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Konstantin K. Likharev 

Essential Graduate Physics 

Lecture Notes and Problems 
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Part CM: 

Classical Mechanics 

© 2013-2016 K. Likharev 

Essential Graduate Physics 

CM: Classical Mechanics 

Table of Contents 

Chapter 1. Review of Fundamentals (14 pp.) 

1.1. Mechanics and dynamics 

1.2. Kinematics: Basic notions 

1.3. Dynamics: Newton laws 

1.4. Conservation laws 

1.5. Potential energy and equilibrium 

1.6. OK, can we go home now? 

1.7. Self-test problems (12) 

Chapter 2. Lagrangian Formalism (14 pp.) 

2.1. Lagrange equation 

2.2. Examples 

2.3. Hamiltonian 

2.4. Other conservation laws 

2.5. Exercise problems (10) 

Chapter 3. A Few Simple Problems (20 pp.) 

3.1. One-dimensional and ID-reducible systems 

3.2. Equilibrium and stability 

3.3. Hamiltonian ID systems 

3.4. Planetary problems 

3.5. 2 nd Kepler law 

3.6. 1 st and 3 ld Kepler laws 

3.7. Elastic scattering 

3.8. Exercise problems (16) 

Chapter 4. Oscillations (34 pp.) 

4.1. Free and forced oscillations 

4.2. Weakly nonlinear oscillations 

4.3. RWA equations 

4.4. Self-oscillations and phase locking 

4.5. Parametric excitation 

4.6. Fixed point classification 

4.7. Numerical approach 

4.8. Hannonic and subharmonic oscillations 

4.9. Exercise problems (14) 

Chapter 5. From Oscillations to Waves (22 pp.) 

5.1. Two coupled oscillators 

5.2 . N coupled oscillators 

5.3. ID Waves in periodic systems 

5.4. Interfaces and boundaries 

5.5. Dissipative, parametric, and nonlinear phenomena 

5.6. Exercise problems (15) 

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Chapter 6. Rigid Body Motion (30 pp.) 

6.1. Angular velocity vector 

6.2. Inertia tensor 

6.3. Fixed-axis rotation 

6.4. Free rotation 

6.5. Torque-induced precession 

6.6. Non-inertial reference frames 

6.7. Exercise problems (22) 

Chapter 7. Deformations and Elasticity (38 pp.) 

7.1. Strain 

7.2. Stress 

7.3. Hooke’s law 

7.4. Equilibrium 

7.5. Rod bending 

7.6. Rod torsion 

7.7. 3D acoustic waves 

7.8. Elastic waves in restricted geometries 

7.9. Exercise problems (15) 

Chapter 8. Fluid Mechanics (26 pp.) 

8.1. Hydrostatics 

8.2. Surface tension effects 

8.3. Kinematics 

8.4. Dynamics: Ideal fluids 

8.5. Dynamics: Viscous fluids 

8.6. Turbulence 

8.7. Exercise problems (19) 

Chapter 9. Deterministic Chaos (14 pp.) 

9.1. Chaos in maps 

9.2. Chaos in dynamic systems 

9.3. Chaos in Hamiltonian systems 

9.4. Chaos and turbulence 

9.5. Exercise problems (4) 

Chapter 10. A Bit More of Analytical Mechanics (14 pp.) 

10.1. Hamiltonian equations 

10.2. Adiabatic invariance 

10.3. The Hamilton principle 

10.4. The Hamilton-Jacobi equation 

10.5. Exercise problems (9) 

* * 

Additional file (available upon request): 

Exercise and Test Problems with Model Solutions (136 + 40 = 176 problems; 227 pp.) 

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Table of Contents 

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Essential Graduate Physics 

CM: Classical Mechanics 

Chapter 1. Review of Fundamentals 

After elaborating a bit on the title and contents of the course, this short introductory chapter lists the 
basic notions and facts of the classical mechanics, that are supposed to be known to the reader from 
undergraduate studies . 1 Due to this reason, the explanations are very brief 

1.1. Mechanics and dynamics 

A more fair title of this course would be Classical Mechanics and Dynamics, because the notions 
of mechanics and dynamics, though much intertwined, are still somewhat different. Term mechanics, in 
its narrow sense, means deriving the equations of motion of point-like particles and their systems 
(including solids and fluids), solution of these equations, and interpretation of the results. Dynamics is a 
more ambiguous tenn; it may mean, in particular: 

(i) the part of mechanics that deals with motion (in contrast to statics)', 

(ii) the part of mechanics that deals with reasons for motion (in contrast to kinematics)', 

(iii) the part of mechanics that focuses on its two last tasks, i.e. the solution of the equations of 
motion and discussion of the results. 

The last definition invites a question. It may look that mechanics and dynamics are just two 
sequential steps of a single process; why should they be considered separate disciplines? The main 
reason is that the many differential equations of motion, obtained in classical mechanics, also describe 
processes in different systems, so that their analysis may reveal important features of these systems as 
well. For example, the famous ordinary differential equation 

x + o)qX = 0 ( 1 . 1 ) 

describes sinusoidal ID oscillations not only of a mass on a spring, but also of an electric or magnetic 
field in a resonator, and many other systems. Similarly, the well-known partial differential equation 

_L cf 

v 2 df 


f(r,t) = 0. 

( 1 . 2 ) 

where v is a constant and V 2 is the Laplace operator, 2 describes not only acoustic waves in an elastic 
mechanical continuum (solid or fluid), but also electromagnetic waves in a non-dispersive media, 
certain chemical reactions, etc. Thus the results of analysis of the dynamics described by these equations 
may be reused for applications well beyond mechanics. 

1 The reader is advised to perform a self-check by solving a few problems of the dozen listed in Sec. 1.7. If the 
results are not satisfactory, it may make sense to start from some remedial reading. For that, I could recommend, 
for example (in the alphabetical order): G. R. Fowles and G. L. Cassiday, Analytical Mechanics, 7 th ed., Brooks 
Cole, 2004; K. R. Symon, Mechanics, 3 rd ed., Addison-Wesley, 1971; or J. B. Marion and S. T. Thornton, 
Classical Dynamics of Particles and Systems, 4 th ed., Saunders, 1995. 

2 This series assumes reader’s familiarity with the basic calculus and vector algebra. The formulas most important 
for this series are listed in the Selected Mathematical Formulas appendix, referred below as MA. In particular, a 
reminder of the definition and the basic properties of the Laplace operator may be found in MA Sec. 9. 

© 2013-2016 K. Likharev 

Essential Graduate Physics 

CM: Classical Mechanics 

To summarize, term “dynamics” is so ambiguous 3 that, after some hesitation, I have opted to 
using for this course the traditional name Classical Mechanics, implying its broader meaning, which 
includes (similarly to Quantum Mechanics and Statistical Mechanics ) studies of dynamics of some non- 
mechanical systems. 

1.2. Kinematics: Basic notions 

The basic notions of kinematics may be defined in various ways, and some mathematicians pay a 
lot of attention to analyzing such systems of axioms and relations between them. In physics, we 
typically stick to less rigorous ways (in order to proceed faster to particular problems), and end debating 
a definition as soon as everybody in the room agrees that we are all speaking about the same thing. Let 
me hope that the following notions used in classical mechanics do satisfy this criterion: 

(i) All the Euclidean geometry notions, including the geometric point (the mathematical 
abstraction for the position of a very small object), straight line, etc. 

(ii) The orthogonal, linear (“Cartesian”) coordinates 4 rj of a geometric point in a particular 
reference frame - see Fig. I. 5 

Fig. 1.1. Cartesian coordinates and 
radius-vector of a point/particle. 

The coordinates may be used to define the point’s radius-vector 6 

3 Another important issue is: Definition (iii) of dynamics is suspiciously close to the part of mathematics devoted 
to the differential equation analysis; what is the difference? To answer, we have to dip, for just a second, into the 
philosophy of physics. Physics may be described as an art (and a bit of science :-) of description of Mother Nature 
by mathematical means; hence in many cases the approaches of a mathematician and a physicist to a problem are 
very similar. The main difference is that physicists try to express the results of their analysis in terms of system ’s 
motion rather than function properties, and as a result develop some sort of intuition (“gut feeling”) about how 
other, apparently similar, systems may behave, even if their exact equations of motion are somewhat different - or 
not known at all. The intuition so developed has an enormous heuristic power, and most discoveries in physics 
have been made through gut-feeling-based insights rather than by plugging one formula into another one. 

4 In these notes the Cartesian coordinates are denoted either as either {r\, r 2 , r 3 } or {x, y, zj, depending on 
convenience in the particular case. Note that axis numbering is important for operations like the vector (“cross”) 
product; the “correct” (meaning generally accepted) numbering order is such that rotation ni — > n 2 — > n 3 — > ni... 
looks counterclockwise if watched from a point with all r, > 0 - see Fig. 1. 

5 In references to figures, formulas, problems and sections within the same chapter of these notes, the chapter 
number is dropped for brevity. 

6 From the point of view of the tensor theory (in which the physical vectors like r are considered the rank-1 
tensors), it would be more natural to use superscripts in the components r ; - and other “contravarianf ’ vectors. 
However, the superscripts may be readily confused with the power signs, and I will postpone this notation (as 
well as the implied summation over the repeated indices) until the discussion of relativity in EM Chapter 9. 

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( 1 - 3 ) 

where n, ,n, ,n 3 are the unit vectors along coordinate axis directions, with the Euclidean metric : 


which is independent, in particular, of the distribution of matter in space. 

(iii) The time - as described by a continuous scalar variable (say, t), typically considered an 
independent argument of various physical observables, in particular the point’s radius-vector r(t). By 
accepting Eq. (4), and an implicit assumption that time t runs similarly in all reference frames, we 
subscribe to the notion of the absolute (“Newtonian”) space/time, and hence abstain from a discussion 
of relativistic effects. 7 

(iv) The (instant) velocity of the point, 

and its acceleration : 



Since the above definitions of vectors r, v, and a depend on the chosen reference frame (are 
“reference-frame-specific”), there is a need to relate those vectors as observed in different frames. 
Within the Euclidean geometry, for two reference frames with the corresponding axes parallel in the 
moment of interest (Fig. 2), the relation between the radius-vectors is very simple: 

in O' 

= r 

in O r 0 in O’ 


Fig. 1.2. Coordinate transfer between two 
reference frames. 

7 Following tradition, an introduction to special relativity is included into the Classical Electrodynamics (“EM”) 

part of these notes. The relativistic effects are small if all particles velocities are much lower than the speed of 

light, c ~ 3.00xl0 8 * * m/s, and all distances are much larger then the system’s Schwarzschild radius r s = 2 Gm/c 2 , 

where G ~ 6.67x1 O' 11 SI units (m 3 /kg-s) is the Newtonian gravity constant, and m is system’s mass. (More exact 

values of c, G, and some other physical constants may be found in appendix CA: Selected Physical Constants .) 

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If the frames move versus each other by translation only (no mutual rotation!), similar relations 
are valid for velocity and acceleration as well: 

in O' 

= V 

in O ^ O in O' '■ 


1 O' 

= a 

inO + a O in O' 


In the case of mutual rotation of the reference frames, notions like \o\mO’ are not well defined. 
(Indeed, different points of a rigid body connected to frame O may have different velocities in frame 
O ’.) As a result, the transfer laws for velocities and accelerations are more complex than those given by 
Eqs. (8) and (9). It will be more natural for me to discuss them in the end of Chapter 5 that is devoted to 
rigid body motion. 

(v) The particle : a localized physical object whose size is negligible, and shape unimportant for 
the given problem. Note that the last qualification is extremely important. For example, the size and 
shape of a Space Shuttle are not too important for the discussion of its orbital motion, but are paramount 
when its landing procedures are being developed. Since classical mechanics neglects the quantum 
mechanical uncertainties, 8 particle’s position, at any particular instant t, may be identified with a single 
geometric point, i.e. one radius-vector r (t). Finding the laws of motion r (t) of all particles participating 
in the given problem is frequently considered the final goal of classical mechanics. 

1.3. Dynamics: Newton laws 

Generally, the classical dynamics is fully described (in addition to the kinematic relations given 
above) by three Newton laws. 9 In contrast to the impression some textbooks on theoretical physics try to 
create, these laws are experimental in nature, and cannot be derived from purely theoretical arguments. 10 

I am confident that the reader of these notes is already familiar with the Newton laws, in one or 
another formulation. Fet me note only that in some formulations the I s ' Newton law looks just as a 
particular case of the 2 nd law - for the case of zero net force acting on a particle. In order to avoid this 
duplication, the 1 st law may be formulated as the following postulate: 

- There exists at least one reference frame, called inertial, in which any free particle (i.e. a 
particle isolated from the rest of the Universe) moves with v = const, i.e. with a = 0. 

According to Eq. (9), this postulate immediately means that there is also an infinite number of 
inertial frames, because all frames O ’ moving without rotation or acceleration relative to the postulated 
inertial frame O (i.e. having a <? \mO’ = 0) are also inertial. 

8 This approximation is legitimate, crudely, when the product of the coordinate and momentum scales of the 
particle motion is much larger than the Planck’s constant ti ~ 1.054xl0" 34 J-s. A more exact formulation may be 
found, e.g., in the Quantum Mechanics (“QM”) part of these note series. 

9 Due to the genius of Sir Isaac Newton, these laws were formulated as early as in 1687, far ahead of the science 
of that time. 

10 Some laws of Nature (including the Newton laws) may be derived from certain more general postulates, such as 
the Hamilton (or “least action”) principle - see Sec. 10.2 below. Note, however, that such derivations are only 
acceptable because all known corollaries of the postulates comply with all kn own experimental results. 

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On the other hand, the 2 nd and 3 ld Newton laws may be postulated together in the following 
elegant way. Each particle, say number k, may be characterized by a scalar constant (called mass mk), 
such that at any interaction of N particles (isolated from the rest of the Universe), in any inertial system, 

and its 

( 1 . 10 ) 

(Each component of this sum, 


is called the mechanical momentum of the corresponding particle, and the whole sum P, the total 
momentum of the system.) 

Pa m k' v k ’ 

( 1 . 11 ) 

Let us apply this postulate to just two interacting particles. Differentiating Eq. (10), written for 
this case, over time, we get 

Pi =-P2 

( 1 . 12 ) 

3rd Newton 

2nd Newton 

Let us give the derivative p, (i.e., a vector) the name of force Fi excerted on particle 1. In our current 
case, when the only possible source of force is particle 2, the force may be denoted as F, 2 . Similarly, 
F 21 = p 2 , so that we get the 3 rd Newton law 


Now, returning to the general case of several interacting particles, we see that an additional (but 
very natural) assumption that all partial forces F**- acting on particle k add up as vectors, leads to the 
general form of the 2 nd Newton law 11 

™a*a = Pa = X F «' = F a » 



that allows a clear interpretation of the mass as a measure of particle’s inertia. 

As a matter of principle, if the dependence of all pair forces F**’ of particle positions (and 
generally maybe of time as well) is known, Eq. (14) augmented with kinematic relations (4) and (5), 
allows the calculation of the laws of motion rft) of all particles of the system. For example, for one 
particle the 2 nd law (14) gives the ordinary differential equation of the second order, 

mil = F(r,f), 


that may be integrated - either analytically or numerically. 

gravity law 

For certain cases, this is very simple. As an elementary example, the Newton’s gravity field 


(where Rsr-r’is the distance between particles of masses m and m’) n , is virtually unifonn and may 
be approximated as 

11 Of course, for composite bodies of varying mass (e.g., rockets emitting jets, see Problem 1), momentum’s 
derivative may differ from ma. 

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F = mg. 


with the vector g = (Gm ’/r ’ 3 )r’ being constant, for local, relatively small-scale motions, with r « r’. n 
As a result, m in Eq. (15) cancels, it is reduced to just r = g , and may be easily integrated twice: 

gravity field 

r(t) = v(0 = J g dt' + v(0) = g t + v(0), (1.17) 


i j2 

r (t) = | \(t')dt' + r (0) = g — + v(0)/ + r(0) , (1.18) 

thus giving the full solution of all those undergraduate problems on the projectile motion, which should 
be so familiar to the reader. 

All this looks (and indeed is) very simple, but in most other cases leads to more complex 
calculations. As an example, let us consider another simple problem: a bead of mass m sliding, without 
friction, along a round ring of radius R in a gravity field obeying Eq. (16b) - see Fig. 3. 

Suppose we are only interested in bead’s velocity v in the lowest point, after it has been dropped 
from the rest at the rightmost position. If we want to solve this problem using only the Newton laws, we 
have to do the following steps: 

(i) consider the bead in an arbitrary intermediate position on a ring, described, for example by 
the angle 6 shown in Fig. 3; 

(ii) draw all the forces acting on the particle - in our current case, the gravity force mg and the 
reaction force N exerted by the ring; 

12 Note that the fact that the masses participating in Eqs. (14) and (16) are equal, the so-called weak equivalence 
principle, is highly nontrivial, but has been verified experimentally to the relative accuracy of at least 10" 13 . Due 
to its conceptual significance of the principle, new space experiments, such as MISCROSCOPE 
( . are being planned for a substantial, nearly 1 00-fold accuracy improvement. 

13 Of course, the most important particular case of Eq. (1.16b) is the motion of objects near Earth’s surface. In this 
case, using the fact that (1.16a) remains valid for the gravity field created by a heavy sphere, we getg = GM E /R E 2 , 
where M E and Re are the Earth mass and radius. Plugging in their values, M E « 5.92xl0 24 kg, Re ~ 6.37xl0 6 m, we 
get g ~ 9.74 m/s 2 . The effective value of g varies from 9.78 to 9.83 m/s 2 at various locations on Earth’s surface 
(due to the deviations of Earth’s shape from a sphere, and the location-dependent effect of the centrifugal “inertial 
force” - see Sec. 6.5 below), with an average value of g « 9.807 m/s 2 . 

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(iii) write the 2 nd Newton law for two nonvanishing components of the bead acceleration, say for 
its vertical and horizontal components a x and a y ; 

(iv) recognize that in the absence of friction, the force N should be normal to the ring, so that we 
can use two additional equations, N x = -N sin 8 and N y = N cos 8; 

(v) eliminate unknown variables N, N x , and N v from the resulting system of four equations, thus 
getting a single second-order differential equation for one variable, for example 8, 

(vi) integrate this equation once to get the expression relating the velocity 8 and the angle 8 ; 
and, finally, 

(vii) using our specific initial condition ( 8 = 0 at 8 = n I 2), find the final velocity as v = R8 at 

8 = 0 . 

All this is very much doable, but please agree that the procedure it too cumbersome for such a 
simple problem. Moreover, in many other cases even writing equations of motion along relevant 
coordinates is very complex, and any help the general theory may provide is highly valuable. In many 
cases, such help is given by conservation laws', let us review the most general of them. 

1.4. Conservation laws 

(i) Energy conservation is arguably the most general law of physics, but in mechanics it takes a 
more humble form of mechanical energy conservation that has limited applicability. To derive it, we 
first have to define the kinetic energy of a particle as 


and then recast its differential as 14 




( m 2 ') 

— V 

= d 

/ m N 

— V- V 

U 7 

1 2 J 

, d\ -dr 

= my ■ d\ = m 


( 1 . 20 ) 

Now plugging in the momentum’s derivative from the 2 nd Newton law, dp/dt = F, where F is the full 
force acting on the particle, we get relation dT= E-dr. Its integration along particle’s trajectory between 
some points A and B gives the relation that is sometimes called the work-energy principle : 


ET = T(r B )-T(r A ) = \¥-dr, 



where the integral in the right-hand part is called the work of the force F on the path from A to B. 

The further step may be made only for potential (also called “conservative”) forces that may be 
presented as (minus) gradients of some scalar function U( r), called the potential energy. 15 The vector 
operator V (called either del or nabla ) of spatial differentiation 16 allows a very compact expression of 
this fact: 

14 Symbol ab denotes the scalar (or “dot-”) product of vectors a and b - see, e.g., MA Eq. (7.1). 

15 Note that because of its definition via the gradient, the potential energy is only defined to an arbitrary additive 

16 Its basic properties are listed in MA Sec. 8. 

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F = -VC/ . 

For example, for the uniform gravity field (16b), 

U = mgh + const, 

( 1 . 22 ) 


where h is the vertical coordinate directed “up” - opposite to the direction of the vector g. 

Integrating the tangential component F T of the vector F, given by Eq. (22), along an arbitrary 
path connecting points A and B, we get 


i.e. work of potential forces may be presented as the difference of values of function U(r ) in the initial 
and final point of the path. (Note that according to Eq. (24), work of a potential force on any closed 
trajectory, with r A = r B , is zero.) 

Now returning to Eq. (21) and comparing it with Eq. (24), we see that 

T(r B )-T(r A ) = U(r A )-U(r B ), 


so that the total mechanical energy E, defined as 


is indeed conserved: 

E(r A ) = E(r B ), 



but for conservative forces only. (Non-conservative forces, e.g., friction, typically transfer energy from 
the mechanical form into some other form, e.g., heat.) 

The mechanical energy conservation allows us to return for a second to the problem shown in 
Fig. 3 and solve it in one shot by writing Eq. (27) for the initial and final points: 17 

0 + /«gi? = yV 2 +0. (1.28) 

Solving Eq. (28) for v immediately gives us the desired answer. Let me hope that the reader agrees that 
this way of problem solution is much simpler, and I have got his or her attention to discuss other 
conservation laws - which may be equally effective. 

(ii) Momentum . Actually, the conservation of the full momentum of any system of particles 
isolated from the rest of the world, has already been discussed and may serve as the basic postulate of 
classical dynamics - see Eq. (10). In the case of one free particle the law is reduced to a trivial result p = 
const, i.e. v = const. If the system of N particles is affected by external forces F (ext) , we may write 

F.-Ff'+iX.. (1.29) 

k = 1 

17 Here the arbitrary constant in Eq. (32) is chosen so that the potential energy is zero in the finite point. 









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If we sum up the resulting Eqs. (14) for all particles of the system then, due to the 3 ld Newton law (13), 
the contributions of all internal forces to this double sum in the right-hand part cancel, and we get the 


which tells us that the translational motion of the system as the whole is similar to that of a single 
particle, under the effect of the net external force F (cxt) . As a simple sanity check, if the external forces 
have a zero sum, we return to postulate (10). Just one reminder: Eq. (30), just as its precursor Eq. (14), 
is only valid in an inertial reference frame. 

(iii) Angular momentum of a particle 18 is defined as the following vector: 




where axb means the vector (or “cross-“) product of the vector operands. 19 Now, differentiating Eq. (31) 
over time, we get 

L = rxp + rxp. 


In the first product, r is just the velocity vector v which is parallel to the particle momentum p = mv, so 
that this product vanishes, since the vector product of any two parallel vectors is zero. In the second 
product, p equals the full force F acting on the particle, so that Eq. (32) is reduced to 





where vector 

L = T, 

t = rxF, 



is called the torque of force F. (Note that the torque is evidently reference-frame specific - and again, 
the frame has to be inertial for Eq. (33) to be valid.) For an important particular case of a central force 
F that is parallel to the radius vector r of a particle (as measured from the force source point), the torque 
vanishes, so that (in that particular reference frame only!) the angular momentum is conserved: 




L = const. 

For a system of N particles, the total angular momentum is naturally defined as 





Differentiating this equation over time, using Eq. (33) for each L*, and again partitioning each force in 
a accordance with Eq. (29), we get 

18 Here we imply that the internal motions of the particle, including its rotation about its own axis, are negligible. 
(Otherwise it could not be represented by a geometrical point, as was postulated in Sec. 1.) For a body with 
substantial rotation (see Chapter 6 below), vector L retains its definition (32), but is only a part of the total 
angular momentum and is called the orbital momentum - even if the particle does not move along a closed orbit. 

19 See, e.g., MA Eq. (7.3). 

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i v 

Et xF tt' 

+ T 


where t 




k,k '= 1 

k = 1 


The first (double) sum may be always divided into pairs of the type (r/ f xF/ ( /,’ + r^xF^-)- With a natural 
assumption of the central forces (F^- 1| iy - ly ), each of these pairs equals zero. Indeed, in this case both 
components of the pair are vectors perpendicular to the plane passed through positions of both particles 
and the reference frame origin, i.e. to the plane of drawing of Fig. 4. Also, due to the 3 rd Newton law 
(13) the two forces are equal and opposite, and the magnitude of each term in the sum may be presented 
as I Fkk\ hkk', with equal “lever arms” hkk ■ = hk'k ■ As a result, each sum (lyxFytr + r^xF**), and hence the 
whole double sum in Eq. (37) vanish, and it is reduced to a very simple result, 

L = 

T (ext) 


that is similar to Eq. (33) for a single particle, and is the angular analog of Eq. (30). In particular, Eq. 
(38) shows that if the full external torque x <ext) vanishes by some reason (e.g., if the system of particles is 
isolated from the rest of the Universe), the conservation law (35) is valid for the full angular momentum 
L, even if its individual components L k are not conserved due to inter-particle interactions. 

System’ s 

Fig. 1.4. Internal and external forces, and 
the internal torque cancellation in a system 
of two particles. 

From the mathematical point of view, most conservation laws present the first integrals of 
motion which sometimes liberate us from the necessity to integrate the second-order differential 
equations of motion, following from the Newton laws, twice. 

1.5. Potential energy and equilibrium 

Another important role of the potential energy U, especially for dissipative systems whose total 
mechanical energy E is not conserved because it may be drained to the environment, is finding the 
positions of equilibrium (sometimes called the fixed points of the system under analysis) and analyzing 
their stability with respect to small perturbations. For a single particle, this is very simple: force (22) 
vanishes at each extremum (minimum or maximum) of the potential energy. 20 Of those fixed points, 
only the minimums of U( r) are stable - see Sec. 3.2 below for a discussion of this point. 

20 Assuming that the additional, non-conservative forces (such as viscosity) responsible for the mechanical energy 
drain, vanish at equilibrium - as they typically do. (Static friction is one counter-example.) 

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A slightly more subtle case is a particle with potential energy £/(r), subjected to an additional 
external force F (ext> (r). In this case, the stable equilibrium is reached at the minimum of not function 
U( r), but of what is sometimes called the Gibbs potential energy 


which is defined, just as U(r ) is, to an arbitrary constant. The proof of Eq. (39) is very simple: in an 
extremum of this function, the total force acting on the particle, 


F (tot) = F + F (ext) = -VC/ + V jF (ext) (r')-dr' = -VC/ G , (1.40) 

vanishes, as it should. 21 For the simplest (and very frequent) case of the applied force independent on 
particle’s position, the Gibbs potential energy is just 

U G (r ) = U{r ) ■ - F (ext) • r + const . (1.41) 

This is ah very straightforward, but since the notion of Uq is not well known to some students, 22 
let me offer a very simple example. Consider a ID deformation of the usual elastic spring providing the 
returning force (-kk), where x is the deviation from spring’s equilibrium. In order for the force to comply 
with Eq. (22), its potential energy should equal to U= kx 12 + const, so that its minimum corresponds to 
x = 0. This works fine until the spring comes under effect of a nonvanishing external force F, say 
independent of x. Then the equilibrium deformation of the spring, xo = Fire, evidently corresponds not to 
the minimum of U but rather to that of the Gibbs potential energy (41): Uq = U - Fx = ioc /2 - Fx + 


1.6. OK, we’ve got it - can we go home now? 

Not yet. In many cases the conservation laws discussed above provide little help, even in 
systems without dissipation. Consider for example a generalization of the bead-on-the-ring problem 
shown in Fig. 3, in which the ring is rotated by external forces, with a constant angular velocity <x > , 
about its vertical diameter (Fig. 5). 23 In this problem (to which I will repeatedly return below, using it as 

21 Physically, the difference Uq-U specified by Eq. (39) may be considered the r-dependent part of the potential 
energy f/ ext) of the external system responsible for the force F (ext) , so that U G is just the total potential energy U + 
lf ext \ besides the part of f/ ext) which does not depend on r and hence is irrelevant for the fixed point analysis. 
According to the 3 ld Newton law, the force exerted by the particle on the external system equals (-F <ext) ), so that 
its work (and hence the change of l/ ext) due to the change of r) is given by the second term in the right-hand part 
of Eq. (39). Thus the condition of equilibrium, -VU G = 0, is just the condition of an extremum of the total 
potential energy, U + l/ ex, \ of the two interacting systems. 

22 Unfortunately, in most physics teaching plans the introduction of U G is postponed until a course of statistical 
mechanics and/or thermodynamics - where it is a part of the Gibbs free energy?, in contrast to U, which is a part of 
the Helmholtz free energy > - see, e.g., Sec. 1.4 of the Statistical Mechanics (“SM”) part of my notes. However, the 
reader should agree that the difference between U G and U, and hence that between the Gibbs and Helmholtz free 
energies, has nothing to do with statistics or thermal motion, and belongs to the basic mechanics. 

23 This is essentially a simplified model of the famous mechanical control device called the centrifugal (or 
“flyball, or “centrifugal flyball”) governor - see, e.g., governor . 

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an analytical mechanics “testbed”), none of the three conservation laws listed in the last section, holds. 
In particular, bead’s energy, 

E = ^-v 2 + mgh , (1-42) 

is not constant, because the external forces rotating the ring may change it. Of course, we still can solve 
the problem using the Newton laws, but this is even more complex than for the above case of the ring at 
rest, in particular because the force N exerted on the bead by the ring now may have three rather than 
two Cartesian components, which are not simply related. One can readily see that if we could exclude 
the so-called reaction forces such as N, that ensure external constraints of the particle motion, in 
advance, that would help a lot. Such an exclusion may be provided by analytical mechanics, in 
particular its Lagrangian formulation, which will be discussed in the next chapter. 

Fig. 1.5. Bead sliding along a rotating ring. 

An even more important motivation for analytical mechanics is given by dynamics of “non- 
mechanical” systems, for example, of the electromagnetic field - possibly interacting with charged 
particles, conducting bodies, etc. In many such systems, the easiest (and sometimes the only practicable) 
way to find the equations of motion is to derive then from the Lagrangian or Hamiltonian function of the 
system. In particular, the Hamiltonian formulation of the analytical mechanics (to be discussed in 
Chapter 10) offers a direct pathway to deriving Hamiltonian operators of systems, which is the standard 
entry point for analysis of their quantum-mechanical properties. 

1.7. Self- test problems 

1.1 . A bicycle, ridden with velocity v on a wet pavement, has no mudguards on its wheels. How 
far behind should the following biker ride to avoid being splashed over? Neglect the air resistance 

1.2 . Two round disks of radius R are firmly connected with a coaxial 
cylinder of a smaller radius r, and a thread is wound on the resulting spool. 
The spool is placed on a horizontal surface, and thread’s end is being pooled 
out at angle cp - see Fig. on the right. Assuming that the spool does not slip on 
the surface, what direction would it roll? 

(Sometimes the device is called the “Watt’s governor”, after the famous engineer J. Watts who used it in 1788 in 
one of his first steam engines, though it had been used in European windmills at least since the 1600s.) 

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1.3 . Calculate the equilibrium shape of a flexible, heavy rope of 
length /, with a constant mass p per unit length, if it is hung in a 
uniform gravity field between two points separated by a horizontal 
distance d - see Fig. on the right. 

1.4 . A unifonn, long, thin bar is placed horizontally on two similar round cylinders rotating 
toward each other with the same angular velocity co and displaced 
by distance d- see Fig. on the right. Calculate the laws of relatively 
slow horizontal motions of the bar within the plane of drawing for 
both possible directions of cylinder rotation, assuming that the 
friction force between the slipping surfaces of the bar and each 
cylinder obeys the usual simple law \F\ = juN, where N is the nonnal 
pressure force between them, and p is a constant (velocity- 
independent) coefficient. Fonnulate the condition of validity of your result. 

hr, \ f 

1.5 . A small block slides, without friction, down a smooth slide 
that ends with a round loop of radius R - see Fig. on the right. What 
smallest initial height h allows the block to make its way around the 
loop without dropping from the slide, if it is launched with negligible 
initial velocity? 

1.6 . A satellite of mass m is being launched from height H over the 
surface of a spherical planet with radius R and mass M » m - see Fig. on 
the right. Find the range of initial velocities Vo (normal to the radius) 
providing closed orbits above the planet’s surface. 

1.7 . Prove that the thin-uniform-disk model of a galaxy describes 
small harmonic oscillations of stars inside it along the direction nonnal to the disk, and calculate the 
frequency of these oscillations in tenns of the Newton’s gravitational constant G and the average density 
p of the star/dust matter of the galaxy. 

1.8 . Derive the differential equations of motion for small 
oscillations of two similar pendula coupled with a spring (see Fig. on the 
right), within the vertical plane. Assume that at the vertical position of both 
pendula, the spring is not stretched (A L = 0). 



F = -kAL 

6 — W — 6 



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1.9 . One of popular futuristic concepts of travel is digging a straight railway tunnel through the 
Earth and letting a train go through it, without initial velocity - driven only by gravity. Calculate train’s 
travel time through such a tunnel, assuming that the Earth’s density p is constant, and neglecting the 
friction and planet rotation effects. 

1.10 . A small bead of mass m may slide, 
without friction, along a light horizontal string, 
stretched with a force f» mg between two points 
separated by a horizontal distance 2d - see Fig. on the 
right. Find the frequency of small oscillations of the 
bead about its equilibrium position. 

1.11 . Find the acceleration of a rocket due to the working jet motor, and explore the resulting 
equation of rocket’s motion. 

Hint : For the sake of simplicity, you may consider a ID motion. 

1.12 . Prove the following virial theorem: 24 for a set of N particles performing a periodic motion, 


^ k = 1 

where (as everywhere in these notes), the top bar means time averaging - in this case over the motion 
period. What does the virial theorem say about: 

(i) the ID motion of a particle in a confining potential U{x) = ax 2 \ with a > 0 and s > 0, and 

(ii) the orbital motion of a particle moving in a central potential U(r) = - Ctrl 


Hint : Explore the time derivative of the following scalar function of time: G{t) = ^ p # • r k . 

k = 1 

24 It was first stated by R. Clausius in 1870. 

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Chapter 2. Lagrangian Formalism 

The goal of this chapter is to describe the Lagrangian formulation of analytical mechanics, which is 
extremely useful for obtaining the differential equations of motion {and sometimes their first integrals) 
not only for mechanical systems with holonomic constraints, but also other dynamic systems. 

2.1. Lagrange equations 

In many cases, the constraints imposed on 3D motion of a system of N particles may be 
described by N vector (i.e. 3N scalar) algebraic equations 

I 't =r i (?i .?2 ?; ?n0. 1 <k<N, (2.1) 

where qj are certain generalized coordinates which (together with constraints) completely define the 
system position, and J < 3N is the number of the actual degrees of freedom. The constraints that allow 
such description are called holonomic. 1 

For example, for our testbed, bead-on-rotating-ring problem (see Fig. 1.5 and Fig. 1 below) J = 
1, because taking into account the constraints imposed by the ring, bead’s position may be uniquely 
determined by just one generalized coordinate - for example, its polar angle 0 . Indeed, selecting the 
reference frame as shown in Fig. 1 and using the well-known formulas for the spherical coordinates, 2 we 
see that in this case Eq. ( 1 ) in Cartesian coordinates has the fonn 

r = [x, y,z}= {R sin & cos <p,R sin 9 sin cp,R cos &}, where <p - cat + const , (2.2) 

where the constant depends on the exact selection of axes x and y and the time origin. Since (pit) is a 
fixed function of time, and R is a fixed constant, the position of particle in space at any instant t is 
indeed completely determined by the value of its only generalized coordinate 6. Note that the 
dimensionality of the generalized coordinate may be different from that of Cartesian coordinates 

Fig. 2.1. Bead on a rotating ring as a 
example of the system with just one degree 
of freedom: J= 1. 

1 Possibly, the simplest example of a non-holonomic constraint is a set of inequalities describing the hard walls 
confining the motion of particles in a closed volume. Non-holonomic constraints are better dealt with other 
methods, e.g., by imposing proper boundary conditions on the (otherwise unconstrained) motion. 

2 See, e.g., MA Eq. (10.7). 

© 2013-2016 K. Likharev 

Essential Graduate Physics 

CM: Classical Mechanics 

Now returning to the general case of J degrees of freedom, let us consider a set of small 
variations (alternatively called “virtual displacements”) dqj allowed by the constraints. Virtual 
displacements differ from the actual small displacements (described by differentials dqj proportional to 
time variation dt ) in that dqj describes not the system’s motion as such, but rather its possible variation - 
see Fig. 1. 

Generally, operations with variations are the subject of a special field of mathematics, the 
calculus of variations. 3 However, the only math background necessary for our current purposes is the 
understanding that operations with variations are similar to those with the usual differentials, though we 
need to watch carefully what each variable is a function of. For example, if we consider the variation of 
the radius-vectors (1), at a fixed time t, as a function of independent variations dqj, we may use the usual 
formula for the differentiation of a function of several arguments: 4 


K ( 2 - 3 ) 

Now let us break the force acting upon the k - th particle into two parts: the frictionless, 
constraining part N* of the reaction force and the remaining part F* - including the force components 
from other sources and possibly the friction part of the reaction force. Then the 2 nd Newton law for k - th 
particle of the system may be presented as 

m k v k -F k =N k . (2.4) 

Since any variation of the motion has to be allowed by the constraints, its 3V-dimcnsional vector with N 
3D-vector components dr/, has to be perpendicular to the 3V-dimensional vector of the constraining 
forces, also with N 3D-vector components N*. (For example, for the problem shown in Fig. 2.1, the 
virtual displacement vector Piy may be directed only along the ring, while the constraining force N, 
exerted by the ring, has to be perpendicular to that direction.) This condition may be expressed as 


3 For a concise introduction to the field see, e.g., I. Gelfand and S. Fomin, Calculus of Variations, Dover, 2000 or 
L. Elsgolc, Calculus of Variations, Dover, 2007. An even shorter review may be found in Chapter 17 of Arfken 
and Weber - see MA Sec. 16. For a more detailed discussion, using many examples from physics, see R. 
Weinstock , Calculus of Variations, Dover, 2007. 

