that 

dc k /dq, - dc,/dq k = c, x c k . (c) 

This nonintegrability relation shows clearly that the basis {c^} is nonholonomic 
(nongradient); whereas if c k = 89/8q k , then (1(86) = 5{d9) =>- 9 = genuine angular 
coordinate. Simplify (c) if the {c/J are an orthogonal-unit-dextral basis (see also 
Brunk, 1981). 


Example 2.13.13 A Special Rigid-Body Transitivity Equation — Nonholonomic 
Coordinates. Continuing from ex. 2.13.11, let us substitute the (fully non¬ 
holonomic) representations 

(o = y^(da>/duk)wk = 'Y, E k u} k = co(q,u)), d9 = ^^E k d9 k , 59 = s k 59 k , 

(a) 


where, as usual, 9 k = quasi coordinates, u> k = d9 k /dt = quasi velocities, and 
E k = dca/dcok = 8{d9) / 8{dO k ) = 8{59)/8{59 k ) = 89/89 k : nonholonomic basis, (b) 
into the fundamental inertial rotational transitivity equation 

d{59)/dt — So = co x 59. (c) 


We find, successively, 
(i) Left side: 


d(59)/dt— 5co\( 8 (o/ 8 uj k )'59 k + { 8 (o / 8(o k ){59 k )'] 

- Y [{ 8 ( 0 / 8 q k ) 5q k + ( 8 (o/ 8 uj k ) 8 oj k ] 

[and setting 8 q k = E A k i 89/ = ( 8 v k / 8 iO /) 89/ (definition of the A k /)\ 

= Y [( do) / duJ k :)'-Y A ik{ d< °/ dc li) Sd k +Y {dco/8uji)[{89,y-8 uj,) 

[recalling the 8 ... /89 k definition (2.9.30a); and setting (as in pr. 2.10.5) 

{89/)' — 5lo/ = Y; h > k89 k (definition of the h' k )] 

= Y [{ d( °/ d ^k )' ~ 8(o/89 k ] 89 k + YJ2 {8(o/dui)h l k 89 k 

= Y + E h l k ( 8 (o/dui)^ 89 k . (d) 


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§2.13 GENERAL EXAMPLES AND PROBLEMS 


(ii) Right side: 

a, x 59 = a) x f y ^(d(o/duk)b6k) = Yj [to x (do/dw*)] (e) 

and, therefore, equating the right sides of (d) and (e), we obtain the identity 

d / dt(da) / dcj k ) — dm/dd k + ^ h l k (da)/dui/) = ®x (dco/dui k ), (f) 

or, in terms of the quasi vectors s k = e k (q), 

ds k /dt — dco/dO k = cj x ( dco/duj k ) — Yj h l k E/. (g) 

Finally, since d£ k /dt = dE k /dt + a> x s k , (g) takes the body-axes form: 

ds k /dt— dm/d9 k = d/dt(d(o/d6 k ) — dco/d9 k =E k * rel (m) = — Y^ h l k £i, (h) 
which is a special case of the transitivity equation (2.10.25). 

[Here too, we point out the differences between the notation-. 

do)(q, u>)/dd, = Y [dto(q, uf)/dq k ] (dv k /daj,), (i) 

and the vector transformation (by chain rule): 

da)(q,io)/du>i = E [da)(q,v)/dv k \(dv k /duj,) or £, = Y A ki e k] (j) 
See also Papastavridis, 1992. 


Problem 2.13.7 By direct substitution of the representations 

d9 = mdt = Y e k d0 k and 59 = ^^£ k 69 k (a) 

into the earlier inertial rotational transitivity equation [ex. 2.13.11, eq. (b)] 

d(60) - 5{d9) = dO x 59 (b) 

and some simple differentiations, show that 

ds k / 861 - d£,/dd k + Y V b ki = «/ x £ k , (c) 

where these special Hamel coefficients rj h kl are defined by d(69 k ) — 6(d9 h ) = 

E E 1 At d9,69 k . 

Example 2.13.14 Angular Acceleration. Let us consider intermediate axes ♦ - u k 
rotating with inertial angular velocity Q = E ®k- u k- If the inertial angular velocity 
of a rigid body, resolved along these axes, is m = E u k u k then its inertial angular 
acceleration equals 

a = dai/dt = dco/dt+ Q x a) = dm/dt — a>„ x cu, (a) 

where dco/dt = E {dui k /dt)u k , and co 0 = co — Q = angular velocity of body relative 
to the intermediate axes. 

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380 CHAPTER 2: KINEMATICS OF CONSTRAINED SYSTEMS 

Applying this result to the earlier case of semimobile axes ♦ -i'j'k' = +-u n ii n k 
(ex. 2.13.9) where 

co = (i uj s )u„ + (w 0 sin 9)u„i + cos 6)k = Q + k = Q + co 0 , (b) 

[with the customary notations: = d(j>/dt , lo 8 = cW/dt , = di//dt\ 

that is, m 0 = ui^ k, we find, after some straightforward calculations, 

oc = a„u„ + (x n 'U n i + (y.^k. (c) 

where 

cx n = dujflIdt -\- sin d, 

cx n ' = (dui, ; , jdt) sin 0 cos 6 — ujq 

ctk = ( dtOfj, / dt) cos 9 + du>^/dt — co e sin 9. (d) 

Let the reader repeat the above for the semifixed axes +-u„u N K, where 

co = ( ui^K + u)gii n ) + oj^k = (t o^K + uigii u ) + — sin 9 i/.y + cos 9K) 

— {j jJ e) u n + (—ufy sin 0 )tiyr + + ufy cos 0 ) A" 

= ui^K + co 0 = fl + (D 0 . (e) 

[For matrix forms of rigid-body accelerations, see (1.11.9a ff.); also Lur’e (1968, 
pp. 68-72).] 


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3 

Kinetics of Constrained Systems 

(i.e., Lagrangean Kinetics) 


Where we may appear to have rashly and needlessly interfered 
with methods and systems of proof in the present day generally 
accepted, we take the position of Restorers, and not of 
Innovators. 

(Thomson and Tait, 1867-1912, Preface, p. vi) 

[A] work of which the unity of method is one of the most 

striking characteristics_That which most distinguishes the plan 

of this treatise from the usual type is the direct application of the 
general principle to each particular case. 

(Gibbs, 1879, 3rd footnote, emphasis added; the work/treatise 
Gibbs refers to is Lagrange’s Mecanique Analytique, and the 
“general principle” is Lagrange’s Principle (§3.2)) 

[T]he author . . . again and again . . . experienced the 
extraordinary elation of mind which accompanies a 
preoccupation with the basic principles and methods of 
analytical mechanics. 

(Lanczos, 1970, p. vii) 


3.1 INTRODUCTION 

This is the key chapter of the entire book; and since it is based on chapter 2, it should 
be read after the latter. We begin with a detailed coverage of the two fundamental 
principles, or pillars, of Lagrangean analytical mechanics: 

(i) The Principle of Lagrange (and its velocity form known as The Central Equation ); and 

(ii) The Principle of Relaxation of the Constraints. 

From these two, with the help of virtual displacements , and so on (§2.5 ff.), we, 
subsequently, obtain all possible kinetic energy-based (Lagrangean ) and acceleration 
energy—based ( Appellian ) equations of motion of holonomic and/or Pfaffian (possibly 
nonholonomic) systems; in holonomic and/or nonholonomic variables, with/without 
constraint reactions; such as the equations of Routh-Voss, Maggi, Flamel, and 
Appell, to name the most important. 

Next, applying standard mathematical transformations to these equations, we 
obtain the theorem of work-energy in its various forms; that is, in holonomic and/ 
or nonholonomic variables, with/without constraint reactions, and so on. This con¬ 
cludes the first, general, part of the chapter (§3.1-12). The second and third parts 
apply the previous Lagrangean and Appellian methods/principles/equations, 


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CHAPTER 3: KINETICS OF CONSTRAINED SYSTEMS 


respectively, to the rigid body (§3.13—15) and to noninertial frames of reference (or 
moving axes ) (§3.16). The chapter ends with (i) a concise discussion of the servo-, or 
control, constraints of Beghin-Appell (§3.17); and (ii) two Appendices on the historical 
evolution of (some of) the above principles/equations of motion, and their relations to 
virtual displacements and the confusion-laden principle of d’Alembert-Lagrange. 

As with the previous chapters, a large number of completely solved examples and 
problems with their answers and/or helpful hints, many of them kinetic continua¬ 
tions of corresponding kinematical examples and problems of chapter 2, have been 
appropriately placed throughout this chapter. 

For complementary reading, we recommend (alphabetically): Butenin (1971), 
Dobronravov (1970, 1976), Gantmacher (1966/1970), Hamel (1912/1922(b), 1949), 
Kilchevskii (1977), Lur’e (1961/1968/2002), Mei (1985, 1987(a), 1991), Mei and Liu 
(1987), Neimark and Fufaev (1967/1972), Nordheim (1927), Pars (1965), Poliahov et al. 
(1985), Prange (1935), Synge (1960). As with chapter 2, we are unaware of any 
other single exposition, in English, comparable to this one in the range of topics 
covered. Only Hamel (1949), Mei et al. (1991) and Neimark and Fufaev (1967/1972) 
cover major portions of the material treated here. 


3.2 THE PRINCIPLE OF LAGRANGE (LP) 

We begin with a finite mechanical system S consisting of particles {P}; each of mass 
dm, inertial acceleration a = dv/dt = d 2 v/dt 2 , and each obeying the Newton-Euler 
equation of motion (§1.4): 

dma = df, (3.2.1) 

where df = total force acting on P. As explained in chapter 2, the continuum notation 
for particle quantities, employed here, simplifies matters, since it allows us to reserve 
all indices (to be introduced below) for system quantities. 


The Force Classification 

Now, and here we start parting company with the Newton-Euler mechanics, we 
decompose df into two parts: (i) a total physical, or impressed, force dF, and (ii) a 
total constraint force, or constraint reaction, dR : 

df = dF + dR. (3.2.2) 

Let us elaborate on these fundamental concepts: 

(i) By constraint reactions, on our particle P, we shall understand (external and/or 
internal) forces, due solely to the (external and/or internal) geometrical and/or kine¬ 
matical constitution of the system 5; that is, forces caused exclusively by the pre¬ 
scribed (external and/or internal) constraints of S, and whose raison d’etre is the 
preservation of these constraints. As a result, such forces are (a) passive (i.e., they 
appear only when absolutely needed; see below), and (b) expressible only through 
these constraints (since, by their definition, they contain neither physical constants 
nor material functions/coefficients). Therefore, these reactions become fully known 
only after the motion of 5 (under possible additional, nonconstraint forces and 
initial conditions) has been found. Examples of constraint reactions are: inextensible 

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§3.2 THE PRINCIPLE OF LAGRANGE (LP) 


cable tensions, internal forces in a rigid body, normal forces among contacting 
(rolling/sliding/pivoting/nonpivoting) rigid bodies, and rolling (or static) friction. 

(Generally, constraints and their reactions are classified, on the basis of the precise 
physical manner by which they are maintained, as passive, or as active. Except §3.17, 
where the latter are elaborated, this chapter deals only with passive constraints / 
reactions.) 

(ii) By physical or impressed forces, on our particle P, we shall understand all 
other (external andjor internal, nonconstraint) forces acting on it, which means that 
[since the total force on P is determined through variables describing the geometrical! 
kinematical and physical state of the rest of the matter surrounding that particle 
(recalling §1.4)] the impressed forces depend, at least partially , on physical, or mate¬ 
rial, constants, unrelated to the constraints, and which can be determined only 
experimentally. Examples of such constants are: mass, gravitational constant, elastic 
moduli, viscous and/or dry friction coefficients, readings of the scale of a barometer 
or manometer; and examples of physical/impressed forces are gravity (weight), elas¬ 
tic (spring) forces, viscous damping forces, steam pressure, slipping (or sliding, or 
kinetic) friction [see remark (iii) below]. 

In other words, the impressed forces are forces expressed by material, or consti¬ 
tutive, equations, that contain those physical constants, and are assumed to be valid 
for any motion of the system; physical means physically ( functionally) given — it does 
not mean that the values of these forces are necessarily known ahead of time! 

In sum: Impressed forces are given by constitutive equations, white reactions are not; 
but, in general, both these forces require, for their complete determination, knowledge 
of the subsequent motion of the system (which, in turn, requires solution of an initial- 
value problem; namely, that of its equations of motion plus initial conditions). 

Impressed forces are also, variously, referred to as {directly’) applied, active, acting, 
assigned, given, known (where the last two terms have the meaning described above — 
see also remarks (iii) and (iv) below). In addition, the great physicist Planck (1928, pp. 
101-103) calls our impressed forces “ treibende ” (driving, or propelling), while the 
highly instructive Langner (1997-1998, p. 49) proposes the rare but conceptually 
useful terms “ urgente ” (urging) for the impressed forces, and “ cogente ” (cogent, con¬ 
vincing) for the constraint forces. We follow Hamel (1949, pp. 65, 82, 517, 551), who 
calls impressed forces “ physikalisch gegebene ” (physically given) or “ eingeprdgte ”; 
also Sommerfeld (1964, pp. 53-54), who calls them “forces of physical origin.” 

REMARKS 

(i) From the viewpoint of continuum mechanics, practically all forces are physical 
(i.e., impressed); for example, an inextensible cable tension can be viewed as the limit 
of the tension of an elastic cable, or rubber band, whose modulus is getting higher 
and higher (—> oo); and a rigid body can be viewed as a very stiff, practically strain¬ 
less, deformable body. But there is also the exactly opposite viewpoint: kinetic and 
statistical theories of matter explain macroscopic phenomena, such as friction, visc¬ 
osity, rust, by the motion of large numbers of smooth molecules, atoms, and so on. 
Their 19th century forerunners (Kelvin, Helmholtz, et al.) even tried to reduce the 
internal potential energy of bodies to the kinetic energy of a number of spinning 
“molecular gyrostats” strategically located inside them — see, for example, Gray 
(1918, chap. 8). And there is, of course, general relativity, which, continuing tradi¬ 
tions of forceless mechanics, initiated by Hertz et al., set out to geometrize gravity 
completely; that is, replace tactile mechanics by a visual mechanics, albeit in a 
four-dimensional “space.” For the modest purposes of macroscopic earthly 

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CHAPTER 3: KINETICS OF CONSTRAINED SYSTEMS 


mechanics, the impressed/constraint force division is both logically consistent and 
practically useful (economical), and so we uphold it throughout this book. 

(ii) The decomposition (3.2.2), what Hamel (1949, p. 218) calls “d’Alembertsche 
Ansatz” (~ initial proposition), is the hallmark of analytical mechanics. Expressing 
system accelerations as partial/total derivatives of kinetic energies with respect to 
system coordinates, velocities, and time (§3.3) is a welcome but secondary character¬ 
istic of Lagrangean analytical mechanics; the primary one is the decomposition 
(3.2.2) and its consequences with regard to the equations of motion. By contrast, 
the Newton-Euler mechanics decomposes df into (a) a total external force df e 
(= force originating, even partially, from outside of our system S ), and (b) a total 
internal, or mutual, force df t (= force due exclusively to the rest of S, on its generic 
particle P ): 


df = df e + df i . (3.2.3) 

The connection between (3.2.2) and (3.2.3) is easily seen by decomposing dF(dR) 
into an external part dF e (dR e ) and an internal part dF^dR,). and then rearranging a 
la (3.2.3); that is, successively, 

df = dF + dR = (dF e + dF,) + ( dR e + dR t ) 

= ( dF e T dR e ) -f- ( dp,- T d R : - 'j ~ d f e 4- d f i , 

where 

df e = dF e + dR e and d '/, = dF t + d /?,. 

The decompositions (3.2.2) and (3.2.3), although physically different, may, for some 
special systems, coincide. For example, in a free (i.e., externally unconstrained) rigid 
body all external forces are impressed (i.e., external reactions = 0), and all internal 
forces are reactions (i.e., interned impressed forces = 0). The coincidence of external 
forces with impressed forces and of internal forces with reactions in this popular and 
well-known system is, probably, responsible for the frequent confusion and error 
accompanying d’Alembert’s principle (detailed below), even in contemporary 
dynamics expositions. 

(iii) Rolling friction should be counted as a constraint reaction because it is 
expressed by a geometrical/kinematical condition, not by a constitutive equation; 
while slipping friction should be counted as an impressed force because, according to 
the well-known Coulomb-Morin friction “law,” it depends both on the contact 
condition (through the normal force, which is in both cases a constraint reaction) 
and on the physical properties of the contacting surfaces (through the kinetic friction 
coefficient). (That slipping friction is governed by a physical inequality does not affect 
our force classification.) The above apply to the (possible) rolling/slipping and pivot¬ 
ing/non-pivoting couples. 

The difference between rolling and slipping friction, from the viewpoint of analy¬ 
tical mechanics (principle of virtual work, etc.), has been a source of considerable 
confusion and error, even among the better authors on the subject. 

(iv) The force decomposition (3.2.2) is completely analogous to that occurring in 
continuum mechanics. For instance, in an incompressible (i.e., internally con¬ 
strained) elastic solid, the total stress (force) consists of a “hydrostatic pressure” 
or “reaction stress” term (constraint reaction), plus an “elastic stress” term 
(impressed force) expressed by a constitutive equation/function of the elastic moduli 

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(3.2.4) 

(3.2.4a) 


§3.2 THE PRINCIPLE OF LAGRANGE (LP) 


(material constants) and the strains (motion —> deformation), and it is assumed to be 
valid for any motion of that system. In general, the values of the stresses, 
both “incompressible/pressure” and “elastic” parts, for specific initial and boundary 
conditions, are found after solving that particular “initial- and boundary-value 
problem”; namely, the equations of motion of the solid plus its initial and boundary 
conditions. 

HISTORICAL 

The fundamental decomposition (3.2.2) seems to have been first given by Delaunay 
(1856); see, for example (alphabetically): Rumyantsev (1990, p. 268), Stackel (1905, 
p. 450, footnote 11a); also Hamel (1912, pp. 81-82, 301-302, 457^158, 469^170), 
Heun [1902 (a, d)], Pars (1953, pp. 447^148), Webster (1912, pp. 41^12, 63-65). 


Example 3.2.1 Let us Find the Most Important Internal I External and Impressed! 
Constraint Forces in a Diesel-Powered Electric Locomotive, Rolling on 
Rails. These are as follows: 

(i) Gravity and air resistance (drag ) are both external (their cause lies outside the 
system locomotive), and impressed (both depend partially on the physical constants: 
g = acceleration of gravity and p = air density, respectively). 

(ii) Pressure of burnt diesel fuel is internal (it originates within the engine’s cylin¬ 
ders) and impressed (depends on the gas temperature, density, etc.). 

(iii) Forces on connecting rods and other moving parts of the engine: 

(a) If these bodies are considered rigid, the forces are internal reactions; 

(b) If they are considered flexible, say elastic, these forces are internal but impressed (and 
to calculate them we must know their elastic moduli). 

(iv) Forces between axles and their wheel bearings are internal (for obvious rea¬ 
sons) and impressed (due to the relative motion among them — no constraints). 

(v) Friction forces between wheels and real are external (caused, partially, by an 
external body, the rail) and reactions (due to the slippingless rolling of wheels), and 
this holds for both their tangential (friction) and normal components; however, in 
the case of slipping (skidding), the friction changes to an external impressed force (it 
depends, partially, on the wheel-rail friction coefficient). 

Example 3.2.2 Let us Identify and Classify the Key Forces on a Person Walking 
up a Rough Hilly Road. The external forces needed to overcome the (also exter¬ 
nal) forces of gravity and air resistance are those generated by the road friction. 
The latter are reactions, since there is no relative motion (i.e., constraint) between 
the walker’s shoes and the road surface. 


Arguments of the Forces 

In classical (Newtonian) mechanics, the force df on a particle P, of a system S, can 
depend, at most, on its position, velocity, and time; and on those of other particles of 
S, or even outside of S; and also, on material functions/coefficients. But, as an 
independent constitutive equation (i.e., not by some artificial control law), df cannot 
depend on the acceleration a of P (and/or its higher time derivatives). This, however, 
does not preclude the occurrence of such a dependence by elimination; in the course 
of solving the equations of motion, and so on, of a problem, it is possible to relate 

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CHAPTER 3: KINETICS OF CONSTRAINED SYSTEMS 


functionally a force with an acceleration; but that is a mathematical coupling, not an 
independent physical one. 

[Pars (1965, pp. 11-12; also 24-25) has shown that if clf depended on «, then the 
initial state of P, that is, its initial position and velocity, would not determine its 
future uniquely; see also Rosenberg (1977, pp. 10-17); and Hamel (1949, p. 49). But 
in other areas of classical physics, for instance electrodynamics (e.g., radiation damp¬ 
ing), such a non-Newtonian explicit «-dependence does not create inconsistencies.] 


Lagrange's Principle 

Dotting each of (3.2.1) and (3.2.2) with the corresponding particle’s inertial virtual 
displacement Sr (§2.5 IT.) and then summing the resulting equations over all system 
particles, for a fixed generic time, we obtain 

g dm a ■ Sr = g dF • Sr + g dR • Sr , (3.2.5) 

or, rearranging, 

g (dm a - dF) • Sr + $ (-dR) -Sr= 0; (3.2.6) 

where [recall (§2.2.7 IT.)] the material sum S (■ ■ ■) is to be understood as a Stieltjes’ 
integral extending over all the continuously and/or discretely distributed system 
particles and their geometric/kinematic/inertial/kinetic variables. 

Equations (3.2.5, 6) do not contain anything physically new; that is, they result 
from (3.2.1, 2) by purely mathematical transformations. To make further progress 
towards the derivation of reactionless equations of motion, one of the key objectives 
of analytical mechanics, we now postulate that (for bilateral, or equality, or rever¬ 
sible, constraints) 

-S'W R = g (-dR) • Sr = -gdR- Sr = 0; (3.2.7) 

in words: at each instant, the (first-order) total virtual work of the system of (external 
and internal) “lost” (or forlorn, or accessory) forces {—c//f}, — S'W R , vanishes. Then, 
equations (3.2.5, 6) immediately reduce to the new and nontrivial principle of 
d’Alembert in Lagrange’s form , or, simply and more accurately, principle of 
Lagrange ( LP ) for such constraints: 

^ dm a • Sr = ^ dF • Sr or ^ (dm a — dF) • Sr = 0; (3.2.8) 

what Lagrange calls “la formule generate de la Dynamique pour le mouvement d’un 
systeme quelconque de corps.” 

This fundamental differential variational equation states that during the motion of 
a constrained system whose reactions, at each instant, satisfy the physical postulate 
(3.2.7), the total (first-order) virtual work of (the negative of) its “inertial forces” 
—{—dm a) = {dm a}, 

61 = $ dm a-Sr, (3.2.9) 

equals the similar virtual work of its ( external and internal) impressed forces {<7E}, 

S'W= gdF-Sr- (3.2.10) 

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§3.2 THE PRINCIPLE OF LAC RANGE (LP) 


that is, 


S'W R = 0 => SI=S’W. 


(3.2.11) 


The entire Lagrangean kinetics is based on LP, equations (3.2.7—11). Let us, therefore, 
examine them closely. 

• Another, equivalent, formulation of the above is the following: during the 
motion, the totality of the lost forces {—dR = df — dm a} are, at each instant, in 
equilibrium-, not in the elementary sense of zero force and moment, but in that of the 
virtual work equation (3.2.7) (see also chap. 3, appendix 2). 

• Here, we must stress that the above equations, and associated virtual work 
conception of equilibrium, are the contemporary formulation and interpretation of 
d’Alembert’s principle; and they are due, primarily, to Heun and Hamel (early 20th 
century). As such, they bear practically zero resemblance to the original workless 
exposition of d’Alembert (1743). The latter postulated what, again in contemporary 
terms, amounts to equilibrium of the {— dR} in the elementary (i.e., Newton-Euler) 
sense of zero resultant force and moment: 

S (-dR) = 0 => g (dm a - dF) = 0. 

$rx(-dR) = 0 => $rx(dma-dF) = 0. (3.2.12) 


It is not hard to see that for a rigid body (what d’Alembert dealt with) (3.2.7) 
specializes to (3.2.12). Indeed, substituting into (3.2.7) the most general rigid 
virtual displacement, 5r = dr. + 56 x (r — r.) [where ♦ = generic body point, and 
SO = (first-order/elementary) virtual rigid body rotation (recalling §1.10 ff.)] and simple 
vector algebra, we obtain, successively, 


-S'W R = $ (-dR) • [hr* + SO x (r - r .)] 


= [£(-<«) 


Sr. + [^(/--r,) x (-dR) 


■SO 


= (-R) ■ Sr. + M. (-R) -S0 = 0, 


(3.2.13) 


from which, since Sr. and SO are arbitrary, (3.2.12) follows [and if $ (—dR) = 0, 
then M.(—R) = M origin (—R)]. If, further, the rigid body is free, that is, uncon¬ 
strained, then, as explained earlier, all its external (internal) forces are impressed 
(reactions) (i.e., {df e } = {dF} and {dff = {dR}), and the above lead to the 
Eulerian principles of linear and angular momentum (recall §1.8.18): 

S‘<f. = S dm a and s (r-r.) x df e =s (r — r.) x dm a. (3.2.14) 

It follows that, in studying the statics of free rigid bodies via virtual work, we only 
need include their external = impressed forces; and that is why here the methods of 
Newton-Euler and d’Alembert-Lagrange coincide and supply conditions that are 
both necessary and sufficient for equilibrium (see also Hamel, 1949, pp. 80-83). 

This preoccupation of d’Alembert, and many others since him, with the special 
case of (systems of) rigid bodies and elementary vector equilibrium (3.2.12), has 
diverted attention from the far more general scalar virtual work equilibrium 
(3.2.7), which constitutes the essence of LP. 

• In LP it is the sum S' W R = S dR ■ Sr that vanishes, and not necessarily each 
of its terms dR • Sr separately; although this latter may happen in special cases. 


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CHAPTER 3: KINETICS OF CONSTRAINED SYSTEMS 


For example, as explained above, in a free rigid body (3.2.7) reduces to S'W R —> 
(S' ^internal forces = although individually dfj • Sr may not vanish. 

• While the dm a are present wherever a mass is accelerated, the dF may act 
only at a few system particles. 

• In general, S'W R and S'W are not the exact (or perfect, or total) virtual differ¬ 
entials of some system “work/force functions” W R and W, respectively; that is, in 
general, they are quasi variables, and that is the purpose of the accented delta S' 
(recall §2.9 ff.). The same holds for SI, but here, for convenience, we will make an 
exception and leave it unaccented. 

• For unilateral (or inequality, or irreversible ) constraints, LP is enlarged from 
(3.2.7-11) to 

gdR-Sr= S(dma-dF)-Sr> 0 => $ din a-Sr > $ dF • Sr, (3.2.15) 


or 


S'W R > 0 => SI > S'W. (3.2.15a) 

For example, in the case of a block resting under its own weight on a fixed horizontal 
table, the sole impressed force on the block, gravity, cannot perform positive virtual 
work; while the normal table reaction cannot perform negative virtual work: 
S'W R = -S'W> 0. 


Lagrange's Principle as a Constitutive Postulate 

It must be stressed that LP, eqs. (3.2.7-11), is what is known in continuum 
mechanics as a constitutive postulate for the nonphysical part of the df s, namely, 
the constraint reactions {dR}: like Hooke’s law in elasticity, or the Navier-Stokes 
law in fluid mechanics; hence, applying LP to a free (i.e., unconstrained) particle is 
like, say, applying the theory of elasticity to a rigid body! As such, LP is not a law of 
nature, like the Newton-Euler equation (3.2.1) (and its Cauchy form, in continuum 
mechanics), but subservient to them; if (3.2.1) can be likened to a constitution article, 
LP is a secondary law (say, a state law). Just as in continuum mechanics, where not 
all parts of the stress need be elastic, here in analytical mechanics too, not all con¬ 
straint reactions need satisfy (3.2.7) (see §3.17). Those reactions that do, which is 
most of this book, we shall call ideal (or perfect, or passive, or frictionless). 

In view of these facts, the frequently occurring expression “workless, or nonwork¬ 
ing, constraints” must be replaced by the more precise one, virtually workless con¬ 
straints. Indeed, under the most general kinematically admissible/possible particle 
displacement (§2.5) 


dr = ^ e k dq k + e 0 dt , where e k = dr/dq k , e 0 = dr/dt (= e n+x ), 

(3.2.16) 

the corresponding (first-order, or elementary) work of the constraint reactions is 
d'W R = gdR-dr=--- = (d'W R ) x +(d'W R ) 2 , 

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(3.2.16a) 


§3.2 THE PRINCIPLE OF LAGRANGE (LP) 


where 


{d'W R ) x = Y Rkdq k , Rk = S dR ' e k > (3.2.16b) 

{d'W R ) 2 = R 0 dt , R 0 = $dR e o (= i?„ +1 ); (3.2.16c) 

while, under an equally general virtual displacement Sr = e k ^k, the correspond¬ 
ing work is 

S'W R = g dR ■ Sr = • • • = Y R k % = 0; (3.2.16d) 


and therefore, since (d' W R ) ] and S'W R are mathematically equivalent ( dq ~ Sq), 
{d'W R \= 0 => d'W R = (d'W R ) 2 = dR • (dr/dt) dt f 0. (3.2.16e) 


[In view of (3.2.16 If.), it is, probably, better to think of first-order virtual work as 
projection of the forces in certain directions; and to forget all those traditional (and 
confusion-prone) definitions of it like “work of forces for a constraint compatible 
infinitesimal movement of the system.”] 

In sum: in general, the constraint reactions are working; even when virtually 
nonworking. Actually, that is why the whole concept of virtualness was invented in 
analytical mechanics. For example, let us consider a particle P constrained to remain 
on a rigid surface S, which undergoes a given motion. Then, the virtual work of the 
normal reaction exerted by S on P is zero, while the corresponding d'W R is not; 
but, if S is stationary, then both S'W R and d'W R vanish. From the viewpoint of 
continuum mechanics, the need for LP, or something equivalent, for the constraint 
reactions is relatively obvious. 

Below, we present a simple such mathematical argument from the viewpoint of 
discrete mechanics. In an A-particle system with equations of motion [discrete coun¬ 
terparts of (3.2.1)], 

m P ap = Fp + R P (P=l,...,N), (3.2.17a) 


and assuming that the impressed F P 's are completely known functions of t, r, v 
(something that may not always be the case: e.g., sliding friction), we have 
3 N + 3N = 6N unknown scalar functions: (i) the 3N position vector components/ 
coordinates {xp(t),y p (t),zp(t): rectangular Cartesian components of />} —> 
{d~x P /dt~ = a Px , d yp/dt = a Pv , d^zp/dt = a Pz \ rectangular Cartesian com¬ 
ponents of a P = d 2 r P /dt 2 }, plus (ii) the 3N reaction force components 
{R Px ,Rp v: Rp-}. Against these unknowns, we have available: (i) the 3 N scalar 
equations of motion (3.2.17a), and (ii) a total of h + m scalar equations of 
constraint (recall §2.2 If.): 

h geometric: f H (t,r P ) = 0 {H = 1, P = 1,..., N), (3.2.17b) 

m velocity (possibly nonholonomic): fi)(t,rp, v P ) =0 (D = 1,... ,m\ P = 1,..., N); 

(3.2.17c) 


that is, a total of 3TV + h + m (differential) equations. Therefore, to make our 
problem determinate, we need 6 N — (3 N + h + m) = (3TV — h) —m = n — m = / 
(= # DOF in the small) additional scalar equations. And here is where LP comes 
in: as shown later in this chapter, the single energetic but variational equation 


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CHAPTER 3: KINETICS OF CONSTRAINED SYSTEMS 


6' W R = 0 =>■ 61 = 6' W produces precisely these f needed independent scalar equa¬ 
tions (unlike the single actual, nonvariational, work/energy theorem, which always 
produces only one such equation!); and the latter, along with initial/boundary con¬ 
ditions make the above constrained dynamical problem determinate, or closed. 

This simple argument, number of equations = number of unknowns [probably 
originated by Lur’e (1968, pp. 245-248) and Gantmacher (1970, pp. 16-23)], 
shows clearly the impossibility of building a general constrained system mechanics 
without additional physical postulates , like LP, or something equivalent (it would 
be like trying to build a theory of elasticity without Hooke’s law, or something 
similar relating stress to strain!), and thus lays to rest frequent but nevertheless 
erroneous claims that “analytical mechanics is nothing but a mathematically sophis¬ 
ticated rearrangement of Newton’s laws.” 

In sum, analytical mechanics is both mathematically and physically different from 
the momentum mechanics of Newton-Euler. Schematically: 

Lagrangean analytical mechanics = Newton-Euler laws 

+ d’Alembert’s physical postulate. 

As Lanczos puts it: “Those scientists who claim that analytical mechanics is nothing 
but a mathematically different formulation of the laws of Newton must assume that 
[LP] is deducible from the Newtonian laws of motion. The author is unable to see 
how this can be done. Certainly the third law of motion, “action equals reaction,” is 
not wide enough to replace [LP]” (1970, p. 77). 

The above also show clearly that trying to prove LP is meaningless; although, in 
the past, several scientists have tried to do that (like trying to prove Hooke’s law in 
elasticity!). These considerations also indicate that if we choose to decompose the 
total force df according to some other physical characteristic, then we must equip 
that mechanics with appropriate constitutive postulates for (some of) the forces 
involved, so as to make the corresponding dynamical problem determinate. Thus, 
in the Newton-Euler mechanics, where, as we have already seen, df is decomposed 
into external and internal parts, the system equations of motion — that is, the 
principles of linear and angular momentum — thanks to the additional constitutive 
postulate of action-reaction, contain only the external forces (and couples); with¬ 
out that postulate, the equations of motion would involve all the forces, and the 
corresponding problem would be, in general, indeterminate. And in the case of 
matter-electromagnetic field interactions (e.g., electroelasticity, magneto-fluid- 
mechanics), we must, similarly, either know all forces involved, or supplement the 
equations of motion (of Newton-Euler and Maxwell) with special electromechanical 
constitutive equations, so that we end up again with a determinate system of 
equations. 


More on Lagrange's Principle as a Constitutive Postulate 

Here is what the noted mechanics historian E. Jouguet says about the physical nature 
of Lagrange’s Principle (freely translated): 

In sum, therefore, Huygens and Jacob Bernoulli implicitly admit that the forces devel¬ 
oped by the constraints in the case of motion, are, like the forces developed by the con¬ 
straints in the case of equilibrium, forces that do no work in the virtual displacements 

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§3.2 THE PRINCIPLE OF LAGRANGE (LP) 


compatible with the constraints. There is here a new physical postulate. It could be quite 
possible that the property of not doing work be true for the constraint forces during 
equilibrium and not for the constraint forces during motion; the reaction of a fixed surface 
on a point could be normal if the point was in equilibrium, and inclined if the point was 
moving; the reaction of a surface on a point could be normal if the surface was fixed and 
oblique if it was moving or deformable. This new postulate expresses, to use the lan¬ 
guage of Mr. P. Duhem [a French master (1861-1916), particularly famous for his 
contributions to continuum thermodynamics/energetics (in the tradition of Gibbs), 
and the history/axiomatics of theoretical mechanics], that the constraints, that have 
already been supposed [statically] frictionless, are also without viscosity. (1908, pp. 
195-196), 

and 

The dynamics of systems with constraints rests therefore on the property of forces 
generated, during the motion, by the constraints, of not doing work in the virtual 
displacements compatible with the given constraints. This is an experimental property, 
and at the same time an experimental property distinct from those that we have found 
for the forces developed by the constraints in the case of equilibrium, because it intro¬ 
duces the condition that the constraints are without viscosity. (1908, p. 202, emphasis 
added). 


When Are the Methods of Newton-Euler (NE) and 
d'Alembert-Lagrange (AL) Equivalent? 

Since there is only one mechanics, this is a natural question, but not an easy one. To 
begin with, since NE divides forces into external and internal (“apples”), while AL 
divides them into impressed and reactions (“oranges”), we should not be surprised if, 
for general mechanical systems and forces, no such equivalence exists, or should be 
expected, at all stages of the formulation and solution of a problem. 

Equivalence at the highest level of the fundamental principles may exist only for 
special systems and problems: that is, those for which (i) the internal forces (NE) 
coincide with those of constraint (AL), and (ii) the external forces (NE) coincide with 
the impressed ones (AL). The only such system that we are aware of, satisfying both 
(i) and (ii), is the earlier-examined free rigid body, and there we saw that LP leads to 
the NE principles of linear and angular momentum. For other systems where the 
internal forces may be (wholly or partly) impressed, for example, an elastic body, the 
NE principles do not follow from LP; the latter, as an independent axiom , says 
nothing about impressed forces. However, for a given system and forces, both meth¬ 
ods of NE and AL do the job pledged by all classical descriptions of motion, which 
is, given (i) the external (NE) and impressed (AL) forces, along with (ii) the system’s 
state at an “initial” instant (i.e., initial configuration and velocities = initial condi¬ 
tions), and (iii) appropriate constitutive postulates for its internal forces (NE) and 
constraint reactions (AL), respectively (and possibly other additional geometrical/ 
kinematical/physical facts intrinsic to that problem), then both NE and AL are 
theoretically equally capable in predicting the subsequent motion of the system 
and its remaining unknown forces (although both approaches may not be equivalent 
laborwise, or from the important Machian viewpoint of conceptual economy). On 
these fundamental issues, see also the masterful treatment of Hamel (1909; 1927, 
pp. 8-10, 14-18, 23-27, 38-39; 1949, chap. 4 and pp. 513-524). 

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CHAPTER 3: KINETICS OF CONSTRAINED SYSTEMS 


The above can be summarized in the following: 


(0 


Force decomposition: 
Newton-Euler (NE) 



D’Alembert-Lagrange (AL) 




Reactions 



External impressed 
External reactions 
Internal impressed 
Internal reactions 
Impressed external 
Impressed internal 
External reactions 
Internal reactions 


(ii) Whence the need for d’Alembert’s principle: 

Unknown forces 

NE: Internal {dff} — > Discrete: action-reaction s dfi = 0, S r x dfi = 0 

[—> Continuum: Boltzmann’s axiom-, i.c., symmetry of stress tensor] 
AL: Reactions {///?} —► Lagrange’s principle ^ dR • Sr = 0 


(iii) Consequences: 

NE: Linear momentum-. 

Angular momentum: 


AL: Lagrange’s principle: 


S d -f‘+S d f °= S dm a 

- s«.=s dm a =>■ f e = ma G (G = mass center) 
S [r x (dfi + df e )\ = S (r x dm a) 

=> S>'* d fc = d / dt [S ( ‘- X dm v)j 

[S'W R = g dR ■ hr = 0] + [dm a = dF + dR] 

=> ^ dF • br = ^ dm a ■ Sr 


(iv) Unknown force retrieval: 

NE: Principles of rigidification and cut 

AL: Principle of constraint relaxation (Befreiungsprinzip, see below and §3.7) 

(v) Coincidence of NE with cl’AL: free rigid body 

Free: External forces = Impressed forces; i.e., {df e } = {dF} ({dR e } = 0) 

Rigid: Internal forces = Constraint reactions; i.e., {dfi} = {dR} ({dF,} = 0) 

[Briefly (a) Rigidification principle: If a system is in equilibrium under impressed and 
constraint forces, it will remain in equilibrium if additional constraints are imposed 
on it so as to render it partly or wholly rigid; that is, deformable bodies in 
equilibrium can be treated just like rigid ones — both satisfy the same (necessary) 
conditions; (b) Cut principle'. We can replace the action of two contiguous parts 

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§3.2 THE PRINCIPLE OF LAGRANGE (LP) 


of the body by corresponding force systems (=>■ free body diagrams). Both prin¬ 
ciples are due to Euler. For details, see books on statics; also Papastavridis (EM, 
in prep.).] 


Example 3.2.3 Plane Mathematical Pendulum: Comparison Between Principles of 
Moment (Original d'Alembert) and Virtual Work (Lagrange). Let us consider 
the motion of a mathematical pendulum, of length / and mass m, about a fixed 
point O on a vertical plane. 

(i) According to the original formulation of the principle (first by Jakob Bernoulli 
and then by d’Alembert), the string reaction S on the oscillating particle P must be in 
equilibrium; that is, its moment about O must vanish: 

M q Erx5 = fl =>■ S must be parallel to the string OP (r = OP) . (a) 

As a result, the second part of the principle—that is, impressed forces minus inertia 
forces must be in equilibrium , yields (with W = weight of P) 

rxW = rx(ma) => — (JF)(/sin</>) = {ni[l(d 2 (f>/dt 2 )]}(l) 

=> d 2 (j)/dt 2 + (g/l) sin (f> = 0. (b) 

(ii) According to Lagrange’s formulation of the principle, the virtual work of S 
must vanish: 

S'W R = S • <)r = 0 => S must be perpendicular to the virtual displacement of P. 

(c) 

and since the latter is along the instantaneous tangent to P’s circular path about O, 
we conclude that S must be parallel to OP, as before. 

Hence, the second part of the principle—that is, virtual work of impressed forces 
minus that of inertia forces must vanish, yields 

W-6r=(ma)-6r => -(Wsincf)(lScf) = {m[l(d 2 (f/dt 2 )]}(lScf) 

=>■ d 2 (j>/dt 2 + (g/l) sin <^> = 0; (d) 

that is, the moment condition (b) and the virtual work condition (d) differ only by an 
inessential factor 6<p, and thus they produce the same reactionless equation of 
motion. 

In view of the extreme similarity, almost identity, of these two approaches in this 
and other rigid-body problems, we can see how, over the 19th and 20th centuries, 
various scientists came to confuse the zero moment method of James (Jakob) 
Bernoulli-d’Alembert {i.e., S r x (dF — dm a) =0, in our notation} with the zero 
virtual work method of Lagrange {i.e., S br ■ (dF — dm a) = 0}, and to view the 
former as equivalent to the latter. (Also, the fact that the string tension S is not 
zero—that is, that the constraint reaction is in equilibrium, not in the elementary 
sense of zero moment and force, but in the sense of zero virtual work, demon¬ 
strates clearly one of the drawbacks of the original d’Alembertian formulation of 
the principle.) 