4 See, e.g., MA Eq. (4.2). In all formulas of this section, all summations over index j are from 1 to J, while those 
over the particle number k are from 1 to N. 

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where the scalar product of 3/V-dimcnsional vectors is defined exactly as that of 3D vectors, i.e. as the 
sum of the products of the corresponding components of the operands. The substitution of Eq. (4) into 
Eq. (5) results in the so-called D ’Alembert principle : 5 

Z ( m ^k -F* )-£r t =0. 


Now we may plug Eq. (3) into Eq. (6) to get 

m k \ k 




where scalars fj , called generalized forces, are defined as follows: 6 

f, - 2X - 





Now we may use the standard argument of the calculus of variations: in order for the left-hand 
part of Eq. (7) to be zero for an arbitrary selection of independent variations Sqj, the expressions in the 
curly brackets, for every j, should equal zero. This gives us a set of /equations 

2>.'vA--#‘ / =o ; < 2 - 9 > 

T dtf . 

let us present them in a more convenient form. First, using the differentiation by parts to calculate the 
following time derivative: 



dr. d f dr, ^ 
dq j Wk dt\dqjj 

( 2 . 10 ) 

we may notice that the first term in the right-hand part is exactly the scalar product in the first term of 
Eq. (9). 

Second, let us use another key fact of the calculus of variations (which is, essentially, evident 
from Fig. 3): the differentiation of a variable over time and over the generalized coordinate variation (at 
fixed time) are interchangeable operations. 

5 It had been spelled out in a 1743 work by J.-B. le Rond d’Alembert, though the core of this result has been 
traced to an earlier work by J. Bernoulli (1667 - 1748). 

6 Note that since the dimensionality of generalized coordinates may be arbitrary, that of generalized forces may 
also differ from the newton. 

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As a result, in the second tenn on the right-hand part of Eq. (10) we may write 



( , ^ 
8r k 


(dr A 





, dt y 


Finally, let us differentiate of Eq. (1) over time: 

dr k ^ or, . dr k 

v ‘ E — =Z^/ + - 




( 2 . 11 ) 


This equation shows that particle velocities \y may be considered as linear functions of the generalized 
velocities q , considered as independent variables, with proportionality coefficients 

d\ k _ dr k 

oq, dqj 

With the account of Eqs. (10), (11), and (13), Eq. (9) turns into 




8q i 


8y k 

8q , 



This result may be further simplified by making, for the total kinetic energy of the system, 


t =-T 2 

k z 

z k 


the same commitment as for v*, i.e. considering T a function of not only the generalized coordinates qj 
and time t, but also of the generalized velocities q, - as variables independent of qj and t. Then we may 
calculate the partial derivatives of T as 



8y k 
8q , ’ 



Z /n * v * 

8v k 

8q / 


and notice that they are exactly the two sums participating in Eq. (13). As a result, we get a system of J 
Lagrange equations, 1 

d ST 
dt dqj 

0, for j=l, 2, 



Their big advantage over the initial Newton law equations (4) is that the Lagrange equations do not 
include the constraining forces Nk. 

This is as far as we can go for arbitrary forces. However, if all the forces may be expressed in the 
form similar but somewhat more general than Eq. (1.31), Fa = -VaC/( iq, r 2 ,...,i>, t), where U is the 

7 They were derived in 1788 by J.-L. Lagrange who pioneered the whole field of analytical mechanics - not to 
mention his key contributions to number theory and celestial mechanics. 




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effective potential energy of the system, 8 and sign V/ ( denotes differentiation over coordinates of k - th 
particle, we may recast Eq. (8) into a simpler form: 


Sr k 



dU dx 

k + du d y. 

dx k dq, dy k 

dU dz 


dq j dz 


j y 

dq , 


Since we assume that XJ depends only on particle coordinates (and possibly time), but not velocities, 
dU / dq j =0, with the substitution of Eq. (18), the Lagrange equation (17) may be presented in its 

canonical form 




= o, where L = T-U. 

dt dqj dq . 


where L is called the Lagrangian function (or just the “Lagrangian”), defined as 


It is crucial to distinguish this function from the mechanical energy ( 1 .26), E = T + U. 

Using the Lagrangian formalism in practice, the reader should always remember that: 



(i) Each system has only one Lagrange function L, but is described by J >\ Lagrange equations 
of motion (for j = 1,2 ,..., J). 

(ii) Differentiating T, we have to consider the generalized velocities q . as independent variables, 
ignoring the fact they are actually the time derivatives of q j . 

2.2. Examples 

As the first, simplest example, consider a particle constrained to move along one axis (say, x): 

T = ^x 2 , U = U{x,t). (2.20) 

In this case, it is natural to consider x as the (only) generalized coordinate, and x as the generalized 
velocity, so that 

L=T-U = yi 2 — U(x, t). 

( 2 . 21 ) 

Considering x an independent variable, we get dLl dx = mx , and dLl dx = -dU I dx, so that the 
Lagrange equation of motion (only one equation in this case of the single degree of freedom!) yields 



' dU ' 
V dx J 

= 0, 

( 2 . 22 ) 

8 Note that due to the possible time dependence of U, Eq. (17) does not mean that forces F* have to be 
conservative - see the next section for more discussion. With this understanding, I will still use for function U the 
convenient name of “potential energy”. 

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evidently the same result as the x-component of the 2 nd Newton law with F x = -dU/dx. This is a good 
sanity check, but we see that the Lagrange formalism does not provide too much advantage in this 
particular case. 

This advantage is, however, evident for our testbed problem - see Fig. 1. Indeed, taking the polar 
angle 9 for the (only) generalized coordinate, we see that in this case the kinetic energy depends not 
only on the generalized velocity, but also on the generalized coordinate: 9 

T = — r 2 (o 2 
2 V 

L = T -U = 

+ or sin 2 #), 

— R 2 (o 2 + or sin 
2 V 

U = -mgz + const = -mgR cos 9 + const, 
2 9)+ mgR cos 9 + const. 


Here it is especially important to remember that at substantiating the Lagrange equation, 9 and 9 have 
to be treated as independent arguments of L, so that 

2 * V. 1^ 2 2 

— r = mR ~ 9 , — = mR ~ oo~ sin 9 cos 9 - mgR sin 9 , 
89 89 

giving us the following equation of motion: 



( mR9 )- (mR 

or sin 9 cos 

9 - mgR sin #) = 0. 



As a sanity check, at co = 0, Eq. (25) is reduced to the correct equation of the usual pendulum: 

9 + Q. 1 sin# = 0, 

where Q = 



We will explore the full dynamic equation (25) in more detail later, but please note how simple its 
derivation was - in comparison with writing the Newton laws and then excluding the reaction force. 

Next, though the Lagrangian formalism was derived from the Newton law for mechanical 
systems, the resulting equations (19) are applicable to other dynamic systems, especially those for which 
the kinetic and potential energies may be readily expressed via some generalized coordinates. As the 
simplest example, consider the well-known connection (Fig. 4) of a capacitor with capacitance C to an 
inductive coil with self-inductance 10 (Electrical engineers frequently call it the LC tank circuit.) 


0 = 





Fig. 2.4. LC ta nk circuit. 

^ This expression for T = (m/2)(x 2 + y 2 +z 2 ) may be readily obtained either by the formal differentiation of Eq. 
(2) over time, or just by noticing that the velocity vector has two perpendicular components: one along the ring 
(with magnitude R9 ) and another one normal to the ring plane (with magnitude cop = coR sin 9 - see Fig. 1). 

10 Let me hope that this traditional notation would not lead to the confusion between the inductance and the 
Lagrange function. 

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As the reader certainly knows, at relatively low frequencies we may use the so-called lumped- 
circuit approximation, in which the total energy of the system as the sum of two components, the electric 
energy Ec localized inside the capacitor, and the magnetic energy El localized inside the inductance coil 


c ~ 

2C ’ 


Since the electric current / through the coil and the electric charge Q on the capacitor are connected by 
the charge continuity equation dQ/dt = I (evident from Fig. 4), it is natural to declare the charge a 
generalized coordinate, and the current, the generalized velocity. With this choice, the electrostatic 
energy Ec ( Q ) should may be treated as the potential energy U of the system, and the magnetic energy 
£) (/), as its kinetic energy T. With this attribution, we get 

d -Ljl± = LI = LO = 0 dU = dEc - Q 

dq dl dq dQ dq dQ C ’ 

so that the Lagrange equation of motion is 



Note, however, that the above choice of the generalized coordinate and velocity is not unique. 
Instead, one can use as the generalized coordinate the magnetic flux ® through the inductive coil, 
related to the common voltage V across the circuit (Fig. 4) by Faraday’s induction law V = - dQ/dt. With 
this choice, (-V) becomes the generalized velocity, E L = 072 L should be understood as the potential 
energy, and Ec = CV 2 / 2 treated as the kinetic energy. It is straightforward to verity that for this choice, 
the resulting Lagrange equation of motion is equivalent to Eq. (29). If both parameters of the circuit, L 
and C, are constant in time, Eq. (29) is just the hannonic oscillator equation similar to Eq. (1.1), and 
describes sinusoidal oscillations with frequency 




This is of course a very well known result that may be derived in the more standard way by 
equating the voltage drops across the capacitor (V = Q/C) and the inductor (V = -Ldl/dt = -Ld 2 Q/dt 2 ). 
However, the Lagrangian approach is much more convenient for more complex systems, for example, 
for the description of electromagnetic field and its interaction with charged relativistic particles. 11 

2.3, Hamiltonian function and energy 

The canonical form (19) of the Lagrange equation has been derived using Eq. (18), which is 
formally similar to Eq. (1.22) for a potential force. Does this mean that the system described by Eq. (19) 
always conserves energy? Not necessarily, because the “potential energy” U, that participates in Eq. 
(18), may depend not only on the generalized coordinates, but on time as well. Let us start the analysis 
of this issue with the introduction of two new (and very important!) notions: the generalized momenta 
corresponding to each generalized coordinate qj. 

11 See, e.g., EM Sec. 9.8. 

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and the Hamiltonian function 




In order to see whether the Hamiltonian function is conserved, let us differentiate its definition 
(32) over time: 





( dL 'l 

dL .. 


8i h) 


+ q : 

dq, Hj _ 




If we want to make use of the Lagrange equation (19), the last derivative has to be calculated 
considering L as a function of independent arguments q . , q t , and t: 






q H q 

dq, J dq* J 

V 1 J 

+ - 

j J 




where the last term is the derivative of L as an explicit function of time. We see that the last term in the 
square brackets of Eq. (33) immediately cancels with the last tenn in the parentheses of Eq. (34). 
Moreover, using the Lagrange equation (19) for the first term in the square brackets of Eq. (33), we see 
that it cancels with the first tenn in the parentheses of Eq. (34). Thus we arrive at a very simple and 
important result: 

dH _ dL 
dt dt 


The most important corollary of this fonnula is that if the Lagrangian function does not depend 
on time explicitly (dL/ dt = 0), the Hamiltonian function is an integral of motion: 

H = const. (2.36) 

Let us see how it works, using the first two examples discussed in the previous section. For a ID 
particle, definition (3 1) of the generalized momentum yields 

dL n 

p x = — = mv, (2.37) 


so that it coincides with the usual momentum - or rather with its x-component. According to Eq. (32), 
the Hamiltonian function for this case (with just one degree of freedom) is 

H = px — L = mx - 

f m , 2 
— x —U 
V 2 j 

m -2 , j r 

= — x +U . 


12 It is sometimes called just the “Hamiltonian”, but it is advisable to use the full term “Hamiltonian function” in 
classical mechanics, in order to distinguish it from the Hamiltonian operator used in quantum mechanics. (Their 
relation will be discussed in Sec. 10.1.) 








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and coincides with particle’s mechanical energy E = T + U. Since the Lagrangian does not depend on 
time explicitly, both H and E are conserved. 

However, it is not always that simple! Indeed, let us return again to our testbed problem (Fig. 1). 
In this case, the generalized momentum corresponding to the generalized coordinate 6 is 

p g = =mR 2 0, (2.39) 


and Eq. (32) yields: 

H = Pg e-L = mR 1 e 1 - 


R 2 [d 2 + or sin 2 0) + mgR cos 0 

+ const 

= R 2 [p 2 -co 2 sin 2 &)- mgR cos 6 + const. 


This means that (as soon as co ^ 0 ), the Hamiltonian function differs from the mechanical energy 

E = T + U = ^-R 2 (& 2 + co 2 sin 2 d)- mgR cos 6 + const. (2.41) 

The difference, E - H— mR co sin # (besides an inconsequential constant), may change at bead’s motion 
along the ring, so that although H is an integral of motion (since dL/dt = 0), energy E is not conserved. 

Let us find out when do these two functions, E and H, coincide. In mathematics, there is a notion 
of a homogeneous function f(x 1 ,x 2 ,...) of degree A, defined in the following way: for an arbitrary 
constant a , 

f(ax l ,ax 2 ,...) = a A f (x l ,x 2 ,...). (2.42) 

Such functions obey the following Euler theorem : 13 

f-x, = Xf , (2.43) 


that may be readily proven by differentiating both parts of Eq. (42) over a and then setting this 
parameter to the particular value a = 1. Now, consider the case when the kinetic energy is a quadratic 
form of all generalized velocities q t : 

T = (2.44) 

with no other terms. It is evident that such T satisfies the definition of a homogeneous function of the 
velocities with A = 2, 14 so that the Euler theorem (43) gives 

v -1 dT . 

=2 T. 

T d qj 


13 This is just one of many theorems bearing the name of the mathematics genius L. Euler (1707-1783). 

14 Such functions are called quadratic-homogeneous. 

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But since U is independent of the generalized velocities, dL! dcj , = dT / dq f , and the left-hand part of 
Eq. (45) is exactly the first term in the definition (32) of the Hamiltonian function, so that in this case 

H = 2T -L = 2T -(T -U) = T + U = E. (2.46) 

So, for the kinetic energy of the type (44), for example a free particle with the kinetic energy 
considered as a function of its Cartesian velocities, 

T m ( 2 , 2 , 2 ' 

T = -\y x +v y +v Zi 


the notions of the Hamiltonian function and mechanical energy are identical. (Indeed, some textbooks, 
very regretfully, do not distinguish these notions at all!) However, as we have seen from our bead-on- 
the-rotating-ring example, this is not always true. For that problem, the kinetic energy, in addition to the 
tenn proportional to 9 2 , has another, velocity-independent tenn - see the first of Eqs. (23) - and hence 
is not a quadratic-homogeneous function of the angular velocity. 

Thus, Eq. (36) expresses a new conservation law, generally different from that of the mechanical 
energy conservation. 

2.4. Other conservation laws 

Looking at the Lagrange equation (19), we immediately see that if L = T - U as a whole is 
independent of some generalized coordinate qj, dL/dqj = 0, 15 then the corresponding generalized 
momentum is an integral of motion: 16 


p= = const. (2.48) 


For example, for a ID particle with Lagrangian (21), momentum p x is conserved if the potential 
energy is constant (the x-component of force is zero) - of course. As a less obvious example, let us 
consider a 2D motion of a particle in the field of central forces. If we use polar coordinates r and cp in 
the role of the generalized coordinates, the Lagrangian function, 17 

L = T-U = ^{r 2 + r 2 <p 2 )-U(r), (2.49) 

is independent of cp and hence the corresponding generalized momentum, 

dL 2 • cn\ 

P tp = — = mr <p , (2.50) 


15 Such coordinates are frequently called cyclic, because in some cases (like in the second example considered 
below) they represent periodic coordinates such as angles. However, this terminology is misleading, because 
some “cyclic” coordinates (e.g., x in our first example) have nothing to do with rotation. 

16 This fact may be considered a particular case of a more general mathematical statement called the Noether 
theorem (named after its author, A. E. Nother, sometimes called the “greatest woman mathematician ever lived”). 
For its discussion see, e.g., Sec. 13.7 in H. Goldstein et al., Classical Mechanics, 3 rd ed. Addison Wesley, 2002. 

17 Note that here r 2 is just the square of the scalar derivative r , rather than the square of vector r = v. 

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is conserved. This is just a particular (2D) case of the angular momentum conservation 
Indeed, for the 2D motion within the [x, y ] plane, the angular momentum vector, 







Lsrxp = 







see Eq. (1.24). 


has only one nonvanishing component, perpendicular to the motion plane: 

L, = x(my) - y(mx). (2.52) 

Differentiating the well-known relations between the polar and Cartesian coordinates, 

x = rcos(p, y = rsirup, (2.53) 

over time, and plugging the result into Eq. (52), we see that L, = mr 2 ('p = p 

Thus the Lagrangian formalism provides a powerful way of searching for non-evident integrals 
of motion. On the other hand, if such conserved quantity is evident or known a priori, it is helpful for 
the selection of the most appropriate generalized coordinates, giving the simplest Lagrange equations. 
For example, in the last problem, if we have known in advance that p v had to be conserved, this could 
provide a motivation for using angle cp as one of generalized coordinates. 

2,5. Exercise problems 

In each of Problems 2.1-2.10: 

(i) introduce a set of convenient generalized coordinate(s) qj of the system, 

(ii) write down Lagrangian L as a function of q r q , , and (if appropriate) time, 

(iii) write down the Lagrangian equation(s) of motion, 

(iv) calculate the Hamiltonian function //; find out whether it is conserved, 

(v) calculate energy E; is E = HI; is energy conserved? 

(vi) any other evident integrals of motion? 

2.1 . Double pendulum - see Fig. on the right. Consider only the motion 
confined to a vertical plane containing the suspension point. 

2.2 . Stretchable pendulum (i.e. a mass hung on an elastic cord that exerts force F 
= -k{1 - /o), where /rand / 0 are positive constants), confined to a vertical plane: 

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2.3 . Fixed-length pendulum hanging from a horizontal support whose motion law 
xo(t) is fixed. (No vertical plane constraint here.) 


2.4 . A pendulum of mass m hung on another point mass m ’ that may slide, 
without friction, along a straight horizontal rail (see Fig. on the right). Its motion is 
confined to the vertical plane that contains the rail. 

2.5 . A bead of mass m, sliding without friction along a light 
string stretched by fixed force T, between two horizontally displaced 
points - see Fig. on the right. Flere, in contrast to the similar Problem 
1.10, string tension T may be comparable with bead’s weight mg, and 
the motion is not restricted to the vertical plane. 

2.6 . A bead of mass m, sliding without friction along a light string of 
fixed length 21, which is hung between two points, horizontally displaced by 
distance 2d < 21 - see Fig. on the right. As in the previous problem, the motion 
is not restricted to the vertical plane. 

2.1 . A block of mass m that can slide, without friction, along the 
inclined plane surface of a heavy wedge with mass m ’. The wedge is free to 
move, also without friction, along a horizontal surface - see Fig. on the right. 
(Both motions are within the vertical plane containing the steepest slope line.) 


2.8 . The two-pendula system that was the subject of Problem 1.8 - 
see Fig. on the right. 

1 ' 

| F | = tcAx 

m 6 — / \/\/ s — 6 m 

2.9 . A system of two similar, inductively-coupled LC 
circuits - see Fig. on the right. 

2.10 .* A small Josephson junction, i.e. a system of two superconductors 
coupled by Cooper-pair tunneling through a thin insulating layer that separates 
them (see Fig. on the right). 

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(i) At not very high frequencies (whose quantum hco is lower than the binding energy 2 A of the 
Cooper pairs), the Josephson effect may be described by coupling energy 

U{fp) = -Ej cos (p + const, 

where constant Ej describes the coupling strength, and variable (p (called the Josephson phase 
difference ) is related to voltage V across the junction via the famous frequency-to-voltage relation 

dcp _ 2e 
dt h 

where e ~ 1.6xl0~ 19 C is the fundamental electric charge and h « 1.054xl0' 34 Js is the Plank constant. 18 

(ii) The junction (as any system of two close conductors) has a substantial electric capacitance C. 

18 More discussion of the Josephson effect and the physical sense of the variable (p may be found, for example, in 
EM Sec. 6.4 and QM Secs. 2.3 and 2.8 of this lecture note series. 

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1 D system 

Chapter 3. A Few Simple Problems 

In this chapter, I will review the solutions of a few simple but very important problems of particle 
motion, that may be reduced to one dimension, including the famous “ planetary ” problem of two 
particles interacting via a spherically-symmetric potential. In the process, we will discuss several 
methods that will be useful for the analysis of more complex systems. 

3.1. One-dimensional and lD-reducuble systems 

If a particle is confined to motion along a straight line (say, axis x), its position, of course, is 
completely defined by this coordinate. In this case, as we already know, particle’s Lagrangian is given 
by Eq. (2.21): 

L = T(x)-U(x,t), T(x)=^x 2 , 


so that the Lagrange equation of motion (2.22) 

mix = - 

8U (x,t) 

is just the x-component of the 2 nd Newton law. 


It is convenient to discuss the dynamics of such really ID systems in the same breath with that of 
effectively ID systems whose position, due to holonomic constraints and/or conservation laws, is also 
fully determined by one generalized coordinate q, and whose Lagrangians may be presented in a form 
similar to Eq. (1): 

L - T ef (q) -U ef (q,t), T a =^-q 2 , 


where m s f is some constant which may be considered as the effective mass of the system, and the 
function C/ ef its effective potential energy. In this case the Lagrange equation (2.19) describing the 
system dynamics has a form similar to Eq. (2): 

"kf q = 

du e f(qd) 



As an example, let us return again to our testbed system shown in Fig. 1.5. We have already seen 
that for that system, having one degree of freedom, the genuine kinetic energy T, expressed by the first 
of Eqs. (2.23), is not a quadratically-homogeneous function of the generalized velocity. However, the 
system’s Lagrangian (2.23) still may be presented in fonn (3), 

L = R 2 d 2 + t ^-R 2 co 2 sin 2 0 + mgR cos 0 + const = T e{ -U ef , (3.5) 

if we take 

T e{ =^-R 2 0 2 , U e{ = - r ^-R 2 (o 2 sin 2 mg/? cos # + const. (3.6) 

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In this new partitioning of function L, which is legitimate because U e f depends only on the generalized 
coordinate 6 \ but not on the corresponding generalized velocity, T e f includes only a part of the full 
kinetic energy T of the bead, while U e f includes not only the real potential energy U of the bead in the 
gravity field, but also an additional term related to ring rotation. (As we will see in Sec. 6.6, this term 
may be interpreted as the effective potential energy due to the inertial centrifugal “force”.) 

Returning to the general case of effectively ID systems with Lagrangian (3), let us calculate their 
Hamiltonian function, using its definition (2.32): 


H = —q-L = m ef q 2 ~(T cf -U a ) = T ef +U cf . (3.7) 


So, H is expressed via T e f and U e f exactly as the mechanical energy E is expressed via genuine T and U. 

3.2. Equilibrium and stability 

Autonomous systems are defined as the dynamic systems whose equations of motion do not 
depend on time. For ID (and effectively ID) systems obeying Eq. (4), this means that their function U e f, 
and hence the Lagrangian function (5) should not depend on time explicitly. According to Eqs. (2.35), in 
such systems the Hamiltonian function (7), i.e. the sum T ef + U e f, is an integral of motion. However, be 
careful! This may not be true for system’s mechanical energy E; for example, as we already know from 
Sec. 2.2, for our testbed problem, with the generalized coordinate q = 6 (Fig. 2.1), H* E. 

According to Eq. (4), an autonomous system, at appropriate initial conditions, may stay in 
equilibrium at one or several stationary (alternatively called fixed) points q n , corresponding to either the 
minimum or a maximum of the effective potential energy (see Fig. 1): 


Fig. 3.1. Effective potential energy profile 
near stable (y 0 , q 2 ) and unstable (q) fixed 
points, and its quadratic approximation (10) 
near point q 0 - schematically. 

In order to explore the stability of such fixed points, let us analyze the dynamics of small 

q(t) = q(t)-q n (3.9) 

from the equilibrium. For that, let us expand function f/ e f(q) in the Taylor series at a fixed point, 

U e f ( q ) = U ef (q n ) + (q n )q+^~ ( q„ )q 2 +.... 


2 dq 1 


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The first term in the right-hand part, U t \{q n ), is arbitrary and does not affect motion. The next term, 
linear in deviation q , is equal zero - see the fixed point definition (8). Hence the fixed point stability is 
determined by the next term, quadratic in q , more exactly by its coefficient, 

d 2 U f 



which plays the role of the effective spring constant. Indeed, neglecting the higher terms of the Taylor 
expansion (10), 1 we see that Eq. (4) takes the familiar form - cf. Eq. (1.1): 

m e{ q+K e{ q=0. (3.12) 

I am confident that the reader of these notes knows everything about this equation, but since we 
will soon run into similar but more complex equations, let us review the formal procedure of its 
solution. From the mathematical standpoint, Eq. (12) is an ordinary, linear differential equation of the 
second order, with constant coefficients. The theory of such equations tells us that its general solution 
(for any initial conditions) may be presented as 

q{t) = c + e X+t + c_e X ~ , (3.13) 

where constants c± are determined by initial conditions, while the so-called characteristic exponents A± 
are completely defined by the equation itself. In order to find the exponents, it is sufficient to plug just 
one partial solution, exp {/if}, into the equation. In our simple case (12), this yields the following 
characteristic equation : 

m ef A 2 +/r ef =0. (3.14) 

If the ratio k e f/m e f is positive, 2 i.e. the fixed point corresponds to the minimum of potential energy 
(e.g., points qo and qi_ in Fig. 1), the characteristic equation yields 

( Y /2 

A ± =±ico 0 , o)q = 5 (3.15) 

l m ef J 


(where i is the imaginary unity, i = -1), so that Eq. (13) describes sinusoidal oscillations of the system, 

q(t) = c + e u +c_e u = c c cosco 0 t + c s smco 0 t, (3.16) 

with eigenfrequency (or “own frequency”) ay, about the fixed point which is thereby stable. On the 
other hand, at the potential energy maximum (k e f < 0, e.g., at point q\ in Fig. 1), we get 

(\k iY /2 

A + =±A, A= ^ , q(t) = c + e +Xt +c e~ 2t . (3.17) 


Since the solution has an exponentially growing part, 3 the fixed point is unstable. 

1 Those terms may be important only in the very special case then K ef is exactly zero, i.e. when a fixed point is an 
inflection point of function U cl (q). 

2 In what follows, I will assume that the effective mass m ei is positive, which is true in most (but not all!) dynamic 
systems. The changes necessary if it is negative are obvious. 

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Note that the quadratic expansion of function U e f(q), given by Eq. (10), is equivalent to a linear 
expansion of the effective force: 

F = _ dU « 



immediately resulting in the linear equation (12). Hence, in order to analyze the stability of a fixed point 
q n , it is sufficient to linearize the equation of motion in small deviations from that point, and study 
possible solutions of the resulting linear equation. 

As an example, let us return to our testbed problem (Fig. 2.1) whose function U e f we already 
know - see the second of Eqs. (6). With it, the equation of motion (4) becomes 

mRrd = = mR 2 [co 2 cos 6* -£2 2 ] sin <9, i.e. 0 = [co 2 cos6 , -f2 2 ]sin6 l , (3.19) 


where Q = (g/R ) 1 2 is the frequency of small oscillations of the system at co = 0 - see Eq. (2.26). 4 From 
requirement (8), we see that on any 2^-long segment of angle 0 , 5 the system may have four fixed points: 

Q . 1 

9 0 = 0, 9 x =n, 6f 3 =±arccos — T , (3.20) 


The last two fixed points, corresponding to the bead rotating on either side of the ring, exist only if the 
angular velocity co of ring rotation exceeds Q. (In the limit of very fast rotation, co » Q, Eq. (20) yields 
02 2 — » ±/r/2, i.e. the stationary positions approach the horizontal diameter of the ring - in accordance 
with physical intuition.) 

In order to analyze the fixed point stability, similarly to Eq. (9), we plug 6 = 0 n +6 into Eq. 
(19) and Taylor-expand the trigonometric functions of 6 up to the first term in 6 : 

! (cos — sin G n $ ) — Q " (sin^ + cos#,, o\ (3.21) 

0 = 


Generally, this equation may be linearized further by purging its right-hand part of the tenn proportional 
to 6 2 ; however in this simple case, Eq. (21) is already convenient for analysis. In particular, for the 
fixed point do = 0 (corresponding to the bead position at the bottom of the ring), we have cos do = 1 and 
sin#o = 0, so that Eq. (21) is reduced to a linear differential equation 

9=(co 2 -Q 2 )d , (3.22) 

whose characteristic equation is similar to Eq. (14) and yields 

A 2 =(o 2 -Q 2 , for d ~ d 0 . (3.23a) 

3 Mathematically, the growing part vanishes at some special (exact) initial conditions which give c+ = 0. However, 
the futility of this argument for real physical systems should be obvious for anybody who had ever tried to 
balance a pencil on its shaip point. 

4 Note that Eq. (19) coincides with Eq. (2.25). This is a good sanity check illustrating that the procedure (5)-(6) of 
moving of a term from the potential to kinetic energy within the Lagrangian function is indeed legitimate. 

5 For this particular problem, the values of 9 that differ by a multiple of 2 n, are physically equivalent. 

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This result shows that if co < Cl, when both roots A are imaginary, this fixed point is stable. However, 
the roots become real, A± = (of - Q ) ", with one of them positive, so that the fixed point becomes 
unstable beyond this threshold, i.e. as soon as fixed points 02,3 exist. An absolutely similar calculations 
for other fixed points yield 

A 2 =Q 2 +co 2 > 0, for 9 ~ 6 X , (3.23b) 

A 2 =Q 2 -g) 2 , for 6 « # 2 3 . (3.23c) 

These results show that fixed point 6 \ (bead on the top of the ring) is always unstable - just as we could 
foresee, while the side fixed points 6 * 2,3 are stable as soon as they exist (at 00 > Q). 

Thus, our fixed-point analysis may be summarized in a simple way: an increase of the ring 
rotation speed co beyond a certain threshold value, equal to Cl (2.26), causes the bead to move on one of 
the ring sides, oscillating about one of the fixed points 62 , 3 . Together with the rotation about the vertical 
axis, this motion yields quite a complex spatial trajectory as observed from a lab frame, so it is 
fascinating that we could analyze it qualitatively in such a simple way. 

Later in this course we will repeatedly use the linearization of the equations of motion for the 
analysis of stability of more complex systems, including those with energy dissipation. 

3.3, Hamiltonian ID systems 

The autonomous systems that are described by time-independent Lagrangians, are frequently 
called Hamiltonian, because their Hamiltonian function H (again, not necessarily equal to the genuine 
mechanical energy E\) is conserved. In our current ID case, described by Eq. (3), 

H = 


ef -2 

q~ + U ef (q) = const . 


This is the first integral motion. Solving Eq. (24) for q , we get the first-order differential equation, 




= + 


[H-U e{ (q)]\ 



which may be readily integrated: 

^/« ef A 

V ^ J 

1/2 q(t) 



dq ' 

[H-U tf (q')\ 

,2 =t ~ t 0 ' 


Since constant H (as well as the proper sign before the integral - see below) is fixed by initial 
conditions, Eq. (26) gives the reciprocal form, t = t(q), of the desired law of system motion, q(t). Of 
course, for any particular problem the integral in Eq. (26) still has to be worked out, either analytically 
or numerically, but even the latter procedure is typically much easier than the numerical integration of 
the initial, second-order differential equation of motion, because at addition of many values (to which 
the numerical integration is reduced 6 ) the rounding errors are effectively averaged out. 

6 See, e.g., MA Eqs. (5.2) and (5.3). 

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Moreover, Eqs. (24)-(25) also allow a general classification of ID system motion. Indeed: 

(i) If H > U c t{q) in the whole range of interest, the effective kinetic energy T e f (3) is always 
positive. Hence derivative dqldt cannot change sign, so that the effective velocity retains the sign it had 
initially. This is the unbound motion in one direction (Fig. 2a). 

(ii) Now let the particle approach a classical turning point A where H = U e f(x) - see Fig. 2b. 7 
According to Eqs. (25), (26), at that point the particle velocity vanishes, while its acceleration, 
according to Eq. (4), is still finite. Evidently, this corresponds to the particle reflection from the 
“potential wall”, with the change of velocity sign. 

(iii) If, after the reflection from point A, the particle runs into another classical turning point B 
(Fig. 2c), the reflection process is repeated again and again, so that the particle is bound to a periodic 
motion between two turning points. 

(a) (b) (c) 

Fig. 3.2. Graphical representation of Eq. (25) for three different cases: (a) unbound motion, with the 
velocity sign conserved, (b) reflection from the “classical turning point”, accompanied with the velocity 
sign change, and (c) bound, periodic motion between two turning points - schematically, (d) Effective 
potential energy (6) of the bead on the rotating ring (Fig. 1.5) for a>> D., in units of 2 mgR. 

The last case of periodic oscillations presents large practical interest, and the whole next chapter 
will be devoted to a detailed analysis of this phenomenon and numerous associated effects. Here I will 
only note that Eq. (26) immediately enables us to calculate the oscillation period: 



(m ef Y 2 i dq 

l 2 J l[H-U ef (q)] 1/2 ’ 




7 This terminology comes from quantum mechanics which shows that actually a particle (or rather its 
wavefunction) can, to a certain extent, penetrate the “classically forbidden range” where H < U e ^x). 

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where the additional upfront factor 2 accounts for two time intervals: for the motion from B to A and 
back (Fig. 2c). Indeed, according to Eq. (25), in each classically allowed point q the velocity magnitude 
is the same, so that these time intervals are equal to each other. 8 

Now let us link Eq. (27) to the fixed point analysis carried out in the previous section. As Fig. 2c 
shows, if H is reduced to approach U mm , the oscillations described by Eq. (27) take place at the very 
bottom of “potential well”, about a stable fixed point qo. Hence, if the potential energy profile is smooth 
enough, we may limit the Taylor expansion (TO) by the quadratic term. Plugging it into Eq. (27), and 
using the mirror symmetry of this particular problem about the fixed point qo, we get 

r = 4 


. 1/2 A 




+ *tf q 

/2 r 

= —I, with / = [ 

ox. { 

(i -er 


where £= q I A, with A = (2//c e f) 1/2 [77 - 17 m i n ] 1/2 being the classical turning point, i.e. the oscillation 
amplitude, and coo is the eigenfrequency given by Eq. (15). Taking into account that the elementary 
integral / in that equation equals n! 2, 9 we finally get 



as it should be for harmonic oscillations (16). Note that the oscillation period does not depend on the 
oscillation amplitude A, i.e. on the difference ( H - U mm ) - while it is small. 

3.4. Planetary problems 

Leaving a more detailed study of oscillations for the next chapter, let us now discuss the so- 
called planetary systems 10 whose description, somewhat surprisingly, may be also reduced to an 
effectively ID problem. Consider two particles that interact via a conservative, central force F 2 i = - Fj 2 
= n,F(r), where r and n r are, respectively, the magnitude and direction of the distance vector r = iq - r 2 
connecting the two particles (Fig. 3). 

m j 

Fig. 3.3. Vectors in the “planetary” problem. 

8 Note that the dependence of points A and B on the “energy” H is not necessarily continuous. For example, for 
our testbed problem, whose effective potential energy is plotted in Fig. 2d (for a particular value of a>> Q), a 
gradual increase of H leads to a sudden jump, at H = H h of point B to position B\ corresponding to a sudden 
switch from oscillations about one fixed point 02,3 to oscillations about two adjacent fixed points (before the 
beginning of a persistent rotation along the ring at H > H 2 ). 

9 Introducing a new variable C, by relation £, = sin £ we get dq = cos C dC= (1 - q 2 ) 12 dq , so that the function 
under the integral is just d£. 

10 This name is very conditional, because this group of problems includes, for example, charged particle scattering 
(see Sec. 3.7 below). 

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Generally, two particles moving without constrains in 3D space, have 3 + 3 = 6 degrees of 
freedom that may described, e.g., by their Cartesian coordinates {xi, yi, z\, x 2 , yi, z i) However, for this 
particular form of interaction, the following series of tricks allows the number of essential degrees of 
freedom to be reduced to just one. 

First, the central, conservative force of particle interaction may be described by time- 
independent potential energy U (r) . Hence the Lagrangian of the system is 

L = T-U(r) = ^ + ^-U(r). 


Let us perform the transfer from the initial six scalar coordinates of the particles to six generalized 
coordinates: three Cartesian components of the distance vector 

r = ri - r 2 , 


and three components of vector 

R _ "'Ll + "LG 

M = m , + in 2 , 


which defines the position of the center of mass of the system. Solving the system of two linear 
equations (3 1) and (32) for the iq and r 2 , we get 

,, m 2 

R + — r. 

R L r. 