Example 3.2.4 Motion of an Unconstrained System Relative to its Mass Center 
G, via Lagrange’s Principle (Adapted from Williamson and Tarleton, 1900, 


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CHAPTER 3: KINETICS OF CONSTRAINED SYSTEMS 


pp. 242-293). Substituting r = r G + v/ G =>■ a = a G + a/ G into LP, (3.2.8), and 
regrouping, we obtain 


0 = Sr G • ( dm a G - dF) J + br G • dm a /G 

+ «g • (S dmSr /G \ + § ( dma /G - dF) • Sr /G , 


(a) 


from which, since S dm r/ G = 0=> S dm 8r j G = 0 and S dm a / G = 0, and the br G , 
8r/ G are unrelated, we obtain 


(i) 


Sr G 


[S ( dm a ° ~ dF "> 


= 0 


S dm a G = S dF ’ 


that is, 


m a G = F (Principle of linear momentum ), (b) 

if 6r G is unconstrained, and 

(ii) ^ (dm a/ G — dF) • br / G = 0, under the constraint dm br/ G = 0. (c) 

Combining, or adjoining, the second of (c) into the first of (c) with the vectorial 
Lagrangean multiplier k = k(t) (see §3.5), we readily get 

dma/ G = dF + kdm, (d) 

and, summing this over the system, we obtain 


S dma /a = s dF + dmj 


0 = F + km => k = —F/m , 


so that, finally, (d) becomes 

dma/ G = dF ~ dm(F/m) 


(= dF — dma G ,as expected). 


(e) 

(f) 


Example 3.2.5 Sufficiency of the Statical Principle of Virtual Work (PVW) for 
the Equilibrium of Ideally Constrained Systems Deduced from LP. In analytical 
statics (i.e., LP with a = 0), the PVW states that in a bilaterally constrained and 
originally motionless system (in an inertial frame), the vanishing of b'W is a neces¬ 
sary and sufficient condition for it to remain in equilibrium in that frame. In con¬ 
crete applications, what we really employ is the sufficiency of the principle; that is, 
if b'W = 0, then the originally motionless system remains in equilibrium. 

Here, we will start with LP as the basic axiom, set S' W = 0, and then derive 
sufficient conditions to maintain equilibrium; that is, go from kinetics to statics. 
Most authors proceed inversely—that is, go from statics to kinetics—and that 
makes the detection of the importance of the various constraints more difficult. 

(i) Necessary conditions'. If the system is in (inertial) equilibrium, then a = 0, and 
therefore 


6’W= $dF-6r = 0 (^S'W R =^dR-6r=0), (a) 

for tj < t < tf, where tftf) = initial (final) time and 1/ — t t = r. 

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§3.2 THE PRINCIPLE OF LAGRANGE (LP) 


(ii) Sufficiency conditions : If S'W = 0, for U<t< tf, then LP gives 

SI = dm a ■ Sr = 0, for < t < tf. (b) 


Let us investigate the consequences of (a, b) for equilibrium. Substituting into (b) the 
particle displacement dr — e 0 dt = (v — e 0 ) dt =[v — (dr/dt)]dt, which is mathemati¬ 
cally equivalent to its virtual displacement, and cancelling dt(ffi 0), we obtain 

dm a • v = ^ dm a • e 0 , (c) 

and since 2T = S dmv-v =>• dT/dt = S dm a ■ v, we are readily led to the following 
rheonomic-type power equation : 


dT/dt = ^ dm a • e {) = ^ dF • e 0 + ^ dR ■ e 0 . (d) 

Integrating the above between t t and t{< tf), and setting T, = T(t/), T =T(t) yields 


AT=T - 

which also follows from S'W dt 
sions: 


T = 


^ dm a- e 0 ) dt ; 


(e) 


= 0. Equation (e) leads to the following conclu- 


(a) If e 0 = 0, then AT = 0, and since v,- = v{t/) = 0 =>■ T, = 0, it follows that T = 0 

for some time 1 — /,■ (< t f - ti) ; and from this, since T = positive definite in the 

v • v = v 2 , we conclude that then all the v’s vanish for / — /,■ (< tf — t { ); that is, the 

system remains in equilibrium in that time interval. 

Conversely, if AT = 0 for any t > t h then (e) leads, for arbitrary systems, to e 0 = 0. 
In this case, (c) gives v = 0; that is, equilibrium [while (a) yields 
S dF ■ v = S dF ■ e 0 = 0], The consequences of this in the presence of additional 

Pfaffian constraints are discussed below. 

(b) If e 0 f 0, then, in general, AT f 0; that is, the system moves away from its original 
equilibrium configuration, even though 6'W = 0, for n<t< tf, and v,- = 0. 
Weaker special assumptions for equilibrium result for the following conditions: 

(c) If e 0 f 0, but tf [S dm a ■ e 0 ) dt = 0; or 

(d) If e 0 f 0 but a ■ e 0 =0, for r ; < t < tf. 


Comparison with Gantmacher 

Gantmacher’s formulation of the PVW is as follows: “For some position (compa¬ 
tible with constraints) of a system to be an equilibrium position, it is necessary and 
sutficient that in this position the sum of the works of effective forces [our impressed 
forces] on any virtual displacements of the system be zero” and “If the constraints 
are nonstationary, then the term ‘compatible with constraints’ signifies that they are 
satisfied for any t if in them we put [our notation] r = r,- and v = 0” and “It is then 
assumed that [our] equation (a) holds for any value of t if in the expression for dF we 
put all r = r t and all v = 0” (1970, p. 25). Let us relate this formulation with ours. By 
(2.5.2) v = J2 e k v k + e 0 . Hence, if v = 0: 

(i) If the v k ’s are unconstrained, and since e k f 0, then v k = 0 =>■ q k = constant and 
e 0 = 0—that is, the constraints are stationary —then the system will remain in equi¬ 
librium. 

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CHAPTER 3: KINETICS OF CONSTRAINED SYSTEMS 


(ii) If, on the other hand, the v*/s are constrained , then, invoking the convenient repre¬ 
sentations (2.11.9, 13c, e), we have 

v —> v 0 = ^2 Pi v i + /?o = 0 =>• v,- = 0 => c[, = constant , and /? 0 = 0; 

P 0 = e 0 + ^2 b D e D = 0 => e Q = 0, and b D = 0; 

v D = b DI v, + b D => v D = 0 => q D = constant (=> q k = constant). 

In the light of the above, the PVW can be reformulated as follows: An originally 

motionless system remains in equilibrium if and only if (i) S'W = 0 and (ii) its 

holonomic constraints are stationary (e Q = dr/dt = 0) and its Pfaffian constraints 
are catastatic (a D = 0 or b D = 0). (The latter, however, may be nonstationary; and 
this explains Gantmacher’s statement: “Note that in this case the virtual displace¬ 
ments ... may also be different for different t.”) 

REMARKS 

(i) That “compatibility with constraints (during equilibrium)” leads to the above 
conclusions about them can be seen more clearly as follows. Let our system be 
subject to h holonomic constraints and m Pfaffian (holonomic and/or nonholonomic 
constraints): 

<t>H{t,r) = 0, f D = $B D (t,r)-v + B D (t,r) = 0 {H = 1,..., h\ D = 1,..., m). 

(f) 

By d/dt(.. ^-differentiating the above, to make them explicit in both velocities and 
accelerations, we readily obtain [recalling dot-product-of-tensor-definition [(see 
1.1.12d ff.)], in the first sum in (g2) below]: 

(a) d<j) H /dt = $ (d(/> H /dr) • v + dcp H /dt = 0, (gl) 

(b) d 2 cj) H /dt 2 = \(d 2 (j) H /dr dr): (v ® v) + ( dcj) H /dr) •a + 2(9 2 cpu/dt dr) • v] 

+ d 2 ^ H /dt 2 = 0, (g2) 

(c) df D /dt = SiK dB D/dr) • v + (dB D /dt)] ■v + B D -a } 

+ S (^B D /dr) ■ v + dB n /dt = 0. (g3) 

Now, since compatibility requires that, for t, < t < tf, eqs. (f-g3) should hold with 
v = 0 and a = 0 in them (just like the equations of motion), we readily obtain from 
the above the following conditions on these constraints: 

(pH = 0, (hi) 

d<t> H /dt = 0 => d(P H /dt = 0, (h2) 

d 2 ct> H /dt 2 = 0 =► d 2 cf) H /dt 2 = 0; (h3) 

Jd = 0 =>■ B d = 0, (h4) 

df D /dt = 0 => dB D /dt = 0; (h5) 

that is, for t, < t < tf , the holonomic constraints must be stationary, and the Pfaffian 
ones must be catastatic, as found earlier. 

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§3.2 THE PRINCIPLE OF LAGRANGE (LP) 


(ii) If we assume Earth to be inertial , then an Earth-bound system is scleronomic. 
But if we assume it to have a given motion, then our system is rheonomic. In both 
cases, the contact (nongravitational) forces from the Earth to that system are exter¬ 
nal reactions. If, finally, the Earth interacts with our system, then the two taken 
together constitute a scleronomic system whose internal forces are impressed (see 
also Nordheim, 1927, pp. 47-49). 

Example 3.2.6 Nonideal Constraints. Let us consider a particle P of mass m, 
moving under an impressed force F and subject to the velocity constraint 

fit, r, v ) = 0. (a) 

If the reaction created by (a) is R , then the equation of motion of P is 

m a = F + R. (b) 

To relate the constraint equation to the reaction, so as to incorporate (a) into (b), we 
d/dt{.. .)-differentiate the former: 

/ = 0 => df/dt = df/dt + ( df/dr ) • v + {df/dv) • a = 0, (c) 

and, therefore, 

ma- {df/dv) = — m[df/dt + (df/dr) • v]; (d) 

but, also, from (b), 

m a • (df/dv) = F • ( df/dv) + R • ( dfjdv). (e) 

Equating the right sides of (d, e), thus eliminating the acceleration, and rearranging, 
we obtain 


R ■ (df/dv) = - [m{df / dt) + m{df/dr) • v + F • {df,/dv)\. (f) 

Now, the most general solution of (f), for R , is 

R = -{df/dv) [m{df/dt) + m{df/dr) ■ v + F ■ {df/dv)}/{df/dvf + T, (g) 

where T = arbitrary vector orthogonal to df/dv. The above shows that, generally, the 
constraint reaction consists of two parts: (i) one parallel to df/dv. 

N = -{df/dv) [/ n{df/dt) + m{df/dr) v + F- {df/dv)]/{df/dvf = A {df/dv) (h) 

[where A = Lagrange an multiplier — see Lagrange’s equations of the first kind, 
(§3.5)]; and (ii) one normal to it, T. 

If T = 0, the constraint (a) is called ideal ; and in that case, clearly, the equation of 
motion of the particle (b), under (a), becomes 

ma = F— [F • {df/dv) + m{df/dr)-v + m{df/dt)] [{df/ dv) / {df/ dv) 1 ]. (i) 

To make the problem determinate, we, usually, introduce a constitutive equation 
between N and T. For example, in the common case of dry (solid/solid) sliding 
friction, we postulate the following relation between their magnitudes: 

T = pN = p\\{df/dv)\, p = coefficient of kinetic friction. (j) 

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CHAPTER 3: KINETICS OF CONSTRAINED SYSTEMS 


Then, and with (g, h), eq. (b) becomes 

ma = F + X (df/dv) — p\X(df/ dv)\u, (k) 

where u = v/\v\. 

For further details and applications of (k) see Poliahov et al. (1985, pp. 152-170). 

Problem 3.2.1 Continuing from the preceding example, show that if the con¬ 
straint (a) has the holonomic form 

<l>(t,r)= 0, (a) 

then (h) and (i) reduce, respectively, to 

N = —(df/dr) [m(d<j>/dt) + m(df/dr) • v + F • (d<j>/dr)] /(df/dr) 2 = X(df/dr) (b) 

and 

ma = F— [F • (df/dr) + m(d<j>/dr) • v + m(d<fi/dt)] [(9</>/<9r)/(<9</>/dr)~]. (c) 

(See also Lagrange’s equations of the first kind, in §3.5.) 


Introduction to the Principle of Relaxation of the 
Constraints (PRC) 

Before we embark into a detailed quantitative discussion of Lagrange’s Principle 
(LP) and its derivative equations of motion, let us discuss briefly the second pillar of 
analytical mechanics, the principle of relaxation of the constraints ( Befreiungsprinzip ; 
Hamel, 1917). LP allows us to get rid of the constraint forces and, eventually, obtain 
reactionless equations of motion; and, historically, this has been considered (and is) 
one of the advantages of the method, especially in physics. However, in many 
engineering problems we do need to calculate these reactions, and thus the question 
arises: How do we achieve this with such a reaction-eliminating Lagrangean form¬ 
alism? 

Here is where PRC comes in: to retrieve a(ny) particular, external and/or internal, 
“lost” reaction we, hypothetically, free, or relax, the system of its particular, external 
and/or internal, geometrical and/or motional, constraint(s) causing that reaction; 
that is, we, mentally, allow the formerly rigid, or unyielding, constraint(s) to deform, 
or become flexible, relaxed, so that the former reaction becomes an impressed 
force that depends on the deformation of the violated constraint via some constitutive 
equation. Then we calculate its virtual work, add it to S'W, and apply LP: 
(SI = S'W) relaxed tem ; and so on and so forth, for as many reactions as needed 
(one, or more, or all, at a time). Last, since in our model the constraints are rigid, 
we enforce them in the final stage of the differential equations of motion. The 
mathematical expression of PRC is the very well-known and widely applied method 
of “ undetermined ,” or Lagrangean, multipliers (§3.5). 

REMARKS 

(i) Another, mixed, method is, first, to use LP to calculate the reactionless equa¬ 
tions (and from them the motion), and to then use the method of Newton-Euler (NE) 
to calculate the external and/or internal reactions. This may be practically expedient, 

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§3.3 VIRTUAL WORK OF INERTIAL FORCES (SI), AND RELATED KINEMATICO-INERTIAL IDENTITIES 


but it is not logically/conceptually satisfactory; it makes Lagrangean mechanics look 
incomplete. 

(ii) The counterpart of PRC in the NE method is the following: if, for example, we 
want to calculate an internal force — that is, one that, due to the action-reaction 
postulate, drops out of the force/moment side in the NE principles of linear/angular 
momentum — then, applying Euler’s cut principle, we choose an appropriate new 
free-body diagram so that the former internal force(s)/moment(s) becomes external, 
and then apply to these new subsystems, the NE principles. 


3.3 VIRTUAL WORK OF INERTIAL FORCES (SI), AND RELATED 
KINEMATICO INERTIAL IDENTITIES 

Elere we transform (3.2.9), 61 = S dm a • 6r, from particle variables to system vari¬ 
ables; both holonomic and nonholonomic. (Actually, 61 is the negative of the virtual 
work of the “inertial forces” {—dm a}. We hope that this slight deviation from 
traditional terminology will not cause any problems.) Understandably, this relies 
critically on the kinematical results of chapter 2 and, therefore knowledge of that 
material is absolutely necessary. To obtain the most general system equations of 
motion from LP, we must use the most general expressions for a and 6r. We recall 
(§2.5 ff.) that these are (with k, l = 1,...,«) 

6r = ^ e k 6q k = holonomic variable representation 

= ^ £/ 69/ = nonholonomic variable representation (= 6r *) 

^ ^ £i66[, under the constraints 69 D = 0: D + 1,..., m\ I = m+ 1 

(3.3.1) 

where the fundamental mixed basis vectors { e k } and {s,} are related by 

e k = dr/dq k = ^ a, k £, o £, = dr/89, = ^ A kl e k . (3.3.1a) 


1. Holonomic System Variables 

Substituting the first of (3.3.1) into 61 we obtain, successively, 

SI = S dm a ' = S dm a - e k 6q,^j = ■ ■ ■ = ^ E k 6q k , (3.3.2) 

where E k = ^ dm a • e k : holonomic (Ar)th component of system inertial “force” 


= ^ dm a - ( dr/dq k ) = ^ dm a - ( dv/dq k ) = ^ dm a - ( da/dq k ) 
= ^ dm a • ( dr/8q k ) = ^ dm a • ( dv/dv k ) = ^ dm a ■ ( da/dw k ) 

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(3.3.3) 




CHAPTER 3: KINETICS OF CONSTRAINED SYSTEMS 


Now, E k transforms, successively, as follows: 

E k = dm a • e k = ^ dm(dv/dt) • (dv/dv k ) 

= d/dt^£jdm v • (dv/dv k ) — ^ [dm v • (d/dt)(dv/dv k )\ 

[recalling identity (2.5.10): E k (v) = d/dt(dv/dv k ) - dv/dq k = 0] 

= d/dt^^Jdm v • (dv/dv k ) — ^dm v ■ (dv/dq k ), (3.3.4) 

or, finally, with the help of the (inertial) kinetic energy 

T = S ( l / 2 )(dmv-v) = T(t,q,q) = T(t,q,v ) [since v = v(t,q,v)\, (3.3.5) 

we obtain 

E k = d/dt(dT/dv k ) - dT/dq k = d/dt{dT/dq k ) - dT/dq k = E k (T), (3.3.6) 
where 


E k (...) = d/dt{d... /dv k ) — d.. ■ /dq k = d/dt(d... /dq k ) — d... /dq k . 

(holonomic Euler-Lagrange operator) /f . (3.3.6a) 

Equation (3.3.6) is a kinematico-inertial identity, that is, it holds always, indepen¬ 
dently of any possible additional constraints, as long as the q’s are holonomic coor¬ 
dinates. Its cardinal importance to Lagrangean mechanics lies in the fact that it 
expresses system accelerations in terms of the partial and total derivatives of a scalar 
energetic function of the system coordinates and velocities, T(t,q,v), AS IF the q's and 
q’s = v’s (and t) were independent variables. That is why we have reserved the special 
notation E k {T) = E k when that operator is applied to the kinetic energy, even though 
E k (.. .) can be applied to any function of the q’s, v’s, and t. Also, (3.3.1-6a) clearly 
show the indispensability of virtual displacements (i.e., the e k vectors) to Lagrangean 
mechanics/equations of motion [i.e., the particular T-based expression for the system 
inertia/acceleration given by (3.3.6)], whether the constraint reactions are ideal or not. 

In sum: no e k s, no Lagrangean equations, that is, for an arbitrary particle/system 
vector z k f e k . 


Qclma-Zkf {d/dt){dT/dv k ) — dT/dq k . (3.3.6b) 

This should put to rest once and for all false claims that “one can build Lagrangean 
mechanics without virtual displacements.” The <5(...) is not the issue; the 
e k (—> projections) are! 

Let us collect the key kinematico-inertial identities involved here: 

(a) g dm v • e k = g dm v ■ ( dv/dv k ) = dT/dv k = dT /dq k = p k {t, q , v) = p k : 

Holonomic (k)th component of system momentum-, (3.3.7a) 

(b) g dm v • ( de k /dt ) = g dm v • ( dv/dq k ) = dT/dq k = r k (t, q , v) = r k : 

Holonomic (k)th component of “associated, or momental, inertial force”; 

(3.3.7b) 

(c) E k = dp k /dt - r k . (3.3.7c) 


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§3.3 VIRTUAL WORK OF INERTIAL FORCES (SI), AND RELATED KINEMATICO-INERTIAL IDENTITIES 

[p k = dT / dq k is the only kind of momentum that there is in analytical (Lagrangean 
and Hamiltonian) mechanics; and, as shown later, it comprises both the linear and 
angular momentum of the Newton-Euler mechanics.] 