M M 

Plugging these relations into Eq. (30), we may reduce it to 

L = — R 2 + — f 2 -U(r), 
2 2 



where m is the so-called reduced mass : 

m , m , 

, 1 



m~ 1 2 , 

so that — 

= 1 . 



m ] 

m 1 


Note that according to Eq. (35), the reduced mass is lower than that of the lightest component of the 
two-body system. If one of mi |2 is much less that is counterpart (like it is in most star-planet or planet- 
satellite systems), then with a good precision m = min [mi, m 2 \. 

Since the Lagrangian function (34) depends only on R rather than R itself, according to our 
discussion in Sec. 2.4, the Cartesian components of R are cyclic coordinates, and the corresponding 
generalized momenta are conserved: 

P : = 


8R : 

= MR . = const, y = l,2, 3. 


Physically, this is just the conservation law for the full momentum P = MR of our system, due to 
absence of external forces. Actually, in the axiomatics used in Sec. 1.3 this law is postulated - see Eq. 
(1.10) - but now we may attribute momentum P to a certain geometric point, the center of mass R. In 
particular, since according to Eq. (36) the center moves with constant velocity in the inertial reference 

Center of 



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frame used to write Eq. (30), we may create a new inertial frame with the origin at point R. In this new 
frame, R = 0, so that vector r (and hence scalar r) remain the same as in the old frame (because the 
frame transfer vector adds equally to iq and n, and cancels in r = iq — n), and the Lagrangian (34) is 
now reduced to 

Z = yf 2 -f/(r). (3.37) 

Thus our initial problem has been reduced to just three degrees of freedom - three scalar 
components of vector r. Moreover, Eq. (37) shows that dynamics of vector r of our initial, two-particle 
system is identical to that of the radius-vector of a single particle with the effective mass m, moving in 
the central potential field U(r). 

3.5. 2 nd Kepler law 

Two more degrees of freedom may be excluded from the planetary problem by noticing that 
according to Eq. (1.35), the angular momentum L = rxp of our effective particle is also conserved, both 
in magnitude and direction. Since the direction of L is, by its definition, perpendicular to both of r and v 
= p/m, this means that particle’s motion is confined to a plane (whose orientation in space is determined 
by the initial directions of vectors r and v). Hence we can completely describe particle’s position by just 
two coordinates in that plane, for example by distance r to the center, and the polar angle cp In these 
coordinates, Eq. (37) takes the form identical to Eq. (2.49): 

Z = y(f 2 +r 2 <p 2 )-U(r). (3.38) 

Moreover, the latter coordinate, polar angle (p , may be also eliminated by using the conservation of 
angular momentum’s magnitude, in the form of Eq. (2.50): 11 

L z = mr 2 cp = const. (3.39) 

A direct corollary of this conservation is the so-called 2 nd Kepler law. 12 the radius-vector r 
sweeps equal areas A in equal times. Indeed, in the linear approximation in dA«A, the area differential 
dA equals to the area of a narrow right triangle with the base being the arc differential rdcp, and the 
height equal to r - see Fig. 4. As a result, according to Eq. (39), the time derivative of the area, 

dA _ r {rdcp) 1 2 
dt dt 

1 2 . L z 
—r<b = — — . 

2 2m 


remains constant. Integration of this equation over an arbitrary (not necessarily small!) time interval 
proves the 2 nd Kepler law. 

11 Here index z stands for the coordinate perpendicular to the motion plane. Since other components of the angular 
momentum are equal zero, the index is not really necessary, but I will still use it, just to make a clear distinction 
between the angular momentum L z and the Lagrangian function L. 

12 One of three laws deduced almost exactly 400 years ago by J. Kepler (1571 - 1630), from the extremely 
detailed astronomical data collected by T. Brahe (1546-1601). In turn, the set of three Kepler laws were the main 
basis for Isaac Newton’s discovery of the gravity law (1.16). That’s how physics marched on.. . 

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Fig. 3.4. Area differential in the polar 

Now note that since dL/ dt = 0 , the Hamiltonian function H is also conserved, and since, 
according to Eq. (38), the kinetic energy of the system is a quadratic-homogeneous function of the 
generalized velocities r and <p , H = E , so that the system energy E, 


is also a first integral of motion. 13 But according to Eq. (39), the second tenn of Eq. (41) may be 
presented as 



2 • 2 

r cp 


2 mr 

2 ’ 


so that energy (41) may be expressed as that of a ID particle moving along axis r, 

E=™r 2 +U e <(r), 


in the following effective potential: 

U e{ (r) = U(r) + 


So the planetary motion problem has been reduced to the dynamics of an effectively ID system. 


Now we may proceed just like we did in Sec. 3, with due respect for the very specific effective 
potential (44) which, in particular, diverges at r — » 0 - possibly besides the very special case of an 
exactly radial motion, L z = 0. In particular, we may solve Eq. (43) for dr/dt to get 

dt = 

f V /2 

m ' 



\E - U ef {r )] 

1/2 - 


The integration of this relation allows us not only to get a direct relation between time t and distance r, 
similar to Eq. (26), 

13 One may claim that this fact should have been evident from the very beginning, because the effective particle 
of mass m moves in a potential field U(r) which conserves energy. 

14 Note that this reduction has been done in a way different from that used for our testbed problem 
(shown in Fig. 2.1) in Sec. 2 above. (The reader is encouraged to analyze this difference.) In order to 
emphasize this fact, I will keep writing E instead of H here, though for the planetary problem we are 
discussing now these two notions coincide. 




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t = ± 

f V /2 

m I 


= + 

A \ 1/2 

777 | 


2 - 11/2 ’ 

.2; J [£ - C/(r) - / 2/777- ] 

but also do a similar calculation of angle (p . Indeed, integrating Eq. (39), 

(p = \(pdt=^\%. 
j m j r 

and plugging dt from Eq. (45), we get an explicit expression for particle’s trajectory cp (r): 

(p = ±- 




= +- 

(2m) U2 J r 2 [E -U ef (r)] V2 (2m) 12 J r 2 [ii - t/(r) -Z 2 /2mr 2 ] 



2 -| 1 / 2 * 




Note that according to Eq. (39), derivative d(p/dt does not change sign at the reflection from any 
classical turning point r ^ 0, so that, in contrast to Eq. (46), the sign in the right-hand part of Eq. (48) is 
uniquely determined by the initial conditions and cannot change during the motion. 

Let us use these results, valid for any interaction law U(r ), for the planetary motion’s 
classification. The following cases should be distinguished. (Following a good tradition, in what follows 
I will select the arbitrary constant in the potential energy in the way to provide U e f — > 0 at r — » 00 .) 

If the particle interaction is attractive, and the divergence of the attractive potential at r — > 0 is 
faster than Hr 2 , then U C {(r) — » -00 at r — » 0, so that at appropriate initial conditions (E < 0) the particle 
may drop on the center even if L z ^ 0 — the event called the capture. On the other hand, with U(r) either 
converging or diverging slower than Mr at r — > 0, the effective energy profile U c \(r) has the shape 
shown schematically in Fig. 5. This is true, in particular, for the very important case 


which describes, in particular, the Coulomb (electrostatic) interaction of two particles with electric 
charges of the opposite sign, and Newton’s gravity law (1.16a). This particular case will be analyzed in 
the following section, but now let us return to the analysis of an arbitrary attractive potential U{r) < 0 
leading to the effective potential shown in Fig. 5, when the angular-momentum term dominates at small 
distances r. 


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According to the analysis of Sec. 3, such potential profile, with a minimum at some distance ro, 
may sustain two types of motion, depending on the energy E (which is of course determined by the 
initial conditions): 

(i) If E > 0, there is only one classical turning point where E = U e f, so that distance r either grows 
with time from the very beginning, or (if the initial value of r was negative) first decreases and then, 
after the reflection from the increasing potential U e f, starts to grow indefinitely. The latter case, of 
course, describes scattering. 

(ii) On the opposite, if the energy is within the range 

U ef {r 0 )<E< 0, (3.50) 

the system moves periodically between two classical turning points r min and r max . These oscillations of 
distance r correspond to the bound orbital motion of our effective particle about the attracting center. 15 

Let us start with the discussion of the bound motion, with energy within the range (50). If energy 
has its minimal possible value, 

^ = t/ ef( r o) = min [ t/ ef( r )]> (3.5 i ) 

the distance cannot change, r = ro = const, so that the orbit is circular, with the radius ro satisfying the 
condition dUJdr = 0. Let us see whether this result allows for an elementary explanation. Using Eq. 
(44) we see that the condition for ro may be written as 

L] _dU 
mi~l dr 

Since in a circular motion, velocity v is perpendicular to the radius vector r, L z is just mrov, the left-hand 
part of Eq. (52) equals mv /r 0 , while its right-hand part is just the magnitude of the attractive force, so 
that this equation expresses the well-known 2 nd Newton law for the circular motion. Plugging this result 
into Eq. (47), we get a linear law of angle change, tp = cot + const, with angular velocity 



co = 


and hence the rotation period T v = In! co obeys the elementary relation 

r _ 

* v 


Now, let the energy be above its minimum value. Using Eq. (46) just as in Sec. 3, we see that 
distance r now oscillates with period 


r , =(2 ,»r j 


\E — U (r) - L z ! 2mr ] 

2-11/2 • 


15 In the opposite case when the interaction is repulsive, U(r ) > 0, the addition of the positive angular energy term 
only increases the trend, and only the scattering scenario is possible. 

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and its 

This period is, in general, different from Indeed, the change of angle <p between two sequential 
points of the nearest approach, that follows from Eq. (48), 

\\(p\ = 2 

L z r T dr 

( 2m ) 12 r min r 2 [e - U ( r ) - L\ / 2 mr 2 } 2 


is generally different from 2 n . Hence, the general trajectory of the bound motion has a spiral shape - 
see, e.g., an illustration in Fig. 6. 

Fig. 3.6. Typical open orbit of a particle 
moving in a non-Coulomb central field. 

3.6. 1 st and 3 rd Kepler laws 

The situation is special, however, for a very important particular case, namely that of the 
Coulomb potential described by Eq. (49). Indeed, plugging this potential into Eq. (48), we get 


= +- 




(2 m) V2 J r 2 (E + a/r-L]/2mr 2 ]' 2 ’ 

This is a table integral, 16 equal to 

(p = ±arccos- 

Z? / m ar — 1 

~ !2 

(l + 2 EL]/ma 2 ) 1 

The reciprocal function, r((p), is 2 n- periodic: 

+ const. 

so that at E < 0, the orbit a closed line, 17 characterized with the following parameters: 



, 2 EL 2 

1+ 2 






16 See, e.g., MA Eq. (6.3a). 

17 It may be proved that for the power-law interaction, U cc r v , the orbits are closed line only if v= -1 (i.e. our 
current case of the Coulomb potential) or v= +2 (the 3D harmonic oscillator) - the so-called Bertrand theorem. 

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The physical meaning of these parameters is very simple. Indeed, according to the general Eq. 
(52), in the Coulomb potential, for which dU/dr = air , we see that p is just the circular orbit radius 18 for 
given L z \ tq = L : hna = p, and 

min[t = (3.61) 

Using this equality, parameter e (called eccentricity ) may be presented just as 

e = 

min[U cf (r)]J 


Analytical geometry tells us that Eq. (59), with e < 1, is one of canonical forms for presentation 
of an ellipse, with one of its two focuses located at the origin. This fact is known as the 1 st Kepler law. 
Figure 7 shows the relation between the main dimensions of the ellipse and parameters p and e. 19 

y = r sin cp 

In particular, the major axis a and minor axis b are simply related to p and e and hence, via Eqs. 
(60), to the motion integrals E and L z \ 

P _ a , _ P _ L z 
1-e 2 2|c|’ (l-e 2 )" 2 (: i m \E\] r 


As was mentioned above, at E — » min [U cl (r)] the orbit is almost circular, with r{(p) = r o = p. On 
the contrary, as E is increased to approach zero (its maximum value for the closed orbit), then e — > 1, so 
that the aphelion point r max = p!( I - e) tends to infinity, i.e. the orbit becomes extremely extended. If the 
energy is exactly zero, Eq. (59) (with e = 1) is still valid for all values of cp (except for one special point 

cp = n where r becomes infinite) and describes a parabolic (i.e. open) trajectory. At E > 0, Eq. (59) is 
still valid within a certain sector of angles cp (in that it yields positive results for r), and describes an 
open, hyperbolic trajectory - see the next section. 

For E < 0, the above relations also allow a ready calculation of the rotation period 7= 7^= 7^. . 
(In the case of a closed trajectory, /T and 7^ have to coincide.) Indeed, it is well known that the ellipse 
area A = nab. But according to the 2 nd Kepler law (40), dA/dt = LJ2m = const. Hence 

18 Mathematicians prefer a more solemn terminology: parameter 2 p is called the l at us rectum of the elliptic 
trajectory - see Fig. 7. 

19 In this figure, the constant participating in Eqs. (58)-(59) is assumed to be zero. It is evident that a different 
choice of the constant corresponds just to a constant turn of the ellipse about the origin. 

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T = 


dA / dt 

L_ / 2m 

Using Eqs. (60) and (63), this result may be presented in several other forms: 

6™V /2 

r = 


(\-e 2 f n (L z /2m) 

= na\ 

r Y /2 



= 2 na 






Since for the Newtonian gravity (1.16a), a = Gm \ m 2 = GmM, at m \ « m 2 (i.e. m « M) this 
constant is proportional to m, and the last form of Eq. (64b) yields the 3 ld Kepler law. periods of motion 
of different planets in the same central field, say that of our Sun, scale as T oc a . Note that in contrast 
to the 2 nd Kepler law (that is valid for any central field), the 1 st and 3 ld Kepler laws are potential- 

3.7. Classical theory of elastic scattering 

If E > 0, the motion is unbound for any interaction potential. In this case, the two most important 
parameters of the particle trajectory are the scattering angle 9 and impact parameter b (Fig. 8), and the 
main task for theory is to find the relation between them in the given potential U(r). For that, it is 
convenient to note that b is related to two conserved quantities, particle’s energy 20 E and its angular 
momentum L z , in a simple way: 21 

\i / 2 

= b{2mE )' 

Hence the angular contribution to the effective potential (44) may be presented as 


= E- 



2 mr r 

Second, according to Eq. (48), the trajectory sections from infinity to the nearest approach point (r = 
r min ), and from that point to infinity, have to be similar, and hence correspond to equal angle changes (fX) 
- see Fig. 8. 

Fig. 3.8. Main geometric parameters of the scattering problem. 

2(1 The energy conservation law is frequently emphasized by calling this process elastic scattering. 

21 Indeed, at r » b , the definition L = rx(/r/v) yields L z = bmv x , where w, = (2Elm) ijl is the initial (and hence the 
final) velocity of the particle. 

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Hence we may apply the general Eq. (48) to just one of the sections, say [r mm , go], to find the 
scattering angle: 

0 = n - 2 <p 0 = n - 2 



(2 rnj 12 

r . r 


[e - U ( r ) — l} z ! 2 mr 2 ] 

= n - 

2 1 


r 2 [l - U (r) / E - b 1 / r 2 ] 

- (3.67) 

In particular, for the Coulomb potential (49), now with an arbitrary sign of a, we can apply the same 
table integral as in the previous section to get 22 

0 = 

_ a / 2 Eb 

71-2 arccos T ^ 

[l + (a/2Eb) 2 ] 

This result may be more conveniently rewritten as 

\ 6 \ F 

tan— = 1 

2 2 Eb 



Very clearly, the scattering angle’s magnitude increases with the potential strength a, and decreases as 
either the particle energy or the impact parameter (or both) are increased. 

The general equation (67) and the Coulomb-specific relations (68) present a formally complete 
solution of the scattering problem. However, in a typical experiment on elementary particle scattering 
the impact parameter b of a single particle is random and unknown. In this case, our results may be used 
to obtain statistics of the scattering angle 6, in particular the so-called differential cross-section 23 


(3.69) cross- 


where n is the average number of the incident particles per unit area, and dN is the average number of 
particles scattered into a small solid angle range <7Q. For a spherically-symmetric scattering center, 
which provides an axially-symmetric scattering pattern, da/dQ. may be calculated by counting the 
number of incident particles within a small range db of the impact parameter: 

dN = n2nbdb. (3.70) 

da _ 1 dN 
dCl n dCl 

and hence scattered into the corresponding small solid angle range <70 = 2n sin# dO. Plugging these 
relations into Eq. (69), we get the following general geometric relation: 








In particular, for the Coulomb potential (49), a straightforward differentiation of Eq. (68) yields 
the so-called Rutherford scattering formula 

22 Alternatively, this result may be recovered directly from Eq. (59) whose parameters, at E >0, may be expressed 
via the same dimensionless parameter (2 Eb/a): p = b(2Eb/a), e = [1 + (2 Eb/a) 2 ]' 12 > 1. 

23 This terminology stems from the fact that an integral of da/dQ over the full solid angle, called the full cross- 
section a, has the dimension of area: a= N/n, where A is the total number of scattered particles. 

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a ' 

2 1 


f4 Ej 

sin 4 (0/ 2) 


This result, which shows very strong scattering to small angles (so strong that the integral that 
expresses the full cross-section a is formally diverging at 6 — » 0), 24 and weak backscattering (scattering 
to angles 6 « jf) was historically extremely significant: in the early 1910s its good agreement with a- 
particle scattering experiments carried out by E. Rutherford’s group gave a strong justification for 
“planetary” models of atoms, with electrons moving about very small nuclei. 

Note that elementary particle scattering is frequently accompanied with electromagnetic 
radiation and/or other processes leading to the loss of the initial mechanical energy of the system, 
leading to inelastic scattering, that may give significantly different results. (In particular, a capture of an 
incoming particle becomes possible even for a Coulomb attracting center.) Also, quantum-mechanical 
effects may be important at scattering, so that the above results should be used with caution. 

3.8. Exercise problems 

3.1 . For the system considered in Problem 2.5 (a bead sliding 
along a string with fixed tension T, see Fig. on the right), analyze 
small oscillations of the bead near the equilibrium. 

3.2 . Calculate the functional dependence of period /"of oscillations of a ID particle of mass m in 
potential U(q) = aq n (where a > 0, and n is a positive integer) on energy E. Explore the limit n — » oo. 

3.3 . Explain why the term mr 2 <p 2 / 2 , recast in accordance with Eq. (42), cannot be merged with 
U(r) in Eq. (38), to form an effective ID potential energy U(r) - L z Hmr , with the second term’s sign 
opposite to that given by Eq. (44). We have done an apparently similar thing for our testbed, bead-on- 
rotating-ring problem in the very end of Sec. 1 - see Eq. (3.6); why cannot the same trick work for the 
planetary problem? Besides a formal explanation, discuss the physics behind this difference. 

3.4 . A dumbbell, consisting of two equal masses m on a light rod of length 
/, can slide without friction along a vertical ring of radius R, rotated about its 
vertical diameter with constant angular velocity oo - see Fig. on the right. Derive 
the condition of stability of the lower horizontal position of the dumbbell. 

24 This divergence, which persists at the quantum-mechanical treatment of the problem, is due to particles with 
large values of b, and disappears at an account, for example, of a finite concentration of the scattering centers. 

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3.5. 25 Analyze the dynamics of the so-called spherical pendulum - a point mass hung, in a 
uniform gravity field g, on a light cord of length /, with no motion’s confinement to a vertical plane. In 

(i) find the integrals of motion and reduce the problem to a ID one, 

(ii) calculate the time period of the possible circular motion around the vertical axis, 

(iii) explore small deviations from the circular motion. (Are the pendulum orbits closed?) 

3.6. The orbits of Mars and Earth around the Sun may be well approximated as circles, with a 
radii ratio of 3/2. Use this fact, and the Earth year duration (which you should know :-), to calculate the 
time of travel to Mars spending least energy, neglecting the planets' size and the effects of their 
gravitational fields on the spacecraft. 

3.7 . Derive first-order and second-order differential equations for u = Hr as a function of cp, 
describing the trajectory of particle’s motion in a central potential U{r). Spell out the latter equation for 
the particular case of the Coulomb potential (3.49) and discuss the result. 

3.8 . For motion in the central potential 

... , a (5 

U (r) = + 

r r 

(i) find the orbit r(cp), for positive a and fl and all possible ranges of energy E; 

(ii) prove that in the limit (3 — > 0, and for energy E < 0, the orbit may be represented as a slowly 
rotating ellipse; 

(iii) express the angular velocity of this slow orbit rotation via parameters a and /? of the 
potential, particle’s mass m, its energy E, and the angular momentum L z . 

3.9 . A particle is moving in the field of an attractive central force, with potential 

U(r) = — where an > 0 . 


For what values of n is a circular orbit stable? 

3.10 . Determine the condition for a particle of mass m, moving under the effect of a central 
attractive force 

where C and R are positive constants, to have a stable circular orbit. 

3.11 . A particle of mass m, with angular momentum L z , moves in the field of an attractive central 
force with a distance-independent magnitude F. If particle's energy E is slightly higher than the value 
E m i n corresponding to the circular orbit of the particle, what is the time period of its radial oscillations? 
Compare the period with that of the circular orbit at E = E mm . 

25 Solving this problem is a very good preparation for the analysis of symmetric top rotation in Sec. 


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3.12 . For particle scattering in a repulsive Coulomb field, calculate the minimum approach 
distance r mm and velocity v m ; n at that point, and analyze their dependence on the impact parameter b (see 
Fig. 3.8 of the lecture notes) and the initial velocity Voo of the particle. 

3.13 . A particle is launched from afar, with impact parameter b, toward an attracting center with 
central potential 


with n > 2, a > 0. 

(i) Express the minimum distance between the particle and the center via b, if the initial kinetic 
energy E of the particle is barely sufficient for escaping the capture by the attracting center. 

(ii) Calculate capture’s full cross-section; explore the limit n — » 2. 

3.14 . A meteorite with initial velocity Voo approaches an atmosphere-free planet of mass M and 
radius R. 

(i) Find the condition on the impact parameter b for the meteorite to hit planet’s surface. 

(ii) If the meteorite barely avoids the collision, what is its scattering angle? 

3.15 . Calculate the differential and full cross-sections of the classical, elastic scattering of small 
particles by a hard sphere of radius R. 

3.16 . The most famous 26 continuation of Einstein’s general relativity theory has come from the 
observation, by A. Eddington and his associates, of light’s deflection by the Sun, during the May 1919 
solar eclipse. Considering light photons as classical particles propagating with the light speed vo — » c « 
2.998x10 m/s, and the astronomic data for Sun’s mass, M s « 1.99x10 kg, and radius, R s ~ 
0.6957xl0 9 m, calculate the nonrelativistic mechanics’ prediction for the angular deflection of the light 
rays grazing the Sun’s surface. 

26 It was not the first confirmation, though. The first one came 4 years earlier from A. Einstein himself, who 
showed that his theory may qualitatively explain the difference between the rate of Mercury orbit’s precession, 
known from earlier observations, and the nonrelativistic theory of this effect. 

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Chapter 4. Oscillations 

In this course, oscillations in ID ( and effectively ID) systems are discussed in detail, because of their 
key importance for physics and engineering. We will start with the so-called “linear ” oscillator whose 
differential equation of motion is linear and hence allows the full analytical solutions, and then proceed 
to “nonlinear’’ and parametric systems whose dynamics may be only explored by either approximate 
analytical or numerical methods. 

A A. Free and forced oscillations 

In Sec. 3.2 we briefly discussed oscillations in a very important Hamiltonian system - a ID 
harmonic oscillator described by a simple ID Lagrangian 1 

L=T(q)-U(q) = ™q 2 -^q 2 , 


whose Lagrangian equation of motion, 



mq + Kq = 0, i.e. q + a> 0 q = 0, with of = — > 0 



is a linear homogeneous differential equation. Its general solution is presented by Eq. (3.16), but it is 
frequently useful to recast it into another, amplitude-phase form: 


q(t) = u cos co 0 t + vsin a> 0 t = A cos (a> 0 t - <p) . 






where A is the amplitude and cp the phase of the oscillations, which are determined by the initial 
conditions. Mathematically, it is frequently easier to work with sinusoidal functions as complex 
exponents, by rewriting Eq. (3a) in one more form: 2 












1 COr\ t 




where a is the complex amplitude of the oscillations: 

a = Ae l<p , lal = A, Rea = Acoscp = u, \ma = AAx\(p = v. 


Equations (3) represent the so-called free oscillations of the system, that are physically due to 
the initial energy of the system. At an account for dissipation, i.e. energy leakage out of the system, such 
oscillations decay with time. The simplest model of this effect is represented by an additional viscosity 
force that is proportional to the generalized velocity and directed opposite to it: 

1 For the notation simplicity, in this chapter I will drop indices “ef’ in the energy components T and U, and 
parameters like m, k, etc. However, the reader should still remember that T and U do not necessarily coincide with 
the real kinetic and potential energies (even if those energies may be uniquely identified) - see Sec. 3.1. 

2 Note that this is the so-called physics convention. Most engineering texts use the opposite sign in the imaginary 
exponent, exp {-icot} — > exp {icot}, with the corresponding sign implications for intermediate formulas, but (of 
course) similar final results for real variables. 

© 2013-2016 K. Likharev 

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f v = -m » 


where constant rj is called the viscosity coefficient? The inclusion of this force modifies the equation of 
motion (2) to become 

mcj + r/q + tcq = 0 . 


This equation is frequently presented in the form 

q + 2 8q + a>lq = 0, with 8 = 




where parameter 8 is called the damping coefficient. Note that Eq. (6) is still a linear homogeneous 
second-order differential equation, and its general solution still has the form of the sum (3.13) of two 
exponents of the type exp{2i}, with arbitrary pre-exponential coefficients. Plugging such an exponent 
into Eq. (4), we get the following algebraic characteristic equation for A: 

A 2 + 28A + co 2 = 0. 


Solving this quadratic equation, we get 

A ± = -8 ± icof where co 0 ' = (cOq -8 2 J' 2 , (4.8) 

so that for not very high damping (8< of) 3 4 we get the following generalization of Eq. (3): 

<7 free (0 = c + e^ +t + c _e^~ t = (u Q cos co Q ’t + v 0 sin co 0 ’t)e St = A 0 e St cos (co 0 't - tp 0 ). (4.9) 

The result shows that, besides a certain correction to the free oscillation frequency (which is very small 
in the most interesting case of low damping, 8 « co d), the energy dissipation leads to an exponential 
decay of oscillation amplitude with time constant t= 1 /S: 

A = A 0 e 

-t! r 

where r = — 





A convenient, dimensionless measure of damping is the so-called quality factor Q (or just Q- 
factor ) which is defined as 00)128, and may be rewritten in several other useful forms: 

3 Here I treat Eq. (5) as a phenomenological model, but in statistical mechanics such dissipative term may be 
derived as an average force exerted on a body by its environment whose numerous degrees of freedom are in 
random, though possibly thermodynamically-equilibrium states. Since such environmental force also has a 
random component, the dissipation is fundamentally related to fluctuations, and the latter effects may be 
neglected (as they are in this course) only if the oscillation energy is much higher than the energy scale of random 
fluctuations of the environment - in the thermal equilibrium at temperature T, the larger of k B T and tuajl - see, 
e.g., SM Chapter 5 and QM Chapter 7. 

4 Systems with very high damping (8 > coq) can hardly be called oscillators, and though they are used in 
engineering and physics experiment (e.g., for the shock, vibration, and sound isolation), for their discussion I have 
to refer the interested reader to special literature - see, e.g., C. Harris and A. Piersol, Shock and Vibration 
Handbook, 5 th ed., McGraw Hill, 2002. Let me only note that at very high damping, S» a>o, the system may be 
adequately described with just one parameter: the relaxation time 1/2+ ~ 28/ of » coq. 








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6) 0 _ m<o 0 
^ 28 ?! 

X _co^ 

71 — — , 

T 2 


where T= 2jr/a>o is the oscillation period in the absence of damping - see Eq. (3.29). Since the 
oscillation energy E is proportional to their amplitude squared, i.e. decays as cxp[-2f/r}, with time 
constant r/2, the last form of Eq. (1 1) may be used to rewrite the 0-factor in one more form: 

Q = o) 0 


= co n 



where F is the dissipation power. (Two other useful ways to measure Q will be discussed in a minute.) 

The range of 0-factors of important oscillators is very broad, all the way from Q ~ 10 for a human leg 

(with relaxed muscles), to 0 ~ 10 4 of the quartz crystals used in “electronic” clocks and watches, all the 

way up to 0 ~ 10 for microwave cavities with superconducting walls. 





In contrast to the decaying free oscillations, the forced oscillations, induced by an external force 
F(t), may maintain their amplitude infinitely, even at nonvanishing damping. This process may be 
described by a still linear but now inhomogeneous differential equation 

mcj + r/q + Kq = F(t), 


or, more conveniently, by the following generalization of Eq. (6b): 

q + 2Sq + a>lq = f(t), where f(t) = F(t)/m. 


For a particle of mass m, confined to a straight line, Eq. (12a) is just an expression of the 2 nd Newton 
law (or rather one of its Cartesian component). More generally, according to Eq. (1.41), Eq. (13) is valid 
for any dissipative ID system whose Gibbs potential energy (1.39) has the form Uc(q, t ) = icq 12 - F(t)q. 

The forced-oscillation solutions may be analyzed by two mathematically equivalent methods 
whose relative convenience depends on the character of function /(/). 

(i) Frequency domain. Let us present function ff) as a Fourier sum of sinusoidal harmonics: 5 



Then, due to linearity of Eq. (13), its general solution may be presented as a sum of the decaying free 
oscillations (9) with frequency ax> independent of function F(t), and forced oscillations due to each of 
the Fourier components of the force: 6 

of Eq. (13) 

Plugging Eq. (15) into Eq. (13), and requiring the factors before each e uot in both parts to be equal, we 

?(0 = *7 free (0 + ^forced ( 0 » Screed (0 = Tj a lo e 

- iwt 


5 Operator Re, used in Eq. (3), may be dropped here, because for any physical (real) force, the imaginary 
components of the sum compensate each other. This imposes the following condition on the complex Fourier 
amplitudes: f a =fa>*, where the star means the complex conjugation. 

6 In physics, this mathematical property of linear equations is frequently called the linear superposition principle. 

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a* = fcoZ(®l 

where complex function /(co), in our particular case equal to 

*(®) = 

(a>Q - co 2 )-2icoS 



is called either the response function or (especially for non-mechanical oscillators) the generalized 
susceptibility. From here, the real amplitude of oscillations under the effect of a sinusoidal force that 
may be represented by just one Fourier harmonic of the sum (15), is 





This formula describes, in particular, an increase of the oscillation amplitude A 0) at co — » coo - see 
Fig. 1. According to Eqs. (1 1) and (20), at the exact resonance, 

w=co 0 


2 <x> 0 S ’ 


so that, according to Eq. (11), the ratio of the oscillator response magnitudes at co = ay, and at (o = 0 
(\y(co)\ or o = I lop ) is exactly equal to the 0-factor. Thus, the response increase is especially strong in the 
low damping limit (S« a>o, i.e. Q » 1); moreover at Q — > oo and co — > coq the response diverges. (This 
fact is very useful for the approximate methods to be discussed later in this chapter.) This is of course 
the classical description of the famous phenomenon of resonance, so ubiquitous in physics. 

Fig. 4.1. Resonance in a harmonic oscillator 
(13), for several values of the Q- factor. 

Due to the increase of the resonance peak height, its width is inversely proportional to Q. 
Quantitatively, in the most interesting low-damping limit, Q » 1, the reciprocal Q - factor gives the 
normalized value of the so-called FWHM (“full-width at half-maximum”) of the resonance curve: 

A co _ 1 
®o Q 


Indeed, A co is defined as the difference (co+ - co.) between the two values of co at that the square of 

I I 2 

oscillator response function, \ z(co)\ (which is, in particular, proportional to the oscillation energy), 

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equals a half of its resonance value (19). In the low damping limit, both these points are very close to 
cat), so that in the first (linear) approximation in ( co - cod) « cod, co we can take (coo“ - at ) = -(co+coo)(co- 
cod) « (-2 co<^) « (-2cod<^), where 

^ = co-co 0 (4.21) 

is a convenient parameter called detuning. (We will repeatedly use it later in this chapter.) In this 
approximation, the second of Eqs. (18) is reduced to 

\x(v )\ 2 = 


4 C0 2 {s 2 +f-y 


2 2 

As a result, points co+ correspond to ^ = 3 , i.e. co± = co 0 ± S = co 0 (l ± 1/20, so that A co = co+ - co. = 
coo/Q, thus proving Eq. (20). 

(ii) Time domain. Returning to the general problem of linear oscillations, one may argue that 
Eqs. (9), (15)-(17) provide a full solution of the forced oscillation problem. This is formally correct, but 
this solution may be very inconvenient if the external force is far from sinusoidal function of time. In 
this case, we should first calculate the complex amplitudes f (0 participating in the Fourier sum (14). In 
the general case of non-periodic fit), this is actually the Fourier integral, 

m = \f.e~ iM clt, 


so that f co should be calculated using the reciprocal Fourier transform, 

■t +oo 

/. = — J oty m ‘dt' . 


Now we can use Eq. (16) for each Fourier component of the resulting forced oscillations, and rewrite the 
last of Eqs. (15) as 

+oo +oo +oo , +oo 

9 forced (0 = \a (0 e~ uot dco= \x{co)f C0 e~ U0t dco = J dm Z(^)~ J dt' f (t')e iw{t 





-t-QU -f-OU 

= jdt'f(t') — ^dco%(co)e 

ia>{t' - 1) 


with the response function j^co) given, in our case, by Eq. (17). Besides requiring two integrations, Eq. 
(25) is conceptually uncomforting: it seems to indicate that the oscillator’s coordinate at time t depends 
not only on the external force exerted at earlier times t’ < t, but also in future times. This would 
contradict one of the most fundamental principles of physics (and indeed, science as a whole), the 
causality, no effect may precede its cause. 

Fortunately, a straightforward calculation (left for reader’s exercise) shows that the response 
function (17) satisfies the following rule: 7 

7 This is true for all systems in which /fi) represents a cause, and q(t) its effect. Following tradition, I discuss the 
frequency-domain expression of this causality relation (called the Kramers-Kronig relations) in the Classical 
Electrodynamics part of this lecture series - see EM Sec. 7.3. 

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jl(a)e lC0T da>= 0, 

for r < 0. 


This fact allows the last form of Eq. (25) to be rewritten in either of the following equivalent forms: 


where G(t), defined as the Fourier transform of the response function, 


is called the ( temporal) Green ’s function of the system. According to Eq. (26), G(r) = 0 for all r < 0. 







While the second form of Eq. (27) is more convenient for calculations, its first form is more clear 
conceptually. Namely, it expresses the linear superposition principle in time domain, and may be 
interpreted as follows: the full effect of force /(/) on an oscillator (actually, any linear system 8 ) may be 
described as a sum of effects of short pulses of duration dt’ and magnitude /!/ ’): 

9,^(0 = £G(t-t')f(0 At'. (4.29) 

t'=- CO 

- see Fig. 2. The Green’s function G(r) thus describes the oscillator response to a unit pulse of force, 
measured at time t = t — t ’ after the pulse. 

Fig. 4.2. Presentation of the force as a 
function of time as a sum of short pulses. 

Mathematically, it is more convenient to go to the limit <//’—» 0 and describe the elementary, 
unit-area pulse by Dirac’s (/-function, 9 thus returning to Eq. (27). This line of reasoning also gives a 
convenient way to calculate the Green’s function. Indeed, for the particular case, 

f(t) = <5(t-t 0 ), with/ 0 </, (4.30) 

Eq. (27) yields q(t ) = G(t - to). In particular, if t > 0, we may take to = 0; then q{t) = G{t). Hence the 
Green’s function may be calculated as a solution of the differential equation of motion of the system, in 
our case, Eq. (13), with the (/-functional right-hand part: 

d 2 G(i) 

dr 2 

+ 2 S^^- + o>qG(t) = S(t), 


8 Thi s is a very unfortunate, but common jargon, meaning “the system described by linear equations of motion”. 

9 For a reminder of the basic properties of the d-function, see MA Sec. 14. 

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and zero initial conditions: 

G(-0) = ^(-0) = 0, 


where t = -0 means the instant immediately preceding / = 0. 

This calculation may be simplified even further. Let us integrate both sides of Eq. (31) over a 
infinitesimal interval including the origin, e.g. \-dr 12, +r/r/2], and then follow the limit dr— > 0. Since 
Green’s function has to be continuous because of its physical sense as the (generalized) coordinate, all 
terms in the left hand part but the first one vanish, while the first term yields dG/dr | +o - dG/dr | _o. Due 
to the second of Eqs. (32), the last of these two terms equals zero, while the right-hand part yields 1. 
Thus, G(r) may be calculated for r > 0 (i.e. for all times when G( r) ^ 0) by solving the homogeneous 
version of system’s equation of motion for r > 0, with the following special initial conditions: 

G(0) = 0, ^(0) = 1. (4.33) 


This approach gives us a convenient way for calculation of Green’s functions of linear systems. 
In particular for the oscillator with not very low damping ( 8 > ox, i.e. Q > Vi), imposing boundary 
conditions (33) on the general free-oscillation solution (9), we immediately get 10 





Equations (27) and (34) provide a very convenient recipe for solving most forced oscillations 
problems. As a very simple example, let us calculate the transient process in an oscillator under the 
effect of a constant force being turned on at t = 0 : 


t < 0, 
t > 0, 


provided that at t < 0 the oscillator was at rest, so that qf Tee (t) = 0. Then the second form of Eq. (27) 

oo * \ 

q(t) = J/(f - r)G(r)dr = f 0 J — -e~ Sr sin<y 0 Y dr. 



The simplest way to work out such integrals is to present the sine function as the imaginary part of 
exp | ioX) ’t}, and merge the two exponents, getting 

q(t) = f 0 — Im 

CO n 

8 + ico 0 ' 

- St - ia> 0 'r 






S 5 



L v 

, 5 . , 

cos co Q t H sinner 



This result, plotted in Fig. 3, is rather natural: it describes nothing more than the transient from 


the initial equilibrium position q = 0 to the new equilibrium position q o =fo/oxf = Fq/k, accompanied by 

10 The same result may be obtained from Eq. (28) with the response function /((») given by Eq. (19). This, more 
cumbersome, way is left for reader’s exercise. 

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decaying oscillations. For this particular simple function J[t), the same result might be also obtained by 
introducing a new variable q (t ) = q(t) - q 0 and solving the resulting homogeneous equation for q (with 
appropriate initial condition q (0) = - q 0 ), but for more complicated functions fit) the Green’s function 
approach is irreplaceable. 


Fig. 4.3. Transient process in a linear 
oscillator, induced by a step-like force f(t), for 
the particular case Slap = 0.1 (i.e., Q = 5). 

Note that for any particular linear system, its Green’s function should be calculated only once, 
and then may be repeatedly used in Eq. (27) to calculate the system response to various external forces - 
either analytically or numerically. This property makes the Green’s function approach very popular in 
many other fields of physics - with the corresponding generalization or re-definition of the function. 11 

4.2. Weakly nonlinear oscillations 

In comparison with systems discussed in the last section, which are described by linear 
differential equations with constant coefficients and thus allow a complete and exact analytical solution, 
oscillations in nonlinear systems generally present a complex and, generally, analytically intractable 
problem. Let us start a discussion of such nonlinear oscillations 12 from an important case that may be 
explored analytically. In many important ID oscillators, higher terms in the potential expansion (3.10) 
cannot be neglected, but are small and may be accounted for approximately. If, in addition, damping is 
low (or negligible), the equation of motion may be presented as a slightly modified Eq. (13): 

q + co 2 q =f(t,q,q,...). 


where co ~ op is the anticipated frequency of oscillations (whose choice is to a certain extent arbitrary - 
see below), and the right-hand part / is small (say, scales as some small dimensionless parameter s « 
1), and may be considered as a perturbation. 

Since at s = 0 this equation has the sinusoidal solution given by Eq. (3), one might naively think 
that at nonvanishing but small s, the approximate solution to Eq. (38) should be sought in the form 

q(t) = q {0) +q {1) +q (2) +..., where q {n) oc s n , 


with q {0) = A cos (apt - <p) oc s. This is a good example of an apparently impeccable mathematical 
reasoning that would lead to a very inefficient procedure. Indeed, let us apply it to the problem we 

11 See, e.g., EM Sec. 2.7, and QM Sec. 2.2. 

12 Again, “nonlinear oscillations” is a generally accepted slang term for oscillations in systems described by 
nonlinear equations of motion. 







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already know the exact solution for, namely the free oscillations in a linear but damped oscillator, for 
this occasion assuming the damping to be very low, 8/ ay, ~ s « 1 . The corresponding equation of 
motion, Eq. (6), may be presented in fonn (38) if we take a>= a>o and 

/ = -28 q, Socs. (4.40) 

The naive approach described above would allow us to find small corrections, of the order of 8 to the 
free, non-decaying oscillations Acos(ayt - (p). However, we already know from Eq. (9) that the main 
effect of damping is a gradual decrease of the free oscillation amplitude to zero, i.e. a very large change 
of the amplitude, though at low damping, 8 « too, this decay takes large time t ~ r » Hay. Hence, if 
we want our approximate method to be productive (i.e. to work at all time scales, in particular for forced 
oscillations with established, constant amplitude and phase), we need to account for the fact that the 
small right-hand part of Eq. (38) may eventually lead to essential changes of oscillation amplitude A 
(and sometimes, as we will see below, also of oscillation phase cp ) at large times, because of the slowly 
accumulating effects of the small perturbation. 13 

This goal may be achieved by the account of these slow changes already in the “0 th 
approximation”, i.e. the basic part of the solution in expansion (39): 

0 th order 

The approximate methods based on Eqs. (39) and (41) have several varieties and several names, 14 but 
their basic idea and the results in the most important approximation (41) are the same. Let me illustrate 
this approach on a particular, simple but representative example of a dissipative (but high-0 pendulum 
driven by a weak sinusoidal external force with a nearly-resonant frequency: 

q + 2 Sq + a>l sin q = f 0 cos cot, (4.42) 

with | co - coo\, S « coo, and the force amplitude fo so small that \q\ « 1 at all times. From what we know 
about the forced oscillations from Sec. 1, it is natural to identify co in the left-hand part of Eq. (38) with 
the force frequency. Expanding sin q into the Taylor series in small q, keeping only the first two terms 
of this expansion, and moving all the small terms to the right-hand part, we can bring Eq. (42) to the 
canonical form (38): 15 



Here a = coo /6 in the case of the pendulum (though the calculations below will be valid for any a), and 
the second tenn in the right-hand part was obtained using the approximation already employed in Sec. 1 : 

q + co 2 q = -2 8q + 2 %coq + aq 3 + / 0 cos cot = f ( t , q, q) . 


q i0) = A{t)cos[cot-cp{t)\, with A,cp 0 at£— >0. 


13 The same flexible approach is necessary to approximations used in quantum mechanics. The method discussed 
here is close in spirit (but not identical) to the WKB approximation (see, e.g., QM Sec. 2.4) rather to the 
perturbation theory varieties (QM Ch. 6). 

14 In various texts, one can meet references to either the small parameter method or asymptotic methods. The list 
of scientists credited for the development of this method and its variations notably includes J. Poincare, B. van der 
Pol, N. Krylov, N. Bogolyubov, and Yu. Mitroplolsky. Expression (41) itself is frequently called the Rotating- 
Wave Approximation - RWA. (The origin of the term will be discussed in Sec. 6 below.) In the view of the 
pioneering role of B. van der Pol in the development of this approach, in some older textbooks the rotating-wave 
approximation is called the “van der Pol method”. 

15 This equation is frequently called the Duffing equation (or the equation of the Duffing oscillator), after G. 
Duffing who was the first one to carry out its (rather incomplete) analysis in 1918. 

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2 2 

(of - cof)q « 2 oi 0 ) - coo)q = 2 cocq, where g = co - op is the detuning parameter that was already used 
earlier - see Eq. (21). 

Now, following the general recipe expressed by Eqs. (39) and (41), in the 1 st approximation in / 
oc a, 1 6 we may look for the solution to Eq. (43) in the form 

q(t) = A cos^ + q [l) (t), where 'P = cot - cp, q m ~ a . (4.44) 

Let us plug this assumed solution into both parts of Eq. (43), leaving only the terms of the first order in 
a. Thanks to our (smart :-) choice of co in the left-hand part of that equation, the two zero-order terms in 
that part cancel each other. Moreover, since each term in the right-hand part of Eq. (43) is already of the 
order of a, we may drop q (l) cc a from the substitution into that part at all, because this would give us 
only terms 0(f) or higher. As a result, we get the following approximate equation: 

q {1) + co 2 q (l] = / <0) = -28 — (^cos'E) + 2£&>.Tcos v E + a(ylcos'F) 3 + / 0 cos<z>E (4.45) 


According to Eq. (41), generally A and cp should be considered as (slow) functions of time. 
However, let us leave the analyses of transient process and system stability until the next section, and 
use Eq. (45) to find stationary oscillations in the system, that are established after the initial transient. 
For that limited task, we may take A = const, cp = const, so that q {0) presents sinusoidal oscillations of 
frequency co. Sorting the terms in the right-hand part according to their time dependence, 17 we see that it 
has terms with frequencies co and 3 co: 

/ <0) = (2 tgcoA + ^-aA 3 + / 0 cos^>)cos v P + (2 ScoA - f 0 sin^)sin v F + ^-aA 3 cosSTf (4.46) 

Now comes the main trick of the rotating-wave approximation: mathematically, Eq. (45) may be 
viewed as the equation of oscillations in a linear, dissipation-free harmonic oscillator of frequency co 
(not <x>o!) under the action of an external force represented by the right-hand part of the equation. In our 
particular case, it has three terms: two quadrature components at that very frequency co, and the third 
one at frequency 3 co. As we know from our analysis of this problem in Sec. 1, if any of the first two 
components is nonvanishing, q a) grows to infinity - see Eq. (19) with 8= 0. At the same time, by the 
very structure of the rotating-wave approximation, q (l) has to be finite - moreover, small! The only way 
out of this contradiction is to require that amplitudes of both quadrature components of / 0) with 
frequency <x>are equal to zero: 

2%coA + — a A 3 + f 0 costp = 0, 

28coA - f 0 sin cp = 0. 


These two harmonic balance equations enable us to find both parameters of the forced 
oscillations: their amplitude A and phase cp. In particular, the phase may be readily eliminated from this 

16 For a mathematically rigorous treatment of the higher approximations, see, e.g., Yu. Mitropolsky and N. Dao, 
Applied Asymptotic Methods in Nonlinear Oscillations, Springer, 2004. A more laymen (and somewhat verbose) 
discussion of various oscillatory phenomena may be found in the classical text A. Andronov, A. Vitt, and S. 
Khaikin, Theory’ of Oscillators, Dover, 2011. 

17 Using the second of Eqs. (44), cos cot may be rewritten as cos (T + cp) = cos ¥ cos cp - sinT sin cp. Then using 
the trigonometric identity cos 3l P = (3/4)cos T + (l/4)cos 34* - see, e.g., MA Eq. (3.4) results in Eq. (46). 

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system (most easily, by expressing si n and cos cp from the corresponding equations, and then requiring 

2 2 

the sum sirrr/) + cos cp to equal 1), and the solution for amplitude A presented in the following implicit 
but convenient form: 

A 2 

fo 1 

4 co 2 £ 2 (A) + S 2 ’ 

where g(A) = E, + — 

3 aA 7 

8 co 

= CO- 


3 aA 2 ' 
8 co j 


This expression differs from Eq. (22) for the linear resonance in the low-damping limit only by the 
replacement of the detuning E, with its effective amplitude-dependent value <*(A) or, equivalently, of the 
eigenfrequency coo of the resonator with its effective, amplitude-dependent value 

3 a A 2 

°o 0 (A) = co 0 . (4.49) 

8 oo 

The physical meaning of coo(A) is simple: this is just the frequency of free oscillations of amplitude A in 
a similar nonlinear system, but with zero damping. Indeed, for 8 = 0 and fo = 0 we could repeat our 
calculations, assuming that co is an amplitude-dependent eigenfrequency 000 (A), to be found. Then the 
second of Eqs. (47) is trivially satisfied, while the second of them gives Eq. (49). 

Expression (48) allows one to draw the curves of this nonlinear resonance just by bending the 
linear resonance plots (Fig. 1) according to the so-called skeleton curve expressed by Eq. (49). Figure 4 
shows the result of this procedure. Note that at small amplitude, co(A) — > coo, and we return to the usual, 
“linear” resonance (22). 

0.8 0.9 1 1.1 


Fig. 4.4. Nonlinear resonance in the Duffing 
oscillator, as described by the rotating-wave 
approximation result (48), for the particular 
case a= cof/6, S/a> = 0.01 (i.e. Q = 50), and 
seven values of parameter f/cof increased 
bv eaual stens from 0 to 0.035. 

To bring our solution to its logical completion, we should still find the first perturbation q V \t) 
from what is left of Eq. (45). Since the structure of this equation is similar to Eq. (13) with the force of 
frequency 3 co and zero damping, we may use Eqs. (16)-(17) to obtain 

<A0 = 

JL i 

-aA cos 3 (cot-cp). 

32 co 


Adding this perturbation (note the negative sign!) to the sinusoidal oscillation (41), we see that as the 
amplitude A of oscillations in a system with a > 0 (e.g., a pendulum) grows, their waveform become a 
bit more “blunt” near the maximum deviations from the equilibrium. 

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Expression (50) also allows an estimate of the range of validity of the rotating- wave 

approximation: since it has been based on the assumption | q { j « |g <0| | < A, for this particular problem 

2 2 2*2 

we have to require aA illco « 1. For a pendulum (with a = coq / 6), this condition becomes A' « 

1/192. Though numerical coefficients in such strong inequalities should be taken with a grain of salt, the 
smallness of this particular coefficient gives a good hint that the method should give very good results 
even for relatively large oscillations withal ~ 1. In Sec. 7 below, we will see that this is indeed the case. 

From the mathematical viewpoint, the next step would be to calculate the next approximation 

q(t) = Acos x ¥ + q m (t) + q (2 \t), q (2) ~ s 1 , (4.51) 

and plug it into the Duffing equation (43), which (thanks to our special choice of q {0) and q 1 1 ') would 
retain only q {2) +a> 2 q {2) in its left-hand part. Again, requiring that amplitudes of two quadrature 
components of frequency <x> in the right-hand part to be zero, we may get the second-order corrections to 
A and (p. Then we may use the remaining part of the equation to calculate q <2 \ and then go after the 
third-order terms, etc. However, for most purposes the sum q {0) + q (l \ and sometimes even just the 
crudest approximation q {0) alone, are completely sufficient. For example, according to Eq. (50), for a 
simple pendulum (a = coo / 6) swinging as much as between the opposite horizontal positions (A = n! 2), 
the 1 st order correction q (l) is of the order of 0.5%. (Soon beyond this value, completely new dynamic 
phenomena start - see Sec. 7 below, but these phenomena cannot be covered by the rotating-wave 
approximation, at least in our current form.) Due to this reason, higher approximations are rarely 
pursued either in physics or engineering. 

4.3. RWA equations 

A much more important issue is the stability of solutions described by Eq. (48). Indeed, Fig. 4 
shows that within a certain range of parameters, these equations give three different values for the 
oscillation amplitude (and phase), and it is important to understand which of these solutions are stable. 

Since these solutions are not the fixed points in the sense discussed in the Sec. 3.2 (each point in Fig. 4 
represents a nearly-sinusoidal oscillation), their stability analysis needs a more general approach that 
would be valid for oscillations with amplitude and phase slowly evolving in time. This approach will 
also enable the analysis of non-stationary (especially the initial transient) processes that are of key 
importance for some dynamic systems. 

First of all, let us formalize the way the harmonic balance equations, such as Eqs. (47), are 
obtained for the general case (38) - rather than for the particular Eq. (43) considered in the last section. 

After plugging in the 0 th approximation (41) into the right-hand part of equation (38) we have to require 
the amplitudes of its both quadrature components of frequency a> to be zero. From the standard Fourier 
analysis we know that these requirements may be presented as 


( 4 . 52 ) balance 

where symbol ... means time averaging - in our current case, over the period In/co of the right-hand part 
of Eq. (52), with the arguments calculated in the 0 th approximation: 

/ (0) = f(t,q (0 \q i0) ,...) = f {t, A cosT*,- A co sin ¥,...), with x ¥ = a>t-(p. (4.53) 

/ (0) sin'F = 0, / (0) cosT^O, 

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Now, for a transient process the contribution of c/ 0) to left-hand part of Eq. (38) is not zero any 
longer, because both amplitude and phase may be slow functions of time - see Eq. (41). Let us calculate 
this contribution. The exact result would be 

<7 (0) +®y o> = 


. dt 2 

■ + co 

A cos(nrf - cp) 


= (2 + 2 dpcoA - <p 2 A)cos{cot -cp)- 2 A(co - <p)sm(cot - cp). 

However, in the first approximation in a, we may neglect the second derivative of A, and also the 
squares and products of the first derivatives of A and cp (that are all of the second order in a), so that Eq. 
(54) is reduced to 

q (0) + co 2 q (0) « 2Acpcocos(cot -cp)-2Acosin(cot -cp) . (4.55) 

In the right-hand part of Eq. (52), we can neglect the time derivatives of the amplitude and phase at all, 
because this part is already proportional to the small parameter. Hence, in the first order in a, Eq. (38) 

q m +co 2 q m =/ e ( f 0) =/ (0) -(^Acpcocos'i* -2 Acosin'i'). (4.56) 

Now, applying Eqs. (52) to function fj°\ and taking into account that the time averages of sirrT' 
and cos 'E are both equal to Vi, while the time average of the product sin'Pcos'P vanishes, we get a pair 
of so-called RWA equations (alternatively called “the reduced equations” or sometimes “the van der Pol 
equations”) for the time evolution of the amplitude and phase: 

A = 

~~f W) sin'P, cp = — — / (0) cos'P . 
co coA 


forms of 

Extending the definition (4) of the complex amplitude of oscillations to their slow evolution in time, a(t) 
= /fificxp j/V/f/)} , and differentiating this relation, we see that two equations (57a) may be also re-written 
in the form of either one equation for a: 

a=L f <v e W + <P) 

J_y( vjcot 


or two equations for the real and imaginary parts of a(t ) = u(t) + iv(t): 


The first-order harmonic balance equations (52) are evidently just the particular case of the RWA 
equations (57) for stationary oscillations (A = cp = 0). 18 

Superficially, the system (57a) of two coupled, first-order differential equations may look more 
complex than the initial, second-order differential equation (38), but actually it is usually much simpler. 

18 One may ask why cannot we stick to the just one, most compact, complex-amplitude form (57b) of the RWA 
equations. The main reason is that when function f{q,q,t ) is nonlinear, we cannot replace its real arguments, 
such as q = A cos( cut - cp), with their complex-function representations like aexp {-icot} (as could be done in the 
linear problems considered in Sec. 4.1), and need to use real variables, such as either {A, cp } or {u, v}, anyway. 

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For example, let us spell them out for the easy case of free oscillations a linear oscillator with damping. 
For that, we may reuse the ready Eq. (46) with a=fo = 0, turning Eqs. (4.57a) into 

A = - — / (0) sin'F = - — (2<iyu4cos v F + 28coA sin *P) sin 'F = —8 A, 
co co 

cp = — / <0) cos'F = — (2^&t4cos'F + 2ScoA sin'F) cos 'F = £ ) . 
coA coA 

The solution of Eq. (58a) gives us the same “envelope” law A(t) = A(0)e a as the exact solution 
(10) of the initial differential equation, while the elementary integration of Eq. (58b) yields (fit) = c t + 
cp{ 0) = cot - coot + cp(0). This means that our approximate solution, 

q {0) (t) = A(t) cos[<ztf - cp(t)\ = A( 0)e~ St cos[<u 0 f - ^>(0)], (4.59) 

agrees with the exact Eq. (9), and misses only correction (8) to the oscillation frequency, that is of the 
second order in S, i.e. of the order of s - beyond the accuracy of our first approximation. It is 
remarkable how nicely do the RWA equations recover the proper frequency of free oscillations in this 
autonomous system - in which the very notion of co is ambiguous. 

The situation is different at forced oscillations. For example, for the (generally, nonlinear) 
Duffing oscillator described by Eq. (43) with / 0 ^ 0, Eqs. (57a) yield the RWA equations, 

A = -8 A + ^-siny?, Acp = £(A)A + ^-cosy? , (4.60) 

2 co 2 co 

which are valid for an arbitrary function 2;(A), provided that the nonlinear detuning remains much 
smaller than the oscillation frequency. Here (after a transient), the amplitude and phase tend to the 
stationary states described by Eqs. (47). This means that cp becomes a constant, so that q W) — » Acos(cot - 
const), i.e. the RWA equations again automatically recover the correct frequency of the solution, in this 
case equal to that of the external force. 

Note that each stationary oscillation regime, with certain amplitude and phase, corresponds to a 
fixed point of the RWA equations, so that the stability of those fixed points detennine that of the 
oscillations. In what follows, we will carry out such an analysis for several simple systems of key 
importance for physics and engineering. 



4,4, Self-oscillations and phase locking 

The rotating-wave approximation was pioneered by B. van der Pol in the late 1920s for analysis 
of one more type of oscillatory motion: self-oscillations. Several systems, e.g., electronic rf amplifiers 
with positive feedback, and optical media with quantum level population inversion, provide convenient 
means for the compensation, and even over-compensation of the intrinsic energy losses in oscillators. 
Phenomenologically, this effect may be described as the change of sign of the damping coefficient 8 
from positive to negative. Since for small oscillations the equation of motion is still linear, we may use 
Eq. (9) to describe its general solution. This equation shows that at 8< 0, even infinitesimal deviations 
from equilibrium (say, due to unavoidable fluctuations) lead to oscillations with exponentially growing 
amplitude. Of course, in any real system such growth cannot persist infinitely, and shall be limited by 

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this or that effect - e.g., in the above examples, respectively, by amplifier saturation or electron 
population exhaustion. 

In many cases, the amplitude limitation may be described reasonably well by nonlinear 
damping : 

28q — > 28q + J3q 3 , (4.61) 

with J3 > 0. Let us analyze this phenomenon, applying the rotating- wave approximation to the 
corresponding homogeneous differential equation: 

q + 2Sq + Jdq 3 + a>lq = 0. (4.62) 

Carrying out the dissipative and detuning terms to the right hand part as f we can readily calculate the 
right-hand parts of the RWA equations (57a), getting 19 

A = -8(A) A, where 8(A) = 8 + -j3m 2 A 2 , (4.63a) 


Arp = 4 A. (4.63b) 

The second of these equations has exactly the same form as Eq. (58b) for the case of decaying 
oscillations and hence shows that the self-oscillations (if they happen, i.e. if A ^ 0) have frequency coo of 
the oscillator itself- see Eq. (59). Equation (63a) is more interesting. If the initial damping fi'is positive, 
it has only the trivial fixed point, Aq = 0 (that describes the oscillator at rest), but if 8 is negative, there is 
also another fixed point, 

A = 

' 8|<5l V/2 

3j3co 2 


which describes steady self-oscillations with a non-zero amplitude. 

Let us apply the general approach discussed in Sec. 3.2, the linearization of equations of motion, 
to this RWA equation. For the trivial fixed point Ao = 0, the linearization of Eq. (63a) is reduced to 
discarding the nonlinear term in the definition of the amplitude-dependent damping 8(A). The resulting 
linear equation evidently shows that the system’s equilibrium point, A = A o = 0, is stable at 8> 0 and 
unstable at 8< 0. (We have already discussed this self-excitation condition above.) The linearization of 
Eq. (63a) near the non-trivial fixed point A\ requires a bit more math: in the first order in 

A = A- A 1 — > 0 , we get 

A = A = -8(A 1 + A)~- (3co 2 (A l + A) 3 &-SA-— J3co 2 3A 2 A = (-8 + 3 8)2 = 28A , (4.65) 

8 8 

where Eq. (64) has been used to eliminate A\. We see that fixed point A \ (and hence the whole process) 
is stable as soon as it exists (8 < 0) - similar to the situation in our “testbed problem” (Fig. 2.1). 

Now let us consider another important problem: the effect of a external sinusoidal force on a 
self-excited oscillator. If the force is sufficiently small, its effects on the self-excitation condition and 
the oscillation amplitude are negligible. However, if frequency co of such weak force is close to the 

19 For that, one needs to use the trigonometric identity sin 3l P = (SAfisinT - (L4)sin3 l F - see, e.g., MA Eq. (3.4). 

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eigenfrequency a>o of the oscillator, it may lead to a very important effect of phase-locking (also called 
“synchronization”). At this effect, oscillator’s frequency deviates from ax >, and becomes exactly equal to 
the external force’s frequency a>, within a certain range 

- A< (o- co 0 < +A . (4.66) 

In order to prove this fact, and also to calculate the phase locking range width 2A, we may repeat 
the calculation of the right-hand parts of the RWA equations (57a), adding term focos cot to the right- 
hand part of Eq. (62) - cf. Eqs. (42)-(43). This addition modifies Eqs. (63) as follows: 20 

A = -8{A)A + -^-smcp, (4.67a) 

2 co 

Acp = E, A + 


2 co 

cos cp. 


If the system is self-excited, and the external force is weak, its effect on the oscillation amplitude is 
small, and in the first approximation in fo we can take A to be constant and equal to the value A\ given 
by Eq. (64). Plugging this approximation into Eq. (67b), we get a very simple equation 21 

cp = £ + Acos cp , 

where in our current case 

A = 

2coA ] 



Within the range - |A| < <^< + |A|, Eq. (68) has two fixed points on each 2;r-segment of variable cp: 

cp ± — i arccos 

+ 27m . 


It is easy to linearize Eq. (68) near each point to analyze their stability in our usual way; 
however, let me this case to demonstrate another convenient way to do this in ID systems, using the so- 
called phase plane - the plot of the right-hand part of Eq. (68) as a function of cp - see Fig. 5. 

Fig. 4.5. Phase plane of a phase- 
locked oscillator, for the particular 
case £ = A/2,/o > 0. 

20 Actually, this result should be evident, even without calculations, from the comparison of Eqs. (60) and (63). 

21 This equation is ubiquitous in phase locking systems, including even some digital electronic circuits used for 
that purpose. 




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Since the positive values of this function correspond to the growth of cp in time, and vice versa, 
we may draw the arrows showing the direction of phase evolution. From this graphics, it is clear that 
one of these fixed points (for /o >0, cp+) is stable, while its counterpart is unstable. Hence the magnitude 
of A given by Eq. (69) is indeed the phase locking range (or rather it half) that we wanted to find. Note 
that the range is proportional to the amplitude of the phase locking signal - perhaps the most important 
feature of phase locking. 

In order to complete our simple analysis, based on the assumption of fixed oscillation amplitude, 
we need to find the condition of validity of this assumption. For that, we may linearize Eq. (67a), for the 
stationary case, near value A\, just as we have done in Eq. (65) for the transient process. The stationary 

A =A 

A , = 


2\S\ 2 co 

sin <p ± « A l 

2 S 

sin <p ± , 


shows that our assumption, A \ « A\. and hence the final result (69), are valid if the phase locking 
range, 2 A, is much smaller than 4| 8\. 

4.5. Parametric excitation 

In both problems solved in the last section, the stability analysis was easy because it could be 
carried out for just one slow variable, either amplitude or phase. Generally, such analysis of the RWA 
equations involves both these variables. The classical example of such situation is provided by one 
important physical phenomenon - the parametric excitation of oscillations. An elementary example of 
such oscillations is given by a pendulum with an externally-changed parameter, for example length l(t) - 
see Fig. 6. Experiments (including those with playground swings :-) and numerical simulations show 
that if the length is changed ( modulated) periodically, with frequency 2 co that is close to 2 coq and a 
sufficiently large swing A/, the equilibrium position of the pendulum becomes unstable, and it starts 
swinging with frequency co equal exactly to the half of the length modulation frequency (and hence only 
approximately equal to the average eigenfrequency ay of the oscillator). 

Fig. 4.6. Parametric excitation of pendulum oscillations. 

For an elementary analysis of this effect we may consider the simplest case when the oscillations 
are small. At the lowest point (6 = 0), where the pendulum moves with the highest velocity v max , 
string’s tension T is higher than mg by the centripetal force: 7j iax = mg + mv max //. On the contrary, at 
the maximum deviation of the pendulum from the equilibrium, the force is weakened by string’s tilt: T mm 

= /«gcos$ max . Using the energy conservation, E = /nv max 12 = mgl( 1 - cos^x), we may express these 
values as T max = mg + 2 EH and T m in = mg - EH. Now, if during each oscillation period the string is pulled 

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up sharply and slightly by A / (|A/| « l) at each of its two passages through the lowest point, and is let to 
go down by the same amount at each of two points of the maximum deviation, the net work of the 
external force per period is positive: 

W' = 2(r,„-r min )A/«6y£, (4.72) 

and hence results in an increase of the oscillator’s energy. If the so-called modulation depth Al/21 is 
sufficient, this increase may be sufficient to overcompensate the energy drained out by damping. 
Quantitatively, Eq. (10) shows that low damping ( 8 « cod) leads to the following energy decrease, 

AE*-4n — E, (4.73) 

co 0 

per oscillation period. Comparing Eqs. (72) and (73), we see that the net energy flow into the 
oscillations is positive, W + A E > 0, i.e. oscillation amplitude has to grow if 22 

A / 2nd _ n 
/ 3a> 0 3 Q 


Since this result is independent on E, the growth of energy and amplitude is exponential (for sufficiently 
low E), so that Eq. (74) is the condition of parametric excitation - in this simple model. 

However, this result does not account for the possible difference between the oscillation 
frequency co and the eigenfrequency coq, and also does not clarify whether the best phase shift between 
the parametric oscillations and parameter modulation, assumed in the above calculation, may be 
sustained automatically. In order to address these issues, we may apply the rotating-wave approximation 
to a simple but reasonable linear equation 

q + 2dq + C 0 q (l + /j cos 2cot)q = 0, (4.75) 

describing the parametric excitation for a particular case of sinusoidal modulation of coo (t). Rewriting 
this equation in the canonical form (38), 

q+a> 2 q = f(t,q,q) = -2dq + 2^coq - pa>lqcos2cot, (4.76) 

and assuming that the dimensionless ratios d/co and c\! co, and the modulation depth // are all much less 
than 1, we may use general Eqs. (57a) to get the following RWA equations: 

A = -d A- — - A sin 2 cp, 

Acp = Ad~^-Acos2cp. 


These equations evidently have a fixed point A 0 = 0, but its stability analysis (though possible) is 
not absolutely straightforward, because phase cp of oscillations is undetermined at that point. In order to 

22 A modulation of pendulum’s mass (say, by periodic pumping water in and out of a suspended bottle) gives a 
qualitatively similar result. Note, however, that parametric oscillations cannot be excited by modulating any 
oscillator’s parameter - for example, oscillator’s damping coefficient (at least if it stays positive at all times), 
because its does not change system’s energy, just the energy drain rate. 

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avoid this (technical rather than conceptual) technical difficulty, we may use, instead of the real 
amplitude and phase of oscillations, either their complex amplitude a = A exp {itpj, or its Cartesian 
components w and v - see Eqs. (4). Indeed, for our function f Eq. (57b) gives 

a = (-8 + i^)a - i^-a *, 

while Eqs. (57c) yield 






u = —8 u — v , 


e /UOJ 

v = -ov + cu a . 




We see that in contrast to Eqs. (77), in Cartesian coordinates {u, v} the trivial fixed point ao = 0 
(i.e. wo = vo = 0) is absolutely regular. Moreover, equations (78)-(79) are already linear, so they do not 
require any additional linearization. Thus we may use the same approach as was already used in Secs. 
3.2 and 4.1, i.e. look for the solution of Eqs. (79) in the exponential form exp {A t}. However, now we are 
dealing with two variables, and should allow them to have, for each value of A, a certain ratio u/v. For 
that, we should take the partial solution in the form 

u = c,e 


v = c„e 



where constants c u and c v are frequently called the distribution coefficients. Plugging this solution into 
Eqs. (79), we get for them the following system of two linear algebraic equations: 

(S-A)c u +(-£-^-)c v = 0, 
(t-tf)c„+(-S-A)c, = 0. 

The characteristic equation of this system, 

has two roots: 

-8- A 


* 4~ 

„ ua> 


-8- A 

= A 2 +28A + 8 2 +f~ 


V 4 j 

= 0, 

A ± 

- 8 ± 

f iiccf 2 





Requiring the fixed point to be unstable, Re/l + > 0 , we get the parametric excitation condition 

^->{8 2 +<f)‘ /2 . (4.84) 

Thus the parametric excitation may indeed happen without any artificial phase adjustment: the arising 
oscillations self-adjust their phase to pick up energy from the external source responsible for the 
parameter variation. 

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Our key result (84) may be compared with two other calculations. First, in the case of negligible 
damping (8 = 0), Eq. (84) turns into condition /uco/4 > |£| . This result may be compared with the well- 
developed theory of the so-called Mathieu equation whose canonical form is 

7 2 

— ^- + {a -2hcos2v)v = 0. (4.85) 


It is evident that with the substitutions y — » q, v — > cot, a — » ( oxj cd) y b — » -jul 2, this equation is just a 
particular case of Eq. (75) for 8= 0. In terms of Eq. (85), the result of our approximate analysis may be 
re-written just as b > I a - 1 I , and is supposed to be valid for b « 1. This condition is shown in Fig. 7 
together with the numerically calculated 23 stability boundaries of the Mathieu equation. 

Fig. 4.7. Stability boundaries of the Mathieu 
equation (85), as calculated: numerically 

(curves) and using the rotating-wave 
approximation (dashed straight lines). In the 
regions numbered by various n the trivial 
solution y = 0 of the equation is unstable, i.e. its 
general solution v(v) includes an exponentially 
growing term. 

One can see that the rotating-wave approximation works just fine within its applicability limit 
(and beyond :-), though it fails to predict some other important features of the Mathieu equation, such as 
the existence of higher, more narrow regions of parametric excitation (at a ~ n , i.e. a>o ~ coin, for all 
integer n ), and some spill-over of the stability region into the lower half-plane a < 0. 24 The reason of 
these failures is the fact that, as can be seen in Fig. 7, these phenomena do not appear in the first 
approximation in the parameter modulation amplitude ju oc q, that is the RWA applicability realm. 

In the opposite case of finite damping but exact tuning (£= 0, co « coq), Eq. (84) gives 



co n 

2 _ 



23 Such calculations may be substantially simplified by the use of the so-called Floquet theorem, which is also 
the mathematical basis for the discussion of wave propagation in periodic media - see the next chapter. 

24 This region describes, for example, the counter-intuitive stability of an inverted pendulum with the periodically 
modulated length, within a limited range of the modulation depth /a. 

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This condition may be compared with Eq. (74), taking A III = 2//.. The comparison shows that though the 
structure of these conditions is similar, the numerical coefficients are different by a factor close to 2. 
The first reason of this difference is that the instant parameter change at optimal moments of time is 
more efficient then the smooth, sinusoidal variation described by (75). Even more significantly, the 
change of pendulum’s length modulates not only its eigenfrequency co o, as Eq. (75) implies, but also its 
mechanical impedance Z = ( gl ) “ - the notion to be discussed in detail in the next chapter. (Due to the 
time restrictions, I have to leave the analysis of the general case of the simultaneous modulation of op 
and Z for reader’s exercise.) 

Before moving on, let me summarize the most important differences between the parametric and 
forced oscillations: 

(i) Parametric oscillations completely disappear outside of their excitation range, while the 
forced oscillations have a non-zero amplitude for any frequency and amplitude of the external force - 
see Eq. (18). 

(ii) Parametric excitation may be described by a linear homogeneous equation - e.g., Eq. (75) - 
which cannot predict any finite oscillation amplitude within the excitation range, even at finite damping. 
In order to describe stationary parametric oscillations, some nonlinear effect has to be taken into 
account. (Again, I am leaving analyses of such effects for reader’s exercises.) 

One more important feature of the parametric oscillations will be discussed in the end of the next 


4,6. Fixed point classification 

RWA equations (79) give us a good pretext for a brief discussion of fixed points of a dynamic 
system described by two time-independent, first-order differential equations. 25 After their linearization 
near a fixed point, the equations for deviations can always be presented in the form similar to Eq. (79): 

M ll q l +M\ 2 q 2 , 
q 2 = M 2x q j + M 22 q 2 , 


where Mjy (with jj ’ =1,2) are some real scalars that may be understood as elements of a 2x2 matrix M. 
Looking for an exponential solution of the type (80), 

q l = c x e ^ , q 2 =c 2 e‘ ^ , (4.88) 

we get a more general system of two linear equations for the distribution coefficients c\£. 

(M n - A)c l + M l2 c 2 = 0, 

M 2l c 1 + (M 22 - A)c 2 = 0. 


These equations are consistent if 

25 Autonomous systems described by a single second-order differential equation, say F(q,q,q) = 0, also belong 
to this class, because we may treat velocity q = v as a new variable, and use this definition as one first-order 
differential equation, and the initial equation, in the form F(q,v,v) = 0 , as the second first-order equation. 

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M n -A 

m 21 

M u 

m 22 -a 

giving us a quadratic characteristic equation 

A 2 -a(m n + M 22 )+(M u M 22 -M 12 M 21 ) = 0. 

Its solution, 26 

-M 22 ) 2 +4M 12 M 21 ] 1/2 , 




shows that the following situations are possible: 

A. The expression under the square root, (Mu- M 22 ) + 4M 1 2 M 2 1 , is positive. In this case, both 
characteristic exponents A± are real, and we can distinguish three sub-cases: 

(i) Both A+ and A. are negative. In this case, the fixed point is evidently stable. Because of 
generally different magnitudes of exponents A±, the process presented on the phase plane [ q t , q 2 ] (Fig. 
8a) may be seen as consisting of two stages: first, a faster (with rate |/l+|) relaxation to a linear 
asymptote , 27 and then a slower decline, with rate \A.\, along this line, i.e. at the virtually fixed ratio of the 
variables. Such fixed point is called the stable node. 