2. Nonholonomic System Variables 

Substituting the second of (3.3.1) into 61, we obtain, successively with a = a* = 
particle acceleration in nonholonomic variables (and similarly for other quantities): 

51 = ^ dm a- 6v = Qdm a* ■ s k 69 ^ = • • • = "Y, 4 59 k , (3.3.8) 

where I k = ^ dm a* • s k = nonholonomic (k) th component of system inertial force 

= ^ dm a* • ( dv*/duj k ) = ^ dm a * • (da* /du> k ), recalling (2.9.35,43) 

= 4 (t,q,u,u) [since a* = a*(t,q,u,ij) and e k = s k (t,q)). (3.3.9) 

From the invariance of 61: E k 6q k = J2 4 69 k , and [recalling (2.9.11, 12)] 

6q k = A kl 69 1 69/ = ai k 5q k , we readily obtain the basic (covariant vector¬ 
like) transformation equations: 

4 = E ^/ k E/ E k = Y, a ft //. (3.3.10) 

The above expresses the nonholonomic inertial components in holonomic variables. 
To express them in terms of nonholonomic variables, we transform (3.3.9), succes¬ 
sively, as follows: 

I k = S d m a * ‘ E k = S dm(dv*/dt) • (dv*/du) k ) 

= d/dt(^^J dm v* • (dv*/du> k )^j - ^ [dm v* • d / dt(dv* / dco k )\ 

[adding and subtracting ^ dm v* • ( dv*/d9 k ), and regrouping] 

= d/dt(^^J dm v* • (dv*/du k )^j - ^ dm v* • (dv*/d9 k ) 

— ^ dm v* • [(d/dt)(dv*/duj k ) — dv*/d9 k \; 

(3.3.11a) 

or, invoking the nonintegrability identity (2.10.24, 25) [Greek subscripts run from 1 
to n + 1 (time)], 

E k *(v*) = d/dt(dv*/duj k ) — dv*/d9 k = dr. k /dt - dv*/d9 k 

= -EE i\i w/£,- - E V* [since ui„ +l = uj 0 = dt/dt = 1] 

= “EE VtaHA = ~EE yk a u a (dv*/duj r ), (3.3.11b) 

introducing the (inertial) kinetic energy in cpiasi variables 

T = ^ l/2(dm v* • v*) = T(t,q,ul) = T* [since v* = v*(t, q, w)] (3.3.11c) 

and recalling the symbolic cpiasi chain rule (2.9.32a, 44a) 

dT*/d9 k = Y (dT*/d qi )(dv,/duj k ) = Y A lk (dT*/dq,), (3.3.1 Id) 

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CHAPTER 3: KINETICS OF CONSTRAINED SYSTEMS 


and the (nonholonomic Euler—Lagrange operator) k 

E k *{...) = d/dt{d.../du k )-d.../dO k , (3.3.1 le) 

we finally obtain the nonholonomic (system) variable counterpart of E k \ 

4 = d / dt{dT* / duj k ) - dT*/d6 k + EE 7 r kl (dT*/dLj r )u, + Y Y k (dT*/du> r ) 

= d/dt(dT*/du k .) - dT*/dO k + YY. YUdTVdu, K 
= E k *{T*) — r k = E k * — r k [note difference from (3.3.6)], (3.3.12) 

where [recalling (2.10.25a)] 

-r k = dm v * - E k*( v *) = E E 7 r ka(dT*/duj r )u a = Y h r k (dT*/du r ) 

= —(System nonholonomic deviation , or correction, term) fc . (3.3.12a) 

We summarize the key kinematico-inertial identities below: 

(a) g dm v* ■ s k = g dm v* ■ (dv*/du> k ) = dT*/du k = P k (t, q, u) = P k : 

Nonholonomic (A’)th component of system momentum , (3.3.13a) 

(b) y k = E k *(v*) = d& k /dt — dv*/dO k = ■■■ = EEto 

= —(Particle nonholonomic deviation , or correction, term)^, (3.3.13b) 

Q dm v* • (de k /dt) = ^ dm v* • (dv*/d9 k ) + ^ dm v* • y k 

= dT*/d6 k + EE 7 r ak(dT*/du r )u a = dT*/dO k - EE 7 r ka (dT*/dLo r )io a 


= dT*/d0 k + r k [note difference from (3.3.7b)], (3.3.13c) 

-f t = EE 7 r ka {dT*/du r )u a = -r M - A,o, (3.3.13d) 

where 

- T Kn = E E 7 ^{dT'/dw^wu (3.3.13e) 

- r k0 = -r M+1 = ^ 7 r k (dT*/du> r y. 

“nonholonomic rheonomic force”. (3.3.13f) 

With the help of the above, 4 , (3.3.12), can be rewritten in the momentum form: 

4 = dP k /dt - dT*/de k + YY, 7 r kaP r w«. (3.3.14) 

[Originally due to Hamel [1904(a),(b)], but for stationary/scleronomic transforma¬ 
tions; that is, with a replaced by, say, / = 1,...,«.] 

(c) SI =S dm a ' 6r = ^2 E k % = E 7/ - ^ (3.3.15) 


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§3.3 VIRTUAL WORK OF INERTIAL FORCES (SI), AND RELATED KINEMATICO-INERTIAL IDENTITIES 


where 

E k = gdm a-e k = d/dt(dT/dv k ) - dT/dq k = E k (T) = ^ a, k I h (3.3.15a) 

4 = S dm a *' £ k = d/dt(dT*/du k ) - dT*/dd k + ^ ^ Y ka (dT* / du r )u a 
= E k *(T*) - r k = E k * -r k = Y J A i k Ei\ (3.3.15b) 

that is, it is E k = E k (T) and I k that transform like covariant vectors; the 
E k * = E k *(T*) do not (or, the terms E k * and r k , considered separately, do not 
transform as covariant vectors; but taken together , as E k * — T k = I k , they do!). 


3. Acceleration, or Appellian, Forms 

The above expressions for the inertia vector E k (or I k ) are based on the kinetic energy 
T (or T*), because for their derivation we used the velocity identities e k = dv/dv k (or 
s k = dv*/du k ). Let us now find expressions for these vectors using the acceleration 
identities e k = da/dq k = da/dw k (or e k = da*/du k ). The results will turn out to be 
based on a scalar function that depends on the accelerations in a similar way that T 
(or T*) depend on the velocities. [The choice e k = dr/dq k does not seem to lead to 
any useful expression for E k \ while the choice z k = dr*/dd k = 
A lk e I = A /k (dv/dvi) will be examined later.] 


(i) Holonomic variables 
We have, successively, 

E k = ^ din a ■ e k = Q dm a • ( da/dq k ) = dS/dq k = dS/dw k , (3.3.16a) 

where 

S = ^ (1/2 )(dm a ■ a) = ^ (1/2 )(dm a 2 ) = S(t, q, q , q) = S(t, q , v, w): 

“Gibbs-Appell function,” or simply Appellian, in holonomic variables 
[or “acceleration energy” (Saint-Germain, 1901)]. (3.3.16b) 


(ii) Nonholonomic variables 
Similarly, we obtain 

4 = ^ dm a* • s k = dm a* • (da*/du> k ) = dS*/dui k , (3.3.17a) 

where 

S* =^l (1/2 )(dm a * • a*) = ^ (1/2 )[dm(a*) 2 } = S*(t , q,uj,u): 

Appellian, in nonholonomic variables. (3.3.17b) 

To relate the above, we apply chain rule to 5 = S*. We obtain, successively, 

dS/dq k = dS/dw k = ^ ( dS*/du,)(du l /dq k ) = a lk (dS*/du,), (3.3.17c) 

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CHAPTER 3: KINETICS OF CONSTRAINED SYSTEMS 
and, inversely, 

dS*/du k = Y ( dS/dq,)(dq,/dw k ) = Y A, k {dS/dq,) = Y A ik(dS/dw,)- 

(3.3.17d) 

which are none other than the transformation equations (3.3.10). 

In sum, we have the following theoretically equivalent expressions for E k and I k : 

(i) E k = £ dm a ■ e k (= Y a ik I l) 

= d/dt(dT/dv k ) — dT/dq k [Lagrange (1780)] 

= Y aik[E,*(T*)-r,] 

= Y a ik(dS*/dd},) = dS/dq k = dS/dw k [Appell (1899)]; (3.3.18a) 

(ii) 4 = S dm a * ' Ek ( = E A ‘ kE >) 

= d/dt(dT*/du k ) - dT*/d9 k + EE 7 r ka {dT*/du, r )u )a 

= E k *(T*) - r k [Volterra (1898), Hamel (1903/1904)] 

= Y A ik[d/dt{dT/dv t ) - dT/dq ,\ [Maggi (1896, 1901, 1903)] 

= Y A ik{dS/dq,) = dS*/du k [Gibbs (1879)]. (3.3.18b) 


REMARKS 

(i) We can define (n + l)th, or (0)th, “temporal” holonomic and nonholonomic 
components of the system inertia vector by (with dq n+ \/dt = dq 0 /dt = dt/dt = 

v„+i = v 0 = 1 ) 

E„ + i = E 0 = ^ dm a • e„ +1 = ^ dm a • e 0 = ^ dm a • ( dr/dt ) 

= • • • = d/dt(dT/dv 0 ) - dT/dq 0 = d/dt(dT/di) - dT/dt, (3.3.19a) 

I n+l = I 0 = ^ dm a* • £„ +1 = Q dm a* • s 0 = ^ dm «* • (dr*/d9 0 ) = ■ ■ ■. 

(3.3.19b) 

However, such nonvirtual components will not be needed in the equations of motion; 
they could play a role in the formulation of “partial work/energy rate” equations 
(§3-9). 

(ii) Here, as throughout this book [e.g. (2.9.38ff.), ch. 5], superstars (.. .)* denote 
functions of t, q, u>, u ,...: 

f(t, d, ?>?>•• •) =f[t, Cl, g(t, q, w), q{t, q,u,Lb),...} =f*(t, q, 

A Special Case 

Let us find E k and I k for the following special quasi-velocity choice (recalling 2.11.9 ff.) 

Vo = Y h r)r( t ;C])v ! + b n (t,q), v, = Y s rr v r = v i, (3.3.20a) 

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§3.4 VIRTUAL WORKS OF FORCES: IMPRESSED (SW) AND CONSTRAINT REACTIONS (5W R ) 


and its inverse 


^D = ^b DI ( t ,q) Vl + b D (t,q), 0Jf = v,. (3.3.20b) 

Here, clearly [recalling (2.11.12b), and with 6. = Kronecker delta ]; 

Add' = A DI = b D j, A ID = 0, A n i = 6 n i , (3.3.20c) 

and so the Maggi form 4 = S A lk E) specializes to I k = A Dk E D + A lk Ej 

=^4>' = ^2 b DD 'E D + )Ej = E d ,, (3.3.20d) 

4 ' = ^ b DI 'E D + ^ bjf'E, = E,' + ^2 b DI 'E D . (3.3.20e) 

In sum, for the special choice (3.3.20a, b) I k takes the following form, in terms of 
holonomic Lagrangean ( T ) and Appellian (S) variables (with D=l,...,m; 
I = m + 1 ,..., n as usual): 

I D — E D = ( dT/dv D y - dT/dq D = dS/dv D = dS/dw D - (3.3.20f) 

i i = e i + Y1 h Dt E D = [{dT/dv,)' - dT/dq,] + ^ b DI [{dT/dv D )' - dT/dq D ] 

[Chaplygin (1895, publ. 1897), Hadamard (1895)] (3.3.20g) 

= dS/dq, + ^2 b DI {dS/dq D ) = dS/dwj + ^ b DI {dS/dw D ). (3.3.20h) 


The specialization of I k , for (3.3.20a, b), to nonholonomic variables [due to Chaplygin 
(1895/1897), in addition to his equations (3.3.20g); and Voronets (1901)] and other 
related results, are given in §3.8. 

We have expressed the {total, first order) virtual work of the {negative of the) 
inertial “forces,” 61, in system variables. The kinematico-inertial identities obtained 
are central to analytical mechanics, and that is why they were deliberately presented 
before any discussion of system forces and constraints; because, indeed, they are 
independent of the latter. These identities also show clearly the importance of the 
kinetic energy (primarily) and the Appellian (secondarily) to our subject, and so these 
quantities are examined in detail later (§3.9.11, 13-16). 

Now, let us proceed to express the virtual works of the real forces, namely, 6'W 
and 6'W r , in system variables. This will be considerably easier than the task just 
completed. 


3.4 VIRTUAL WORKS OF FORCES: 

IMPRESSED (<5'I/V) AND CONSTRAINT REACTIONS (<5'l/V R ) 

1. Holonomic Variables 

Substituting 6r=J2 e kbqk into the earlier expressions for 6'W and 6'W R (3.2.7, 
10), we readily obtain 

5'W= SdF.8v= SdF.^ekbqk) = ■■■ = ^ Q k Sq k , (3.4.1a) 

s ' Wr - S dR • Sr =S dR ' (H Ck Sqk ) = " ■ = X! Rk 6qk ' 

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(3.4.1b) 


CHAPTER 3: KINETICS OF CONSTRAINED SYSTEMS 


where 

Q k = s dF -e k : Holonomic (Ar)th component of system impressed force, (3.4.1c) 

R k = s dR • e k : Holonomic (A:)th component of system constraint reaction. 

(3.4.Id) 


2. Nonholonomic Variables 

Substituting Sr = J2 s k S9 k (= <$*'*) into S'W and S'W R , we, similarly, obtain 

6’ W = S dF ■ Sr* = S dF - (E S °k) = • • ■ = E 0 a- 60 k , (3.4.2a) 

S' W R = $ dR ■ Sr* = $ dR ■ (J2 s k 60 k ) = ■ ■ ■ = E A k 69 k , (3.4.2b) 

where 

© k = s dF • e. k : Nonholonomic (/c)th component of system impressed force, 

(3.4.2c) 

A k = s dR ■ s k : Nonholonomic (Ar)th component of system constraint force. 

(3.4.2d) 

Here too, these are ever valid definitions/results, no matter how many constraints 
may be imposed on the system later. 


3. Transformation Relations 

From the invariance of the virtual differentials S'W and S'W R , we obtain the follow¬ 
ing transformation formulae for the various system forces; that is, from 


s'w = Y Qk % - E 2a- (E A * S6 >) 

we conclude 

@1 = E A klQk 

and, inversely, 

Qk = E a ‘k 


£ = £ ®'(E 

a lk foil)) 

'y ~ Qk &qk, 

(3.4.3a) 

E Qk(9v k /duJi) 

(3.4.3b) 

E ( 9 w // 9 v a -) 6 >/ ; 

(3.4.3c) 


and, similarly, from S'W R = ■ ■ ■, we conclude 


d/ — E AkiRk 

and, inversely, 

R k = E a ik^i 


E R k( dv k/dui) 

(3.4.3d) 

Y( d ui/dv k )A, . 

(3.4.3e) 


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§3.4 VIRTUAL WORKS OF FORCES: IMPRESSED (SW) AND CONSTRAINT REACTIONS (SW R ) 


[These formulae can also be obtained from the e k s k transformation equations 
(2.9.25a, b) as follows: 

&, = SdF-s, = $dF. (J2 A kl s k ) = E A kl (SdF-e k ) = E A kl Q k , 

Qk = S dF • e k = S dF ' (E a ' kEl ) = E a ‘ k (S dF • £ /) = E a lk & A 

Rheonomic, or “temporal,” (n+ 1 )th nonvirtual force components can also be 
defined by 

Qn+i — Qo = f/T • C f; . ] = s dF • e 0 = ^ dF • ( dr/dt ) (holonomic impressed), 

(3.4.4a) 

^71+1 = ^0 e n+\ = S dR • e 0 = S dR • {dr/dt) (holonomic reaction); 

(3.4.4b) 

@ n+ 1 = ©o = S dF • £ «+i = S dF ■ £ o = S dF • (9r*/dd n+ 1 ) 

(nonholonomic impressed), (3.4.4c) 

A n+ \ =A 0 = ^dR- £„+i = ^dR-s 0 = S dR - {dr*/dB n+x ) 

(nonholonomic reaction); (3.4.4d) 

and, recalling (2.9.26a, b), we can easily deduce the following transformation equa¬ 
tions among these components: 

Q/i+i — Qo — ^ AF • e n +i ^ dF • ^ (i k ,n+\£ k + 

= E a >^+ 1 £a ) + S dF ' £ »+! = E a k,n+\ 0 k + ®n+ 1, (3.4.4e) 

-^n +1 — *0 = S dF * ^ n +l S dF ' . ‘"Tv.'i ■ I + £72+1^ 

^ dR * T ^dR'Efj^ i ^ ' ^k,n} i -4 / c T /I l i, (3.4.4f) 

and, conversely, 

0 o = S dF ' s ° = S dF ' (E + e ») 

= E (5 dF ■ e k) +S dF ' e o = E + 2o, (3.4.4g) 

= iS ^ • £ o = ^ dR ' (E ^ AeA + e °) 

= E Ak {S dR • ca ) + S dR • e ° = E ^+j«o- ( 3 - 4 - 4h ) 

Problem 3.4.1 With the help of the second of each of (2.9.3a, b), prove the addi¬ 
tional forms of the above transformation equations: 

&,,+ 1 = -E a k,n+\®k +Qn+ 1 , or, simply, 00 = -E Uk@k + G°’ ( a ) 

A „+1 = - E a k,n+ \ A k + ^,,+ 1 , or, simply, A 0 = - E a k A k + Rq- (b) 


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CHAPTER 3: KINETICS OF CONSTRAINED SYSTEMS 


REMARK 


A little analytical reflection will show that all these transformations can be con¬ 
densed in the formulae [with Greek subscripts running from 1 to n+ 1, recall 
(2.9.6a, b)]: 



®ot — ^2 ^ Qfi ~ U ap®ai 

(3.4.5a) 


A-a. ^ ^ -^-fia^-fi R-fi ^ ^ &a.fiA-or 

(3.4.5b) 

A Special Case 



For the earlier particular case (3.3.20a IF.: A dd i = 8 DD ', A DI = b DI , 
A n i = 8jp), the above transformation equations specialize to 

5 

II 

© 

@D' — 55 A kD 'Q k 

— 55 ^dd'Qd + 55 (0)2/ = Qd'i i- e -> @ d = Qd, 

(3.4.6a) 

&r = A ki'Qk 

= 55 ^ di'Qd +55 = Qf + 55 ^dvQd-, 



i.e., &! = Qi + 55 b diQd = Qi,o = Qio\ 

(3.4.6b) 

and, similarly, 

@o = ■ ■ ■ = 55 boQo + Qo = Qo,o■ 

(3.4.6c) 


Example 3.4.1 Virtually Workless Forces. The following are examples of forces 
that do zero virtual work: 

(i) Forces among the particles of a rigid body; generally, the forces among rigidly 
connected particles and/or bodies. 

(ii) Forces on particles that are either at rest (e.g., a fixed pivot, or hinge, about which a 
system body may turn, or a joint between two system bodies), or are constrained to 
move in prescribed ways; that is, their (inertial) motion is known in advance as a 
function of time. 

(iii) Forces from completely smooth curves and/or surfaces that are either at (inertial) 
rest or have prescribed (inertial) motions. 

(iv) Forces from perfectly rough curves and/or surfaces, either at rest or having pre¬ 
scribed motions. See also Pars (1965, pp. 24-25), Whittaker (1937, pp. 31-32). 

Example 3.4.2 If z is a virtual displacement, then S dR-z = 0. Let us show the 
converse'. If for a kinematically admissible/possible vector z we have S dR • z = 0, 
then z is a virtual displacement; that is, 

z = ^2eiS9, (I = m+ 1. (a) 

The proof is by contradiction: Let z = Sr + y Sr), where the relaxed part y may 
be, at most, 

y = s D 6'9 d + s 0 6't (. D=l,...,m ; 6'6 D , 8't: components of j). (b) 

Substituting (b) into LP we get, successively, 

0= gdR-z = $dR 6r+ $dR y= ■■■ = 0 + 55 A D 8'0 D + A 0 6't, 


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§3.5 EQUATIONS OF MOTION VIA LAGRANGE'S PRINCIPLE: GENERAL FORMS 


from which, since the m + 1 6'6 D and 6't are independent, we obtain A D , A 0 = 0. But, 
clearly, due to the constraints 66 D = 0 and 6t = 0, this is impossible. Thus, if we 
assume that y ^ 0, we are led to a contradiction. Hence, y = 0, and z is a virtual 
displacement expressible by (a). 


3.5 EQUATIONS OF MOTION VIA LAGRANGE'S PRINCIPLE: 

GENERAL FORMS 

Let us now proceed to the final synthesis; that is, the formulation of equations of 
motion in general system variables. We begin with the “constraint reaction part” of 
LP, eq. (3.2.7), and so on: 

S'W R = SdR-8r = Y, R k 6q k = Y J A k S6 k = 0. (3.5.1) 

If the n 6q s are unconstrained (or independent , or free), so are the n 66' s. Then, (3.5.1) 
leads to 


Rk = 0, A k = 0. (3.5.2) 

If, however, the n 5q s are constrained by the m (< n) Pfaffian, holonomic and/or 
nonholonomic, constraints 

66 D = E ° Dk 6q k = 0 (3.5.3) 

then, introducing m Lagrangean (hitherto) undetermined multipliers —A R = —A o(t) 
(the minus sign is only for algebraic convenience—see multiplier rule, below), and 
invoking (3.5.3), we can replace (3.5.1) with 

6'Wr + Y, (-^) S0 D = E A k M k + E (-**) 60 d 

= e R k %+ E E (—X D )a Dk 6q k — 0, (3.5.4) 

where, now, the n 6q's and 66's (can be treated as if they) are free. Therefore, (3.5.4) 
leads immediately to the following: 

(i) in holonomic variables, 

E (E - E An = 0 => R k = E X D a Dk [= R k {q , t)]; (3.5.5) 

(ii) in nonholonomic variables (with I = m + 1,..., n), 

E A k b ° k - E An S0 D = E ( a d ~ An) 60 D + Y {A ’~ °) 60 ' = °- ( 3 - 5 - 6 ) 

and from this to the nonholonomic counterpart of (3.5.5), 

A D = X D (1 • 66 d = 0) and Aj = 0 (0 • SOj = 0); (3.5.7) 

that is, the m Lagrangean multipliers associated with the m “equilibrium” constraints 
lo d = 0 or 66 d = 0 are, in effect, the first m nonholonomic (covariant) components of 
the system reaction vector in configuration space. We also notice that when¬ 
ever 66 k = 0, A k ^ 0, and vice versa (k=l,...,n; and even «+l), that is, 

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CHAPTER 3: KINETICS OF CONSTRAINED SYSTEMS 


A d 89 d = (Yl Z ))(0) = 0 and A t 86 } = (0 ){88 I ) = 0, so that 

8'W R = 80 d + ^2Ai66 i = 0 + 0 = 0, (3.5.8) 

in accordance with LP. 