(ii) Both A+ and A. are positive. This case (rarely met in actual physical systems) of the 
unstable node differs from the previous one only by the direction of motion along the phase plane 
trajectories (see dashed arrows in Fig. 8a). Here the variable ratio is also approaching a constant soon, 
but now the one corresponding to the larger of the rates A 

(iii) Finally, in the case of a saddle (A+ > 0, A. < 0) the system dynamics is different (Fig. 
8b): after the rate-1 A+ 1 relaxation to the /L-asymptotc, the perturbation starts to grow, with rate A., along 
one of two opposite directions. (The direction is determined on which side of another straight line, 
called separatrix, the system has been initially.) It is evident that the saddle 28 is an unstable fixed point. 

B. The expression under the square root, (Mu- M 22 ) + 4 M \ 2 M 2 1 , is negative. In this case the 
square root in Eq. (92) is imaginary, making the real parts of both roots equal, Re/l± = (Mi 1 + M 2 2 )/2, 
and their imaginary parts equal but sign-opposite. As a result, here there can be just two types of fixed 

(i) S table focus, at (Mu + M 22 ) < 0. The phase plane trajectories are spirals going to the 
center (i.e. toward the fixed point) - see Fig. 8c with solid arrow. 

(ii) Unstable focus, taking place at (Mu + M 22 ) > 0, differs from the stable one only by 
the direction of motion along the phase trajectories - see the dashed arrow in Fig. 8c. 

26 In terms of linear algebra, A± are the eigenvalues, and the corresponding sets [ci, c 2 ]± , the eigenvectors of 
matrix M with elements My. 

27 The asymptote direction may be found by plugging the value A+ back into Eq. (89) and finding the 
corresponding ratio C\/c 2 . 

28 The term “saddle” is due to the fact that system’s dynamics in this case is qualitatively similar to those of 
particle’s motion in the 2D potential U( f , q 2 ) having the shape of a horse saddle (or a mountain pass). 

equation of 
system (87) 

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(a) (b) 

r -1 -0.6 

M = 

[-2.6 -1 

i 7 7 

A ± = 



-1 ±1.25 



asymptote —f — N 
/ \ 



separatrix >/ 



/ / 

7 \ 

\ -1 -0.25] 

M = 

-1.25 -4 

4 = 


-1 ± 0.75 

unstable / 

7 \ 

stable \ 





(c) (d) 

(b) saddle, (c) focus, and (d) center. The particular values of the matrix M used in the first three 
panels correspond to the RWA equations (81) for parametric oscillators with £, - S, and three 
different values of parameter ) uco/Ad . : (a) 1.25, (b) 1.6 and (c) 0. 

C. Sometimes the border case, M\\ + M 22 = 0, is also distinguished, and the corresponding fixed 
point is refereed to as the center (Fig. 8d). Considering centers a special category makes sense because 
such fixed points are typical for Hamiltonian systems whose first integral of motion may be frequently 
presented as the distance of the from a fixed point. For example, a harmonic oscillator without 
dissipation may be described by the system 

q = —, P = ~m<x>lq, (4.94) 


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that is evidently a particular case of Eq. (87) with Mu = M 22 = 0, M\ 2 M 2 \ = - <Dq <0, and hence (Mu- 
M 22 ) + 4 M 12 M 21 = -4a>o < 0, and M\ \ + M 22 = 0. The phase plane of the system may be symmetrized by 
plotting q vs. the properly nonnalized momentum p/mco 0 . On the symmetrized plane, sinusoidal 
oscillations of amplitude A are represented by a circle of radius A about the center-type fixed point A = 
0. Such a circular trajectory correspond to the conservation of the oscillator’s energy 

mq 1 ma>lq 2 

h = 1 

2 2 



\ mco Q ) 

+ q- 


This is a convenient moment for a brief discussion of the so-called Poincare (or “slow-variable”, 
or “stroboscopic”) plane. 29 From the point of view of the rotating-wave approximation, sinusoidal 
oscillations q(t) = Acos(cot - cp ), in particular those described by a circular trajectory on the real (or 
“fast”) phase plane (Fig. 8c) correspond to a fixed point {A, cp}, which may conveniently presented by a 
steady geometrical point on a plane with these polar coordinates (Fig. 9a). (As follows from Eq. (4), the 
Cartesian coordinates on that plane are u and v.) The quasi-sinusoidal process (41), with slowly 
changing A and cp , may be represented by a slow motion of that point on this Poincare plane. 

Fig. 4.9. (a) Presentation of a 

sinusoidal oscillation (point) and a 
slow transient (line) on the Poincare 
plane, and (b) transfer from the “fast” 
phase plane to the “slow” (Poincare) 

Figure 9b shows one possible way to visualize the relation between the “real” phase plane of an 
oscillator, with symmetrized Cartesian coordinates q and phn op, and the Poincare plane with Cartesian 
coordinates u and v: the latter reference frame rotates relative to the former one about the origin 
clockwise, with angular velocity co? {) Another, “stroboscopic” way to generate the Poincare plane 
pattern is to have a fast glance at the “real” phase plane just once during the oscillation period T= 2 n! co. 

In many cases, such presentation is more convenient than that on the “real” phase plane. In 
particular, we have already seen that the RWA equations for such important phenomena as phase 
locking and parametric oscillations, whose original differential equations include time explicitly, are 
time-independent - cfi, e.g., (75) and (79) describing the latter effect. This simplification brings the 

29 Named after J. H. Poincare (1854-1912) who is credited, among many other achievements, for his contributions 
to special relativity (see, e.g., EM Chapter 9) and the idea of deterministic chaos (to be discussed in Chapter 9 

30 This notion of phase plane rotation is the basis for the rotating-wave approximation’s name. (Word “wave” has 
sneaked in from this method’s wide application in classical and quantum optics.) 

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equations into the category considered in this section, and enables the classification of their fixed points, 
which may shed additional light on their dynamic properties. 

In particular, Fig. 10 shows the classification of the trivial fixed point of a parametric oscillator, 
which follows from Eq. (83). As the parameter modulation depth // is increased, the type of the trivial 
fixed point A\ = 0 on the Poincare plane changes from a stable focus (typical for a simple oscillator with 
damping) to a stable node and then to a saddle describing the parametric excitation. In the last case, the 
two directions of the perturbation growth, so prominently featured in Fig. 8b, correspond to the two 
possible values of the oscillation phase cp, with the phase choice determined by initial conditions. 

This double degeneracy of the parametric oscillation’s phase could already be noticed from Eqs. 
(77), because they are evidently invariant with respect to replacement (p — » (p + n. Moreover, the 
degeneracy is not an artifact of the rotating- wave approximation, because the initial Eq. (75) is already 
invariant with respect to the corresponding replacement q(t) — > q(t - nlco). This invariance means that all 
other characteristics (e.g., the amplitude) of the parametric oscillations excited with either of two phases 
are absolutely similar. At the dawn of the computer age (in the late 1950s and early 1960s), there were 
substantial attempts, especially in Japan, to use this property for storage and processing digital 
information coded in the phase-binary form. 

4,7. Numerical approach 

If the amplitude of oscillations, by whatever reason, becomes so large that the nonlinear tenns in 
the equation describing a system are comparable to its linear terms, numerical methods are virtually the 
only avenue available for their study. In Hamiltonian ID systems, such methods may be applied directly 
to integral (3.26), but dissipative and/or parametric systems typically lack first integrals of motion 
similar to Eq. (3.24), so that the initial differential equation has to be solved. 

Let us discuss the general idea of such methods on the example of what mathematicians call the 
Cauchy problem (finding the solution for all moments of time, starting from known initial conditions) 
for first-order differential equation 

q = f(t,q). (4.96) 

(The generalization to a set of several such equations is straightforward.) Breaking the time axis into 
small, equal steps h (Fig. 9) we can reduce the equation integration problem to finding the function 
value in the next time point, q n +\ = q{t n +i) = q(t n + h ) from the previously found value q n = q(t n ) - and, if 

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necessary, the values of q at other previous time steps. In the generic approach (called the Euler 
method), q n +\ is found using the following fonnula: 

<ln + 1 = + k, 

k = hf(t n ,q n ). 


It is evident that this approximation is equivalent to the replacement of the genuine function q(t), on the 
segment [t n , t„+ i], with the two first tenns of its Taylor expansion in point t n : 

q(t„ + h) « q(t n ) + q(t n )h = q(t n ) + hf (t n ,q n ). (4.98) 

Fig. 4.11. The basic notions used at numerical 
integration of ordinary differential equations. 


Such approximation has an error proportional to h . One could argue that making the step h 
sufficiently small the Euler’s method error might be done arbitrary small, but even with the number- 
crunching power of modern computers, the computation time necessary to reach sufficient accuracy may 
be too high for large problems. 31 Besides that, the increase of the number of time steps, which is 
necessary at /) — > 0 , increases the total rounding errors, and eventually may cause an increase, rather 
than the reduction of the overall error of the computed result. 

A more efficient way is to modify Eq. (97) to include the terms of the second order in h. There 
are several ways to do this, for example using the 2 nd -order Runge-Kutta method: 

<ln + 1 = 7 „ +k 2 , 

k 2 = h f(t n + y q„ + y), K = h f(t n ,q n ). 


One can readily check that this method gives the exact result if function q(t) is a quadratic polynomial, 
and hence in the general case its errors are of the order of h . We see that the main idea here is to first 
break the segment [ t„ , t n +\] in half (Fig. 1 1), then evaluate the right-hand part of the differential equation 
(96) at the point intermediate (in both t and q) between points n and (n + 1), and then use this 
information to predict q n +\. 

The advantage of the Runge-Kutta approach is that it can be readily extended to the 4 th order, 
without an additional breaking of the interval [t n , t n +\].: 

31 In addition, the Euler method is not time-reversible - the handicap which may be essential for integration of 
Hamiltonian systems described by systems of second-order differential equations. However, this drawback may 
be readily overcome by the so-called leapfrogging - the overlap of time steps h for a generalized coordinate and 
the corresponding generalized velocity. 

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<7„+l “ Vn + T(^l + 2^2 + 2^3 + ^ 4 )’ 



k A = h fi t n + h ’<ln + k ll k 3 = h f if n 

+ 7r)» ^2 = h f^n +7’^» + ^)’ k l = h f dn^n)- 


This method reaches much lower error, 0(h 5 ), without being not too cumbersome. These features have 
made the 4 th -order Runge-Kutta the default method in most numerical libraries. Its extension to higher 
orders is possible but requires more complex fonnulas and is justified only for some special cases, e.g., 
very abrupt functions q{t)? 2 The most frequent enhancement of the method is the automatic adjustment 
of step h to reach the specified accuracy. 

Figure 12 shows a typical example of application of that method to the very simple problem of a 
damped linear oscillator, for two values of fixed time step h (expressed in terms of the number N of such 
steps per oscillation period). Black lines connect the points obtained by the 4 th -order Runge-Kutta 
method, while the points connected by green straight lines present the exact analytical solution (22). A 
few -percent errors start to appear only at as few as ~10 time steps per period, so that the method is 
indeed very efficient. I will illustrate the convenience and handicaps of the numerical approach to the 
solution of dynamics problems on the discussion of the following topic. 



Fig. 4.12. Results of the fixed-point Runge-Kutta solution to the equation of linear oscillator with damping 
(with S/a>o = 0.03) for: (a) 30 and (b) 6 points per oscillation period. The results are shown by points; lines 
are only the guide for the eye. 

4,8. Higher harmonic and subharmonic oscillations 

Figure 13 shows the numerically calculated 33 transient process and stationary oscillations in a 
linear oscillator and a very representative nonlinear system, the pendulum described by Eq. (42), both 
with the same resonance frequency coo for small oscillations. Both systems are driven by a sinusoidal 

32 The most popular approaches in such cases are the Richardson extrapolation, the Bulirsch-Stoer algorithm, and 
a set of prediction-correction techniques, e.g. the Adams-Bashforth-Moulton method - see the literature 
recommended in MA Sec. 16 (iii). 

33 All numerical results shown in this section have been obtained by the 4 th -order Runge-Kutta method with the 
automatic step adjustment which guarantees the relative error of the order of 10' 4 - much smaller than the pixel 
size in the plots. 

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external force of the same amplitude and frequency - in this illustration, equal to the small-oscillation 
eigenfrequency ox of both systems. The plots show that despite a very substantial amplitude of the 
pendulum oscillations (an angle amplitude of about one radian) their waveform remains almost exactly 
sinusoidal. 34 On the other hand, the nonlinearity affects the oscillation amplitude very substantially. 
These results illustrate that the validity of the small-parameter method and its RWA extension far 
exceeds what might be expected from the formal requirement \q\ « 1 . 



Fig. 4.13. Oscillations induced by a similar sinusoidal external force (turned on at t = 0) in two 
systems with the same small-oscillation frequency co 0 and low damping - a linear oscillator (two 
top panels) and a pendulum (two bottom panels). d/co 0 = 0.03,/i = 0.1, and a> = co 0 . 

The higher harmonic contents in the oscillation wavefonn may be sharply increased 35 by 
reducing the external force frequency to ~ oxln , where integer n is the number of the desirable hannonic. 
For example, Fig. 14a shows oscillations in a pendulum described by the same Eq. (42), but driven at 
frequency roo/3. One can see that the 3 ld harmonic amplitude may be comparable with that of the basic 
harmonic, especially if the external frequency is additionally lowered (Fig. 14b) to accommodate for the 
deviation of the effective frequency <x>o(a ) of own oscillations from its small-oscillation value coo - see 
Eq. (49), Fig. 4 and their discussion in Sec. 2 above. 

Generally, the higher hannonic generation by nonlinear systems might be readily anticipated. 
Indeed, the Fourier theorem tells us that any non-sinusoidal periodic function of time, e.g., an initially 
sinusoidal waveform of frequency co, distorted by nonlinearity, may be presented as a sum of its basic 
harmonic and higher harmonics with frequencies nco. Note that an effective generation of higher 

34 In this particular case, the higher harmonic contents is about 0.5%, dominated by the 3 rd harmonic whose 
amplitude and phase are in a very good agreement with Eq. (50). 

35 This method is used in practice, for example, for the generation of electromagnetic waves with frequencies in 
the terahertz range (10 12 -10 13 Hz) which still lacks efficient electronic self-oscillators. 

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harmonics is only possible with adequate nonlinearity of the system. For example, consider the 
nonlinear term aq used in equations explored in Secs. 2 and 3. If the waveform q(t) is approximately 
sinusoidal, such term can create only the basic and 3 ld harmonics. The “pendulum nonlinearity” sing 
cannot produce, without a constant component is process q(t), any even (e.g., the 2 nd ) harmonic. The 
most efficient generation of harmonics may be achieved using systems with the sharpest nonlinearities - 
e.g., semiconductor diodes whose current may follow an exponential dependence on the applied voltage 
through several orders of magnitude. 

0 5 10 15 20 25 30 

CO 0 t / 2 K 

Fig. 4.14. Oscillations induced in a pendulum with damping SIcoq = 0.03, driven by a sinusoidal 
external force of amplitude /o = 0.75, and frequency coq/3 (top panel) and 0.8&V3 (bottom panel). 

However, numerical modeling of nonlinear oscillators, as well as experiments with their physical 
implementations, bring more surprises. For example, the bottom panel of Fig. 15 shows oscillations in a 
pendulum under effect of a strong sinusoidal force with a frequency close to 3coo. One can see that at 
some parameter values and initial conditions the system’s oscillation spectrum is heavily contributed 
(almost dominated) by the 3 rd .swbharmonic, i.e. a component that is synchronous with the driving force 
of frequency 3 co, but has the frequency co that is close to the eigenfrequency <ao of the system. 

This counter-intuitive phenomenon may be explained as follows. Let us assume that the 
subharmonic oscillations of frequency co ~ coo have somehow appeared, and coexist with the forced 
oscillations of frequency 3 co: 

g(0«^cos v F + ^ sub cos v F sub , where Y = 3 cot-cp, ¥ sub = cat - (p sub . (4.101) 

Then, the first nonlinear term aq 3 of the Taylor expansion of pendulum’s nonlinearity sin q yields 

<7 3 =04 cosT + 4,* cos'F sub ) : 3 

= A i cos 2 Y + 3A A mb cos 2 'Fcos'F sub + 3 AA; ub cosTW x F sub + A s J ub cos J ¥ sub . 


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While the first and the last terms of this expression depend only of amplitudes of the individual 
components of oscillations, the two middle terms are more interesting because they produce so-called 
combinational frequencies of the two components. For our case, the third term, 

3zM 2 ub cos'Fcos 2 T' sub =^ 2 ub cos(¥-2‘F sub ) + ..., (4.103) 

of a special importance, because it produces, besides other combinational frequencies, the subharmonic 
component with the total phase 

^ ~ 2 v F sub = cot ~(p + 2<p sub . (4.104) 

Thus, within a certain range of the mutual phase shift between the Fourier components, this nonlinear 
contribution is synchronous with the subharmonic oscillations, and describes the interaction that can 
deliver to it the energy from the external force, so that the oscillations may be self-sustained. Note, 
however, that the amplitude of the tenn (103) describing this energy exchange is proportional to the 
square of 4 su b, and vanishes at the linearization of the equations of motion near the trivial fixed point. 
This means that the point is always stable, i.e., the 3 ld subharmonic cannot be self-excited and always 
need an initial “kick-off’ - compare the two panels of Fig. 15. The same is evidently true for higher 

Fig. 4.15. Oscillations induced in a pendulum with 5/co 0 = 0.03 by a sinusoidal external force of 
amplitude f 0 = 3 and frequency 3ruox0.8, with initial conditions q( 0) = 0 (the top row) and q(Q) = 1 
(the bottom row). 

Only the second subharmonic presents a special case. Indeed, let us make a calculation similar to 
Eq. (102), by replacing Eq. (101) with 

q(t) * A cosY + A sub cos'P sub , where T* = 2cot - q>, ^ mb =cot- cp sab , (4. 105) 

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for a nonlinear term proportional to q : 

q 2 = (AcosV + A sub cos¥ sub ) 2 = A 2 cos 2 ¥ + 2 AA sub cos'Fco sW sub +A 2 ub cos 2 W sub . (4.106) 

Here the combinational-frequency tenn capable of supporting the 2 nd subharmonic, 

2AA sub cos'Fcos x F sub = AA sub cos ('F - l F sub ) = AA sub cos (cot -<p + <p mb )+... , (4. 107) 

is linear in the subharmonic amplitude, i.e. survives the equation linearization near the trivial fixed 
point. This mean that the second subharmonic may arise spontaneously, from infinitesimal fluctuations. 

Moreover, such excitation of the second subharmonic is very similar to the parametric excitation 
that was discussed in detail in Sec. 5, and this similarity is not coincidental. Indeed, let us redo 
expansion (4.106) at a somewhat different assumption that the oscillations are a sum of the forced 
oscillations at the external force frequency 2 a>, and an arbitrary but weak perturbation: 

q(t) = A cos(2 cot - cp) + q(t), |g| « A . (4.108) 

Then, neglecting the small term proportional to q 2 , we get 

q 2 « A 2 cos 2 (2cot - cp) + 2q(t) A cos(2cot - cp). (4.109) 

Besides the inconsequential phase cp, the second tenn in the last formula is exactly similar to the term 
describing the parametric effects in Eq. (75). This fact means that for a weak perturbation, a system with 
a quadratic nonlinearity in the presence of a strong “pumping” signal of frequency 2co is equivalent to a 
system with parameters changing in time with frequency 2 <x> . This fact is broadly used for the 
parametric excitation at high (e.g., optical) frequencies where the mechanical means of parameter 
modulation (see, e.g., Fig. 5) are not practicable. The necessary quadratic nonlinearity at optical 
frequencies may be provided by a noncentrosymmetric nonlinear crystal, e.g., the /kphase barium borate 
(BaB 2 0 4 ). 

Before finishing this chapter, let me elaborate a bit on a general topic: the relation between the 
numerical and analytical approaches to problems of dynamics (and physics as a whole). We have just 
seen that sometimes numerical solutions, like those shown in Fig. 15b, may give vital clues for 
previously unanticipated phenomena such as the excitation of subharmonics. (The phenomenon of 
deterministic chaos, which will be discussed in Chapter 9 below, presents another example of such 
“numerical discoveries”.) One might also argue that in the absence of exact analytical solutions, 
numerical simulations may be the main theoretical tool for the study of such phenomena. These hopes 
are, however, muted by the problem that is frequently called the curse of dimensionality , 36 in which the 
last word refers to the number of input parameters of the problem to be solved. 37 

Indeed, let us have another look at Fig. 15. OK, we have been lucky to find a new phenomenon, 
the 3 ld subharmonic generation, for a particular set of parameters - in that case, five of them: S! cy = 
0.03, 3 col (Do = 2.4, fo = 3, q(0) = 1, and dq/dt (0) = 0. Could we tell anything about how common this 
effect is? Are subharmonics with different n possible in the system? The only way to address these 

36 This term had been coined in 1957 by R. Bellman in the context of the optimal control theory (where the 
dimensionality typically means the number of parameters affecting the system under control), but gradually has 
spread all over quantitative sciences using numerical methods. 

37 In EM Sec. 1.2, I discuss implications of the curse implications for a different case, when both analytical and 
numerical solutions to the same problem are possible. 

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questions computationally is to carry out similar numerical simulations in many points of the d- 
dimensional (in this case, d= 5) space of parameters. Say, we have decided that breaking the reasonable 
range of each parameter to N = 100 points is sufficient. (For many problems, even more points are 
necessary - see, e.g., Sec. 9.1.) Then the total number of numerical experiments to carry out is IC 1 = 
(10 2 ) 5 = 10 10 - not a simple task even for the powerful modern computing facilities. (Besides the pure 
number of required CPU cycles, consider storage and analysis of the results.) For many important 
problems of nonlinear dynamics, e.g., turbulence, the parameter dimensionality d is substantially larger, 
and the computer resources necessary for one numerical experiment, are much greater. 

In the view of the curse of dimensionality concerns, approximate analytical considerations, like 
those outlined above for the subharmonic excitation, are invaluable. More generally, physics used to 
stand on two legs, experiment and (analytical) theory. The enormous progress of computer perfonnance 
during a few last decades has provided it with one more point of support (a tail? :-) - numerical 
simulation. This does not mean we can afford to cut and throw away any of the legs we are standing on. 

4.9. Exercise problems 

4.1 . * Prove Eq. (26) for the response function given by Eq. (17). 

Hint : You may like to use the Cauchy integral theorem for analytical functions of complex 
variable. 38 

4.2 . A square-wave pulse of force (see Fig. on the right) is exerted on a 

linear oscillator with eigenfrequency ay (no damping), initially at rest. Calculate y 

the law of motion q{t), sketch it, and interpret the result. 

4.3 . At t = 0, a sinusoidal external force F(t) = Focos at, with constant A ® 2 k / co Q t 

and co, is applied to a linear oscillator with eigenfrequency ay and damping 8, which was at rest at t < 0. 

(i) Calculate the general expression for the time evolution of the oscillator’s coordinate, and 
interpret the result. 

(ii) Spell out your result for the case of the resonance (oj = ay) in a system with low damping (8 
« ay), and, in particular, explore the limit c7— » 0. 

4.4 . A pulse of external force F(t), with a finite duration T, is exerted on a harmonic oscillator, 
initially at rest in the equilibrium position. Neglecting dissipation, calculate the change of oscillator’s 
energy, using two different methods, and compare the results. 

4,5 . For a system with the following Lagrangian function: 

T • l L , £ • 2 2 

y q ~i q T q ■ 

calculate the frequency of free oscillations as a function of their amplitude A, at A — > 0, using two 
different approaches. 

38 See, e.g., MAEq. (15.1). 

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4.6 . For a system with the Lagrangian function 

L = 

+ sq\ 

with small parameter s, use the rotating-wave approximation to find the frequency of free oscillations as 
a function of their amplitude. 

4,7 . Find the regions of real, time-independent parameters a\ and a% in which the fixed point of 
the following system of equations, 

q x = a { (q 2 -q x ), 

q 2 ~ f?2 ’ 

is unstable. On the [a\, ai] plane, sketch the regions of each fixed point type - stable and unstable nodes, 
focuses, etc. 

4.8 . Solve Problem 4.3(ii) using the rotating-wave approximation, and compare the result with 
the exact solution. 

4.9 . Use the rotating-wave approximation to analyze forced oscillations in an oscillator with 
weak nonlinear damping, described by equation 

q + 2 8q + co 2 q + J3q 3 = f 0 cos cot, 

with co ~ coo; (3, 8> 0; and fJooA « 1. In particular, find the stationary amplitude of forced oscillations 
and analyze their stability. Discuss the effect(s) of the nonlinear term on the resonance. 

4.10 . * Analyze stability of the forced nonlinear oscillations described by Eq. (43). Relate the 
result to the slope of resonance curves (Fig. 4). 

4.1 1. Use the rotating-wave approximation to analyze parametric excitation of an oscillator with 
weak nonlinear damping, described by equation 

q + 2 8q + / 3 cj 3 + co 0 2 ( 1 + /jcos2cot)q = 0, 

with co ~ coo; ft, 8> 0; and //, fiooA « 1. In particular, find the amplitude of stationary oscillations and 
analyze their stability. 

4.12 . Adding nonlinear term aq to the left-hand part of Eq. (76), 

(i) find the corresponding addition to the RWA equations, 

(ii) find the stationary amplitude A of parametric oscillations, 

(iii) sketch and discuss the A(c) dependence, 

(iv) find the type and stability of each fixed point of the RWA equations, 

(v) sketch the Poincare phase plane of the system in main parameter regions. 

4.13 . Use the rotating-wave approximation to find the conditions of parametric excitation in an 
oscillator with weak modulation of both the effective mass m(t) = /«o( I + ju m cos 2 cot) and spring constant 
K{t) = K( } [\ + J u K cos(2cot -i//)]. with the same frequency 2co~ 2 aoo, but arbitrary modulation depths ratio 

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/uj /4 and phase shift i//. Interpret the result in terms of modulation of the instantaneous frequency coif) = 

1/9 * 1 j'y 

[K(t)/m(f)\ and mechanical impedance Z(t) = [idf)m(f)\ of the oscillator. 

4,14 .* Find the condition of parametric excitation of a nonlinear oscillator described by equation 

q + 2 Sq + <x>lq + yq 2 = / 0 cos 2 cot, 
with sufficiently small l\ y, f u and g = co - coo. 

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Chapter 5. From Oscillations to Waves 

In this chapter, the discussion of oscillations is extended to systems with two and more degrees of 
freedom. This extension naturally leads to another key notion - waves. The discussion of waves ( at this 
stage, in ID systems ) is focused at such key phenomena as their dispersion and reflection from 

5.1. Two coupled oscillators 

Let us move on to discuss oscillations in systems with more than one degree of freedom, starting 
from the simplest case of two linear, dissipation-free oscillators. If the Lagrangian of the system may be 
presented as a sum of those for two harmonic oscillators, 

T f j l -Z>2 , 

T - T -II - 

^ 1,2 1 1,2 1,2 


1,2 . 2 *" l ,2 

# 1,2 I - 


# 1,2 ’ 


(plus arbitrary, inconsequential constants if you like), then according to Eq. (2.19), the equations of 
motion of the oscillators are independent of each other, and each one is similar to Eq. (1.1), with its 
partial frequency Oi,2 equal to 

m 1 2 

This means that in this simplest case, the arbitrary motion of the system is just a sum of independent 
sinusoidal oscillations at two frequencies equal to the partial frequencies (2). 

Hence, in order to describe the oscillator coupling (i.e. interaction), the full Lagrangian L should 
contain an additional mixed tenn L mt depending on both generalized coordinates q\ and q 2 and/or 
generalized velocities. The simplest, and most frequently met type of such interaction term is the 
following bilinear form U m t = - icq\q 2 , where k is a constant, giving L mt = -U int = kzpq 2 . Figure 1 shows 
the simplest example of system with such interaction. 1 In it, three springs, keeping two massive 
particles between two stiff walls, have generally different spring constants. 

Fig. 5.1. Simple system of two 
coupled harmonic oscillators. 

Indeed, in this case the kinetic energy is still separable, T = T\ + T 2 , but the total potential 
energy, consisting of elastic energies of three springs, is not: 

U = 

( q\ -<h )" + 


1 Here it is assumed that the particles are constrained to move in only one dimension (shown horizontal). 

© 2013-2016 K. Likharev 

Essential Graduate Physics 

CM: Classical Mechanics 

where q 1.2 are the horizontal displacements of particles from their equilibrium positions. It is convenient 
to rewrite this expression as 

^ = y^i 2 +Y ?2 -W\ q i> where k x = k l + k m , k 2 =k r +k m , k = k m , (5.3b) 

showing that the Lagrangian L = T—U of this system indeed contains a bilinear interaction term: 

L = L,+L 2 + L ml , L ml = Jcq x q 2 . (5.4) 

The resulting Lagrange equations of motion are 

m l q l +m l D.;q l =Kq 2 , 
m 2 q 2 +m 2 Q 2 2 q 2 = Kq v 

Thus the interaction energy describes effective generalized force i<cq 2 exerted on subsystem 1 by 
subsystem 2, and the reciprocal effective force Kq\. Note that in contrast to real physical forces (these 
effective forces (such as F ]2 = - F 2 \ = icviqi - q\) for the system shown in Fig. 1) the effective forces in 
the right-hand part of Eqs. (5) do obey the 3 ld Newton law. Note also that they are proportional to the 
same coefficient tc, this feature is a result of the general bilinear structure (4) of the interaction energy 
rather than of any special symmetry. 

We already kn ow how to solve Eqs. (5), because it is still a system of linear and homogeneous 
differential equations, so that its general solution is a sum of particular solutions of the fonn similar to 
Eqs. (4.88), 

q l =c l e^' t , q 2 =c 2 e^. (5.6) 

for all possible values of X. These values may be found by plugging Eq. (6) into Eqs. (5), and requiring 
the resulting system of two linear algebraic equations for the distribution coefficients c \ j2 , 

m^X 2 Cj +m l Q 2 c ] =xc 2 , 
m 2 X 2 c 2 + m 2 Q 2 2 c 2 = kc 2 , 


to be self-consistent. In our particular case, we get a characteristic equation, 

m ] (X 2 +Q 2 ) 


- K 

m 2 (X 2 +Q 2 ) 

= 0, 

that is quadratic in X 2 , and thus allows a simple solution: 

(x 2 ) ± =-^{n 2 +al)+ 

= --fe +f2;)+ 

-(q" +f2“) 2 -n 2 £i 2 7 


+ - 



2 K 



According to Eqs. (2) and (3b), for any positive values of spring constants, product QiQ 2 = {k l + k m ){k r 
1 /2 • 1/2 1/2 
+ /Cy/)/(/»i/» 2 ) “ is always larger than /o / (/«i/n 2 ) = K M l{m\m 2 ) , so that the square root in Eq. (9) is 

2 2 2 

always less than (Qi +Q 2 )/2. As a result, both values of X are negative, i.e. the general solution to Eq. 




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(5) is a sum of four terms, each proportional to exp{±z'®T}, where both eigenfrequencies ox = iA± are 




A plot of these eigenfrequencies as a function of one of the partial frequencies Q (say, Qi), with 
the other partial frequency fixed, gives the famous anticrossing (also called “avoided crossing” or non- 
crossing”) diagram (Fig. 2). One can see that at weak coupling, frequencies ox are close to the partial 
frequencies everywhere besides a narrow range near the anticrossing point Qi = Q 2 . Most remarkably, at 
passing through this region, ox smoothly “switches” from following Q 2 to following Qi and vice versa. 

Fig. 5.2. Anticrossing diagram for two values of 
the oscillator coupling strength 
0.3 (red lines) and 0.1 (blue lines). In this plot, 
Qi is assumed to be changed by varying k x 
rather than m u but in the opposite case the 
diagram is qualitatively similar. 

The reason for this counterintuitive behavior may be found by examining the distribution 
coefficients cq 2 corresponding to each branch of the diagram, which may be obtained by plugging the 
corresponding value of A± = -iox back into Eqs. (7). For example, at the anticrossing point Qi = Q 2 = Q, 
Eq. (10) is reduced to 

col = Q 2 ± 


( m i m 2 ) V2 

= Q 

1 ± 


{ K \ K l ) 12 


Plugging this expression back into any of Eqs. (7), we see that for the two branches of the anticrossing 
diagram, the distribution coefficient ratio is the same by magnitude but opposite by sign: 2 

C 1 

m 2 

= + 

V C 2 ) 


l m i J 

at Q, = Q 2 . 


In particular, if the system is symmetric (m\ = m 2 , /<? = k r ), then at the upper branch, corresponding to 
ox > ci = - c 2 . This means that in this hard mode, 3 masses oscillate in anti-phase: q\(t) = -qiU). The 

2 It is useful to rewrite Eq. (12) as Z\C\ = +Z 2 C 2 , where Z\i = (/q. 2 / 771 . 2 )' 2 are of the partial oscillator impedances - 

the notion already mentioned in Chapter 4, and to be discussed in more detail in Sec. 4 below. 

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resulting substantial extension/compression of the middle spring yields additional returning force which 
increases the oscillation frequency. On the contrary, on the lower branch, corresponding to co., the 
particle oscillations are in phase: c\ = ci, qft) = qiif), so that the middle spring is never stretched at all. 
As a result, the soft mode oscillation frequency co. is lower than co+ and does not depend on rc 

2 0 2 « *L 

CO =12 = 







Note that for both modes, the oscillations equally engage both particles. 

Far from the anticrossing point, the situation is completely different. Indeed, an absolutely 
similar calculation of c\q shows that on each branch of the diagram, one of the distribution coefficients 
is much larger (by magnitude) than its counterpart. Hence, in this limit any particular mode of 
oscillations involves virtually only one particle. A slow change of system parameters, bringing it 
through the anticrossing, results, first, in a maximal delocalization of each mode, and then in the 
restoration of the localization, but in a different partial degree of freedom. 

We could readily carry out similar calculations for the case when the systems are coupled via 
their velocities, Z int = mq x q 2 , where m is a coupling coefficient - not necessarily a certain physical mass. 

(In mechanics, with q standing for actual particle displacements, such coupling is hard to implement, 
but there are many dynamic systems of non-mechanical nature in which such coupling is the most 
natural one.) The results are generally similar to those discussed above, again with the maximum level 
splitting at Qi = 0,2 = Cl: 

co 2 = 


1+ m 2 

O 2 



( m ! m 2 ) 12 


the last relation being valid for weak coupling. The generalization to the case of both coordinate and 
velocity coupling is also straightforward - see the next section. 

The anticrossing diagram shown in Fig. 2 may be met not only in classical mechanics. It is even 
more ubiquitous quantum mechanics, because, due to the time-oscillatory character of the Schrodinger 
equation solutions, weak coupling of any two quantum states leads to a qualitatively similar behavior of 
eigenfrequencies co± and hence of the eigenenergies (“energy levels”) E± = hco±. 4 

5.2, N coupled oscillators 

The calculations of the previous section may be readily generalized to the case of arbitrary 
number (say, N) coupled harmonic oscillators, with arbitrary type of coupling. It is evident that in this 
case Eq. (4) should be replaced with 

3 In physics, term “mode” is typically used for a particular type of variable distribution in space (in our current 
case, a certain set of distribution coefficients c 12 ), that sustains oscillations at a single frequency. 

4 One more property of weakly coupled oscillators, a periodic slow transfer of energy from one oscillator to the 
other and back, is more important for quantum rather than for classical mechanics. This is why I refer the reader 
to QM Secs. 2.5 and 5.1 for a detailed discussion of this phenomenon. 

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j = 1 ;',/=! 


Moreover, we can generalize the above relations for the mixed tenns Ly >, taking into account their 
possible dependence not only on the generalized coordinates, but on the generalized velocities, in a 
bilinear fonn similar to Eq. (4). The resulting Lagrangian may be presented in a compact form, 


j,j '= 1 

111 , 

K jf 


where the off-diagonal terms are index-symmetric: /%■■ = nijj, kjj • = Kyj, and the factors Vi compensate the 
double counting of each tenn with j ^ j ’, taking place at the summation over two independently running 
indices. One may argue that Eq. (16) is quite general is we still want the equations of motion to be linear 
- as they have to be if the oscillations are small enough. 

Plugging Eq. (16) into the general fonn (2.19) of the Lagrange equation, we get N equations of 
motion of the system, one for each value of index j - 1,2 

X('M, • *>?;)= °- (5-17) 

j = i 

Just as in the previous section, let us look for a particular solution to this system in the fonn 

q j =c/ t . (5.18) 

As a result, we are getting a system of N linear, homogeneous algebraic equations, 



for the set of N distribution coefficients q. The condition that this system is self-consistent is that the 
determinant of its matrix equals zero: 

T>et(m jr A 2 +k ]] )=0. (5.20) 

2 2 

This characteristic equation is an algebraic equation of degree N for A , and so has N roots (A )„. For any 
Hamiltonian system with stable equilibrium, matrices /% ■ and Ky ensure that all these roots are real and 
negative. As a result, the general solution to Eq. (17) is the sum of 2 N terms proportional to exp {±ico n t}, 
n= 1, 2,. . ., N, where all N eigenfrequencies co n are real. 