The advantage of the nonholonomic (3.5.7, 8) over the holonomic (3.5.5) is that, 
in the former, constraints and reactions decouple naturally; whereas in the latter they 
are coupled; that is, in general, 

Sq k ^0, R k ^ 0 ^ R k 6q k ^ 0, but R k 8q k = 0. (3.5.9) 

Finally, substituting the first of (3.5.7) into (3.5.5), we recover the earlier transfor¬ 
mation equations (3.4.3e): R k = Y A D a Dk = Y A i a ik > as expected. 

REMARK 

In the case of unilateral constraints 89 D > 0 (if, originally, they have the form 
80 D < 0, we replace 88 D with — 80 D ), from the “unilateral LP” 8'W R > 0 and 
(3.5.4) we conclude that Y ^d 80 d > 0; and since the 80 D are positive or zero, the 
X D must also be positive or zero. 

In sum: If the unilateral constraints are chosen so that 86 * > 0 is possible/admis¬ 
sible, then the corresponding reaction A* is positive or zero (see also §3.7). 


The Lagrangean Multiplier Rule, 
or Adjoining of Constraints 

This fundamental mathematical theorem [one of the most useful mathematical 
results of the 18th century, initiated by Euler, but brought to prominence by 
Lagrange — see Hoppe (1926(a), p. 62)] states that: 

The single (differential) variational equation 

8'M = ^2 M k 8q k = 0, (3.5.10a) 

where M k = M k (q , q, q,..., t) and the n 8q’s are restricted by the m (<«) indepen¬ 
dent Pfaffian constraints 

80 d = Y^ a D k 8q k = 0 [rank (a Dk ) = m\, (3.5.10b) 

is completely equivalent to the new variational equation 

i’m + Y. (~X D ) 80 d = 8'M + EE ( — A D a Dk) bdk ~ 0) 

or 

E (^k ~ Y, X D a Dk ) 8q k = 0, (3.5.10c) 

where the n 8q’s are (better, can be viewed as) unconstrained; that is, (3.5.10a, b) are 
equivalent to the n equations 


M k — 22 ^ ° a Dk 

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(3.5. lOd) 



§3.5 EQUATIONS OF MOTION VIA LAGRANGE'S PRINCIPLE: GENERAL FORMS 
which, along with the m constraints (3.5.10b), in velocity form 

^E a D k<lk + a D = 0, (3.5.lOe) 

make up a system of n + m equations for the n + m unknown functions q(t) and A (t). 
INFORMAL PROOF 

Let us define the m X D ’s by the m nonsingular equations 

M D , = J2 X D a DD' (D,D'= (3.5.lOf) 

that is, eqs. (3.5.lOd) with k —> D'. For such A’s eq. (3.5.10c) reduces to 

E ( M/ _ H x d<*di) Hi = °> (3.5.lOg) 

where the (n — m) Sqfs are now free. From the above, we immediately conclude that 

Mi = E A d ci di'i (3-5. lOh) 


that is, eqs. (3.5. lOd) with k —> I. 

[References on the multiplier rule: Gantmacher (1970, pp. 20-23), Hamel (1949, 
pp. 85-91), Osgood (1937, pp. 316-318), Rosenberg (1977, pp. 132, 212-214). For a 
linear algebra based proof, see, for example, Woodhouse (1987, pp. 114-115).] 


Example 3.5.1 Lagrange’s Equations of the First Kind. The multiplier rule 
applied to 

6'W r = gdR-Sr = 0, (a) 

where the Sr are restricted by (i) the h holonomic constraints (// = 1..... /;) 

‘MbO = 0 => Sfif = g (dfn/dr) ■ Sr = 0, (bl) 

and (ii) the m Pfaffian (possibly nonholonomic) constraints [D=l,...,m 
(< n = 3N— h) and B D = B D (t , r)] 

Sb d -v + B d = 0 =» SB D -Sr = 0, (b2) 

leads, with the help of the h + m Lagrangean multipliers \i H = /i//(t) and \ D — \d( 0, 
to 

S{ dR -H dH(d 0 H /dr) - E A d Bd) -Sr= 0, (cl) 

from which, since the Sr can now be treated as free, we obtain the constitutive 
equation for the total constraint reaction on the typical particle P due to all system 
constraints: 

dR = E dHi^H/dr) + E ^pBp- (°2) 

Then, the Newton-Euler/d’Alembert particle equation dm a = dF + dR becomes the 
famous Lagrange’s equation of the first kind : 

dma = dF + fi H (d(j) H /dr) X d B d . (c3) 

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411 



CHAPTER 3: KINETICS OF CONSTRAINED SYSTEMS 


In the more common discrete notation, the constraints (bl,2) become (with 
P = 1,..., N = number of system particles) 

S(p H = Y (d^H/drp) • 6r P = 0 and Y B DP • 6r P = 0, (c4) 

respectively, while the equation of constrainted motion (c3) assumes the form 

m P a P = Fp + Rp = F P + ^ dttid&it/dr P ) + ^ X D B DP . (c5) 

To understand the relation between the particle reactions dR, R P and their system 
counterparts R k , A k , we insert (c2) into their corresponding definitions (3.4.Id, 2d). 
We find, successively, 

(i) Rk ~S dR - e k = S dr-rid&Ff/dr) + Yj X D B D ^j ■ e k 

(dc/) H /dr) • ( dr/dq k )j + *■>■«) 

~ Yj dHidrf’H/dqk) + Ys ^D^Dki (dl) 

and, comparing with the second of (3.5.5), R k = J2 ^D a Dk , we readily conclude that 
q)/dq k = S i d Ml, r)/dr\ ■ [dr{t, q)/dq k \ = 0, (d2) 

and 

Bpk = S B ° 4 e k = a Dk ; (d3) 


recall ex. 2.4.1 and (2.6. Iff.). 

REMARK 

The above also show that, as long as the quasi variables are chosen so that 

0 = S B °' Sr = S9d = Y a ° k Sqk ^ S B '>' ek = aDk '- ( el ) 

the multipliers X D in (c2, 3) coincide with those in the second of (3.5.5). Indeed, 
e k — dotting (c3) and then S -summing, we obtain the “Routh-Voss” equations [see 
(3.5.15) below]: 

S dm a - ek = S dF ' ek+ J2 m(S (d(/) H /dr) • e,)j + 5>(S B D -e^j, (e2) 
or 

Ek = Qk + 0 + Yj X D a Dk ■ ( e 3) 

(ii) A k = S dR ' Ek = S (5Z dH^H/dr) + Y 

= Y ^niS ( df H /dr) • (dr/dd k )^j + Y Xd {S B °' E k ) 

= "Yj dH(d(j) H /d(j) k ) + Y^ Wok, 

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<fi) 


§3.5 EQUATIONS OF MOTION VIA LAGRANGE'S PRINCIPLE: GENERAL FORMS 


and, comparing with (3.5.7), A D — A D and Aj = 0, we readily conclude that 
d&n/dOk = £ ( d(j) H /dr) -s k = g (d(p H /dr) ■ (E A lk e^j 

= • • • = E A ik(d ( t > H /dqi) = 0, (f2) 

and (with k = 1 D, D' = 1,. ... in; I = m+ 1 

B'nk — S ^ D ‘ Ek ~ S • (E A ik e i) = E ^ lk (S ' e i ) = E A ik a Di = 

that is, 

B'dd' = $ ' £ d' = Sdd'i B'di = $ B d • £j = b D[ = 0. (G) 

Some of the above can also be obtained from the virtual forms of the constraints. 
Thus, we find, successively, 

(a) 

0 = ty H = S Wtf/dr) ■ hr = g {d(p H /dr) • (E e k bO^j = • • • = E (d<p H /dd k ) bQ k 

= E (WH/dd D ) b0 D + E (dcpn/dd,) SBj = 0 + E (d<P H /de,) 66, =► d0 H /dO, = 0. 

(gl) 

(b) 0 = S0 D = S b d Sr = S B d • (E «* «**) = • • • = E ^ 

= E B'od 1 69 d’ + E! B'di 69, = 0 + E B'di 69, => B di = 0. (g2) 


HISTORICAL 

The terms Lagrange’s equations of the first kind (and second kind—see below) seem 
to have originated in Jacobi’s famous Lectures on Dynamics (winter 1842/1843, publ. 
1866), and have been widely used in the German and Russian literature. They are not 
too well known among English and French authors (see, e.g., Voss, 1901, p. 81, 
footnote #220). 

Example 3.5.2 Lagrange’s Principle and Multipliers: Particle on a Surface (Kraft, 
1885, vol. 2, pp. 194-195). Let us consider a particle P of mass m moving on a 
smooth surface (p(x,y, z, t) = 0, where x,y,z are inertial rectangular Cartesian 
coordinates of P , under a total impressed force with rectangular Cartesian compo¬ 
nents (V, Y,Z). According to LP, the motion is given by (with the customary 
notations dx/dt = v x , d 2 x/dt 2 = dv x /dt = a x ,...) 

(m a x — X) Sx + (m a v — Y) by + (m a z — Z) bz = 0, (a) 

under the (virtual form of the surface) constraint 

bcf> = 0: (<9</>/ dx) bx + (dcp/dy) by + (dcp/dz) bz = 0. (b) 

By Lagrange’s multipliers, (a) and (b) combine to the unconstrained variational 
equation, 


[m a x — X — X(d(p/dx)\ bx + [m a y ~Y — \{d(p/dy)\ by 

+ [m a z — Z — A (d(p/dz)\ bz = 0, 


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(c) 


CHAPTER 3: KINETICS OF CONSTRAINED SYSTEMS 


and this leads directly to the three Lagrangean (Routh-Voss) equations of the first 
kind: 

ma x = X + \(d(f)/dx), ma y = Y + X(df/dy), ma z = Z + X(d<j)/dz). (d) 

Eliminating the multiplier A among (d) we obtain the two reactionless equations 
(with subscripts denoting partial derivatives): 

(ma x — X)/<f> x = {ma y — Y)/f y = ( ma : — Z)!f z (= A). (e) 


Next: 

(i) either we solve the system consisting of any two of (e), plus the constraint <f) = 0, 
for the three unknown functions x(t), y(t ), z(t), and then calculate A — > A(?) from 
(d), or (e), if needed; 

(ii) or we solve the system consisting of (d) and (j> = 0 for the four unknown functions 
x(t), y(t), z(t), and X(t). 

Equations (e) can also be obtained as follows: in view of (b), only two out of the three 
virtual displacements are independent, here n = 3 and m = 1. Taking Sx as the depen¬ 
dent virtual displacement, and solving (b) for it in terms of the other two (assuming 
that (j) x f 0), we obtain 

Sx = ~{(j)y/(j) x )5y - (<j) z /(j) x )5z, (f) 

and substituting this into (a), and regrouping terms, we get the new unconstrained 
variational equation of motion 

[ma y - Y - ( ma x - X)(f y /f x )\ 6y 

+ [m a z — Z — (m a x — X) (cp z / ff)] Sz = 0. (g) 

The above, since Sy and Sz are now free, leads immediately to the two reactionless = 
kinetic equations, 

ma y = Y + (nia x — X)(f v /f x ), ma z = Z + ( ma x — X)((j) z /f x ), (h) 

which are none other than the earlier eqs. (e). 

REMARKS 

(i) Equations (h) can be, fairly, called “Maggi —*■ Hadamard equations of the first 
kind”; and the extension of this idea to holonomic system variables and correspond¬ 
ing Pfaffian constraints yields “ELadamard’s equations (of the second kind)” (§3.8). 

(ii) Equations (b-d = “ adjoining of constraints”) and equations (f h = “em¬ 
bedding of constraints”) embody the two available ways of handling constrained 
stationary problems in differential calculus; although, there, the former is discussed 
much more frequently than the latter! 


Specialization 

Let the reader verify that if the surface constraint has the special form z = f(x,y), 
then: 

(i) eqs. (d, e) reduce, respectively, to 

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§3.5 EQUATIONS OF MOTION VIA LAGRANGE'S PRINCIPLE: GENERAL FORMS 


Routh-Voss equations: 

ma x = X + X(df/dx), ma y = Y + \(df/dy), ma z = Z — X, (i) 
Kinetic Maggi —> Hadamard equations (of the first kind): 

(ma x - X)/f x = ( ma y - Y)/f y = ( ma z - Z)/(-1) (= A), (j) 


=> (772 a x — X) + (m a : — Z)f x = 0, (m a v — Y) + (m a z — Z)f y = 0. (k) 

(ii) Substituting into (k): a z = z = ■ ■ ■ = xf x + yf y + (A) 2 /™ + (fflyy + 2 xyf xy ; 
that is, using the constraint and its (...)'-derivatives to eliminate 2 and its derivatives 
from them, we obtain the two kinetic equations in x,y and their derivatives alone: 

*(1 +fx 2 ) + yfxfy + itffxfxx + 2-Xyfxfxv + (jffxfyy = i X + fx Z)/m, (1) 

+fy 2 ) + xfxfy + ( yffyfyy + 2.xyf.J\, + {x) 2 f y f xx = (7 +f y Z)/m, (m) 

[which are the “Chaplygin-Voronets”-type equations of the problem (see §3.8)]. 

(iii) Solving the last of (j) for A, and then using into it the earlier expression 
z = • • •, we obtain the following (kinetostatic) expression: 


A = Z — 
= Z — 


m 


xfx + yfy + ( x) 2 f XX + (yffvy + Xyf 


(n) 


which, once the motion has been found: x = x(t), y = y(t), yields the constraint 
reaction A = A(t). 

(iv) Finally, substituting (n) into the first and second of (i), we recover (k, 1), 
respectively. 


Example 3.5.3 Let us apply the results of the preceding example to a particle P 
of mass m moving under gravity on a smooth vertical plane that spins about a 
vertical of its straight lines, Oz (positive upward), with constant angular velocity <u 
(Kraft, 1885, vol. 2, pp. 194-195). Choosing inertial axes O-xyz so that Ox coin¬ 
cides with the original intersection of the spinning plane and the horizontal plane 


O-xy through the origin, we have, for the impressed forces, 

X = 0, 7 = 0, Z = +mg\ (a) 

and, for the constraint, 

y/x = sin(uV)/cos(uV) =>■ cj>{t, x,y, z) = ycos(ut) — vsin(wt) = 0. (b) 

Therefore, equations (e) of the preceding example yield 

(. mix. — 0)/[— sin(wt)] = (my — 0)/ cos (cot) = (mz — mg)/ 0 (= —A), 

or, rearranging (to avoid the singularity caused by f z = 0), 

(mx—0) cos(otf) = (my — 0)[— sin(utf)] => xcos(wt) +y sin(wt) = 0, (c) 

(mx — 0)(0) = (mz — mg) [— sin(u;t)] => z = g, (d) 

(my — 0)(0) = (mz — mg) cos(wt) =>■ z = g. (e) 

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CHAPTER 3: KINETICS OF CONSTRAINED SYSTEMS 

Let us, now, solve (c-e). Using plane polar coordinates (r, 4>)\ x = rcos(wt) =>■ 
x = ■ ■ • => x = • • • and y = r sin(wt) => y = •••=> y = •• •, we can rewrite (c) in the 
simpler form 

r — ui 2 r = 0. (f) 

The solution of (d) = (e), with initial conditions 2 ( 0 ) = z 0 and z(0) = v 0 , is 

z = (l/2)gt 2 + v 0 t + z 0 , (g) 

while that of (f), with initial conditions r(0) = r 0 and r(0) = v r o , is 

2 ui r = (u r 0 + v, ;0 )e w ' + (ur 0 - v ri0 )e~ wt . (h) 

Equations (g, h) locate P on the spinning plane at time t, and, with the help of (b), 
specify its inertial position at the same time. (See also Walton, 1876, pp. 398-411.) 


Example 3.5.4 Lagrange’s Equations of the First Kind; Particle on Two Surfaces. 
Let us calculate the reactions on a particle P moving in space under the two con¬ 
straints (where x,y,z are the inertial rectangular Cartesian coordinates of P) 

(f>i = x 2 + y 2 + z 2 — / 2 = 0 and <j) 2 = z — y tan 9 = 0; (a) 


that is, n = 3 — 2=1: for example, the bob of spherical pendulum of (constant) 
length /, forced to remain on the plane (j> 2 = 0, that makes an angle 9 with the 
plane z = 0. Using commas followed by subscripts to denote partial (coordinate) 
derivatives, we find, from (a), 

64>i = <t>\ iX + </>i r Sy + 4>\~ 6z = 0, S(f> 2 = <\> 2 x & x + 02,y + 02,z &z = 0. (b) 

Solving (b) for the two excess virtual displacements in terms of the third, say 6y and 
8z in terms of 8x, we obtain 


where 


Sy = —(2x/J) Sx and Sz = —(2xtan 9/J) 6x, 


01, y 01,z 

02,y 02,z 


2(_v + ztan 9) (f0, assumed). 


(c) 

(d) 


Substituting Sy and Sz from (c) into the principle of d’Alembert-Lagrange for the 
particle reaction — that is, 

R x Sx + R y Sy + R~ Sz = 0, (e) 

results in 


[R x - (2 x/J)R y - (2x tan 9/ J)R Z ] Sx = R' x Sx = 0, (f) 

from which, since Sx is independent, we obtain R' x = 0; that is, 

RJ (R y + tan 9R-) = x/(v + z tan 9). (g) 

Further, the ideal reaction postulate for R: 

R= (dfi /dr) + A 2 (df 2 /dr), (h) 


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§3.5 EQUATIONS OF MOTION VIA LAGRANGE'S PRINCIPLE: GENERAL FORMS 


with (a) and in components, yields 

R X = + A 2 </> 2 , JC = • • • = 2\ { X , (i) 

R y = A| (f> ly + = • • • = 2A l v — A 2 tan0, (j) 

R~ = A[(/>i z + A 2 </ , 2 ,z = • • • = 2A(Z + A 2 ; (k) 

which, of course, are consistent with (g). Finally, since / ^ 0, we can use any two of 

(i-k) to express A ] 2 uniquely, in terms of R xyz l for instance, solve (j, k) for A ] 2 in 
terms of R r z . (See also Routh, 1891, p. 35.) 


Example 3.5.5 Lagrange’s Equations of the First Kind; and Elimination of Reactions. 
Let us consider a system of N particles, moving under the h + m = M (possibly 
nonholonomic but ideal) constraints 

Mt,r,v) = 0 (D=l,...,M<3N), (a) 

and, therefore (recalling ex. 3.2.6), having Lagrangean equations of motion of the 
first kind (we revert to continuum notation for convenience), 

dm a = dF + E A D {df D /dv). (b) 

Now, to obtain reactionless = kinetic equations of motion, we will combine (b) 
with the acceleration form of (a). To this end, we (...)'-differentiate (a) once, thus 
obtaining 

dfo/dt = dfo/dt + £ [(df D /dr) • v + (df D /dv) ■ a] = 0, (c) 

and from this, rearranging, we get 

S ( d/n/dv ) • a = ( df D /dr) • v - df D /dt. (d) 

Now, to be able to use (d) in (b), so as to eliminate a, we dot the latter with df D /dv 
and sum over the particles (with D,D' = 1,..., M ): 

S ( d fo/9v) a = S [( dF/dm) • (q f D /dv)\ + ^ V (S ( d fo'/ dv ) • ( df D /dv)/dm S j, 

(e) 


and, comparing the right sides of the above with (d), we readily conclude that 
EMS [dfn'/d(dmv)\ • (df D /dv)} 

= S { (iF ' l d fD/d(dmv)}} - £ (df D /dr) • v - dfo/dt. (f) 

Since rank[df D /d(dmv)\ = rank(df D /dv) = M, and therefore 

Det lS [df d'/ d{dmv)\ • ( df D /dv ) | f 0, (g) 

the M linear nonhomogeneous equations (f) can supply uniquely (locally, at least) 
the A/)’s as functions of the r’s, v’s, and t. Finally, substituting the so-calculated \ D ’s 
back into (b), we obtain N second-order equations for the v = r(t). (See also exs. 
3.10.2, 5.3.5, and 5.3.6; and Voss, 1885.) 

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CHAPTER 3: KINETICS OF CONSTRAINED SYSTEMS 


Problem 3.5.1 Continuing from the preceding example, find the form that (f) 
takes if the constraints (a) have the holonomic form 


<t>H{t,r) = 0 . 


(a) 


HINT 

Calculate f D = d(f) D /dt = 0, and then show that df D /dv = dcj) D /dr. 

Let us now turn to the second, and more important, “reactionless part” of LP, (eqs. 
3.2.8, 11), and express it in system variables. 


1. Holonomic Variables 

In this case LP, SI = S'W, assumes the form 

E Ek Sqk = E & k 6qk ’ 


or, explicitly, 

J2{[d/d t (d T /dq k ) - dT/dq k ] - Q k }Sq k 

= 22 { [d/dt{dT / dv k ) - dT/dq k \ - Q k } Sq k = 0. (3.5.11) 

This differential variational equation is fundamental to Lagrangean analytical 
mechanics; all conceivable/possible Lagrangean equations of motion are based on 
it and flow from it. 

(a) Now, if the n Sq’s are independent (i.e., m = 0 =>f = n DOF), (3.5.11) leads 
immediately to Lagrange’s equations of the second kind'. E k = Q k , or explicitly (re¬ 
calling the kinematico-inertial results of §3.3 in holonomic variables), 

E k = d/dt(dT/8q k ) - dT/dq k 

= d/dt(dT/dv k ) — dT/dq k = Q k [Lagrange (1780)], (3.5.12) 

= dS/dq k = dS/dv k = dS/dw k = Q k [Appell (1899)]. (3.5.13) 

Further, substituting 6q k = 22 A k i S0 t (k,l = 1,..., n) into (3.5.11) readily yields 

Z A klEk — E AkiQ k (i.e., 7/ = 0/, hut in holonomic variables) 

[Maggi (1896,1901,1903)]. (3.5.14) 

However, in this unconstrained case, neither Appell’s equations, (3.5.13), nor Maggi’s 
equations, (3.5.14), offer any particular advantages over those of Lagrange, (3.5.12); 
their real usefulness/advantages over eqs. (3.5.12) lie in the constrained case (see 
below). Equations (3.5.12) are rightfully considered among the most important 
ones of the entire mathematical physics and engineering; we shall call them simply 
Lagrange’s equations. 

(b) If the n Sq’s are constrained by (3.5.3): ffa Dk 6q k = 0 (D=\,...,m\ 
k = 1,...,«), that is, / = n — m = number of DOF, then application of the multiplier 

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§3.5 EQUATIONS OF MOTION VIA LAGRANGE'S PRINCIPLE: GENERAL FORMS 


rule, between these constraints and (3.5.11), leads immediately to the Routh-Voss 
equations 

E k — Qk + E ^ D a Dk (— Qk + Rk )> (3.5.15) 

or, explicitly, as in (3.5.12, 13), 

E k = d/dt(dT/dq k ) - dT/dq k = Q k + E X D a Dk 

[Routh (1877), Voss (1885)], (3.5.16) 

= dS/dq k = Q k + A o a Dk (Appellian form of the Routh-Voss eqs.). (3.5.17) 

The corresponding Maggi form is presented below. 

[Equations (3.5.15) are not to be confused with the other, more famous, equations 
of Routh of steady motion, etc. (§8.3 ff.)] 

CAUTION 

Some authors (e.g., Elaug, 1992, pp. 169-170) state, falsely, that if the n cf s are 
independent, the n bq's are arbitrary, and then (3.5.12, 13) follow from (3.5.11). 
But as we have seen (§2.3, §2.8, and §2.12), if the additional constraints (3.5.3, 
10b) are nonholonomic the cf s remain independent, whereas, obviously, the bq' s 
are no longer arbitrary, that is, (3.5.12, 13) do not always hold for independent q s. 


2. Holonomic —> Nonholonomic Variables 

In this case LP, 61 = b'W, assumes the form 

£ 4 M/t = £>* M/t- (3-5.18) 

(a) If the n <5(9’s are unconstrained (i.e., if m = 0=^f=n — m = n = # DOF), 
then (3.5.18) leads to I k = 0 k , or, due to the kinematico-inertial identities (3.3.10 
If.) for I k , to the following three general forms: 

4 = ^ A ik Ei = A ik Qi , 

or, in extenso, 

^ [, d/dt(dT/dq ,) - dT/dq,]A lk = ^ A lk Q, 

{Maggi form: holonomic variables), (3.5.19a) 

= E Ai k {dS/dqi) = E ^ikQi 

{Appellian form of Maggi form: holonomic variables), (3.5.19b) 

= dS*/ddi k = 0 k 

[Gibbs (1879): nonholonomic variables, but no constraints!], (3.5.19c) 
= d/dt{dT*/du k ) - dT*/d6 k + EE 7 r UdT*ldw r )u> a = 0, 

= E k *(T*) -r k = 0 , 

[Volterra (1898), Hamel (1903-1904)]. (3.5.19d) 

Equations (3.5.19a, b) have no advantages over Lagrange’s equations (3.5.12); 
but equations (3.5.19c, d) may be truly useful for unconstrained systems in quasi 

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CHAPTER 3: KINETICS OF CONSTRAINED SYSTEMS 


variables, for example, a rigid body moving about a fixed point [—> Eulerian rota¬ 
tional equations (§1.17)]. 

(b) If the 89' s are constrained by 89 D = 0, but 69, ^ 0 (i.e., if f = n — m = 
# DOF), then the multiplier rule applied to (3.5.18) yields the following two groups 
of equations: 

Kinetostatic (i.e., reaction containing) equations: 

Id = + A d [=&d + \d {D = 1,..., m)\, (3.5.20a) 

Kinetic (i.e., reactionless) equations: 

I, = 0, (/ = m + (3.5.20b) 

[and in view of the constraint 1 • 89 n+l = 1 • 6t = 0, we also have 7„ +1 = 0 n+ \ + A n+1 , 
but that nonvirtual relation is more of an energy rate-like equation (as in §3.9)]. 


Alternative Derivation of Equations (3.5.20a, b) 

First, with the help of the Kronecker delta (hopefully, not to be confused with the 
virtual variation symbol 8 ...), we rewrite the constraints 89 D = 0 as 

0 = 89 n = ^ 6 dd . 89 d , = ^ 8[)[)' 89 d , + ^ 8 DI 89, = ^ S Dk 86 k . (3.5.20c) 

Then, using the method of Lagrangean multipliers, we combine them with (3.5.18): 
(1) we multiply each constraint 89 D (y= 0) with —\ D {^ 0) and sum over D ; (2) we 
multiply each “nonconstraint” 89,{^= 0) with — X,(= 0) and sum over /; and, (3), we 
add the so-resulting two zeros to (3.5.18), thus obtaining 

E (A - & k - E 6 ™) 6 °k = 0- (3.5.20d) 

Since the 89 k can now be viewed as unconstrained, (3.5.20d) decouples to the two 
sets of equations: 

k = D'\ I d > — & D ' = ^ X D 8 DD i = A D i (Kinetic equations ), (3.5.20e) 

k = I: I, — ©i = ^2 X D 8 d / = X, = 0 (Kinetostatic equations). (3.5.20f) 

Here, too, as with the unconstrained case (3.5.19a-d), we have the following three 
general forms for (3.5.20a) and (3.5.20b): 

• Kinetic equations (with /,/' = m + 1,..., n; k = 1,..., n; 7 / = 7 /«+i): 

I I = 2L = '22 AklQki 

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§3.5 EQUATIONS OF MOTION VIA LAGRANGE'S PRINCIPLE: GENERAL FORMS 


or, in extenso, 

E [d/dt(dT/dq k ) - dT/dq k ]A kI = Y A kiQk 

[Maggi (1896, 1901, 1903): holonomic variables], (3.5.21a) 

= E A ki{dS/dq k ) = Y A kiQk 

(Appellian form of Maggi form: holonomic variables), (3.5.21b) 

= dS*/dcb, = 0j 

[Appell (1899-1925): special cases of nonholonomic variables], (3.5.21c) 

= d/dt{dT 7<9w 7 ) - dT*/dO, + EE Yn'{dT*/dui k )uj r 

+ Y 7 k i(dT*/du k ) = 0, 

[Hamel (1903-1904): “Lagrange-Euler equations”]; (3.5.21d) 

• Kinetostatic equations (with D = 1,..., m; I = m + 1,...,«; A* = 1 
Yd = 7 Vi) : 

Id = E A kDEk = E A kDQk + A D ! 

or, in extenso, 


Y [d/dt(dT/dq k ) - dT/dq k ]A kD = Y A koQk + a d, (3.5.22a) 

= E A ko(dS/dq k ) = Y^ A kDQk + a Di (3.5.22b) 

= dS* / du>£) = 0 d + Ad 

[Cotton (1907): special variables] (3.5.22c) 

= d/dt(dT*/dcj D ) - dT*/d0 D 

+EE YDiidT^d^io, + Y 7 k D (.dT*/du k ) = 0 d + A d 

[Stiickler (1955); special case by Schouten (late 1920s, 1954)]. (3.5.22d) 


REMARKS 

(i) In the absence of constraints, the above n equations in the ca’s, plus the n 
transformation equations q k = J2 a u IjJ i + A k- constitute a system of 2 n first-order 
equations in the 2 n unknown functions ui k = u> k (t) and q k = q k (t). [Or, after using 
the lo <-> q equations in them, thus expressing the ui k s in terms of the q k s, they 
constitute a set of n second-order equations for the n unknowns q k {t)-] In the presence 
of m constraints oj d = 0, the n — m kinetic equations plus the n transformation 
equations q k = J^ A ki U} i + A k constitute a system of 2n — m first-order equations 
in the 2 n — m functions u I = tu I (t) and q k = q k {t). Or, equivalently, substituting 
tjjj = a Ik q k + fl/ (fi 0) into the n — in kinetic equations, we obtain a system of 
n — m second-order equations for the q k = q k (t); and then, pairing them with the m 
constraints a nfik + «n = 0. we hnally obtain a system of (n — in ) + in = n 
second-order reactionless equations for the q k = q k (t). Further, it can be shown 
that there exists a nonsingular linear transformation q k = A k /LO/ + A k , or 

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CHAPTER 3: KINETICS OF CONSTRAINED SYSTEMS 


q a = Y^ (recalling that dq n+] /dt = w„ +1 = dt/dt = 1) that brings the non¬ 

negative kinetic energy 2T = M a pq a qp to the following sum of squares 
form: 


2 T -2 = u « 2 = E + w„+i 2 ; (3.5.23a) 

in which case, since P k = dT*/du> k = uj k and P„+ 1 = dT*/doj n+l = w„ +1 = 1, 
dT*/dd k = {dT*/dqi)A Ik = 0, the nonholonomic system inertia assumes the 
Eulerian form (recall inertia side of Eulerian rigid-body rotational equations, §1.17): 

I k dui k /dt T 'y ( y ( 'y ka ut r cu a , (3.5.23b) 

and that is why Hamel called his equations “Lagrange-Euler equations.” (However, 
by choosing the u> k s so as to nullify the dT*/dd k s, we probably end up complicat¬ 
ing the Y ka ’s.) 

(ii) The advantage of nonholonomic variables in the Hamel “equilibrium form” 
co D = 0 is that then both constraints and equations of motion decouple naturally into 
n — m purely kinetic (i.e., reactionless) equations (69j f 0; Aj = 0 =>- 7/ = 0f) and m 
reaction-containing, or kinetostatic, equations (S0 D = 0; A D f 0 => Id — 0d + A D ). 
In holonomic variables, by contrast, both (Pfaffian) constraints and (Routh-Voss 
and Appell) equations of motion are coupled. Solving the n — m kinetic equations 
(plus constraints, etc.) constitutes the lion’s share of the difficulty of the problem. 
Once this has been achieved, then the reactions A D follow immediately from the 
(now) algebraic equations: A D = A D (t) = I D (t) — 0 D (t). 

(iii) When using Hamel’s equations under the constraints uj d = 0, we must 
enforce the latter after all partial differentiations have been carried out, not before; 
otherwise, we would not, in general, calculate correctly the key nonholonomic terms 
(k,r = 1 a = 1+ 1; /' = m + 1,...,«) 

-r k = YY1 7 r ka(dT*/du r )u a = YY1 7 r ki'(dT*/du r )uj r + Y Y k (dT*/duj r )-, 

(3.5.24a) 

and, unfortunately, this drawback holds for both kinetic and kinetostatic equations. 
Let us see why. Expanding T* a la Taylor around lu d = 0, we obtain 

T* = T* 0 + ^2 (9T*/8uj d ) 0 uj d + quadratic terms in iv D , (3.5.24b) 

where 

T* a = T*(q, u> D = 0 ,u} u t) = T* 0 (^w/H); (3.5.24c) 

and, generally, (.. .) 0 = (... ,uj d = 0,...) (a useful notation, to be utilized frequently, 
for extra clarity); and, therefore, 

{dT*/du} D ) 0 f dT\/dco D = 0, (3.5.24d) 

(dT*/doj I ) 0 = dT* 0 /diu 1 d/dt[(dT*/du,) 0 ] = d/dt(dT* 0 /du I ), (3.5.24e) 
{dT*/d6 k ) 0 = Y A rk (dT*/dq,) 0 = Y A rk {dT* 0 /dq r ). (3.5.24f) 

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§3.5 EQUATIONS OF MOTION VIA LAGRANGE'S PRINCIPLE: GENERAL FORMS 


In view of these results, — T k , (3.5.24a), transforms to 

l D ki'{dT*/du} D ) 0 u r + EE 7 7 ki'(dT* 0 /duj I »)uj I ' 

+ E 7 D k(dT*/du D ) 0 + 7 MdT^/diUj)-, (3.5.24g) 

an expression that shows clearly that the presence of the first ( double ) and third 
(, single ) sums generally necessitates the use of T*, instead of T* a . 

However, with the help of the above expression we can obtain conditions that tell 
us when we can use the constrained kinetic energy T* a in Hamel’s equations right 
from the start. Let us do this, for simplicity, for the common case of the kinetic such 
equations of a scleronomic system. Then, 

-r^-r 7 = EE 7 D II ,(dT*/dco D ) 0 u I ,+ EE T* a /duj I »)u> I r, 

(3.5.24h) 

and (3.5.24d, e) make it clear that the sought conditions will result from the (iden¬ 
tical) vanishing of the first sum in (3.5.24h); that is, 

EE 1 D II ,(dT*/du D ) o u> r = 0. (3.5.241) 

But (as made clear in §3.9), 

1T * = E E M *kM)u k u, = E E ( d2r */ du} k dui)uj k uj, 

=> (dT*/dLe D ) 0 = J2(d 2T */ d “Ddu; I )u I , (3.5.24j) 

and so (3.5.24i) reduces to 

EEE 7 D IV {d 1 T*/du} D dujr'jujjnoj,, = 0; (3.5.24k) 

and from this we easily conclude that the necessary and sufficient conditions for the 
use of T*„ in Hamel’s equations are 

E 7 D IV {d 2 T*/dLo D dur) = 0. (3.5.241) 

For example, in the case of a single Pfafhan constraint, uq = 0 (i.e., m = 1), (3.5.241) 

yields 


y l w (d 2 T*/duj { dwj») = 0 (/,/',/" = 2,... ,n), (3.5.24m) 

which means that either all “nonholonomic inertial coefficients” d 2 T*/du l dujj" 
= M* |/» vanish [i.e., T* consists of an wpfree part and an ^-proportional part; 
or 7)// = 0, which means that constraint is holonomic (by Frobenius’ theorem, 
§2.12)]. The consequences of (3.5.241) are detailed in Hamel (1904(a), pp. 22-29); 
see also Hadamard (1895). 

In sum, in using the Hamel equations, even if we are not interested in 
constraint reactions, we must begin with the unconstrained kinetic energy 
T* = T*(q,uj D ,Lc I ,t), carry out all required differentiations, and then enforce the 
constraints wo = 0, at the end; and, a constraint uid = 0 can be enforced ahead of 
time in T*-terms that are quadratic in that uj d \ namely, in (.. .)to D 2 terms. 

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CHAPTER 3: KINETICS OF CONSTRAINED SYSTEMS 


This inconvenience is a small price to pay for such powerful and conceptually 
insightful equations. Similarly, a detailed analysis of (3.5.24g) shows that it is 
possible to have r k = 0 (i.e., Hamel equations —> Lagrange’s equations) even 
though not all 7 ’s are zero. An analogous situation occurs in the Maggi equations, 
even in the kinetic case—that is, 

1 i=Yl Ak,Ek = [ d /dt(dT/dq k ) - dT/dq k \A kI = A kI Q k , (3.5.24n) 

since k= 1we have to calculate T = TfijCpqnfij)', the “reduced,” or con¬ 
strained, kinetic energy 


To = T{t,q,q D = b nl q, + b n ,q,) = T 0 (t,q,q >), (3.5.24o) 

obviously will not do. This seems to be a drawback of all T-based (i.e., Lagrangean) 
equations. No such problems appear for the kinetic Appellian equations: there, with 
the convenient notation 


S* = S*(t,q,u} D ,u> I ,u> D ,u)i) = original, or unconstrained, or relaxed, Appellian 
-*■ S*(t,q,u D = 0 ,u,,w D = 0, w/) = 

= S* 0 = constrained Appellian, (3.5.25a) 

and the help of the Taylor expansion (with some obvious calculus notations) 

S* = S* a + ^2 [(9S*/duj D ) 0 uj D + (dS*/du D ) 0 tjj D \ + quadratic terms in uj d ,lj d , 

(3.5.25b) 


we get the general results [similar to (3.5.24c, e)] 

(dS*/du I ) 0 = dS* 0 /du I and (dS*/dcu D ) 0 ± dS* 0 /dcu D = 0. (3.5.25c) 

Therefore, if we are not interested in finding constraint reactions, we can enforce the 
constraints lo d = 0 and lu d = 0 into S* right from the beginning; that is, start work¬ 
ing with S* 0 , and thus save a considerable amount of labor. This property, due to the 
first of (3.5.25c), marks a key difference between the equations of Appell and Hamel, 
and their corresponding special cases. 


Special Case 

If all constraints on the q's are holonomic and have the equilibrium form 
0d c 1d = constant = q Do , then 

{dS*/d^) 0 -+ Ej{T)\ 0 - Ej(T 0 ), 

where T a = T(t,q D = constant, q h q D = 0, qj) = T 0 (t, q h q,), and, similarly for the 
impressed forces, Qj = Q\(t,q, q) —> — Qioi aQ d so the kinetic equations 

become E, = E,(T n ) = dS 0 /dq, = Q Io . 

(iv) Comparison between Lagrange’s equations of the first and second kind, and 
their respective constraints. Those of the first kind, eq. (ex. 3.5.1: c3), constitute a 
set of 


3 N + (h + m) = [(3iV — h) + m\ + 2h = (n + m) + 2 h 

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§3.5 EQUATIONS OF MOTION VIA LAGRANGE'S PRINCIPLE: GENERAL FORMS 


scalar equations, for the 3 N + (.h + m) unknown functions: 

{x P ,y P ,z P ; P=l,...,N}, {p ff ; H = 1,..., h} and {\ D ; D = 1,... ,m}. 

Once the positions (and hence accelerations) and multipliers become known functions 
of time, (ex. 3.5.1: c2) supply the reactions. 

Those of the second kind, actually the Routh-Voss equations (3.5.15, 16), con¬ 
stitute a set of 


n + m = (3 N — h) + m 

equations for the n + m unknowns: 

{q k ; k=l,...,n} and {X D ; D = l,... ,m}. 

Once the p’s and A’s have been found as functions of time, then r P = r P (t, q) —* r P (t) 
[—> a P = a P {t)], and, again, (ex. 3.5.1: c2) supply the reactions. From the latter and 
the (now) known A’s, we can calculate the p’s. 

In sum, in the second-kind case we have 2 h fewer equations, which is the result of 
having absorbed the h holonomic constraints into that description with the 
n = 31V — h q s [see remark (v) below]. Also, even in the presence of additional 
holonomic and/or nonholonomic constraints, we still work with the unconstrained 
kinetic energy T. 

However, and this is a general comment, the ultimate judgement regarding the 
relative merits of various types of equations of motion must be shaped by several, 
frequently intangible/nonquantifiable considerations (in the sense of the famous 
Machian principle of Denkokonomie), in addition to the mere tallying of their num¬ 
ber of equations, and so on (“bean counting”). 

(v) Purpose for appearance of the multipliers. That the multipliers p. H , of the h 
holonomic constraints <j> H (t,r) = 0, are not present in Lagrange’s equations of the 
second kind (and in the Routh-Voss equations) is no accident: the m A^’s (and this 
is a general remark) express the reactions of whatever constraints have not been 
taken care of by our chosen p’s; that is, they are due to the additional holonomic and/ 
or nonholonomic constraints not yet built in (or embedded, or absorbed) into our 
particular q’s description. Then, the multipliers appear as coefficients in the virtual 
work of the reactions of these additional constraints. 