Plugging each of these 2 N values of A = ±ia>„ back into the set of linear equations (17), one can 
find the corresponding set of distribution coefficients cj±. Generally, the coefficients are complex, but in 
order to keep c/ } (t) real, the coefficients cj+ corresponding to A = +ia>„ and q. corresponding to A = -ia> n 
have to be complex conjugate of each other. Since the sets of the distribution coefficients may be 
different for each A n , they should be marked with two indices, j and n. Thus, at general initial conditions, 
the time evolution of j - th coordinate may be presented as 

Vj = - X V fjn ex P ! +/ (f) ,A + c jn exp {-/ C 0 J\ j = Re X c Jn exp {/ coJ\ . (5.21) 

^ n = 1 n - 1 

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This formula shows very clearly again the physical sense of the distribution coefficients c jn : a set 
of these coefficients, with different values of index j but the same n, gives the complex amplitudes of 
oscillations of the coordinates for the special choice of initial conditions, that ensures purely sinusoidal 
motion of the system, with frequency co„. Moreover, these coefficients show how exactly such special 
initial conditions should be selected - within a common constant factor. 

Calculation of the eigenfrequencies and distribution coefficients of a coupled system with many 
degrees of freedom from Eq. (20) is a task that frequently may be only done numerically. 5 Let us discuss 
just two particular but very important cases. First, let all the coupling coefficients be small (|/%- 1 « nij 
= nijj and t< u ■ |« /q = /qy, for all j ^j), and all partial frequencies Qy= ( /q/nq) “ be not too close to each 

l/cJ m jr \ 

, — , for all j*j\ (5.22) 

a j K J m , 

(Such situation frequently happens if parameters of the system are “random” in the sense that they do 
not follow any special, simple rule.) Results of the previous section imply that in this case the coupling 
does not produce a noticeable change of oscillation frequencies: {co n } ~ {Q,}. In this situation, 
oscillations at each eigenfrequency are heavily concentrated in one degree of freedom, i.e. in each set of 
the distribution coefficients cj„ (for a given n), one coefficient’s magnitude is much larger than all 

Now let the conditions (22) be valid for all but one pair of partial frequencies, say Qi and Q 2 , 
while these two frequencies are so close that coupling of the corresponding partial oscillators becomes 
essential. In this case the approximation { co n } « {Q ; } is still valid for all other degrees of freedom, and 
the corresponding terms may be neglected in Eqs. (19) for j = 1 and 2. As a result, we return to Eqs. (7) 
(perhaps generalized for velocity coupling) and hence to the anticrossing diagram (Fig. 2) discussed in 
the previous section. As a result, an extended change of only one partial frequency (say, Qi) of a 
weakly coupled system produces a series of eigenfrequency anticrossings - see Fig. 3. 

Fig. 5.3. Level anticrossing in a system of N 
weakly coupled oscillators - schematically. 

5 Fortunately, very effective algorithms have been developed for this matrix diagonalization task - see, e.g., 
references in MA Sec. 16(iii)-(iv). For example, the popular MATLAB package was initially created for this 
purpose. (“MAT” in its name stands for “matrix” rather than “mathematics”.) 

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5.3. ID waves in periodic systems 

For coupled systems with considerable degree of symmetry, the general results of the last section 
may be simplified, some with very profound implications. Perhaps the most important of them are 
waves. Figure 4 shows a classical example of a wave-supporting system - a long ID chain of massive 
particles, with the elastic next-neighbor coupling. 








o i > q h i o i > q, o i > q H 

I I I 

(j - 1 )d jd (j + \)d 

Fig. 5.4. Uniform ID chain of 
elastically coupled particles. 

Let us start from the case when the system is so long (formally, infinite) that the boundary 
effects may be neglected; then its Lagrangian may be represented by an infinite sum of similar terms, 
each including the kinetic energy of y'-th particle, and the potential energy of the spring on one (say, 
right) side of it: 

] L 

111 . 2 




From here, the Lagrange equations of motion (2.19) have the same form for each particle: 

mqj - K(q j+1 -qj) + K{ qj - q M ) = 0 . (5.24) 

Apart from the (formally) infinite size of the system, this is evidently just a particular case of Eq. (17), 
and thus its particular solution may be looked in the fonn (18), with A — > -of < 0. With this 
substitution, Eq. (24) gives the following simple form of the general system (17) for the distribution 
coefficients c/. 

(- mar + 2 k)c } - kc j+1 - KCj_ x = 0 . (5.25) 

Now comes the most important conceptual step toward the wave theory: the translational 
symmetry of Eq. (23), i.e. its invariance to the replacement j — > j + 1, allows it to have a particular 
solution of the following form: 

c i = ae 



where coefficient a may depend on a> (and system’s parameters), but not on the particle number j. 
Indeed, plugging Eq. (26) into Eq. (25) and cancelling the common factor e mj , we see that it is 
identically satisfied, if a obeys the following algebraic equation: 

{-ma> 2 +2 k)- K e + ,a -Ke~ ia =0 . (5.27) 

The physical sense of solution (26) becomes clear if we use it and Eq. (18) with A = +ia> to write 

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cij (t) = Rc 

i i 




1 1 








1 1 



where wave number k is defined as k = aid, and zj=jd is the equilibrium position of y-th particle - the 
notion that should not be confused with particle’s displacement qj from that equilibrium position - see 
Fig. 4. Relation (28) describes nothing else than a sinusoidal traveling wave of particle displacements 
(and hence of spring extensions/constrictions), that propagates, depending on the sign before v ph , to the 
right or to the left along the particle chain with phase velocity 


Perhaps the most important characteristic of a wave is the so-called dispersion relation, i.e. the 
relation between its frequency co and wave number k - essentially between the temporal and spatial 
frequencies of the wave. For our current system, this relation is given by Eq. (27) with a = kd. Taking 
into account that (2 - e ia - e iu ) = 2(1 - cos or) = 4sin 2 (o72), it may be rewritten in a simpler form: 

.a . kd 

co = ±co 0 sin — = ±<0 O sin — , 

where co 0 



This result, frequently called the Debye dispersion relation , 6 is sketched in Fig. 5, and is rather 
remarkable in several aspects. 

Fig. 5.5. The Debye 
dispersion relation. 

First, if the wavelength X = 2jzl\k\ is much larger than the spatial period a of the structure, i.e. if 
\kd\ « 1 (so that | a\ « coo), the dispersion relation is approximately linear: 

co = ±oo n — = ±vk 


where parameter v is frequency-independent: 

a> 0 d 


/ y/2 

K ' 




Comparison of Eq. (31) with Eq. (28) shows that this constant plays, in the low-frequency region, the 
role of phase velocity for any frequency component of a waveform created in the system - say, by initial 
conditions. As a result, low-frequency waves of arbitrary form can propagate in the system without 

6 Named after P. Debye who developed this theory in 1912, in the context of specific heat of solids at low 
temperatures (beating nobody else than A. Einstein on the way :-) - see, e.g., SM Sec. 2.6. 






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1 D wave 

1 D wave 

deformation (called dispersion). Such waves are called acoustic, 1 and are the general property of any 
elastic continuous medium. 

Indeed, the limit \kd\ « 1 means that distance d between adjacent particles is much smaller than 
wavelength A = 2jd\k\, i.e. that the differences q r \(t) - qft) and q/t) - qj-\{t), participating in Eq. (24), are 
relatively small and may be approximated with dq/dj = GqlG(zld) = d(dq/dz), with the derivatives taken 
at middle points between the particles: respectively, z+ = (z r \ - zj)/! and z.= (zj - zj. i)/2. Here z is now 
considered as a continuous argument (and hence the system, as a ID continuum), and q{z,t), as a 
continuous function of space and time. In this approximation, the sum of the last two terms of Eq. (24) is 
equal to - i<d\Gql Gz(z+)-Gq/ Gz(z.)], and may be similarly approximated by -rcd~(8~q/8z~), with the second 
derivative taken at point (z+ - z_)/2 = z 7 , i.e. exactly at the same point as the time derivative. As the result, 
the ordinary differential equation (24) is reduced to a partial differential equation 


Kd 2 


Using Eqs. (30) and (32), we may present this equation in a more general form 

' 1 d 2 


d 2 ' 
dz 2 J 

q(z,t ) = o, 


which describes a scalar acoustic wave (of any physical nature) in a ID linear, dispersion- free 
continuum - cf. Eq. (1.2). In our current simple model (Fig. 4), direction z of the wave propagation 
coincides with the direction of particle displacements q; such acoustic waves are called longitudinal . 
However, in Chapter 7 we will see that 3D elastic media may also support different, transverse waves 
that also obey Eq. (33b), but with a different acoustic velocity v. 

Second, when the wavelength is comparable with the structure period d (i.e. the product kd is not 
small), the dispersion relation is not linear, and the system is dispersive. This means that as a wave, 
whose Fourier spectrum has several essential components with frequencies of the order of a>o, travels 
along the structure, its waveform (which may be defined as the shape of a snapshot of all qj, at the same 
time) changes. 8 This effect may be analyzed by presenting the general solution of Eq. (24) as the sum 
(more generally, an integral) of components (28) with different complex amplitudes a: 

, J (0 = Re)V[ fe '-^ ) 'U. (5.34) 

This notation emphasizes the dependence of the partial wave amplitudes ak and frequencies on 
the wave number k. While the latter dependence is given by the dispersion relation, in our current case 
by Eq. (30), function ak is determined by the initial conditions. For applications, the case when ak is 
substantially different from zero only is a narrow interval, of width Ak « ko around some central value 
ko, is of special importance. (The Fourier transform reciprocal to Eq. (34) shows that this is true, in 
particular for a so-called wave packet - a sinusoidal wave modulated by an envelope with a large width 

7 This term is purely historical. Though the usual sound waves in air belong to this class, the waves we are 
discussing may have frequency both well below and well above human ear’s sensitivity range. 

8 The waveform deformation due to dispersion (which we are considering now) should be clearly distinguished 
from its possible change due to attenuation, i.e. energy loss - which is not taken into account is our energy- 
conserving model (23) - cf. Sec. 5 below. 

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A z ~ 1/A k » \/ko - see Fig. 6.) Using that strong inequality, the wave packet propagation may be 
analyzed by expending the dispersion relation co(k) into the Taylor series at point ko, and, in the first 
approximation in A k/ ko, restricting the expansion by its first two terms: 

k=k where co 0 = co(k 0 ), and k = k-k 0 . (5.35) 

co(k ) « CO Q H 


In this approximation, Eq. (34) yields 

<7 y (0~ Re J 

= Re 

a k exp< 7 

[k {) + k ]z 

/ “ ® o + 

exp{/ : {k 0 Zj - n> 0 /)} | a k cxpj ik 




k=k n 

1 dk 





Comparing this expression with the initial form of the wave packet, 


,■ (0) - Re J 

ikz . 

a,.e 1 dk = Re 

+o° , , 

Qxp\ik 0 Zj } J a k exp | ikzj jdk 



and taking into account that the phase factors before the integrals in the last forms of Eqs. (36) and (37) 
do not affect its envelope, we see that in this approximation 9 the envelope sustains its initial fonn and 
propagates along the system with the so-called group velocity 


VgT= ~dk 

k=k 0 ' 


Note that, with the exception of the acoustic wave limit (31), this velocity (that characterizes the 
propagation of waveform’s envelope), is different from the phase velocity (28) that describes the 
propagation of the “carrier” sine wave - for example, one of its zeros - see Fig. 6. 

Fig. 5.6. Phase and group 
velocities of a wave packet. 

Next, at the Debye dispersion law (30), the difference between v p h and v gr increases as the 
average frequency <x> approaches coo, with the group velocity tending to zero, while the phase velocity 
staying virtually constant. The existence of such a maximum for the wave propagation frequency 

9 Taking into account the next term in the Taylor expansion of function a(q), proportional to d 2 coldq 2 , we would 
find that actually the dispersion leads to a gradual change of the envelope form. Such changes play an important 
role in quantum mechanics, so that I discuss them in that part of my notes (see 
QM Sec. 2.1). 



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presents one more remarkable feature of this system. It physics may be readily understood by noticing 
that according to Eq. (30), at co = coo, the wave number k equals n/r/d, where n is an odd integer, and 
hence the phase shift a = kd is an odd multiple of n. Plugging this value into Eq. (28), we see that at the 
Debye frequency, oscillations of two adjacent particles are in anti-phase, for example: 

q 0 (t) = a exp {-i cot}, q ] (t) = a exp {i (71 - cot)} = -a exp {-i cot} = -q 0 (t). (5.39) 

It is clear from Fig. 4 that at such phase shift, all the springs are maximally stretched/compressed (just as 
in the hard mode of the two coupled oscillators analyzed in Sec. 1), so that it is natural that this mode 
has the highest frequency. 

This invites a natural question what happens with the system if it is excited at a frequency co > 
co d, say by an external force applied at the system’s boundary. While the boundary phenomena will be 
considered in the next section, the most essential part of the answer may be obtained immediately from 
Eqs. (26) and (30). Indeed, reviewing the calculations that have led to these results, we see that they are 
valid not only for real but also any complex values of a. In particular, at co > coo the dispersion relation 
(30) gives 

a = nn ± i — , where A 


2arccosh(W co 0 ) 


Plugging this relation into Eq. (26), we see that the wave’s amplitude becomes an exponential function 
of position: 


± /Ima: 
a e J 

±z . / A 
oc e 1 


Physically this means that the wave decays penetrating into the structure (from the excitation point), 
dropping by a factor of e ~ 3 on the so-called penetration depth A. (According to Eq. (40), this depth 
decreases with frequency, but rather slowly, always remaining of the order of the distance between the 
adjacent particles.) Such a limited penetration is a very common property of various waves, including 
the electromagnetic waves in plasmas and superconductors, and quantum-mechanical “de Broglie 
waves” (wavefunctions) in the classically-forbidden regions. Note that this effect of “wave expulsion” 
from the media they cannot propagate in does not require any energy dissipation. 

Finally, one more fascinating feature of the dispersion relation (30) is that if it is satisfied by 
some wave number k 0 (co), it is also satisfied at any k n (co) = kd co) + 2mJd, where n is any integer. This 
property is independent of the particular dynamics of the system: it follows already from Eq. (27), 
before its substitution into Eq. (25), because such a wave number translation by 2 nld, i.e. the addition of 
2n to phase shift a , is equivalent to the multiplication of q/t) by exp { /2/zj = 1. Thus, such (2n/d)- 
periodicity in the wave number space is a common property of all systems that are ^-periodic in the 
usual (“direct”) space. 10 

Besides dispersion, one more key characteristic of any wave-supporting system is its wave 
impedance - the notion strangely missing from many physics (but not engineering) textbooks. It may be 

10 This property has especially important implications for quantum properties of periodic structures, e.g., crystals. 
It means, in particular, that the product hk cannot present the actual momentum of the particle (which is not 
conserved in periodic systems), but rather serves as its quasi-momentum (or “crystal momentum”) - see, e.g., QM 
Sec. 2.5. 

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revealed by calculating the forces in the sinusoidal wave (28). For example, the force exerted by y-th 
particle on its right neighbor, given by the second term in Eq. (24), equals 

F J+ (0 = Adj (0 - q J+ 1 (o] = R e 

x\ \ -e 



i(kz . + cot) 


i(kz +ojt) 


, (5-42) 

where the last form is valid in the most important acoustic wave limit, kd — » 0. Let us compare this 
expression for the wave of forces with that for the corresponding wave of particle velocities: 

djU) = R e 

+ icoae 

i(a j + cot) 


We see that these to waves have the same phase, and hence their ratio does not depend on either time or 
the particle number. Moreover, this ratio, 

F + kdx dx 

q co v 

= ±Z, 


is a real constant independent even on wave’s frequency. Its magnitude is called the wave impedance : 



and characterizes the dynamic “stiffness” of the system for the propagating waves. 

In particular, the impedance scales the power carried by the wave. Indeed, the direct time 
averaging of the instantaneous power if (t) = Fj(t)dqj/dt transferred through particle j to the subsystem 
on the right of it, using Eqs. (42)-(43), yields a position-independent result 


where A = \a\ is the real amplitude of the wave, and, as before, the positive sign corresponds to the wave 
propagating to the right (and vice versa). Note that P is the power flow in the acoustic wave, and its 
spatial and temporal independence means that wave’s energy is conserved - as could be expected from 
our Hamiltonian system we are considering. 11 Hence, the wave impedance Z characterizes the energy 
transfer along the system rather than its dissipation. 

5.4, Interfaces and boundaries 

The importance of the wave impedance notion becomes even more evident when we consider 
waves in non-uniform and finite-size systems. Indeed, our previous analysis assumed that the ID system 
supporting the waves (Fig. 4) is exactly periodic, i.e. macroscopically uniform, and extends all the way 
from -oo to +oo. Now let us examine what happens when this is not true. The simplest (and very 
important) example of such nonuniform systems is an interface, i.e. a point at which system parameters 
experience a change. Figure 7 shows a simple and representative example of such a sharp interface, for 
the same ID wave system that was analyzed in the last section. 

11 The direct calculation of the energy (per unit length) is a simple but useful exercise, left for the reader. 






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Fig. 5.7. ID system with a sharp 
interface at z = 0, and the wave 
components at the partial 
reflection of a wave incident from 
the left. 

Since the parameters k and m are still constant on each side of the interface (put, for 
convenience, at zj = 0), equations of motion (24) are is still valid for j < 0 and j > 0 (in the latter case, 
with the primed parameters), and show that at a fixed frequency co, they can sustain sinusoidal waves of 
the type (28). However, the final jump of parameters at the interface (m ’ ^ m, k’ -t- k) leads to a partial 
reflection of the incident wave from the interface, so that at least on the side of incidence (say, zj < 0), 
we need to assume two such waves, one describing the incident wave and another, the reflected wave: 

q,(t) = Re 


i\ kz . - cot 

+ a^e 

i\ - kz ■ - cot 

a _^e 

i\ k'zj - cot 

for j < 0, 
for j > 0. 


In order to obtain boundary conditions for “stitching” these waves (i.e. getting relations between 
their complex amplitudes) at j = 0, i.e. z ; - = 0, we need to take into account, first, that displacement qft) 
of the interface particle has to be the same whether it is considered a part of the left or right sub-system, 
and hence participates in Eqs. (24) for both j < 0 and j > 0. This gives us the first boundary condition, 

a^+a^=afi. (5.48) 

Second, writing the equation of motion for the special particle with j = 0, 

tn 0 q 0 ~K J (q l -q 0 ) + tc(q 0 - q = 0. 

and plugging into it the solution (47), we getthe second boundary condition 

- co"m n a' —id a' 


1 +/d 

\-e- ikd ) + a. \l-e ikd 

= 0 . 



The system of two linear equations (48) and (50) allows one to express both and a f via 
amplitude a^> of the incident wave, and hence find the reflection (R) and transmission ( T) coefficients 
of the interface: 12 

R = 


The general result for R and T is a bit bulky, but may be simplified in the most important acoustic wave 
limit: k’d, kd — > 0. Indeed, in this limit all three parentheses participating in Eq. (50) may be 
approximated by the first terms of their Taylor expansions, e.g., exp {ik’d} - 1 « ik’d, etc. Moreover, in 
this limit, the first term of Eq. (50) is of the second order in small parameter co/coo- ka « 1 (unless the 

12 Sorry, one more traditional usage of letter T. I do not think there any chance to confuse it with the kinetic 

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interface particle mass mo is much larger than both m and in ’), and hence may be neglected. As a result, 
Eq. (50) takes a very simple form 13 

/dc{a^-a^)= x'k'af. (5.52a) 

According to Eqs. (31), (32) and (45), in the acoustic limit the ratio of factors xk of the waves (with the 
same frequency <x>\) propagating at z < 0 and z > 0 is equal to that of the wave impedances Z of the 
corresponding parts of the system, so that Eq. (52a) may be rewritten as 

Z(a^-a^) = Z'af. (5.52b) 

Now, solving the simple system of linear equations (48) and (52a), we get very important formulas, 

R = 


z + z 7 ’ 

T = 

2 Z 

Z + Z 7 ’ 






which are valid for any waves in ID continua - with the corresponding re-definition of impedance. 14 
Note that coefficients R and T characterize the ratios of wave amplitudes rather than their power. Using 
Eq. (46), for the time-averaged power flows we get relations 

" Z-Z' V 
vZ + Z'J ’ 


(z + Z'Y ' 


(Note that A7 + 'Pf = Af, again reflecting the energy conservation.) 

The first important result of this calculation that wave is fully transmitted through the interface if 
the so-called impedance matching condition Z’ = Z is satisfied, even if the wave velocities v (32) are 
different on the left and the right sides of the interface. On the contrary, the equality of the acoustic 
velocities in two media does not guarantee the full transmission of their interface. Again, this is a very 
general result. 

Now let us consider the two limits in which Eq. (53) predicts a total wave reflection, A7/2A — > 
0: Z7Z — » oo (when R = - 1) and Z7Z — » 0 (when R = 1). According to Eq. (45), the fonner limit 
corresponds to the infinite product x’m so that particles on the right side of the interface cannot move 
at all. This means that this particular case also describes a perfectly rigid boundary (Fig. 8a) for arbitrary 
co, i.e. not necessarily in the acoustic wave limit. The negative sign off? in the relation A = -1 means that 
in the reflected wave, the phase of particle oscillations is shifted by n relative to the initial wave, a = a<_ 
= -a_>, so that the sum of these two traveling waves may be also viewed as a single standing wave 








r . -| 






ae v 

-ae y 




2 iae ’ 0Jt si n kz ■ 

2Asm(cot -(p)smkzj, (5.55) 

where a = a_> = A e l<p . At the boundary (z, = 0) this expression yields qo(t) = 0, i.e., a node of particle 
displacements. On the contrary, the corresponding standing wave of spring forces, described by Eq. 
(42), has a maximum at z = 0. 

13 This equation could be also obtained using Eq. (42), as the condition of balance of the forces exerted on the 
interface particle with j = 0 from the left and right - again, neglecting the inertia of that particle. 

14 See, e.g., corresponding parts of my lecture notes: QM Sec. 2.3 and EM Sec. 7.4. In 2D and 3D systems, Eqs. 
(53) are valid for the normal wave incidence only, otherwise they have to be modified - see, e.g., EM Sec. 7.4. 

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A similar standing wave fonns in the opposite limit Z’/Z — » 0, that describes an “open” boundary 
shown in Fig. 8b. However, in this limit (with R = + 1), the standing wave of displacements has a 
maximum at zj = 0, 

<lj< o(0 = Re 


i\ kz ■ - cot 

+ ae 

;| - kz . - cot 

= Re 

2 ae 

- icot 

cos kzj 

= 2Acos(cot-cp)coskz j , (5.56) 

while the corresponding wave of forces has a node at that point. Most importantly, for both boundaries 
shown in Fig. 8, the standing waves are fonned at any ratio co/coq. 


Fig. 5.8. (a) Rigid and (b) open 
boundaries of a ID chain. 

If the opposite boundary of a finite-length chain also provides a total wave reflection, the system 
may only support standing waves with certain wave numbers k n , and hence certain eigenfrequencies co n 
that may be found from the set of k n and the dispersion relation co„ = oik n ), in our case given by Eq. 
(30). For example, if both boundaries of a chain with length L are rigid (Fig. 8a), then the standing wave 
(54) should have nodes at them both, giving the wave number quantization condition 15 

sink„Z = 0, i.e. k n 


where n is an integer. In order to count the number of different modes in a chain with a finite number N 
of oscillating particles, let us take into account, first, that adding one period A k = 2nld of the dispersion 
relation to any k n leads to the same mode. Moreover, changing the sign of k„ in standing wave (55) is 
equivalent to changing the sign of its amplitude. Hence, there are only N different modes, for example 

. . , r . , TC TC , T TC 

n = 1,2,..., A, i.e. k n = — ,2 — ,..., N — . 



This fact is of course just a particular case of the general result obtained in Sec. 2. 

According to Eq. (56), if both boundaries are open (Fig. 8b), the oscillation modes are different, 
but their wave numbers form the same set (57). Finally, if the types of boundary conditions on the 
chain’s ends are opposite, the wave number set is somewhat different, 



n ( X s 

— n , 

L{ 2) 


15 This result should be very familiar to the reader from freshmen-level “guitar string”-type problems. 
Note, however, that Eqs. (54)-(56) are valid not only for continuous ID systems like a string, but also 
for (uniform) chains with a finite and arbitrary number N of particles - the fact we will use below. 

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but since the distance between the adjacent values of k n is still the same (tt/NcI), the system still has 
exactly N such values within each period 2 n/d of the dispersion law, and hence, again, exactly N 
different oscillation modes. 

This insensitivity of the number of modes and their equal spacing (called equidistance) on the k 
axis, enables the following useful (and very popular) trick. In many applications, it is preferable to 
speak about the number of different traveling, rather than standing waves in a system of a large but 
finite size, with coordinates zq and z,v describing the same particle. One can plausibly argue that the local 
dynamics of the chain of N » 1 particles should not be affected if it is gradually bent into a large closed 
loop of length L = Nd » d. Such a loop may sustain traveling waves, if they satisfy the following 
periodic Born-Karman condition : q 0 (t) = q : \(t). (A popular vivid image is that the wave “catches its own 
tail with its teeth”.) According to Eq. (27), this condition is equivalent to 

ik L 
e " 

= 1 , 


2 7t 

= — n . 


This equation gives a set of wave numbers twice more sparse than that described by Eqs. (57). 
However, now we can use N values of n, giving k n , for example, from -N to +N (strictly speaking, 
excluding one of the boundary values to avoid double counting of the identical modes with n = ±N), 
because traveling waves (28) with equal but opposite values of k„ propagate in opposite directions and 
hence present different modes. As a result, the total number of different traveling-wave modes is the 
same (AO as that of different standing-wave modes, and they are similarly (uniformly) distributed along 
the wave number axis. Since for N » 1 the exact values of k n are not important, the Born-Carman 
boundary conditions and the resulting set (59) of wave numbers are frequently used even for multi- 
dimensional systems whose bending into a ring along each axis is hardly physically plausible. 

5.5. Dissipative, parametric, and nonlinear phenomena 

In conclusion, let us discuss more complex effects in oscillatory systems with more than one 
degree of freedom. Starting from linear systems, energy dissipation may be readily introduced, just as 
for a single oscillator, by adding terms proportional to rj jq ,, to the equations of motion such as Eqs. (5), 

(17), or (24). In arbitrary case, viscosity coefficients r]j are different for different particles; however, in 
many unifonn systems like that shown in Fig. 4, the coefficients are naturally equal, turning Eq. (24) 

mijj + qq . - K(q j+l -qj) + /c(q . - q j _ ] ) = 0 . 


In the most important limit of acoustic waves, we may now repeat the arguments that have led to the 
wave equation (33) to get its generalization 

V v 2 dt 2 v 2 8t 


q(x,t) = 0, 

with S = ^~. 


Such dissipative equation may describe two major particular effects. First, it describes the decay 
in time of the standing waves in an autonomous wave system (say, of a finite length L) that have been 
caused by some initial push, described by non-trivial initial conditions, say, q(z, 0) ^ 0. In order to 
analyze these decaying oscillations, one may look for the solution of Eq. (61) in the form of a sum of 
standing wave modes (that satisfy the given boundary conditions), each with its own, time-dependent 









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amplitude A n (t). For example, for rigid boundary conditions (q = 0) at z = 0 and z = L, we can use Eq. 
(55) as a hint to write 


<?(z,0 = XX(OsinA:„z, (5.62) 

n = 1 

where the set of q n is given by Eq. (57). Plugging this solution into Eq. (61), 16 we get 

-yXfc + 15 ^n + °K A n)^ nk n Z = 0, With k„ = y«. (5.63) 

v“ „= i L 

Since functions sin k„z are mutually orthogonal, Eq. (63) may be only satisfied if all N expressions in 
parentheses are equal to zero. As the result, the amplitude of each mode satisfies an ordinary differential 
equation absolutely similar to that studied in Sec. 4.1, with a similar solution describing the free 
oscillation decay with the relaxation constant (4.23). Here the wave character of the system gives 
nothing new, besides that different modes have different (9-factors: Qn = coj'18. 

More wave-specific is a different situation when the waves are due to their persisting excitation 
by some actuator at one of the ends (say, z = 0) of a very long structure. In this case, an initial transient 
process settles to a wave with a time-independent wavefonn limited by certain envelope A(z) that decays 
at z — » oo. In order to find the envelope, for the simplest case of sinusoidal excitation of frequency ox one 
may look for a particular solution to Eq. (61) in a form very different from Eq. (60): 

q(z,t) = Re 



generally with complex a(z). Plugging this solution into Eq. (61), we see that this is indeed a valid 
solution, provided that q(0,t) = o(0)exp{-/ruf} satisfies the boundary condition (now describing the wave 
excitation), and a(z) obeys an following ordinary differential equation that describes wave’s evolution in 
space rather than in time: 17 

( d 2 } 
d 7 +k 2 

<7 = 0, withk° = 

f 6)^ 

K dz~ ) 


+ v- 

The general solution to such differential equation is 

/ \ ikz , — ikz 

a{x) = a + e +a_e , 
with k now having both real and imaginary parts, k= k’ + ik”, so that the wave (64) is 

, \ i(k'z-cot ) - k" z i(-k' z - cot) k" z 

q{z,t) = a + e e +a_e e 




If our boundary conditions correspond to the wave propagating to the right, we have to keep only the 
first term of this expression, with positive k”. The first exponent of that term describes the wave 
propagating from the boundary into the system (at low damping, with velocity virtually equal to v), 
while the second exponent describes an exponential decay of the wave’s amplitude in space: 

16 Actually, this result may be also obtained from Eq. (60) and hence is valid for an arbitrary ratio coJcoq. 

17 Equation (65), as well as its multi-dimensional generalizations, is frequently called the Helmholtz equation, 
named after H. von Helmholtz (1821-1894). 

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A(z) = ja(z)j = A(0)e~ az/2 , 



where the last, approximate relation is valid in the weak damping limit (8« co, i.e. d/v « k’). Constant 
a is called the attenuation coefficient, and in more general wave systems may depend on frequency co. 
Physically, Ha is the scale of wave penetration into a dissipative system. 18 Note that our simple solution 
(68) is only valid if the system length L is much larger than 2/ a; otherwise we would need to use the 
second term in Eq. (67) to describe wave reflection from the second end. 

Now let me discuss (because of the lack of time, on a semi-quantitative level only), nonlinear 
and parametric phenomena in oscillatory systems with more than one degree of freedom. One important 
new effect here is the mutual phase locking of (two or more) weakly coupled self-excited oscillators 
with close frequencies: if the eigenfrequencies of the oscillators are sufficiently close, their oscillation 
frequencies “stick together” to become exactly equal. Though its dynamics of this process is very close 
to that of the phase locking of a single oscillator by external signal, that was discussed in Sec. 4.4, it is 
rather counter-intuitive in the view of the results of Sec. 1, and in particular the anticrossing diagram 
shown in Fig. 2. The analysis of the effect using the rotating-wave approximation (that is highly 
recommend to the reader) shows that the origin of the difference is oscillator’s nonlinearity, which 
makes oscillation amplitude virtually independent of phase evolution - see Eq. (4.68) and its discussion. 

One more new effect is the so-called non-degenerate parametric excitation. It may be illustrated 
of the example of just two coupled oscillators - see Sec. 1 above. Let us assume that the coupling 
constant k, participating in Eqs. (5), is not constant, but oscillates in time - say with frequency co p . In 
this case the forces acting on each oscillator from its counterpart, described by the right-hand parts of 
Eqs. (5), will be proportional to x^ 2 ,i(l + p cos co p t). Assuming that oscillations of q\ and qi are close to 
sinusoidal, with frequencies co\^, we see that the force acting on each oscillator will contain the so- 
called combinational frequencies 

(Op ± a> 2 l . 


If one of these frequencies in the right-hand part of each equation coincides with its own oscillation 
frequency, we can expect a substantial parametric interaction between the oscillators (on the top of the 
constant coupling effects discussed in Sec. 1). According to Eq. (69), this may happen in two cases: 

c Op = co { ±co 2 . 


The quantitative analysis (also highly recommended for reader’s exercise) shows that in the 
positive sign case, the parameter modulation indeed leads to energy “pumping” into oscillations. As a 
result, sufficiently large //, at sufficiently low damping coefficients A .2 and effective detuning 

4 = <o p -{p. 1 + 0 ,), 


may lead to the simultaneous excitation of two frequency components co\^. These frequencies, while 
being close to corresponding eigenfrequencies of the system, are related to the pumping frequency co p by 
exact relation (70), but otherwise are arbitrary, e.g., incommensurate (Fig. 9a), thus justifying the term 

18 In engineering, the attenuation coefficient of wave-carrying systems is most frequently characterized by a 
logarithmic measure called decibel per meter (or just dB/m): a dB /m = 10 logio a. 






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non-degenerate parametric excitation. (The parametric excitation of a single oscillator, that was 
analyzed in Sec. 4.5, is a particular, degenerate case of such excitation, with a>\ = an = co p /2.) On the 
other hand, for the case described by Eq. (70) with the negative sign, parameter modulation always 
pumps energy from the oscillations, effectively increasing system’s damping. 

Somewhat counter-intuitively, this difference between two cases (70) may be simpler interpreted 
using the notions of quantum mechanics. Namely, equality co p = co\ + op enables a decay of an external 
photon of energy Ticq p into two photons of energies fioo\ and box going into the oscillatory system. (The 
complementary relation, co\ = co p + ox results in the oscillation photon decay.) 








t 1 

\ 1 

\ 1 
\ / 






co = 


co x +co 2 




CO , 



- oo { + a> 2 frequency 0 


Fig. 5.9. Spectra of oscillations at (a) the non-degenerate parametric excitation, and (b) four- 
wave mixing. The arrow directions symbolize the power flows into and out of the system. 

Proceeding to nonlinear phenomena, let us note, first of all, that the simple reasoning, that 
accompanied Eq. (4.109), is also valid in the case when oscillations consist of two (or more) sinusoidal 
components with incommensurate frequencies. Replacing notation 2 co for co p , we see that non- 
degenerate parametric excitation of the type (70a) is possible to implement in a system of two coupled 
oscillators with a quadratic nonlinearity (of the type yq~), “pumped” by an intensive external signal at 
frequency (Op « Qi + Q 2 . In optics, it is often more convenient to have all signals within the same, 
relatively narrow frequency range. A simple calculation, similar to the one made in Eqs. (4.108)-(4.109), 
shows that this may be done using the cubic nonlinearity 19 of the type acf, which allows the similar 
parametric energy exchange at frequency relation (Fig. 9b) 




2co = co x + co 2 , with co « co x « co 2 . 


This process is often called the four-wave mixing (FWM), because it may be interpreted 
quantum-mechanic ally as the transformation of two externally-delivered photons, each with energy h co p , 
into two other photons of energies Tico\ and box. Word “wave” in this term stems from the fact that at 
optical frequencies, it is hard to couple a sufficient volume of a nonlinear medium with lumped-type 
resonators. It is easier to implement the parametric excitation of light (as well as other nonlinear 
phenomena like the higher harmonic generation) in distributed systems of a linear size much larger than 
the involved wavelengths. In such systems, the energy transfer from the incoming wave of frequency co 
to generated waves of frequencies co\ and ox is gradually accumulated at their joint propagation along 
the system. From the analogy between Eq. (65) (describing the evolution of wave’s amplitude in space), 

19 In optics, the nonlinearity is implemented using transparent crystals such as lithium niobate (LiNbCf), with the 
cubic-nonlinear dependence of the electric polarization as a function of the applied electric field: Px 3 + a3 3 . 

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and the usual equation of the harmonic oscillator (describing its evolution in time), it is clear that this 
energy transfer accumulation requires not only the frequencies co, but also wave numbers k be in similar 
relations. For example, the four-wave mixing requires that not only the frequency balance (72a), but also 
a similar relation 

2k = k l +k 2 , (5.72b) 

to be exactly fulfdled. Since all three frequencies are close, this is easy to arrange if the dispersion 
relation ofk) of the media is not too steep. Unfortunately, due to the lack of time/space, for more 
discussion of this interesting subject, nonlinear optics , I have to refer the reader to special literature. 20 

Note that even if the frequencies a>\ and an of the parametrically excited oscillations are 
incommensurate, the oscillations are highly correlated. Indeed, the quantum mechanical theory of this 
effect 21 shows that the generated photons are entangled. This fact makes the parametric excitation very 
popular for a broad class of experiments in several currently active fields including quantum 
computation and encryption, and Bell inequality / local reality studies. 22 

It may look like a dispersion-free media, with a>lk = v = const, is the perfect solution for 
arranging the parametric interaction of waves, because in such media, for example, Eq. (72b) 
automatically follows from Eq. (72a). However, in such media not only the desirable three 
parametrically interacting waves, but also all their harmonics, have the same velocity. At these 
conditions, energy transfer rates between all harmonics are of the same order. Perhaps the most 
important result of such multi-harmonic interaction is that intensive waves, interacting with nonlinear 
media, may develop sharply non-sinusoidal waveforms, in particular those with an almost instant change 
of the field at a certain moment. Such shock waves, especially those of mechanical nature, present large 
interest for certain applications - some not quite innocent, e.g., the explosion of usual and nuclear 
bombs. I will only briefly return to shock waves in Sec. 8. 5. 23 

5.6. Exercise problems 

5J_ . For the system of two elastically coupled pendula, confined to a 
vertical plane, with the parameters shown in Fig. on the right (cf. Problem 1.3), 
find possible frequencies of small sinusoidal oscillations and the corresponding 
distribution coefficients. Sketch the oscillation modes. 