(vi) Apparent indeterminacy of Lagrange’s equations. Let us consider a system 
with equations of motion 

E k = d/dt(dT/dq k ) - dT/dq k = Q k . (3.5.26a) 

Since, as explained earlier, all possible constraints are already built in into the chosen 
p-description, the corresponding system constraint reactions R k have been elimi¬ 
nated from the right side of (3.5.26a); the Q k are wholly impressed. However, occa¬ 
sionally, the latter depend on constraint reactions: for example, the sliding 
Coulomb-Morin friction F on a particle sliding on a rough surface — according 
to our definition, an impressed force — is given by 

-FN(v/\v\), (3.5.26b) 

where N = normal force from surface to particle (clearly, a contact constraint reac¬ 
tion), p = sliding friction coefficient, and v = particle velocity relative to the surface. In 
such a case, if we embed cdl holonomic constraints into our p’s, and hence into our T 

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426 CHAPTER 3: KINETICS OF CONSTRAINED SYSTEMS 

and Qf s, the resulting Lagrangean equations (3.5.26a) will, in general, constitute an 
indeterminate system; that is, the total number of equations, including constitutive 
ones like (3.5.26b), will be smaller than the number of unknowns involved. Such an 
indeterminacy [what Kilmister and Reeve (1966, p. 215) call “failure” of Lagrange’s 
equations] can be easily removed by relaxing the system’s constraints, and thus 
generating the hitherto missing equations (see also “principle of relaxation” in 
§3.7). Similar “failures” would appear if one used minimal quasi velocities to 
embed all nonholonomic constraints (see also Rosenberg, 1977, pp. 152-157). 

(vii) We have presented the four basic types of equations of motion: Routh-Voss, 
Maggi, Hamel, and Appell. They can be classified as follows: 

Kinetic energy- based equations of motion 

Holonomic variables: Routh-Voss (coupled) 

Maggi (uncoupled: kinetic, kinetostatic) 
Nonholonomic variables: Hamel (uncoupled: kinetic, kinetostatic) 

Acceleration- based equations of motion 
Holonomic variables: Appell (coupled) 

Nonholonomic variables: Appell (uncoupled: kinetic, kinetostatic) 

Additional special cases and/or combinations of the above—for example, equations 
of Ferrers, Hadamard, Chaplygin, Voronets, et al.—are presented in §3.8. 

• From all the equations of constrained motion given earlier, only those by Hamel 
(and their special cases—see §3.8), through their 7 -proportional terms (recall 
Hamel’s formulation of Frobenius’ theorem, §2.12), can distinguish between genu¬ 
inely nonholonomic Pfaffian constraints and holonomic ones disguised in Pfaffian/ 
velocity form. All other types, that is the equations of Routh-Voss, Maggi, Appell 
(and their special cases—see §3.8), hold unchanged in form whether their Pfaffian 
constraints are holonomic or nonholonomic; that is, those equations cannot detect 
nonholonomicity, only Hamel's equations can do that. 

• On the other hand, only Appell’s equations preserve their form in both holo¬ 
nomic and nonholonomic variables', and, in the kinetic ones, the nonholonomic con¬ 
straints can be enforced in the Appellian function right from the start. 

(viii) The terms kinetic and kinetostatic, in the particular sense used here (brought 
to mainstream dynamics by Heun and his students, in the early 20th century), and 
observed by some of the best contemporary textbooks on engineering dynamics, for 
example, Butenin et al. (1985, vol. 2, chap. 16, pp. 330-339), Loitsianskii and Lur’e 
(1983, vol. 2, chap. 28, pp. 345-384), Ziegler (1965, vol. 2, pp. 146-152), are not well 
known among English language authors, and so one should be careful in comparing 
various references. 

(ix) Finally, we would like to state that we are not partial to any particular set of 
equations of motion; all have advantages and disadvantages; all are worth learning! 

All such conceivable equations (whose combinations and special cases are practi¬ 
cally endless; see also §3.8) flow out of the differential variational principles of 
analytical mechanics; that is, the principles of Lagrange and of relaxation of the 
constraints, in their various forms (see also §3.6 and §3.7). These principles, being 
invariant, constitute the sole physical and mathematical glue that holds all these 
(coordinate and constraint-dependent) equations of motion together—and they keep 
reminding us that, in spite of appearances, there is only one ( classical ) mechanics! 

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§3.5 EQUATIONS OF MOTION VIA LAGRANGE'S PRINCIPLE: GENERAL FORMS 


Geometrical Interpretation of the Uncoupling of the 
Equations of Motion into Kinetic and Kinetostatic 

The Routh-Voss equations, 

E k = d/dt(dT/dq k ) - dT/dq k = Q k + ^ A D a D k, (3.5.27a) 

represent an equation among (covariant) components of vectors at a point (q) in 
configuration space, or a point (t,q) in event space. Now, we recall from §2.11, eq. 
(2.11.19a IT.), that the n — in vectors A/ T = (A u ,..., A nI ) span, at that point, the 
null, or virtual, hyperplane (or affine space) Nj = V, of the constraint matrix 
A d = (a Dk ); while the m vectors A D T = (A lD ,... ,A nD ) span its orthogonal comple¬ 
ment, the range, or constraint, hyperplane (or affine space) C m . Therefore, multi¬ 
plying (3.5.27a) with A kl (A kD ) and then summing over k, from 1 to n, means 
projecting that equation onto the local virtual ( constraint) space; and since the con¬ 
straint reactions R k = Aare perpendicular to the virtual space, they disap¬ 
pear from the kinetic Maggi equations. Indeed, we have, successively, 

(i) ^ A kI E k = ^ A kI Q k + ^ ^ A D a DkA k i 

= ^ A k iQk + ^ A D 6 di = ^ A kl Q k + 0, 

that is, 

^ A k ,E k = ]T A kf Q k or I, = (3.5.27b) 

(ii) A kD' E k = A kD'Qk + A D a Dk^kD' 

= ^ A k D'Qk + ^ A D Sdd 1 = ^ A kD 'Qk + X D t, 

that is, 

A k oEk = A kD Q k + A D or I D = 0 D + X D . (3.5.27c) 


Tensorial Treatment 

(Kinetic complement of comments made at the end of §2.11; may be omitted in a first 
reading.) In the language of tensors (whose general indicial notation begins to show 
its true simplicity and power here), the //, 0/, A k = 0 ( I D , 0 D ,A D = X D ) are covar¬ 
iant components of the corresponding system vectors along the contravariant basis 
A 1 (A d ), which is dual to the earlier basis Aj(A d ). Dotting the vectorial Routh-Voss 
equations [fig. 3.1(a)] 


E = Q + R, (3.5.28a) 

where (with summation convention) R = R k E k = (A fl a° k )E k (i.e., R is per¬ 

pendicular to the virtual local plane) with Aj = A'^Ek —that is, projecting it onto the 
virtual local plane — yields 


E-A, = Q-A l +R-A Ii (3.5.28b) 

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CHAPTER 3: KINETICS OF CONSTRAINED SYSTEMS 


(a) (b) 




Figure 3.1 (a) Geometrical interpretation of uncoupling of equations of motion ("Method 

of projections" of Maggi); and (b) its application to the planar mathematical pendulum. 

or, since R- A, = X D (A D ■ Aj) = X D 6 D j = 0, finally E ■ Aj = Q- Aj, 

i.e., A k t E k = A k ,Q k (kinetic Maggi), or I I = 0 I - (3.5.28c) 

while dotting them with A D = A k D E k —that is, projecting it onto the constraint local 
plane—yields 

E-A d = Q-A d + R-A d , (3.5.28d) 

or, since R - A D = X D '(A D ■ A D ) = X D : 6 D D = X D , finally 

A k D E k = A k D Q k + X n (kinetostatic Maggi), or I D = 0 D + X D . (3.5.28e) 

For the planar mathematical pendulum of length /, mass m, and string tension 5 
[fig. 3.1(b)], Aj = dr /d(j> = along tangent, A D = dr/dr = along normal , and so 
(3.5.28b, d) become 

E-Aj = Q-Aj\ ml(d 2 (f>/dt 2 ) = —mg sincj) (kinetic Maggi eq.), 

(3.5.28f) 

E ■ A D = Q-Aj) + R- A D : ml(d(f>/dt) = —mg cos cj)+ S (kinetostatic Maggi eq.). 

(3.5.28g) 

These geometrical considerations demonstrate the importance of the method of pro¬ 
jections of Maggi, over and above that of the Maggi equations. His method can be 
applied to any kind of multiplier-containing (mixed) equations. 


Example 3.5.6 Lagrange’s Equations (Williamson and Tarleton, 1900, pp. 437- 
438). Let us consider a scleronomic system described by the Lagrangean equations 

d/dt(dT/dv k ) — dT/dq k = Q k (k = !,...,«). (a) 


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§3.5 EQUATIONS OF MOTION VIA LAGRANGE'S PRINCIPLE: GENERAL FORMS 


Now, the change of the system momentum p k = dT/dv k during an elementary time 
interval dt is ( dp k /dt)dt , and this, according to (a), equals Q k dt + [dT/dq k ) dt. 
Since the system is scleronomic, dT/dq k = quadratic homogeneous function of the 
v k ’s (see also §3.9), and therefore if the system is at rest, it vanishes. Hence, the result: 
The elementary change of a typical component of the system momentum consists of two 
parts: one due to the corresponding impressed force, and one due to the [possible ) 
previous motion. 


Problem 3.5.2 Lagrange’s Equations: 1 DOF. Let us consider the most general 
holonomic and rheonomic 1 DOF system; that is, n = 1 and m = 0, with inertial 
(double) kinetic energy 


2 T = A(t, q)q 2 + 2B(t, q)q + C[t, q), [A, C > 0, always) 
and hence Lagrangean (negative) inertial force 
E q {T) = [dT/dq)' — dT/dq 

= (1/2) [2Aq + [dA/dq)q 2 +2[dA/dt)q + 2[dB/dt) -dC/dq], 


(i) Show that the new Lagrangean coordinate x, defined by 


x = 


[A[t,q)) xl2 dq 


x{t, q) q=q(t,x), 


(a) 

(b) 

(c) 


reduces 27" to 


2T = x 2 + 2b[x, t)x + c(x, t ), (d) 

where 

b[t,x) = (A l/2 [B/A + dq/dt) ) (e) 

k ) evaluated at q=q(t,x) 

c(t,x) = (A(dq/dt) 2 + 2B(dq/dt) + cl , (f) 

) evaluated at q=q(t,x) 

and generates the following (negative) Lagrangean inertial force: 

E X [T) = ( dT/dx)' — dT/dx = d 2 x/dt 2 + db/dt — (l/2)(<9c/c>x); (g) 

that is, no [dx/dt)-proportional (i.e., damping/friction) terms. Such coordinate trans¬ 
formations may prove useful in nonlinear oscillation problems. 

(ii) Show that in the scleronomic case, i.e., when B, C = 0 and hence 2T = A(q)q 2 , 
the inertia forces (b) and (g) reduce, respectively, to 

A[d 2 q/dt 2 ) + [\/2)[dA/dq)[dq/dt) 2 and d 2 x/dt 2 . (h) 


Problem 3.5.3 Lagrange’s Equations: 1 DOF. Let us consider a 1 DOF system 
with kinetic and potential energies 

2 T = A[q)[dq / dt) 2 and V = V(q), (a) 

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CHAPTER 3: KINETICS OF CONSTRAINED SYSTEMS 


respectively, capable of oscillating about its equilibrium position q = 0. Show that 
the period of its small amplitude (i.e., linearized, or harmonic) vibration equals [with 
(••■)' = d(...)/dq] 

2n[A(0)/V"(0)} 1/2 . (b) 


HINT 

Here, A( 0) > 0, V(0) = 0, V'(0) = 0, V"(0) > 0; and, as shown in §3.9 If., 

Q = -dV/dq = - dV/dq = -V’. 

Expand T and V a la Taylor about q = 0, and keep only up to quadratic terms in q 
and q, etc. 

Problem 3.5.4 Lagrange’s Equation: 1 DOF. Continuing from the preceding 
problem, show that if q = q a is an equilibrium position, instead of q = 0, then (b) 
is replaced by 


2n[A(q 0 )/V"(q 0 )] 1 ' 2 . (a) 

Problem 3.5.5 Lagrange’s Equations: Pendulum of Varying Length. Show that 
the planar oscillations of a mathematical pendulum of varying, or variable, length 
/ = /(f) = given function of time, on a vertical plane, are governed by the (variable 
coefficient) equation 

+glsin(f) = 0 => d 2 (f>/dt 2 + 2(j/ l)(d(j> / dt) + (g/l) sin^ = 0, (a) 

where <j> = angle of pendulum string with vertical. 

For the treatment of special cases, see for example, Lamb (1943, pp. 198-199). 


Example 3.5.7 Lagrange’s Equations: Planar Double Pendulum; Work of 
Impressed Forces. Let us consider a double mathematical pendulum in vertical 
plane motion, under gravity [fig. 3.2(a)]. Below we calculate the components of 
the system impressed force by several methods. 


(a) 


(b) 


(C) 





Figure 3.2 (a-c) Double planar mathematical pendulum, under gravity; 
calculation of impressed system forces. 


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§3.5 EQUATIONS OF MOTION VIA LAGRANGE'S PRINCIPLE: GENERAL FORMS 

(!) From the Q k -definitions (§3.4). Here, with q x = 0i, q 2 = 02 and some obvious 
notations, we have 

I*! = (/j cos0i, /[ sin0 l7 0), r 2 = cos0j + / 2 cos <j> 2 , l\ sin0! + l 2 sin0 2 ,0), (a) 

F\ = (™ig,0,0), F 2 = (m 2 g. 0,0), (b) 

and, therefore, we obtain 

Qi = S dF 1 ( dr / d di) = F i • (9fi/dqi) + F 2 • (dr 2 /dq x ) 

= ■ ■ ■ = —m x gl i sin0! - m 2 gl\ sin = —(m, + m 2 )gl\ shn^, (c) 

Q 2 = S dF ' ( dr / d( h) = P\ ■ {dr\/dq 2 ) + F 2 • (, dr 2 /dq 2 ) 

= • • • = —m 2 g l 2 sin 0 2 . (d) 

(ii) Directly from virtual work. Let us find Q 2 , that is, S'W for 8<j>\ =0 and 
<50 2 f 0: (8'W) 2 = Q 2 8(j> 2 . Referring to fig. 3.2(b), we have 

(8'W) 2 = (m 2 g)S(l 2 cos 0 2 ) = —m 2 gl 2 sin 0 2 8(j> 2 => Q 2 = — m 2 gl 2 sin0 2 . (e) 

Similarly, to find Q\ —that is, 8'W for <50] f 0 and <50 2 = 0: (b'W) { = Q\ <50j, 
referring to fig. 3.2(c), we find 

(8'W) { = ( m x g ) 6(li cos 00 + (, m 2 g ) 8(l x cos 00 = (m x + m 2 )g 8(1 x cos 00 

= ~(m\ + m 2 )gl\ sin0! <50] => Q x = ~(m x + m 2 )gli sin0]. (f) 

(iii) From potential energy (see also §3.9). Here, the total potential energy of 
gravity (^impressed forces), V = F(0i,0 2 ), is 

V = —(m\g)(l\ cos 00 - (m 2 g)(l\ cos0, +/ 2 cos0 2 ) 

= -(m x +m 2 )gl[ cos0! -m 2 gl 2 cos0 2 , (g) 

and since S'W = —8V, we obtain 

£?i = -dV/dfi = -(mi +m 2 )gli sin0!, g 2 = -dV/dcf) 2 = -m 2 g/ 2 sin0 2 . 

(M) 


REMARK 

Had we chosen as system positional coordinates (fig. 3.3) 

<7i = #1 = 01 and q 2 = 6» 2 = 0 2 - 0! = 0 2 - 0,, (j) 

then (g) would assume the form 

F = F(0i, 0 2 ) = -(m, + m 2 )gli cos9i - m 2 gI 2 cos (9 X +9 2 ), (k) 

and the corresponding Lagrangean forces would be 

Q x = -dV/d9 x = -(mi + m 2 )gl x sin 0, - m 2 gl 2 sin (0, + 0 2 ), 

Q 2 = —8V/d9 2 = — m 2 g l 2 sin(9i +9 2 ). 

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( 1 ) 

(m) 


CHAPTER 3: KINETICS OF CONSTRAINED SYSTEMS 



I 

Figure 3.3 Double planar mathematical pendulum under gravity; 
alternative coordinates. 


Example 3.5.8 Lagrange’s Equations: Planar Double Pendulum; Derivation of 
Equations of Motion. Continuing from the preceding example (and its figures), 
let us first calculate the kinetic energy of the pendulum. We find, successively, 


jci = /| cos 0! => jcj = —/ 1 sin 0!, (a) 

y\ = l\ sin =>j>i =hh cos0 1; (b) 

X 2 = 1 1 cos (j)\ + 12 COS (j) 2 =>■ x 2 = sin^! — l 2 f 2 Sin(j) 2: ( c ) 

y 2 = !\ sin (f> x + l 2 sin^ 2 => y 2 = h<Pi cos^] + l 2 f 2 cosfy, (d) 

vi 2 = (.t 1 ) 2 + (j,) 2 = ••• = /! W, (e) 


v 2 ~ = (x 2 ) 2 + (yi)~ = ■■■ = i\~{4> i) 2 + 2 / 1 / 2 cos (</>2 — f 2 + 4“(<fe) 2 ; (f) 

2 T = m x vf + m 2 v 2 

= ■ ■ ■ = {m x + m 2 )l\-(<j) i) 2 + 2m 2 l x l 2 cos{<f> 2 — <pi)fi<j> 2 + m 2 / 2 2 (</> 2 ) 2 ; (g) 

and by the preceding example, 

Qi = -( m i+m 2 )glism(j> l , Q 2 =-m 2 gl 2 sin f 2 . (h) 

From the above, we obtain 

dT/dfi = {ni\ + m 2 )l x ~(j)i + m 2 lf 2 cos(cf> 2 — <t>\)<j> 2 , 

(dT/dfaY = {m x +m 2 )l x 2 ij)\ 

+ m 2 l\l 2 cos (</> 2 — f\)cf> 2 — m 2 I\l 2 sin(</> 2 — fi)(4> 2 — 4 >i) 4> 2 , 
dT/dfi = m 2 lf 2 sin (</> 2 — (f> x )f x f 2 . 


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(j) 





§3.5 EQUATIONS OF MOTION VIA LAGRANGE'S PRINCIPLE: GENERAL FORMS 


Therefore, Lagrange’s equaLion for q x = fa\ (dT/dfa)' — dT/dfi = Q x , becomes 
after some simple algebra, 

(in\ T mi)l\~(d(f>\/dt ) —|— 77^2/1 /^z cos ((/>2 — $1 ) (d(f 2 !dt ) 

— m 2 l\l 2 sin (</>2 — 4>\)(dfa/dt) 2 + (in x + m 2 )gl\ sin^q = 0 . (k) 

Similarly, we find Lagrange’s equation for q 2 = fa'. 

m 2 l 2 2 {d 2 (j) 2 /dt 2 ) + m 2 l x l 2 cos(fa — (j)\){d 2 ({)\/ dt 2 ) 

+ m 2 l\l 2 sin (</> 2 — (j)\){d(j)\ / dt) 2 + m 2 gl 2 sin fa = 0 . ( 1 ) 

The above constitute a set of two coupled nonlinear second-order equations for 
fa(t) and <j> 2 (t). 


Constraints 

(i) Assume, next, that we impose on our system the constraint 

/1 =yi = h sin^i =0 [=> fa(t) = 0 =>84>\ = 0]; (m) 

that is, we restrict the upper half OP\ to remain vertical, so that the double pendulum 
reduces to a simple pendulum P\ P 2 oscillating about the fixed point P x . 

Since df/dfa = l\ cos fa = !\ and df/d(j) 2 = 0 [=> <5/j = {l\ cosfa)5fi + {0)S(f>2 = 
(l\)8fa + ( 0 )^ 2 ], the equations of motion in this case are (k) and ( 1 ), but with the terms 
A] /1 cos</>i = Ai l\ and and Ai -0 =0 (where X\ = multiplier corresponding to the con¬ 
straint df\ = 0 ) added , respectively, to their right sides; that is, in general, it is not enough 
to simply set in these two equations f 1 = 0 (=>- fa = 0 , <f>\ = 0 )! Indeed, then the 
equations of the (m)-constrained pendulum motion decouple to the Routh-Voss equations: 


fa: X x = m 2 l 2 [cosfa(d' , fa/df) — sinfa(dfa/dt) ] (kinetostatic) , (n) 

fa : d 2 fa/dt 2 + {g/l 2 ) sin (/> 2 = 0 {kinetic). (o) 

With the initial conditions at, say, t = 0: <p 2 {0) = fa = 0 and fa(0) = fa, equation (o) 
readily integrates, in well-known elementary ways, to (the energy equation) 

(fa) 2 = (faf ~ (2g// 2 )(l - cos fa), (p) 

in which case, (n) yields the constraint reaction in terms of the angle fa = faf) and its 
initial conditions 

Ai = ■ ■ ■ = m 2 [(2 - 3 cos fa)g - l 2 (fa) 2 ]sinfa = Ai (t,fa,fa). (q) 

Finally, since 

6 , W R = [X l (df/dfa)\Sfa = R l 6fa 

= (Xil x cosfa)6fa = X l 6(l l sin fa) = A, 6y x (= 0), (r) 


the multiplier represents the (variable) horizontal force of reaction needed to preserve 
the constraint (m). [Other forms of (m) will result in different, but physically equiva¬ 
lent, forms of the multiplier. See also §3.7: Relaxation of Constraints.] 

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CHAPTER 3: KINETICS OF CONSTRAINED SYSTEMS 

(ii) Similarly, if 0 2 acquires a prescribed motion, say <j) 2 = f{t) = known 
function of time, then, since in that case <50 2 = bf(t) = 0 [= (0) bfi + (1) <50 2 ], we 
must add a term A 2 - 0 = 0 to the right side of the 0]-equation, and a term A 2 • 1 
to the right side of the 0 2 -equation [where A 2 = multiplier corresponding to the 
constraint f 2 = 0 2 — /(/) = 0 => 8f 2 = 0]. The rest of the calculations are left to the 
reader. 


Problem 3.5.6 Constrained Double Pendulum. Continuing from the preceding 
example, assume that we impose on our pendulum the constraint 

fi=y 2 = h sin 0 i + h sin 0 2 = 0 . (a) 

(i) Show that in this case, and for the special simplifying choice l x = l 2 = 
/ [=> sin 0 i + sin 0 2 = 0 => 0 i + 0 2 = 0 ], the equations of motion reduce to 

(m x + m 2 )l 2 (d 2 cj) x /dt 2 ) — m 2 l 2 cos(2</) x ) (d 2 f x /dt 2 ) + ;w 2 / 2 sin( 20 1 )(c/ 0 1 /dt ) 2 

+ (m x + m 2 )g l sin 0! = Ai/cos0 1; (b) 

— m 2 l 2 (d 2 <f>\/dr) + m 2 l 2 cos( 20 i) (d 2 f x /dt 2 ) — m 2 l 2 sin( 20 1 )(c/ 0 1 /(f /) 2 

- m 2 g I sin f x = Ai/cos 0 i. (c) 

(ii) From the above, deduce that [e.g. by adding (b) and (c) etc.]: 

A[ = (m 1 // 2 )(l/cos 0 1 )(c? 2 0 1 /r(t 2 ) + (m 1 g/ 2 )tan 0 1 . (d) 

Interpret the multiplier Aj physically. 

(iii) From the above, deduce that [e.g. by subtracting (b) and (c) from each other etc.]: 

(m!+4m 2 sin 2 0 i)(ii 2 0 i/fl!f 2 )+ 2 m 2 s'm(2(/)i) (dfi / dt) 2 + (m l +2 m 2 )(g/l) sin 0 ! = 0 ; (e) 

i.e., a single pendulum-like, reactionless (kinetic) and nonliner equation. 


Example 3.5.9 Small (Linearized) Oscillations of Double Pendulum. Continuing 
from the preceding example, let us study the small (linearized) amplitude/velocity/ 
acceleration oscillatory motions of our planar double mathematical pendulum 
about its equilibrium configuration 0 ! = 0 , 0 2 = 0 . 

There are two ways to proceed. Either (i) we keep up to quadratic terms in 0 1; 0 2 
and their derivatives in T and V (or up to linear ones in the g’s) so that the 
corresponding Lagrangean equations end up linear in these functions; or (ii) we 
directly linearize the earlier-found equations of motion (for a more general treatment 
of linearized motions, see §3.10). 

Let us begin with the first way; it is not hard to show that the earlier T , V (Q i 2 ) 
approximate to the homogeneous quadratic (linear ) forms: 

2 T = ( m x + m 2 ) I f (fi)" + 2m 2 l i/ 2 0 i 02 + m 2 l 2 2 (ff) 2 (a) 
2V = (m x + m 2 )gl x (f> 2 + m 2 gl 2 (j) 2 2 + constant terms , (b) 

Q\ = -(mi + m 2 )g /[ 0 i, Q 2 = —m 2 g l 2 f 2 . (c) 


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§3.5 EQUATIONS OF MOTION VIA LAGRANGE'S PRINCIPLE: GENERAL FORMS 


Then, with L = T — V = Lagrangean of the system, we easily obtain 

dL/df i = (mi -\- m 2 )l 2 fi + m 2 lil 2 ^> 2 , (d) 

(dL/dfi) = (nii + m 2 )l\~4>\ + m 2 lil 2 (f> 2 , (e) 

dL/dfi = -(mi +m 2 )gh<t>\ (= Q\)\ (f) 

dL/d (}>2 = m 2 l iL^i + m 2 ^i ^21 (g) 

(dL/dff) = m 2 l il 2 cji + m 2 l 2 2 (j> 2 , (h) 

dL/ d(j> 2 = —m 2 g l 2 cj) 2 (=Qi). (i) 

Therefore, Lagrange’s linearized (but still coupled!) equations are 

(nii + m 2 )li(d 2 (j)i/dt 2 ) + m 2 l 2 (d 2 (j) 2 /dt 2 ) + (m x + mfgfi = 0, (j) 

1 1 (d~ fi / dt) + l 2 (d~cf> 2 /dr) + g <fi 2 = 0. (k) 


The reader can verify that (j, k) result by direct linearization of (k, 1) of the preceding 
example, respectively. 


Solution of System of Equations (j, k) 

As the theory of differential equations/linear vibration teaches us, the general solu¬ 
tion of this homogeneous system is a linear combination, or superposition, of the 
following harmonic motions (or modes)'. 

fi = A sin(wt + e) and cj) 2 = Bsin(ujt + e), (1) 

where A,B = mode amplitudes, to = mode frequency, and e = mode phase. Substitu¬ 
ting ( 1 ) into (j, k), we are readily led to the algebraic system for the mode amplitudes: 

[(nii + m i)(g ~ hu 2 )\A + (—m 2 l 2 tj?)B = 0, (m) 

(— liui 2 )A + (g — 1 2 cj 2 )B = 0. (n) 

The requirement for nontrivial A and B leads, in well-known ways, to the determi- 
nantal (secular) equation 


(mi +m 2 )(g- liu?) 

-fui 2 


— / 7 ? 2 12^ 

g - hu 2 


= 0 , 


(o) 


which, when expanded, becomes 

(milffuf - [(mi + m 2 )(li + l 2 )g\ur + (m x + m 2 )g 2 = 0 . (p) 

To simplify the algebra we, henceforth, assume that m x = m 2 = m and l\ = l 2 = l. 
Then (p) reduces to 

tv 4 - A(g/l)u? + 2 (g/l) 2 = 0, 


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CHAPTER 3: KINETICS OF CONSTRAINED SYSTEMS 


and its positive roots can be easily shown to be 

w i = {P - (2) 1/2 ]0//)} 1/2 {lowerfrequency), (rl) 

w 2 = {P+(2 ) 1/2 ](g//)} 1/2 (> uj u higher frequency). (r2) 

For lu = u x , u> 2 , the amplitude ratios 

H^B/A = [l l u?/{g-l 2 u?)}=J/[{g/l)-J} [=n{u?)] (s) 

[obtained from (n), for /[ = l 2 ] are found to be 

Mi = B x /A x = [2 - (2) 1 / 2 ]/[(2 ) 1/2 - 1] = (2) 1 / 2 , (si) 

M 2 = B 2 /A 2 = -[2 + (2)‘/ 2 ]/[l + (2) 1 / 2 ] = —(2) 1 / 2 , (s2) 

that is, B x = {2) l ' 2 A x and B 2 = — (2)^ 2 A 2 , for any initial conditions, and therefore 
the general solution of (j, k) is 

— 4>\,l + ^1,2) = 4*2,\ T 4*2, (t) 


where 


0i,i — A\ sin(c<Di/ + £i), 02,i — Mi^i sin(cDi? + £i), (tl) 

(f> 1,2 = ^2 sin(w 2 t + £ 2 ) , 0>,2 = M 2^2 sin(w 2 t + s 2 )- (t2) 

The above show that, for each frequency u) k {k = 1,2), the ratio of the correspond¬ 
ing mode amplitudes <j> x k and <j> 2k is constant ; that is, independent of the initial 
conditions 


02,i / 0i,i — Mi — (2) 1 / 2 and 02, 2 /0i,2 — M2 — —(2)*^“- (t3) 

The remaining four constants A x , s x , and A 2 , e 2 are determined from the initial 
conditions. 

For example, if at t = 0 we choose = 0, fa = 0, and f 2 = f 0 , <fi 2 = 0, then, 
since 


<j) x — A j lo x cos(cc j t T £ x ) -f- A 2 uj 2 cos(c o 2 t T e 2 ) , (ul) 

<j) 2 = (2) A x uj x cos(uij t-fCj) — (2) *' ~ A 2 lu 2 cos(cc 2 t T c 2 ) , (n2) 


eqs. (t-t 2 ), the above, and the initial conditions lead to the following algebraic 
system: 



<t>\- 

0 = Ai sin^ + A 2 sine 2 , 

(vl) 



</>o = (2 ) X/1 A { sin^ - (2) l//2 A 2 sin e 2 , 

(v2) 


01 : 

0 = A j UJ\ COS £\ ~\~ A 2 (jJ 2 cos e 2 , 

(v3) 


fi- 

0 = ( 2 ) ^A\uj\ cos£i — ( 2 ) t~A 2 uj 2 cose 2 . 

(v4) 

From 

0 => 
[( 2) 1/2 

the last two equations, we readily conclude that cos e x = cos e 2 = 
e x = e 2 = 7 t/ 2 ; and so the first two reduce to A x + A 2 = 0 and A x — A 2 = 
/2]f 0 , and from these we easily find A x = [(2) 1/<2 /4 ]f 0 and A 2 = 


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§3.5 EQUATIONS OF MOTION VIA LAGRANGE'S PRINCIPLE: GENERAL FORMS 




LOWER Frequency ( CO p HIGHER Frequency (tip 

Figure 3.4 Angular modes of planar double pendulum, for its two frequencies: (a) lower 
frequency, (b) higher frequency. The amplitudes of <j >i t, <j>-\ 2 depend on the initial conditions. 


— [(2)'' 2 /4]^ 0 . Hence, the particular solution of our system (j, k), satisfying the 
earlier chosen initial conditions, is 

h = [ < / > o(2) 1/2 /4][cos(tui?) — cos(w 2 ?)], (wl) 

02 = (0 o /2) [cos (uq t) F cos(w 2 t)]; (w2) 

where oj|, u 2 are given by (rl, 2). 

The relative modal contributions for each frequency are shown in fig. 3.4(a, b). 

Problem 3.5.7 Double Pendulum; Noninertial Coordinates. Consider the double 
pendulum of fig. 3.3. 

(i) Show that its (Lagrangean) equations of motion in the angles 0j(= 0p and 0 2 , 
under gravity, are 

[n%il F T m2(/1 - T 2/ \l 2 cos 9 2 F l 2 2 J\(d ^ 9 1 / dt 2 ^) F ni 2 l 2 (l \ cos 9 2 F l2)(d~9 2 /dt 2 ^j 
— ( m 2 l x l 2 sin 9 2 )(d9 2 /dt) 2 — (2m 2 /[/ 2 sin9 2 )(d9i/dt)(d9 2 /dt) 

F (mi F m 2 )l\gsm9\ F m 2 l 2 g %m(6 { F 0 2 ) = 0, (a) 

( m 2 I 2 2 )(d 2 0 2 /dt 2 ) + m 2 l 2 (l\ cos 0 2 F l 2 )(d 2 9i/dt 2 ) 

F (m 2 l 1 l 2 sin9 2 )(d9 l /dt) 2 F m 2 l 2 gsin(9i F 9 2 ) = 0. (b) 

(ii) Obtain its equations of small motion; that is, linearize (a, b). 

(iii) What do (a, b) reduce to for /[ = 0, or / 2 = 0, before and after their linear¬ 
ization? 

Problem 3.5.8 Double Physical Pendulum. A rigid body / of mass M can rotate 
freely about a fixed and smooth vertical axis. A second rigid body II of mass m 
can rotate freely about a second smooth and also vertical axis that is fixed on 
body I (fig. 3.5). 

(i) Show that the (double) kinetic energy of this double planar “physical” pendu¬ 
lum is 

2T = Aip 2 F2r<j>ij)F # 0 2 , 

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CHAPTER 3: KINETICS OF CONSTRAINED SYSTEMS 



where A = MK 2 + ma 2 , B = m(k 2 + b 2 ), T = mab cos(</> — ip) = mab cos (ip — </>) = 
in a b cos x (definition of angle x); K(k) = radius of gyration of /(//) about 0 (G^. 

(ii) Show that, in this (force-free) case, 

dT/dij> + dT/dip = = total angular momentum about G-axis 

= constant = c, (b) 

or 

(A + r){df/dt) + {B + r)(dip/dt) = c-, (bl) 

and 

2T = E ( another constant). (c) 

(iii) Show that, with the help of x — ^ — e q. (bl) can be further transformed 
to 

(A + 2r + B){dip/dt) = c — (A + r)(dx/dt), 
or 

(A + 2r + B)[A(d<p/dt) + r{dip/dt)\ = {A + r)c+(AB - r 2 )(d X /dt). (d) 

(iv) With the help of this integral, show that the energy integral (c) can be 
rewritten as 


(dx/dt){A{d<p / dt) + r(dip/dt)\ + c(dip/dt) = E, 


or 


(, d X /dt) 2 (AB - r 2 ) + c 2 = (A + 2r + B)E. 

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§3.5 EQUATIONS OF MOTION VIA LAGRANGE'S PRINCIPLE: GENERAL FORMS 

(v) Finally, and recalling the F-definition, show that (e) transforms to 

( d\/dt ) 2 = [(A + B + 2mabcosx)E — c 2 ]/[AB — ( mat ) 2 cos 2 x] = /(x)> (f) 

that is, the problem has been led to a quadrature. 

For further discussion of this famous problem, and of its many variations, see, for 
example (alphabetically): Marcolongo (1912, pp. 213-216), Schell (1880, pp. 549- 
551), Thomson and Tait (1912, pp. 310, 324-325), Timoshenko and Young (1948, 
pp. 209-211, 215-216, 249-250, 276-278, 312-314). 

Problem 3.5.9 Double Physical Pendulum: Vertical Axes. Continuing from the 
preceding problem (penduli axes through O and O' vertical), obtain its 
Lagrangean equations of motion. What happens to these equations if the center of 
mass of the entire system I + II is at its maximum / minimum distance from 01 
Assume that O , G, O' are collinear, and OG = h. 

HINT 

Introduce the new angular variables cj\ = (p and q 2 = 9 = ip — <p (= — x) = 
inclination of body II relative to / (positive counterclockwise). Then, 
T —> T(6;<p,9), and so on. 

Problem 3.5.10 Double Physical Pendulum: Horizontal Axes. Consider the pre¬ 
ceding double pendulum problem, but now with both axes through O and O' 
horizontal. In addition, assume that the mass center of body /, G, lies in the plane 
of the axes O and O' , and OG = h. Show that here T is the same, in form , as in 
the previous vertical axes case, but the potential of gravity forces, V, equals 
(exactly) 


V = — M ghcoscp — mg(a cos <p + b cos if) + constant , (a) 

and therefore the corresponding Lagrangean impressed forces are 

Qcj, = -dV/dcp = • • • and = -dV/dtp = ■ ■ ■. (b) 

Then write down Lagrange’s equations for q\= f and q 2 = tp. 

Problem 3.5.11 Double Physical Pendulum: Horizontal Axes; Small Oscillations. 
Continuing front the preceding problem (O and O' horizontal), show that for small 
oscillations about the vertical equilibrium position ip = 0, linearization of the 
exact equations leads to the coupled system 

(Mh + ma)[L(d 2 cp/dt 2 ) + grp] + m ab(d 2 ip/dt 2 ) = 0, 

a(d 2 (p/dt 2 ) + L'(d 2 ip/dt 2 ) + gip = 0 , (a) 

where 

L = (MK 2 + ma 2 )/(Mh +ma) and L'= (k 2 + b 2 )/b. (b) 

Interpret L and L' in terms ofsingle pendulum quantities. Then, assume as solutions 
of (a) 

cp = (p„ sin(uV + e) and ip = ip 0 sin(cut + e), (c) 

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CHAPTER 3: KINETICS OF CONSTRAINED SYSTEMS 


where 4> 0 ,ip 0 = angular amplitudes, e = initial phase, and u> = frequency, and show 
that the ur are real, positive, and unequal, and are the roots of 

(£2 — L)(Q — L 1 ) = (. ma 2 b)/(Mh + ma ), where Q = g/w 2 ; (d) 

say Q l < Q 2 ', and thus conclude that 

0, < min (L,L r ) < ma x(L,L') < Q 2 ■ (e) 

Finally, show that 

f 0 /f 0 = a/(Q-L') = ---, (f) 

and, therefore, (i) for the smaller to (—> larger Q = Qf), • ip > 0 (i.e., in the slower 
mode, the angles have the same sign); while (ii) for the larger 
lo (—> smaller Q = Q { ), f-ip <0 (i.e., in the faster mode, the angles have opposite 
signs). 

[For a discussion of the historically famous case of the nonringing, or ",silent ”, bell 
of Koln (Cologne), Germany (1876; bell + clapper = double pendulum), based on 
(a), see, for example, Flamel ([1922(b)] 1912, 1st ed., pp. 514 If.), Szabo (1977, pp. 
89-90), Timoshenko and Young (1948, p. 278).] 

Problem 3.5.12 General Form of Lagrange’s Equations for a 2 DOF System. 
Consider a 2 DOF holonomic and scleronomic system; for example, a particle on 
a fixed surface, or the previous double pendulum, with (double) kinetic energy 

2T = A(dx/dt)~ + 2r(dx/dt)(dy/dt) + B(dy/dtf , (a) 

and such that 6'W = X8x + Y5y, where A,B,r,X , Y, are functions of x, y. 

(i) Show that its Lagrangean equations of motion in q\= x and q 2 = y are 

A(d 2 x/dt 2 ) + r(d 2 y/dt 2 ) + (l/2)(dA/dx)(dx/dt) 2 + ( dA/dy)(dx/dt)(dy/dt ) 

+ [dr/dy - (1/2 )(dB/dx)](dy/dt) 2 = X , (b) 

B(d 2 y / dt 2 ) + r(d 2 x/dt 2 ) + (1/2 ){dB/dy){dy/dt) 2 + (dB/dx)(dx/dt)(dy/dt) 

+ [dr/dx — (1/2 )(dA/dy)](dx/dt) 2 = Y, (c) 

and ponder over the geometrical/kinematical/inertial meaning and origin of each of 
these terms. 

(ii) Show that these equations linearize to the (still coupled) system: 

A 0 {d 2 x/dt 1 ) + r 0 {d 2 y/dt 1 ) = {dX/dx) 0 x + {dX/dy) a y, (d) 

B 0 (d 2 y/dt 2 ) + r o {d 2 x/dt 2 ) = {dY/dx) a x + (dY/dy) a y, (e) 

where (.. ,) o = (...) evaluated at x,y = 0. 

Example 3.5.10 Lagrange’s Equations, 2 DOF: Elastic Pendulum, or Swinging 
Spring. Let us derive and discuss the equations of plane motion, under gravity, 
of a pendulum consisting of a heavy particle (or bob) of mass m suspended by a 
linearly elastic and massless spring of stiffness k (a positive constant) and 
unstretched (or natural) length b (fig. 3.6). This is a holonomic and scleronomic 

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§3.5 EQUATIONS OF MOTION VIA LAGRANGE'S PRINCIPLE: GENERAL FORMS 


o 



two DOF system; that is, n = 2, m = 0. With Lagrangean coordinates as the polar 
coordinates of the bob: cp = r, q x = <j>, its (double) kinetic energy is 

2T = mv 2 = m(ds/dt ) 2 = m[(f) 2 + r 2 (</>) 2 ], (a) 

while the virtual work of its impressed forces, gravity and spring force, equals 

S'W = —k(r — b) 6r + (mg cos </) 6r — (mg sin (j>)(r 8(j>) = Q r 6r + Q^ 6(j> , (b) 

that is, 


Q r = -k(r-b) +mgcos<t>, Q <j> = -m gr sin cj>. (c) 

Alternatively, the potential energy of the system is 

V = (1/2 )k(r — b) 2 — mgr cos 0 = V(r, (j>), (d) 

and so the corresponding Lagrangean forces are Q,. = — dV/dr = ■ ■ ■, = 

— dV/d(j) = ■ ■ -, equations (c). We also notice that for r > b: Q rsprmg 
= — k(r — b) < 0, as it should; and analogously for r < b. Lagrange’s equations, 
then, are 


E r (T) = E r = Q r : (mr)'— mr(<j>) 2 =—k(r — b) + mgcostj), (e) 

E d T ) = E^> = Qf (mr 2 <j>y =-mgr sin fa (f) 

or, after some simplifications (since r ^ 0), 

d 2 r/dt 2 — r(d(j>/dt ) 2 = —(k/m)(r — b) + geos </, (g) 

r(d 2 (j)/dt 2 ) + 2(dr/dt)(dcf>/dt) = —gsincf). (h) 


The general solution of this nonlinear and coupled system is unknown, and so we will 
limit ourselves to some simple and physically motivated special solutions of it. 

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(i) Equilibrium solution : Setting all (.../'-derivatives in (g, h) equal to zero, we 
find [wi