■f m m 

5.2 . The same task as in Problem 1, for the double pendulum, confined to the 
vertical plane containing the support point (considered in Problem 2.1), with m ’ = m 
and 1 = 1’- see Fig. on the right. 


20 See, e.g., the classical monograph by N. Bloembergen, Nonlinear Optics, 4 th ed., World Scientific, 1996, or a 
more modern treatment by R. W. Boyd, Nonlinear Optics, 3 rd ed.. Academic Press, 2008. 

21 Which is, surprisingly, not much more complex than the classical theory - see, e.g., QM Sec. 5. 5. 

22 See, e.g., QM Secs. 8.5 and 10.1, correspondingly. 

23 The classical (and perhaps still the best) monograph on the subject is Ya. Zeldovich, Physics of Shock Waves 
and High-Temperature Phenomena, Dover, 2002. 

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5.3 / The same tasks as in Problem 5.1, for the triple pendulum shown in 
Fig. on the right, with the motion confined to a vertical plane containing the support 

Hint. You may use any (e.g., numerical) method to calculate the 
characteristic equation roots. 



(i) Explore an approximate way to analyze waves in a continuous ID system with 
parameters slowly varying along its length. 24 

(ii) Apply this method to calculate eigenfrequencies of transverse standing waves 
on a freely hanging heavy rope of length L, with constant mass per unit length - see Fig. 
on the right. 

(iii) For three lowest standing wave modes, compare the results with those 
obtained in the solution of Problem 5.3 for the triple pendulum. 




5.5 . The same tasks as in Problem 1 , for a linear, symmetric system 
of 3 particles, shown in Fig. on the right. Assume that the connections 
between the particles not only act as usual elastic springs (as described by 
their potential energies U = at / 2 / 2 ), but also resist system’s bending, 

giving an additional potential energy U' = k' l 2 6 2 / 2, where 6 is the 
(small) bending angle. 25 

5.6 . Three similar beads, which may slide along a circle of radius R 
without friction, are connected with similar springs with elastic constants k 
and equilibrium lengths /o - see Fig. on the right. Analyze stability of the 
symmetric stationary state of the system, and calculate the frequencies and 
modes of its small oscillations about this state. 

5.7 . An external force F(t) is applied to the 
right particle of system of shown in Fig. 5.1 of the 
lecture notes, with k l = k r = k’ and m\ = m2 = m (see 
Fig. on the right), and the response q\(t) of the left 
particle to this force is being measured. 

5.8 . Calculate the spatial distributions of the kinetic and potential energies in a standing, 
sinusoidal, ID acoustic wave, and analyze its evolution in time. 

24 The reader familiar with the WKB approximation in quantum mechanics (see, e.g., QM Sec. 2.4) is welcome to 
adapt it for this classical application. Another possible starting point is the rotating-wave approximation (RWA), 
discussed in Sec. 4.3 above, which should be translated from the time domain to the space domain. 

25 This is a good model for small oscillations of linear molecules such as CO 2 (for which the values of elastic 
constants k and k’ are well known). 

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5.9 . Calculate the dispersion law oik) and the maximum 
frequency of small longitudinal waves in an infinite line of 
similar, spring-coupled pendula - see Fig. on the right. 

/ K Ik l 

m m m 

5.10 . Calculate and analyze the dispersion relation 
oik) for longitudinal waves in an infinite ID chain of 
coupled oscillators with alternating masses - see Fig. on the 
right. In particular, find and discuss dispersion relation’s 
period A k. 


m m • m m ' 

< > < > < > 

d d d 

5.11 . Calculate the longitudinal wave reflection 
from a “point inhomogeneity”: a single particle with a 
different mass mo ^ m, in an otherwise uniform ID chain - 
see Fig. on the right. Analyze the result. 

m m m 0 m 

5.12 .* Use the rotating- wave approximation to analyze the mutual phase locking of two weakly 
coupled self-oscillators with the dissipative nonlinearity, for the cases of: 

(i) direct coordinate coupling, described by Eq. (5.5) of the lecture notes, and 

(ii) linear but otherwise arbitrary coupling of two similar oscillators. 

5.13 .* Extend the second task of the previous problem to the mutual phase locking of N similar 
oscillators. In particular, explore the in-phase mode’s stability for the case of the so-called global 
coupling via a single force F contributed equally by all oscillators. 

5.14 .* Find the condition of non-degenerate parametric excitation in a system of two coupled 
oscillators, described by Eqs. (5) with time-dependent coupling: k — » k{\ + //cos co p t), with co p « Qi + 
Q 2 , and FE - 0 1 » k/m. 

Hint: Assuming the modulation depth //, static coupling tc, and detuning c = co p - (Q 1 + Q 2 ) 
sufficiently small, use the rotating-wave approximation for each of the coupled oscillators. 

5.15 . Show that the cubic nonlinearity of the type aq 3 indeed enables the parametric interaction 
(“four-wave mixing”) of oscillations with incommensurate frequencies related by Eq. (72a). 

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Chapter 6. Rigid Body Motion 

This chapter discusses the motion of rigid bodies, with a focus on their rotation. Some byproduct results 
of this analysis will enable us to discuss, in the end of the chapter, the description of motion of point 
particles in non-inertial reference frames. 

6.1. Angular velocity vector 

Our study of ID waves in the past chapter has prepared us to for a discussion of 3D systems of 
particles. We will start it with a (relatively :-) simple limit when the changes of distances = |rk -rkj 
between particles of the system are negligibly small. Such an abstraction is called the ( absolutely ) rigid 
body, and is a reasonable approximation in many practical problems, including the motion of solids. In 
this model we neglect deformations - that will be the subject of the next two chapters. 

The rigid body approximation reduces the number of degrees of freedom of the system from 3N 
to just 6 - for example, 3 Cartesian coordinates of one point (say, O), and 3 angles of the system rotation 
about 3 mutually perpendicular axes passing through this point. (An alternative way to arrive at the 
same number 6 is to consider 3 points of the body, which uniquely define its position. If movable 
independently, the points would have 9 degrees of freedom, but since 3 distances r^- between them are 
now fixed, the resulting 3 constraints reduce the number of degrees of freedom to 6.) 

Let us show that an arbitrary elementary displacement of such a rigid body may be always 
considered as a sum of a translational motion and a rotation. Consider a “moving” reference frame, 
firmly bound to the body, and an arbitrary vector A - see Fig. 1 . 

The vector may be represented by its Cartesian components Aj in that reference frame: 

A = i>,n.. (6- 1 ) 


Let us calculate its time derivative in an arbitrary, possibly different (“lab”) frame, taking into account 
that if the body rotates relative to this frame, then the directions of the unit vectors n 7 change in time. 
Hence, we have to differentiate both operands in each product contributing to sum (1): 

dA , 

r in lab 



«, + ±A 


dn . 


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Essential Graduate Physics 

CM: Classical Mechanics 

In this expression, the first sum evidently describes the change of vector A as observed from the moving 
frame. Each of the infinitesimal vectors <7n ; participating in the second sum may be presented by its 
Cartesian components in the moving frame: 


drtj = Y* d( Pjr n r ■ ( 6 - 3 ) 

r = i 

In order to find more about the set of scalar coefficients dq)jj ■, let us scalar-multiply each part of this 
relation by an arbitrary unit vector n, -, and take into account the evident orthogonality condition: 


■n j"=Sj r . 


As a result, we get 

dcp.,. = dn -n . 


Now let us use Eq. (5) to calculate the first differential of Eq. (4): 

dn r ■ n .„ + n ., • dn = dcp rj „ + dcp j7 = 0; 

in particular, 2dn . ■ n . = 2 dcp 7 = 0 . (6.6) 

These relations, valid for any choice of indices j, j\ and j” of the set {1, 2, 3}, mean that the matrix of 
elements d(p u ' is antisymmetric; in other words, there are not 9, but just 3 independent coefficients dcpjj-, 
all with j j ’. Hence it is natural to renumber them in a simpler way: dcp u - = - dcpyj = dcpy, where indices 
j, j\ and j” follow in a “correct” order - either {1,2,3}, or {2,3,1}, or {3,1,2}. Now it is easy to check 
(say, just by a component-by-component comparison) that in this new notation, Eq. (3) may be 
presented just as a vector product: 

ch \ j =d(pxn y , 


where dc p is the infinitesimal vector defined by its Cartesian components dcpj (in the moving frame). 
Relation (7) is the basis of all rotation kinematics. Using it, Eq. (2) may be rewritten as 

dA | _ dA 

in lab 


Z . d(0 UA I . , UW 

dA i 

_ r/tp 




In order to interpret the physical sense of vector co, let us apply Eq. (8) to the particular case when A is 
the radius-vector r of a point of the body, and the lab frame is selected in a special way: its origin moves 
with the same velocity as that of the moving frame in the particular instant under consideration. In this 
case the first term in the right-hand part of Eq. (8) is zero, and we get 



in special lab frame 

coxr , 


were vector r is the same in both frames. According to the vector product definition, the particle 
velocity described by this formula has a direction perpendicular to vectors co and r (Fig. 2), and 
magnitude corsmO . As Fig. 2 shows, this expression may be rewritten as cop, where p = rsin# is the 
distance from the line that is parallel to vector co and passes through point O. This is of course just the 
pure rotation about that line (called the instantaneous axis of rotation), with angular velocity co. Since, 
according to Eqs. (3) and (8), the angular velocity vector co is defined by the time evolution of the 
moving frame alone, it is the same for all points r, i.e. for the rigid body as a whole. Note that nothing in 



in time 

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our calculations forbids not only the magnitude but also the direction of vector 00 , and thus of the 
instantaneous axis of rotation, to change in time (and in many cases it does); hence the name. 

Fig. 6.2. Instantaneous axis of rotation. 

Now let us generalize our result a step further, considering two laboratory reference frames that 
do not rotate versus each other: one arbitrary, and another one selected in the special way described 
above, so that for it Eq. (9) is valid in it. Since their relative motion of these two reference frames is 
purely translational, we can use the simple velocity addition rule given by Eq. (1.8) to write 


where r is the radius-vector of a point is measured in the body-bound (“moving”) frame O. 


in lab V 0 


= V 

in special lab frame O 

• , , + o) x r, 

m lab ’ 


6.2. Inertia tensor 

Since the dynamics of each point of a rigid body is strongly constrained by conditions = 
const, this is one of the most important fields of application of the Lagrangian formalism that was 
discussed in Chapter 2. The first thing we need to know for using this approach is the kinetic energy of 
the body in an inertial reference frame. It is just the sum of kinetic energies of all its points, so that we 
can use Eq. (10) to write: 1 

7 ’ = Zy v2 =Ey( v o + 0)xr ) 2 = Zy v o + X mv o’(° )xr ) + Iy(° )xr ) 2 ’ C 6 - 11 ) 

Let us apply to the right-hand part of Eq. (11) two general vector analysis formulas, listed in the Math 
Appendix: the operand rotation rule MA Eq. (7.6) to the second term, and MA Eq. (7.7b) to the third 
term. The result is 

T = Zy v o +Yj mr ' x ®) + Xy W 1 ' 2 ~ ' r ) 2 ]- ( 6- 12) 

This expression may be further simplified by making a specific choice of point O (from the radius- 
vectors r of all particles are measured), namely if we use for this point the center of mass of the body. 
As was already mentioned in Sec. 3.4, radius-vector R of this point is defined as 

MR = y~y«r, M = 2>, (6.13) 

1 Actually, all symbols for particle masses, coordinates and velocities should carry the particle index, say k, over 

which the summation is carried out. However, for the sake of notation simplicity, this index is just implied. 

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where M is just the total mass of the body. In the reference frame centered as that point, R = 0, so that 
in that frame the second sum in Eq. (12) vanishes, so that the kinetic energy is a sum of two terms: 

T = T +T 

* ± tran 1 ± rot ’ 

T = —V 2 

± tran ^ r ' 

-O' 1 *) 2 ], 


where V = dR/dt is the center-of-mass velocity in our inertial reference frame, and all particle positions 
r have to be measured in the center-of-mass frame. Since the angular velocity vector co is common for 
all points of a rigid body, it is more convenient to rewrite the rotational energy in a form in which the 
summation over the components of this vector is clearly separated from the summation over the points 
of the body: 

1 Vr 
- 2 

z j,f = i 


where the 3x3 matrix with elements 

/.„• =X /,, ( r ' <) V • w) 

is called the inertia tensor of the body. 


Actually, the term “tensor” for the matrix has to be justified, because in physics this name 
implies a certain reference-frame-independent notion, so that its elements have to obey certain rules at 
the transfer between reference frames. In order to show that the inertia tensor deserves its title, let us 
calculate another key quantity, the total angular momentum L of the same body. 2 Summing up the 
angular momenta of each particle, defined by Eq. (1.31), and using Eq. (10) again, in our inertial 
reference frame we get 

L = 2^r x P = ^mr x v = ^mr x (v G + «xr) = ^mr x v 0 +2fflrx(o)xr) . (6.17) 

We see that the momentum may be presented as a sum of two terms. The first one, 

L 0 = ^mrx v 0 =Mx v 0 , (6.18) 

describes possible rotation of the center of mass about the inertial frame origin. This term evidently 
vanishes if the moving reference frame’s origin O is positioned at the center of mass. In this case we are 
left with only the second term, which describes the rotation of the body about its center of mass: 

L = L rot = X mr x (<° x r ) • ( 6 - 19 ) 

Using one more vector algebra formula, the “bac minis cab” rule, 3 we may rewrite this expression as 

L = 2^ ni[(or 2 - r(r • to)] . 
Let us spell out an arbitrary Cartesian component of this vector: 


2 Hopefully, there is a little chance of confusion between the angular momentum L (a vector) and its Cartesian 
components Lj (scalars with an index) on one hand, and the Lagrange function L (a scalar without an index) on the 
other hand. 

3 See, e.g., MA Eq. (7.5). 






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moments of 

and angular 
in principal 


cor" —r. 


r r m r 

= X ffl Z®/( r2 ^' - r j r f) 

j '= 1 

( 6 . 21 ) 

Changing the order of summations, and comparing the result with Eq. (16), we see that the angular 
momentum may be conveniently expressed via the same matrix elements /y as the rotational kinetic 

( 6 . 22 ) 

Since L and o> are both legitimate vectors (meaning that they describe physical vectors 
independent on the reference frame choice), their connection, the matrix of elements Ijp, is a legitimate 
tensor. This fact, and the symmetry of the tensor (/y = /y y) , which is evident from its definition (16), 
allow the tensor to be further simplified. In particular, mathematics tells us that by a certain choice of 
the axis orientation, any symmetric tensor may be reduced to a diagonal fonn 

hr = 7 A” 


where, in our case 

Ij = ^m(r 2 -rj)= + r })= Yu m P) ’ 


pj being the distance of the particle from the /-th axis, i.e. the length of the perpendicular dropped from 
the point to that axis. The axes of such special coordinate system are called the principal axes, while 
the diagonal elements Ij given by Eq. (24), the principal moments of inertia of the body. In such a 
special reference frame, Eqs. (15) and (22) are reduced to very simple forms: 



Both these results remind the corresponding relations for the translational motion, 7) ran = MV 1 12 and P = 
MV, with the angular velocity oo replacing the “linear” velocity V, and the tensor of inertia playing the 
role of scalar mass M. However, let me emphasize that even in the specially selected coordinate system, 
with axes pointing in principal directions, the analogy is incomplete, and rotation is generally more 
complex than translation, because the measures of inertia, 7y ; are generally different for each principal 

Let me illustrate this fact on a simple but instructive system of three similar massive particles 
fixed in the vertices of an equilateral triangle (Fig. 3). 

Fig. 6.3. Principal moments of 
inertia: a simple case study. 

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Due to symmetry of the configuration, one of the principal axes has to pass through the center of 
mass O, perpendicular to the plane of the triangle. For the corresponding principal moment of inertia, 
Eq. (24) readily yields / 3 = 3 mp". If we want to express the result in terms of the triangle side a , we may 
notice that due to system’s symmetry, the angle marked in Fig. 3 equals k/6, and from the corresponding 
right triangle, all = /xos(tz/6) = (A 3/2, giving p = aN 3, so that, finally, / 3 = ma 2 . 

Another way to get the same result is to use the following general axis shift theorem, which may 
be rather useful - especially for more complex cases. Let us relate the inertia tensor components /y and 
I’jj calculated in two reference frames - one in the center of mass O, and another one displaced by a 
certain vector d (Fig. 4a), so that for an arbitrary point, r’ = r + d. Plugging this relation into Eq. (16), 
we get 

I ' J r = X 4 r + d ) 2 s jr - ( r j + d j i r r + d r )] (6 27) 

= + 2r • d + d 2 ^jS jr - (r.r. +r-dj, +r r d j + d jd r )\. 

Since in the center-of-mass frame, all sums 'Emr J equal zero, we may use Eq. (16) to finally obtain 

I',r =I ir +M(S ir d 2 -d l d r ). 





In particular, this equation shows that if the shift vector d is perpendicular to one (say, y-th) of the 
principal axes (Fig. 4b), i.e. dj = 0, then Eq. (28) is reduced to a very simple formula: 

I'j = Ij +Md 2 . 







Fig. 6.4. (a) General reference 
frame shift from the center of mass, 
and (b) a shift perpendicular to one 
of the principal axes. 

Returning to the system shown in Fig. 3, let us perform such a shift so that the new (“primed”) 
axis passes through the location of one of the particles, still perpendicular to particles’ plane. Then the 
contribution of that particular mass to the primed moment of inertia vanishes, and /’ 3 = 2 ma". Now, 
returning to the center of mass and applying Eq. (29), we get / 3 = /’ 3 - Mp 2 = Ima 2 - (3m)(aN3 ) 2 = ma 2 , 
i.e. the same result as above. 

The symmetry situation inside the triangle plane is somewhat less evident, so let us start with 
calculating the moments of inertia for the axes shown vertical and horizontal in Fig. 3. From Eq. (24) we 
readily get: 

/, = Itnh 1 + mp 2 = m 

( V 

. a 


6 \ 2 


2 ^) lV3y 


I 2 = 2 m 



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where I have taken into account the fact that the distance h from the center of mass and any side of the 
triangle is h = psin (tt/6) = p/2 = a/2 V3 . We see that/i = h, and mathematics tells us that in this case any 
in-plane axis (passing through the center of mass O ) may be considered as principal, and has the same 
moment of inertia. A rigid body with this property, I\ = h * h, is called the symmetric top. (The last 
direction is called the main principal axis of the system.) 

Despite the name, the situation may be even more symmetric in the so-called spherical tops, i.e. 
highly symmetric systems whose principal moments of inertia are all equal, 


Mathematics says that in this case the moment of inertia for rotation about any axis (but still passing 
through the center of mass) is equal to the same I. Hence Eqs. (25) and (26) are further simplified for 
any direction of vector co: 

L = Io ) , 


thus making the analogy of rotation and translation complete. (As will be discussed in the next section, 
the analogy is also complete if the rotation axis is fixed by external constraints.) 

An evident example of a spherical top is a uniform sphere or spherical shell; a less obvious 
example is a uniform cube - with masses either concentrated in vertices, or uniformly spread over the 
faces, or uniformly distributed over the volume. Again, in this case any axis passing through the center 
of mass is principal, and has the same principal moment of inertia. For a sphere, this is natural; for a 
cube, rather surprising - but may be confirmed by a direct calculation. 

6.3. Fixed-axis rotation 

Now we are well equipped for a discussion of rigid body’s rotational dynamics. The general 
equation of this dynamics is given by Eq. (1.38), which is valid for dynamics of any system of particles 
- either rigidly connected or not: 

L = t , (6.33) 

where x is the net torque of external forces. Let us start exploring this equation from the simplest case 
when the axis of rotation, i.e. the direction of vector co, is fixed by some external constraints. Let us 
direct axis z along this vector; then eo x = ay, = 0. According to Eq. (22), in this case, the z-component of 
the angular momentum, 

L : =I z: m : , (6.34) 

where I- z , though not necessarily one of the principal momenta of inertia, still may be calculated using 
Eq. (24): 

= X m Pz = X m (* 2 + >’ 2 )’ ( 6 - 35 ) 

with p z being the distance of each particle from the rotation axis z. According to Eq. (15), the rotational 
kinetic energy in this case is just 

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Moreover, it is straightforward to use Eqs. (12), (17), and (28) to show that if the rotation axis is 
fixed, Eqs. (34)-(36) are valid even if the axis does not pass through the center of mass - if only 
distances p z are now measured from that axis. (The proof is left for reader’s exercise.) 

As a result, we may not care about other components of vector L , 4 and use just one component 
of Eq. (33), 

L z =t 2 , (6.37) 

because it, when combined with Eq. (34), completely detennines the dynamics of rotation: 

h A = r z , i- e - = A- (6-38) 

where 6 Z is the angle of rotation about the axis, so that co z =6 . Scalar relations (34), (36) and (38), 
describing rotation about a fixed axis, are completely similar to the corresponding formulas of ID 
motion of a single particle, with a> z corresponding to the usual (“linear”) velocity, the angular 
momentum component L z - to the linear momentum, and I z - to particle’s mass. 

The resulting motion about the axis is also frequently similar to that of a single particle. As a 
simple example, let us consider what is called the physical pendulum (Fig. 5) - a rigid body free to rotate 
about a fixed horizontal axis A that does not pass through the center of mass O, in the unifonn gravity 
field g. 

Fig. 6.5. Physical pendulum. The 
fixed (horizontal) rotation axis A is 
perpendicular to the plane of drawing. 

Let us drop a perpendicular from point O to the rotation axis, and call the corresponding vector 1 
(Fig. 5). Then the torque (relative to axis A) of the forces exerted by the axis constraint is zero, and the 
only contribution to the net torque is due to gravity alone: 

*| in ^ = Z r ! in A X F = X ( ! + r ! in O ) X ™ l 1 X g) ' + £ H O X g = Ml X g . (6.39) 

(For the last transition, I have used the facts that point O is the center of mass, and that vectors 1 and g 
are the same for all particles of the body.) This result shows that the torque is directed along the rotation 

4 Note that according to Eq. (22), other Cartesian components of the angular momentum, L x = I xz a> : and L v = I y: (o : 
may be different from zero, and even evolve in time. (Indeed, if axes x and y are fixed in lab frame, I xz and I vz 
may change due to body’s rotation.) The corresponding torques r T <ext) and r v <ext) , which obey Eq. (33), are 
automatically provided by external forces which keep the rotation axis fixed. 

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axis, and its (only) component r z is equal to -MglsmO, where 6 is the angle between vectors 1 and g, i.e. 
the angular deviation of the pendulum from the position of equilibrium. As a result, Eq. (38) takes the 

I a 6 = -Mgl sin 6, (6.40) 

where, Ia is the moment of inertia for rotation about axis A rather about the center of mass. This 
equation is identical to that of the point-mass (sometimes called “mathematical”) pendulum, with the 
small-oscillation frequency 





n = 


l *A ) 

As a sanity check, in the simplest case when the linear size of the body is much smaller than the 
suspension length /, Eq. (35) yields Ia = Ml , and Eq. (41) reduces to the well-familiar formula Q = 

1 /9 

(g/l) for the mathematical pendulum. 

Now let us discuss the situations when a body not only rotates, but also moves as the whole. As 
we already know from our introductory chapter, the total momentum of the body, 

P = ^ m\ = y' i mr = ^-^mr . 


satisfies the 2 nd Newton law in the form (1.30). Using the definition (13) of the center of mass, the 
momentum may be presented as 

V=MR = MV , 


so Eq. (1.30) may be rewritten as 

law of 

MV = F , 


where F is the vector sum of all external forces. This equation shows that the center of mass of the body 
moves exactly as a point particle of mass M, under the effect of the net force F. In many cases this fact 
makes the translational dynamics of a rigid body absolutely similar to that of a point particle. 

The situation becomes more complex if some of the forces contributing to the vector sum F 
depend on rotation of the same body, i.e. if its rotational and translational motions are coupled. Analysis 
of such coupled motion is rather straightforward if the direction of the rotation axis does not change in 
time, and hence Eqs. (35)-(36) are still valid. Possibly the simplest example is a round cylinder (say, a 
wheel) rolling on a surface without slippage (Fig. 6). 



Fig. 6.6. Round cylinder 
rolling over (a) plane surface 
and (b) concave surface. 

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The no-slippage condition may be presented as the requirement of zero net velocity of the 
particular wheel point A that touches the surface - in the reference frame connected to the surface. For 
the simplest case of plane surface (Fig. 6a), the application of Eq. (10) shows that this requirement gives 
the following relation between the angular velocity co of the wheel and the linear velocity V of its 

V + rco = 0. (6.45) 

Such kinematic relations are essentially holonomic constraints, which reduce the number of 
degrees of freedom of the system. For example, without condition (45) the wheel on a plane surface has 
to be considered as a system with two degrees of freedom, so that its total kinetic energy (14) is a 
function of two independent generalized velocities, say V and co : 

T = T, I , n +T„, = A ^V 2 +U\ (6.46) 

Using Eq. (45) we may eliminate, for example, the linear velocity and reduce Eq. (46) to 

T = ^-{cor) 2 +~® 2 = ~Y <X>2 ’ w here I e{ =I + Mr 2 . (6.47) 

This result may be interpreted as the kinetic energy of pure rotation of the wheel about the instantaneous 
axis A, with 7 e f being the moment of inertia about that axis, satisfying Eq. (29). 

Kinematic relations are not always as simple as Eq. (45). For example, if the wheel is rolling on 
a concave surface (Fig. 6b), we need relate the angular velocities of the wheel rotation about its axis O 
(denoted co) and that of its axis’ rotation about the center (9’ of curvature of the surface (Q). A popular 
error here is to write Q = -(r/R)co [WRONG!]. A prudent way to get the correct relation is to note that 
Eq. (45) holds for this situation as well, and on the other hand the same linear velocity of wheel’s center 
may be expressed as V=(R — r)Q. Combining these equations, we get a (not quite evident) relation 

Q = —ox (6.48) 

R — r 

Another famous example of the relation between the translational and rotational motion is given 
by the “sliding ladder” problem (Fig. 7). Let us analyze it for the simplest case of negligible friction, and 
ladder’s thickness small in comparison with its length / . 

Fig. 6.7. Sliding ladder problem. 

In order to use the Lagrangian formalism, we may write the kinetic energy of the ladder as the 
sum (14) of the translational and rotational parts: 

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T = ^-(x 2 +Y 2 )+^cc 2 , (6.49) 

where X and Y are the Cartesian coordinates of its center of mass in an inertial reference frame, and / is 
the moment of inertia for rotation about the z-axis passing through the center of mass. (For the 
uniformly-distributed mass, an elementary integration of Eq. (35) yields / = Ml 712). In the reference 
frame with the center in the corner O, bothX and Y may be simply expressed via angle a : 

X = —cos a, Y = — since. (6.50) 

2 2 

(The easiest way to obtain these relations is to notice that the dashed line in Fig. 7 has slope a and 
length 1/2.) Plugging these expressions into Eq. (49), we get 

T = ^~a\ h^I + M 

r /\ 2 

= - Ml 2 . 


Since the potential energy of the ladder in the gravity field may be also expressed via the same angle, 

/ . 

U = MgY = Mg — sma, 


a may be conveniently used as the (only) generalized coordinate of the system. Even without writing the 
Lagrangian equation of motion for that coordinate explicitly, we may notice that since the Lagrangian 
function (T - U) does not depend on time explicitly, and the kinetic energy (51) is a quadratic- 
homogeneous function of the generalized velocity a , the full mechanical energy, 

E = T + U 

I e f >2 , , / 

— a + Mg— sm a 
2 2 

Mgl ( la 2 


+ sin a , 



is conserved and gives us the first integral of motion. Moreover, Eq. (53) shows that the system’s energy 
(and hence dynamics) is identical to that of a physical pendulum with an unstable fixed point a\ = n/2, 
stable fixed point at ai = -nil, and frequency 

Q = 

f3gV /2 
l 2 l) 


of small oscillations near the latter point. (Of course, that fixed point cannot be reached in the simple 
geometry shown in Fig. 7, where ladder’s hitting the floor would change its equations of motion). 

6.4. Free rotation 

Now let us proceed to more complex case when the rotation axis is not fixed. A good illustration 
of the complexity arising is this case comes from the simplest case of a rigid body left alone, i.e. not 
subjected to external forces and hence its potential energy U is constant. Since in this case, according to 
Eq. (44), the center of mass moves (as measured from any inertial reference frame) with a constant 
velocity, we can always use an convenient inertial reference frame with the center at that point. From 
the point of view of such frame, the body’s motion is a pure rotation, and 7j ran = 0. Hence, the system’s 
Lagrangian equals just the rotational energy (15), which is, first, a quadratic-homogeneous function of 

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components a>j (that may be taken for generalized velocities), and, second, does not depend on time 
explicitly. As we kn ow from Chapter 2, in this case the energy is conserved. For the components of 
vector to in the principal axes, this means 


(6.55) ener gy’ s 

v ' conservation 

Next, as Eq. (33) shows, in the absence of external forces the angular momentum L of the body 
is conserved as well. However, though we can certainly use Eq. (26) to present this fact as 


(6.56) momentum’s 

where n, are the principal axes of inertia, this does not mean that components 0 )j of the angular velocity 
vector to are constant, because the principal axes are fixed relative to the rigid body, and hence may 
rotate with it. 

Before going after these complications, let us briefly mention two conceptually trivial, but 
practically very important, particular cases. The first is a spherical top (7) = h = h = !)■ In this case Eqs. 

(55) and (56) imply that all components of vector co = L //, i.e. both the magnitude and the direction of 
the angular velocity are conserved, for any initial spin. In other words, the body conserves its rotation 
speed and axis direction, as measured in an inertial frame. 

The most obvious example is a spherical planet. For example, our Mother Earth, rotating about 
its axis with angular velocity co = 2zz/(l day) « 7.3xlO' 5 s , keeps its axis at a nearly constant angle of 
23°27’ to the ecliptic pole, i.e. the axis normal to the plane of its motion around the Sun. (In Sec. 6 
below, we will discuss some very slow motions of this axis, due to gravity effects.) 

Spherical tops are also used in the most accurate gyroscopes, usually with gas or magnetic 
suspension in vacuum. If done carefully, such systems may have spectacular stability. For example, the 
gyroscope system of the Gravity Probe B satellite experiment, flown in 2004-2005, was based on quartz 
spheres - round with precision of about 10 nm and covered by superconducting thin films (which have 
enabled their magnetic suspension and SQUID monitoring). The whole system was stable enough to 
measure that the so-called geodetic effect in general relativity (essentially, the space curving by Earth’s 
mass), resulting in the axis precession by just 6.6 arcseconds per year, i.e. with a precession frequency 
of just -10' 1 's' 1 , agrees with theory with a record -0.3% accuracy. 5 

The second simple case is that of the “symmetric top” (I\ = h h), with the initial vector L 
aligned with the main principal axis. In this case, co = L if = const, so that the rotation axis is 
conserved. 6 Such tops, typically in the shape of a flywheel (rotor) supported by a “gimbal” system (Fig. 

8), are broadly used in more common gyroscopes, core parts of automatic guidance systems, for 


L = V I .co ,n , = const , 

t-j j j j 

J = i 

5 Such beautiful experimental physics does not come cheap: the total Gravity Probe B project budget was about 
$750M. Even at this price tag, the declared main goal of the project, an accurate measurement of a more subtle 
relativistic effect, the so-called frame-dragging drift (or “the Schiff precession”), predicted to be about 0.04 arc 
seconds per year, has not been achieved. 

6 This is also true for an asymmetric top, i.e. an arbitrary body (with, say, I x < I 2 < h), but in this case the 
alignment of vector L with axis n 2 , corresponding to the intermediate moment of inertia, is unstable. 

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example, in ships, airplanes, missiles, etc. Even if the ship’s hull wobbles, the suspended gyroscope 
sustains its direction relative to Earth (which is sufficiently inertial for these applications). 7 

Fig. 6.8. Typical gyroscope. (Adapted from .) 

However, in the general case with no such special initial alignment, the dynamics of symmetric 
tops is more complex. In this case, vector L is still conserved, including its direction, but vector © is not. 
Indeed, let us direct axis n 2 perpendicular to the common plane of vectors L and the instantaneous 
direction n 3 of the main principal axis (in Fig. 9, the plane of drawing); then, in that particular instant, 
L 2 = 0. Now let us recall that in a symmetric top, axis n 2 is a principal one. According to Eq. (26) with j 
= 2, the corresponding component an has to be equal to L 2 // 2 , so it vanishes. This means that vector © 
lies in this plane (the common plane of vectors L and n 3 ) as well - see Fig. 9a. 




Fig. 6.9. Free rotation of a symmetric top: (a) 
the general configuration of vectors, and (b) 
calculating the free precession frequency. 

Now consider any point of the body, located on axis n 3 , and hence within plane [n 3 , L]. Since © 
is the instantaneous axis of rotation, according to Eq. (9), the point has instantaneous velocity v = ©xr 
directed normally to that plane. Since this is true for each point of the main axis (besides only one, with 
r = 0 , i.e. the center of mass, which does not move), this axis as a whole has to move perpendicular to 
the common plane of vectors L, ©, and n 3 . Since such conclusion is valid for any moment of time, it 
means that vectors © and n 3 rotate about the space-fixed vector L together, with some angular velocity 
co^k, at each moment staying in one plane. This effect is usually called the free precession (or “torque- 

7 Much more compact (and much less accurate) gyroscopes used, e.g., in smartphones and tablet computers, are 
based on the effect of rotation on oscillator frequency, and implemented as micro-electromechanical systems 
(MEMS) on silicon chip surface - see, e.g., Chapter 22 in V. Kaajakari, Practical MEMS, Small Gear Publishing, 

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free”, or “regular”) precession, and has to be clearly distinguished it from the completely different effect 
of the torque-induced precession which will be discussed in the next section. 

In order to calculate tUp re , let us present the instant vector co as a sum of not its Cartesian 
coordinates (as in Fig. 9a), but rather of two non-orthogonal vectors directed along 113 and L (Fig. 9b): 

m = oj uA n,+o.) wc n L , n L 




It is clear from Fig. 9b that oy ot has the meaning of the angular velocity of body rotation of the body 
about its main principal axis, while <% e is the angular velocity of rotation of that axis about the constant 
direction of vector L, i.e. the frequency of precession. Now the latter frequency may be readily 
calculated from the comparison of two panels of Fig. 9, by noticing that the same angle 6 between 
vectors L and 113 participates in two relations: 

sin# = — = (6.58) 

L m 




depend on the alignment, and vanishes if L is parallel to 113. 8 Note also that if all principal moments of 
inertia are of the same order, co p re is of the same order as the total angular velocity co= |oo| of rotation. 

Now, let us briefly discuss the free precession in the general case of an “asymmetric top”, i.e. a 
body with I\ h ^ h. In this case the effect is more complex because here not only the direction but 
also the magnitude of the instantaneous angular velocity to may evolve in time. If we are only interested 
in the relation between the instantaneous values of coj and Lj, i.e. the “trajectories” of vectors co and L as 
observed from the reference frame { 111 , 112 , 113 } of the principal axes of the body (rather than an explicit 
law of their time evolution), they may be found directly from the conservation laws. (Let me emphasize 
again that vector L, being constant in an inertial frame, generally evolves in the frame rotating with the 
body.) Indeed, Eq. (55) may be understood as the equation of an ellipsoid in Cartesian coordinates {a>\, 
CO 2 , C 0 i), so that for free body, vector to has to stay on the surface of that ellipsoid . 9 On the other hand, 
since the reference frame rotation preserves the length of any vector, the magnitude (but not direction!) 
of vector L is also an integral of motion in the moving frame, and we can write 

l 2 = Z l ) = Z 12 0)2 = const • ( 6 . 60 ) 

7=1 7=1 

Since axis ni is principal, we may use Eq. (26) for j = 1, i.e. L\ = I\CO\, to eliminate co\ from Eq. (58), and 
get a very simple formula 


This result shows that the precession frequency is constant and independent of the alignment of vector L 
with the main principal axis 113 , while the amplitude of this motion (characterized by angle 6 ) does 

8 For Earth, the free precession amplitude it so small (below 1 0 m of linear displacement on the Earth surface) that 
this effect is of the same order as other, irregular motions of the rotation axis, resulting from the turbulent fluid 
flow effects in planet’s interior and its atmosphere. 

9 It is frequently called the Poinsot ellipsoid, after L. Poinsot (1777-1859) who have made several key 
contributions to the rigid body mechanics. 

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Hence the trajectory of vector co follows the closed curve formed by the intersection of two ellipsoids, 
(55) and (60). It is evident that this trajectory is generally “taco-edge-shaped”, i.e. more complex than a 
plane circle but never very complex either. 

The same argument may be repeated for vector L, for whom the first form of Eq. (60) descries a 
sphere, and Eq. (55), another ellipsoid: 

7 ’~ = Z^- i ?= const - («•«) 

J = 1 11 i 

On the other hand, if we are interested in the trajectory of vector co in an inertial frame (in which 
vector L stays still), we may note that the general relation (15) for the same rotational energy T mt may 
also be rewritten as 

7 'n, - 

2 j = 1 /=! 


But according to the Eq. (22), the second sum in the right-hand part is nothing more than Lj, so that 

2 ^ 


This equation shows that for a free body (Trot = const, L = const), even is vector co changes in time, its 
end point should stay within a plane perpendicular to angular momentum L. (Earlier, we have seen that 
for the particular case of the symmetric top - see Fig. 9b, but for an asymmetric top, the trajectory of the 
end point may not be circular.) 

If we are interested not only in the trajectory of vector co, but also its explicit evolution in time, it 
may be calculated using the general Eq. (33) presented in principal components C0j. For that, we have to 
recall that Eq. (33) is only valid in an inertial reference frame, while the frame {ni, m, 113 } may rotate 
with the body and hence is generally not inertial. We may handle this problem by applying to vector L 
the general relation ( 8 ): 




in mov 

+ 0)xL. 


Combining it with Eq. (33), in the moving frame we get 





= T . 




where x is the external torque. In particular, for the principal-axis components Lj, related to components 
C 0 j by Eq. (26), Eq. (65) is reduced to a set of three scalar Euler equations 

Ij.(bj + (/ .„ -I r )co r (o r =T jt 

( 6 . 66 ) 

where the set of indices {j,j’ ,j” } has to follow the usual “right” order - e.g., {1, 2, 3}, etc . 10 

10 These equations are of course valid in the simplest case of the fixed rotation axis as well. For example, if co = 
n z co, i.e. co x = a>y = 0, Eq. (66) is reduced to Eq. (38). 

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In order to get a feeling how do the Euler equations work, let us return to the case of a free 
symmetric top (t\ = T 2 = T 3 = 0, I\ = h * h). In this case, I\~h = 0, so that Eq. (66) with j = 3 yields co 3 
= const, while the equations for j = I and j = 2 take the simple form 

^l=-^pre«2> ^2=^pre^l> (6-67) 

where Q pre is a constant determined by the system parameters and initial conditions: 

^ P re = ® 3 ^7 — • (6-68) 


Obviously, Eqs. (67) have a sinusoidal solution with frequency Q pre , and describe uniform rotation of 
vector co, with that frequency, about the main axis 113. This is just another presentation of the torque-free 
precession analyzed above, this time as observed from the rotating body. Evidently, Q pre is substantially 
different from the frequency r% e (59) of the precession as observed from the lab frame; for example, the 
former frequency vanishes for the spherical top (with 1 \= h = h), while the latter frequency tends to the 
rotation frequency. 

Unfortunately, for the rotation of an asymmetric top (i.e., an arbitrary rigid body), when no 
component <x>j is conserved, the Euler equations (66) are strongly nonlinear even in the absence of the 
external torque, and a discussion of their solutions would take more time than I can afford. 1 1 

6.5. Torque-induced precession 

The dynamics of rotation becomes even more complex in the presence of external forces. Let us 
consider the most important and counter-intuitive effect of torque-induced precession, for the simplest 
case of an axially-symmetric body (which is a particular case of the symmetric top, I\ = h ^ h) rapidly 
spinning about his symmetry axis, and supported at some point A of that axis, that does not coincide 
with the center of mass O - see Fig. 10. Without external forces, such top would retain the direction of 
its rotation axis that would always coincide with the direction of the angular momentum: 

L = /,o) = / 3 <y rot n 3 . (6.69) 

Fig. 6.10. Symmetric top in the gravity field: 
(a) a side view at the system and (b) the top 
view at the evolution of the horizontal 
component of the angular momentum vector. 

11 Such discussion may be found, e.g. in Sec. 37 of L. Landau and E. Lifshitz, Mechanics, 3 rd ed., Butterworth- 
Heinemann, 1976. 

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The uniform gravity field creates bulk-distributed forces that, as we know from the analysis of 
the physical pendulum in Sec. 3, are equivalent to a single force Mg applied in the center of mass - in 
Fig. 10, point O. The torque of the force relative to the support points is 

T = r o|in^ xM g = Mn 3 x g- (6.70) 

Hence the general equation (33) of the angular momentum (valid in the inertial “lab” frame, in which 
point A rests) becomes 



L = Ml n 3 xg. 


Despite the apparent simplicity of this (exact!) equation, its analysis is straightforward only in 
the limit of relatively high rotation velocity oi G t or, alternatively, very small torque. In this limit, we 
may, in the 0 th approximation, still use Eq. (69) for L. Then Eq. (71) shows that vector Lis 
perpendicular to both n 3 (and hence L) and g, i.e. lies within the horizontal plane, and is perpendicular 
to the horizontal component L xy of vector L - see Fig. 10b. Since the magnitude of this vector is 
constant, | L | = mgl sin <9, vector L (and hence the body’s main axis) rotates about the vertical axis with 
angular velocity 




Thus, very counter-intuitively, the fast-rotating top “does not want to” follow the external, 
vertical force and, in addition to fast spinning about the symmetry axis n 3 , also performs a revolution, 
called the torque-induced precession, about the vertical axis. Note that, similarly to the free-precession 
frequency (59), the torque-induced precession frequency (72) does not depend on the initial (and 
sustained) angle 6 . However, the torque-induced precession frequency is inversely (rather than directly) 
proportional to co, and is typically much lower. This relative slowness is also required for the validity of 
our simple theory of this effect. Indeed, in our approximate treatment we have used Eq. (69), i.e. 
neglected precession’s contribution to the angular momentum vector L. This is only possible if the 
contribution is relatively small, Ia>p re « / 3 co mh where / is a certain effective moment of inertia for the 
precession (to be worked out later). Using our result (72), this condition may be rewritten as 

®ro, » 

f MglI V ' 2 



For a body of not too extreme proportions, i.e. with all linear dimensions of the order of certain length /, 
all inertia moments are of the order of Ml ", so that the right-hand part of Eq. (73) is of the order of 
(g/l) , i.e. comparable with the eigenfrequency of the same body as the physical pendulum, i.e. at the 
absence of fast rotation. 

In order to develop a qualitative theory that could take us beyond such approximate treatment, 
the Euler equations (66) may be used, but are not very convenient. A better approach, suggested by the 
same L. Euler, is to introduce a set of three independent angles between the principal axes {ni,n 2 ,n 3 } 
bound to the rigid body, and axes {n T ,n v ,n-} of an inertial reference frame (Fig. 1 1), and then express the 
basic equation (33) of rotation, via these angles. There are several possible options for the definition of 

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such angles; 12 Fig. 1 1 shows the set of Euler angels, most convenient for discussion of fast rotation. As 
one can see at the figure, the first Euler angle, 0, is the usual polar angle measured from axis n z to axis 
113. The second one is the azimuthal angle cp, measured from axis n A to the so-called line of nodes formed 
by the intersection of planes [n v ,n v ] and [111,112]. The last Euler angle, 1//, is measured within plane 
[111,112], from the line of nodes to axis ni. In the simple picture of the force-induced precession of a 
symmetric top, which was derived above, angle 6 is constant, angle y/ changes very rapidly, with the 
rotation velocity ny ot , while angle cp grows with the precession frequency a>p re (72). 

Now we can express the principal-axes components of the instantaneous angular velocity vector, 
a>\, an, and an,, as measured in the lab reference frame, in terms of the Euler angles. It may be easily 
done calculating, from Fig. 11, the contributions to the change of Euler angles to each principal axis, 
and then adding them up. The result is 

&»j = <^sin 6 1 sin ^ + # cosy/', 
a> 2 = (p sin & cosy/ - Osiny/, 
a > 3 = tpcosO + y/. 


These formulas allow the expression of the kinetic energy of rotation (25) and the angular 
momentum components (26) in terms of the generalized coordinates 6 , cp, and y/, and use then powerful 
Lagrangian formalism to derive their equations of motion. This is especially simple to do in the case of 
symmetric tops (with I\ = /?), because plugging Eqs. (74) into Eq. (25) we get an expression, 

r rot = ^{e 2 + (p 2 sin 2 d)+^{(pcosd + y/) 2 , 


which does not include explicitly either cp or y/. (This reflects the fact that for a symmetric top we can 
always select axis ni to coincide with the line of nodes, and hence take y/ = 0 at the considered moment 
of time. Note that this trick does not mean we can take 1 // = 0 , because axis ni, as observed from the 
inertial reference frame, moves!) Now we should not forget that at the torque-induced precession, the 
center of mass moves as well (see Fig. 10), so that according to Eq. (14), the total kinetic energy of the 
body is the sum of two terms, 

12 Of the several choices more convenient in the absence of fast rotation, the most co mm on is the set of so-called 
Tait-Brian angles (called the yaw, pitch, and roll) that are broadly used in airplane and maritime navigation. 



of co via 

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T = T„+T m , T a =^V 2 =^l 2 (e 2 +<p 2 S m 2 o\ (6.76) 

while the potential energy is just 

U = Mgl cos 9 + const . (6.77) 

Now we could readily write the Lagrangian equations of motion for the Euler angles, but it is 
better to immediately notice that according to Eqs. (75)-(77), the Lagrangian function, T - U, does not 
depend explicitly on “cyclic” coordinates cp and i//, so that the corresponding generalized momenta are 

U 1 . o 

p = — = I aV sin" 9 + I 3 {(pcos9 + \f/)cos9 = const, (6.78) 


p = = / 3 {ip cos 9 + \j/) = const, (6.79) 


where, according to Eq. (29), I A = h +Ml~ is just the body’s moment of inertia for rotation about a 
horizontal axis passing through the support point A. According to the last of Eqs. (74), p ¥ is just A,, the 
angular momentum’s component along the rotating axis n 3 . On the other hand, by its definition p (p is L z , 
the same vector L’s component along the static axis z. (Actually, we could foresee in advance the 
conservation of both these components of L, because vector (70) of the external torque is perpendicular 
to both n 3 and n z .) Using these notions, and solving the simple system of linear equations (78)-(79) for 
the angle derivatives, we get 

<P = 

L z - Z 3 cos 9 
I A sin 2 9 

Z, L_ -Z 3 cos 6 

w = — cos <9 . 

/ 3 I A sin 6 

One more conserved quantity in this problem is the full mechanical energy 13 
E = T + U = ^-(<9 2 +<p 2 sin 2 o)+ -^-((pcosO + \f/) 2 +Mglcos9. 



Plugging Eqs. (80) into Eq. (81), we get a first-order differential equation for angle 9, which may be 
presented in the following physically transparent fonn: 

Z-e 2 +£/, r (6>) = £, ^ (L \ T L '^' + A + Mgfcos^ + const. (6.82) 

2 2/ <1 sin“6' 2 / 3 

Thus, similarly to the planetary problems considered in Sec. 3.5, the symmetric top precession 
has been reduced (without any approximations!) to a ID problem of motion of one of its degrees of 
freedom, the polar angle 9 , in an effective potential U t \{ 9), which is the sum of the real potential energy 
U (77) and a contribution from the kinetic energy of motion along two other angles. In the absence of 
rotation about axes n z and n 3 (i.e., L z = Z 3 = 0), Eq. (82) is reduced to the first integral of the equation 
(40) of motion of a physical pendulum. If the rotation is present, then (besides the case of special initial 

13 Indeed, since the Lagrangian does not depend on time explicitly, H= const, and since the full kinetic energy T 
is a quadratic-homogeneous function of the generalized velocities, E = H. 

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conditions when 6(0) = 0 and L z = LJ), 14 the first contribution to U ct ( 6) diverges at 6— > 0 and n, so that 
the effective potential energy has a minimum at some finite value do of the polar angle 6 . 

If the initial angle #(0) equals this i.e. if the initial effective energy is equal to its minimum 
value C/ ef ( <%), the polar angle remains constant through the motion: 6{t) = do. This corresponds to the 
pure torque-induced precession whose angular velocity is given by the first of Eqs. (80): 

«pre =<P = 

L , - L 3 cos d 0 
I A sin 2 d Q 


The condition for finding do, dU e Jdd = 0, is a transcendent algebraic equation that cannot be solved 
analytically for arbitrary parameters. However, in the high spinning speed limit (73), this is possible. 
Indeed, in this limit the potential energy contribution to U e f is small, and we may analyze its effect by 
successive approximations. In the 0 th approximation, i.e. at Mgl = 0, the minimum of U e f is evidently 
achieved at cos 6*0 = LJL 3, giving zero precession frequency (83). In the next, 1 st approximation, we may 
require that at 9= do, the derivative of first term in the right-hand part of Eq. (82) for U e f over cos6*, 
equal to -L Z (L Z - Z,3COs6)//Ashr 6 1 , 15 is cancelled with that of the gravity-induced term, equal to Mgl. This 
immediately yields co lvc = (L- - L 3 cos6l))// / isirr6 , o = Mgl/L 3, so that taking Z3 = (as we may in the 
high spinning speed limit), we recover the simple expression (72). 

The second important result that readily follows from Eq. (82) is the exact expression the 
threshold value of the spinning speed for a vertically rotating top (6= 0, L z = LJ). Indeed, in the limit 6 
— > 0 this expression may be readily simplified: 

U c[ ( 6 ) « const + 

f L 2 


M gl\ 32 


This formula shows that if 03 = L3//3 (i.e. the angular velocity that was called co m{ in the approximate 
theory) is higher than the following threshold value, 

(6.85) angular 


then the coefficient at 6 in Eq. (84) is positive, so that U e f has a stable minimum at do = 0. On the other 
hand, if an, is decreased below a\ h, the fixed point becomes unstable, so that the top falls down. Note 
that if we take I = Ia in condition (73) of the approximate treatment, it acquires a very simple sense: co mt 

» COth- 

Finally, Eqs. (82) give a natural description of one more phenomenon. If the initial energy is 
larger than U e f( 6b), angle 6 oscillates between two classical turning points on both sides of the fixed 
point 6*o. The law and frequency of these oscillations may be found exactly as in Sec. 3.3 - see Eqs. 

(3.27) and (3.28). At an, » o\\„ this motion is a fast rotation of the symmetry axis n 3 of the body about 
its average position performing the slow precession. These oscillations are called nutations, but 

® th = 2 

r M g n A } 



\ 3 7 

14 In that simple case the body continues to rotate about the vertical symmetry axis: 0(t) = 0. Note, however, that 
such motion is stable only if the spinning speed is sufficiently high - see below. 

15 Indeed, the derivative of the fraction \l2I A sm~6 , taken at the point cos 6 = LJL 3, is multiplied by the nominator, 
( L : - L 3 cos6) 2 , which at this point vanishes. 

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physically they are absolutely similar to the free precession that was analyzed in the previous section, 
and the order of magnitude of their frequency is still given by Eq. (59). 

It may be proved that small energy dissipation (not taken into account in our analysis) leads first 
to a decay of nutations, then to a slower drift of the precession angle do to zero and, finally, to a gradual 
decay of the spinning speed ah, until it reaches the threshold (85) and the top falls down. 

6.6. Non-inertial reference frames 

Before moving on to the next chapter, let us use the results of our discussion of rotation 
kinematics in Sec. 1 to complete the analysis of transfer between two reference frames, started in the 
introductory Chapter 1 - see Fig. 1.2. Indeed, the differentiation rule described by Eq. (8) and derived 
for an arbitrary vector A enables us to relate not only radius-vectors, but also the velocities and 
accelerations of a particle as measured in two reference frames: the “lab” frame O ’ (which will be later 
assumed inertial) and the “moving” (possibly rotating) frame O - see Fig. 12. 

Fig. 6.12. General case of transfer 
between two reference frames. 

As this picture shows, even if frame O rotates relative to the lab frame, the radius-vectors are 
still related, at any moment of time, by the simple Eq. (1.7). In the notation of Fig. 12 it reads 

i lab 

= r 

O in lab 

+ r 

i lab 

( 6 . 86 ) 

However, as was discussed in Sec. 1, for velocities the general addition rule is already more complex. In 
order to find it, let us differentiate Eq. (86) over time: 

d , i d 

in lab 

in lab ' 


The left-hand part of this relation is evidently particle’s velocity as measured in the lab frame, and the 
first term in the right-hand part of Eq. (87) is the velocity of point O, as measured in the same frame. 
The last term is more complex: we need to differentiate vector r that connects point O with the particle 
(Fig. 12), considering how its evolution looks from the lab frame. Due to the possible mutual rotation of 
frames O and O’, that term may not be zero even if the particle does not move relative to frame O. 

Fortunately, we have already derived the general Eq. (8) to analyze situations exactly like this 
one. Taking A = r, we may apply it to the last term of Eq. (87), to get 

of velocity 

V|inlab = V o|inlab + (V + (0 X f), 

( 6 . 88 ) 

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where co is the instantaneous angular velocity of an imaginary rigid body connected to the moving 
reference frame (or we may say, of the frame as such), an v is dr/dt, as measured in the moving frame O, 
(Here and later in this section, all vectors without indices imply their observation from the moving 
frame.) Relation (88), on one hand, is a natural generalization of Eq. (10) for v * 0; on the other hand, if 
co = 0, it is reduced to simple Eq. (1.8) for the translational motion of frame O. 

Now, in order to calculate acceleration, me may just repeat the trick: differentiate Eq. (88) over 
time, and then use Eq. (8) again, now for vector A = v + coxr. The result is 

a| inlab =a 0 | inlab +4(v + o)xr) + cox(v + «xr). (6.89) 


Carrying out the differentiation in the second term, we finally get the goal equation, 

a | in iab =a 0 | inlab +a + d)xr + 2(ox v + tox((oxr), 


where a is particle’s acceleration, as measured in the moving frame. Evidently, Eq. (90) is a natural 
generalization of the simple Eq. (1.9) to the rotating frame case. 

Now let the lab frame O ’ be inertial; then the 2 nd Newton law for a particle of mass m is 

Hinlab= F > (6-91) 

where F is the vector sum of all forces action on the particle. This is simple and clear; however, in many 
cases it is much more convenient to work in a non-inertial reference frames. For example, describing 
most phenomena on Earth’s surface, in is rather inconvenient to use a reference frame resting on the Sun 
(or in the galactic center, etc.). In order to understand what we should pay for the convenience of using 
the moving frame, we may combine Eqs. (90) and (91) to write 

ma = F — ma Q | in lab - nm x(wxr) - 2 /ho) x v - nm x r . 


This result may be interpreted in the following way: if we want to use the 2 nd Newton law’s analog in a 
non-inertial reference frame, we have to add, to the real net force F acting on a particle, four pseudo- 
force terms, called inertial forces, all proportional to particle’s mass. Let us analyze them, while always 
remembering that these are just mathematical terms, not real forces. (In particular, it would be futile to 
seek for the 3 ld Newton law’s counterpart for an inertial force.) 

The first term, -mao\ m iab, is the only one not related to rotation, and is well known from the 
undergraduate mechanics. (Let me hope the reader remembers all these weight-in-the-moving-elevator 
problems.) Despite its simplicity, this term has subtle and interesting consequences. As an example, let 
us consider a planet, such as our Earth, orbiting a star and also rotating about its own axis - see Fig. 13. 
The bulk-distributed gravity forces, acting on a planet from its star, are not quite uniform, because they 
obey the Hr gravity law (1.16a), and hence are equivalent to a single force applied to a points slightly 
offset from the planet’s center of mass O toward the star. For a spherically-symmetric planet, points O 
and A would be exactly aligned with the direction toward the star. However, real planets are not 
absolutely rigid, so that, due to the centrifugal “force” (to be discussed shortly), their rotation about their 
own axis makes them slightly elliptic - see Fig. 13. (For our Earth, this equatorial bulge is about 10 km 
in each direction.) As a result, the net gravity force does create a small torque relative to the center of 
mass O. On the other hand, repeating all the arguments of this section for a body (rather than a point), 
we may see that, in the reference frame moving with the planet, the inertial “force” -Mao (which is of 




2 nd Newton 
law in non- 

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course equal to the total gravity force and directed from the star) is applied exactly to the center of mass 
and does not create a torque. As a result, this pair of forces creates a torque x perpendicular to both the 
direction toward the star and the vector connecting points O and A. (In Fig. 13, the torque vector is 
perpendicular to the plane of drawing). If angle 8 between the planet’s “polar” axis of rotation and the 
direction towards the star was fixed, then, as we have seen in the previous section, this torque would 
induce a slow axis precession about that direction. Flowever, as a result of orbital motion, angle 8 
oscillates in time much faster (once a year) between values (nil + s) and (id 2 - <s), where s is the axis 
tilt, i.e. angle between the polar axis (direction of vectors L and (D ml ) and the normal to the ecliptic plane 
of the planet’s orbit. (For the Earth, £•« 23.4°.) A straightforward averaging over these fast oscillations 16 
shows that the torque leads to the polar axis precession about the axis perpendicular to the ecliptic 
plane, keeping angle s constant. For the Earth, the period, T pve = 2 idcOp K , of this precession of the 
equinoxes (or “precession of the equator”), corrected to the substantial effect of Moon’s gravity, is close 
to 26,000 years. 

Fig. 6.13. Axial precession of 
a planet (with the equatorial 
bulge and the force line offset 
strongly exaggerated). 

Returning to Eq. (92), the direction of the second term of its right-hand part, F c = -mcox(coxr), 
called the centrifugal force, is always perpendicular to, and directed out of the instantaneous rotation 
axis - see Fig. 14. 



Indeed, vector ooxr is perpendicular to both co and r (in Fig. 14, normal to the picture plane and 
directed from the reader) and has magnitude nrsin# = cop, where p is the distance of the particle from 
the rotation axis. Hence the outer vector product, with the account of the minus sign, is normal to the 
rotation axis co, directed out from the axis, and equal to co" rsin# = or p. The “centrifugal force” is of 

16 Details of this calculation may be found, e.g., in Sec. 5.8 of the textbook by H. Goldstein, C. Poole, and J. 
Safko, Classical Mechanics, 3 rd ed., Addison Wesley, 2002. 

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course just the result of the fact that the centripetal acceleration of p, explicit in the inertial reference 
frame, disappears in the rotating frame. For a typical location of the Earth {p - Re ~ 6xl0 6 m), with its 
angular velocity coe ~ 10" 4 s' 1 , the acceleration is rather considerable, of the order of 3 cm/s 2 , i.e. -0.003 
g, and is responsible, in particular, for the largest part of the equatorial bulge mentioned above. 

As an example of using the centrifugal “force” concept, let us return again to our “testbed” 
problem on the bead sliding along a rotating ring - see Figs. 1.5 and 2.1. In the non-inertial reference 
frame attached to the ring, we have to add, to real forces mg and N acting on the bead, the horizontal 
centrifugal “force” 17 directed out of the rotation axis, with magnitude mco p. In the notations of Fig. 2.1, 
its component tangential to the ring equals mofpco s0 = mofRsinOcos0 , and hence the Cartesian 
component of Eq. (92) along this direction is 

ma = -mg sin O + marR sin 6 cos 0 . (6.93) 

With a = RO , this gives us the equation of motion equivalent to Eq. (2.25), which had been derived in 
Sec. 2.2 (in the inertial frame) using the Lagrangian formalism. 

The third term in the right-hand part of Eq. (92) is the so-called Coriolis force , 18 which exists 
only if the particle moves in the rotating reference frame. Its physical sense may be understood by 
considering a projectile fired horizontally, say from the North Pole. From the point of view of the Earth- 
based observer, it will a subject of an additional Coriolis force Fc = -2m cox v, directed westward, with 
magnitude Iiticoev, where v is the main, southward component of the velocity. This force would cause 
the westward acceleration a = 2coev, and the resulting eastward deviation growing with time as d = at'l 2 
= (OEVt - see Fig. 15. (This fonnula is exact only if d is much smaller than the distance r = vt passed by 
the projectile.) On the other hand, from the point of view of the inertial-frame observer, the projectile 
trajectory in the horizontal plane is a straight line, but during the flight time t, the Earth surface slips 
eastward from under the trajectory by distance d = r<p= (vt){oj\_t) = opvt 2 where (p = opt is the azimuthal 
angle of the Earth rotation during the flight). Thus, both approaches give the same result. 

Fig. 6.15. Trajectory of a projectile fired 
horizontally from the North Pole, from the 
point of view of an Earth-bound observer 
looking down. Circles show parallels, 
straight lines mark meridians. 

Hence, the Coriolis “force” is just a fancy (but frequently very convenient) way of description of 
a purely geometric effect pertinent to rotation, from the point of view of the observer participating in it. 
This force is responsible, in particular, for the higher right banks of rivers in the Northern hemisphere, 
regardless of the direction of their flow - see Fig. 16. Despite the smallness of the Coriolis force (for a 

17 For this problem, all other inertial “forces”, besides the Coriolis force (see below) vanish, while the latter force 
is directed perpendicular to the ring and does not affect the bead’s motion along it. 

18 Named after G.-G. Coriolis (1792-1843), who is also credited for the first unambiguous definitions of 
mechanical work and kinetic energy. 



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2 2 5 

typical velocity of the water in a river, v ~ 1 m/s, it is equivalent to acceleration ac ~ 10"" cm/s ~ 10" 
g), its multi-century effects may be rather prominent. 19 

Fig. 6.16. Coriolis “forces” due to 
Earth’s rotation, in the Northern 

The last, fourth tenn of Eq. (92), - nm x r , exists only when the rotation frequency changes in 
time, and may be interpreted as a local-position-specific addition to the first term. 

Equation (92), derived above from the Newton equation (91), may be alternatively obtained from 
the Lagrangian approach, which also gives some important insights on energy at rotation. Let us use Eq. 
(88) to present the kinetic energy of the particle in an inertial frame in terms of v and r measured in a 
rotating frame: 

T = y[ v o|inia b +(v + o)xr)] 2 , (6.94) 

and use this expression to calculate the Lagrangian function. For the relatively simple case of particle 
motion in the field of potential forces, measured from a reference frame that perfonns pure rotation (so 
that Vo|in lab = 0) with a constant angular velocity od, the result is 

L = T-U = yv 2 +mv-(toxr) + -^-(toxr) 2 -U = ~ y2 + m\ • (e> x r) - C/ ef , (6.95) 

where the effective potential energy, 20 

U ef =U-j{<oxr) 2 , (6.96) 

is just the sum of the real potential energy U of the particle and the so-called centrifugal potential 
energy associated with the centrifugal “inertial force”: 

19 The same force causes also the counter-clockwise circulation inside our infamous “Nor’easter” storms, in 
which velocity v, caused by lower atmospheric pressure in the middle of the cyclone, is directed toward its center. 

20 Note again the difference between the negative sign before the (always positive) second term, and the positive 
sign before the similar positive second term in Eq. (3.44). As was already discussed in Chapter 3, this difference 
hinges on different background physics: in the planetary problem, the angular momentum (and hence its 
component L z ) is fixed, while the corresponding angular velocity (p is not. On the opposite, in our current 
discussion, the angular velocity to (of the reference frame) is fixed, i.e. is independent on particle’s motion. 

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F = -mo x (w x r) = -V 

m / \2 

(w x r I 

2 V ' 


Of course, the Lagrangian equations of motion derived from Eq. (95), considering the Cartesian 
components of r and v as generalized coordinates and velocities, coincide with Eq. (92) (with a 0 |i n i a b = 

(0=0, and F = -VC), but it is very informative to have a look at a by-product of this derivation, the 
generalized momentum corresponding to particle’s coordinate r as measured in the rotating reference 
frame, 21 


(6.98) momentum 
at rotation 

According to Eq. (88), with Volin tab = 0, the expression in parentheses is just mv[ m i a t>. However, from the 
point of view of the moving frame, i.e. not knowing about the physical sense of vector p = m\ |i n tab, we 
would have a reason to speak about two different momenta of the same particle, the so-called kinetic 
momentum p = m\ and the canonical momentum p = p + moxr. 22 

Now let us calculate the Hamiltonian function H and energy E as functions of the same moving- 
frame variables: 

dL / \ 

p = — = /?z(v + to x r j . 


H = ^ Vj -L=p \-L = mv • (v + to x r) - 

/=i dv 

m 2 

—v + m\ • (to x r) - C, 



+ C ef , (6.99) 

E = T + U = + m\ ■ (g> x r) + -y (to x r) 7 + U = y v 2 + U e{ + m\ • (to x r) + m(o x r) 2 . (6. 100) 

These expressions clearly show that E and H are not equal. In hindsight, this is not surprising, because 
the kinetic energy (94), expressed in the moving-frame variables, includes a term linear in v, and hence 
is not a quadratic-homogeneous function of this generalized velocity. The difference of these functions 
may be presented as 

E -H = mv - ((0 x r) + m{p x r) 2 = m(\ + g>x r)-(o)x r) = mv| 1Illab • (toxr) . (6.101) 

Now using the operand rotation rule again, we may transform this expression into a even simpler form: 23 

E and H 

(6.102) at rotation 

Let us evaluate this difference for our testbed problem - see Fig. 2.1. In this case, vector to is 

aligned with axis z, so that of all Cartesian components of vector L, only component L : is important for 

2 2 2 

the scalar product (102). This component evidently equals ! z a>= mp co= mcoR sin 6, so that 

E-H = ma> 2 R 2 sin 2 0, (6.103) 

E - H = to • (r 

x m\\ 

i lab 

) = to • (r x p) = to • L| 

in lab 

21 8L/d\ is just a shorthand for a vector with Cartesian components dL/dvj. In a different language, this is the 
gradient of L in the velocity space. 

22 A very similar situation arises at the motion of a particle with electric charge q in magnetic field B. In that case 
the role of the additional term p - p = maxr is played by product qA, where A is the vector-potential of the field 
(B = VxA) - see, e.g., EM Sec. 9.7, and in particular Eqs. (9.183) and (9.192). 

23 Note that by definition (1.36), angular momenta L of particles merely add up. As a result, Eq. (102) is valid for 
an arbitrary system of particles. 

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i.e. the same result that follows from the direct subtraction of Eqs. (2.40) and (2.41). 

The last form of Eq. (99) shows that in the rotating frame, the Hamiltonian function of a particle 
has a very simple physical sense. It is conserved, and hence may serve as an integral of motion, in many 
important situations when L, and hence E, are not - our testbed problem is again a very good example. 

6.1. Exercise problems 

6.1 . Calculate the principal moments of inertia for the following rigid bodies: 

(i) a thin, plane round hoop, 

(ii) a flat uniform round disk, 

(iii) a thin spherical shell, and 

(iv) a uniform solid sphere. 

Compare the results assuming that all the bodies have the same radius R and mass M. 

6.2 . Calculate the principal moments of inertia for the following rigid bodies (see Fig. below): 

(i) an equilateral triangle made of thin rods with a uniform linear mass density //, 

(ii) a thin plate in the shape of an equilateral triangle, with a unifonn areal mass density cr, and 

(iii) a tetrahedral pyramid made of a heavy material with a unifonn bulk mass density p . 

Assuming that the total mass of the three bodies is the same, compare the results and give an 
interpretation of their difference. 

6.3 . Prove that Eqs. (34)-(36) are valid for rotation about a fixed axis, even if it does not pass 
through the center of mass, if all distances p : are measured from that axis. 

6.4 . The end of a unifonn, thin, heavy rod of length 21 and mass m, 
initially at rest, is hit by a bullet of mass m\ flying with velocity vo, which gets 
stuck in the stick - see Fig. on the right. Use two different approaches to calculate 
the velocity of the opposite end of the rod right after the collision. 

m' I 

v„ 1 


< > 


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6.5. A uniform ball is placed on a horizontal plane, while rotating with an angular velocity co o, 
but having no initial linear velocity. Calculate the angular velocity after ball’s slippage stops, assuming 
the usual simple approximation of the kinetic friction force: Ff = juN, where A is a pressure between the 
surfaces, and // is a velocity-independent coefficient. 

6.6 . A body may rotate about fixed horizontal axis A - see Fig. 5. Find the frequency of its small 
oscillations, in a uniform gravity field, as a function of distance / of the axis from body’s center of mass 
O, and analyze the result. 

6.1 . A thin unifonn bar of mass M and length / is hung on a light thread of 
length / ’ (like a “chime” bell - see Fig. on the right). Find: 

(i) the equations of motion of the system (within the plane of drawing); 

(ii) the eigenfrequencies of small oscillations near the equilibrium; 

(iii) the distribution coefficients for each oscillation mode. 

Sketch the oscillation modes for the particular case 1 = 1’. 







6.8 . A solid, uniform, round cylinder of mass M can roll, 
without slipping, over a concave, round cylindrical surface of a block 
of mass M\ in a uniform gravity field - see Fig. on the right. The 
block can slide without friction on a horizontal surface. Using the 
Lagrangian formalism, 

(i) find the frequency of small oscillations of the system near 
the equilibrium, and 

(ii) sketch the oscillation mode for the particular case M’ = M,R’ = 2 R. 

6.9 . A uniform solid hemisphere of radius R is placed on a 
horizontal plane - see Fig. on the right. Find the frequency of its small 
oscillations within a vertical plane, for two ultimate cases: 

(i) there is no friction between the hemisphere and plane surfaces, 

(ii) the static friction is so strong that there is no slippage between 
these surfaces. 

6.10 . For the “sliding ladder” problem started in Sec. 3 (see Fig. 7), find the critical value a c of 
angle a at which the ladder loses contact with the vertical wall, assuming that it starts sliding from the 
vertical position with a negligible initial velocity. 

6.1 1 . A rigid, straight, uniform rod of length /, with the lower end on a pivot, falls 
in a uniform gravity field - see Fig. on the right. Neglecting friction, calculate the 
distribution of the bending torque r along its length, and analyze the result. 

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6.12 . Six similar, uniform rods of length / and mass m are connected by 
light joints so that they may rotate, without friction, versus each other, forming 
a planar polygon. Initially, the polygon was at rest, and had the correct hexagon 
shape - see Fig. on the right. Suddenly, an external force F is applied to the 
middle of one rod, in the direction of hexagon’s symmetry center. Calculate the 
accelerations: of the rod to which the force is applied (a), and of the opposite 
rod ( a ’), immediately after the application of the force. 

6.13 . A rectangular cuboid (parallelepiped) with sides a\, 02 , and « 3 , 
made of a material with constant density p, is rotated, with a constant 
angular velocity co, about one of its space diagonals - see Fig. on the right. 
Calculate the torque x necessary to sustain such rotation. 

6.14. One end of a light shaft of length / is firmly 
attached to the center of a uniform solid disk of radius R « l 
and mass M, whole plane is perpendicular to the shaft. Another 
end of the shaft is attached to a vertical axis (see Fig. on the 
right) so that the shaft may rotate about the axis without friction. 

The disk rolls, without slippage, over a horizontal surface, so 
that the whole system rotates about the vertical axis with a constant angular velocity co. Calculate the 
(vertical) supporting force exerted on the disk by the surface. 

6.15 . An air-filled balloon is placed inside a container filled with water that moves in space, in a 
negligible gravity field. Suddenly, force F is applied to the container, pointing in a certain direction. 
What direction would the balloon move relative to the container? 

6.16 . Calculate the height of solar tides on a large ocean, using the following simplifying 
assumptions: the tide period (V 2 of Earth's day) is much longer than the period of all ocean waves, the 
Earth (of mass Me) is a sphere of radius Re, and its distance r s from the Sun (of mass Ms) is constant and 
much larger than /fi . 

6.17 / A satellite is on a circular orbit, of radius R, around the Earth. 

(i) Write the equations of motion of a small body as observed from the satellite, and simplify 
them for the case when body’s motion is limited to a close vicinity of the satellite. 

(ii) Use the equations to prove that at negligible external forces (in particular, negligible 
gravitational attraction to the satellite) the body may be placed on an elliptical trajectory around 
satellite’s center of mass, within its plane of rotation about Earth. Calculate the ellipse’s orientation and 

6.18 .* A non-spherical shape of an artificial satellite may ensure its stable angular orientation 
relative to Earth’s surface, advantageous for many practical goals. Modeling the satellite as a strongly 
elongated, axially- symmetric body, moving around the Earth on a circular orbit of radius R , find its 

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stable orientation, and analyze possible small oscillations of the satellite’s symmetry 
equilibrium position. 

6.19 . A coin of radius or radius r is rolled, with velocity V, on a horizontal 
surface without slippage. What should be coin's tilt angle 6 (see Fig. on the right) for 
it to roll on a circle of radius R» r ? Modeling the coin as a very thin, uniform disk, 
and assuming that angle 6 is small, solve this problem in: 

6.20 . Two planets are on the circular orbit around their common center of mass. Calculate the 
effective potential energy of a much lighter mass (say, a spaceship) rotating with the same angular 
velocity, on the line connecting the planets. Sketch the plot of function the radial dependence of U e f and 
find out the number of so-called Lagrange points is which the potential energy has local maxima. 
Calculate their position approximately in the limit when one of the planets is much more massive than 
the other one. 

6.21 . A small body is dropped down to the surface of Earth from height h « Re, without initial 
velocity. Calculate the magnitude and direction of its deviation from the vertical, due to the Earth 
rotation. Estimate the effect’s magnitude for a body dropped from the Empire State Building. 

6.22 . Use Eq. (94) to calculate the generalized momentum and derive the Lagrange equation of 
motion of a particle, considering L a function of r and v as measured in a non-inertial but non-rotating 
reference frame. 

axis around this 

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Chapter 7. Deformations and Elasticity 

The objective of this chapter is a brief discussion of small deformations of 3D continuous media, with a 
focus on elastic properties of solids. The reader will see that deformation of solids is nontrivial even in 
the absence of motion, so that several key problems of statics will need to be discussed before 
proceeding to such dynamic phenomena as elastic waves in infinite media and thin rods. 

7.1. Strain 

Rigid bodies discussed in the previous chapter are just a particular case of continuous media. As 
has already been mentioned, these are systems of particles so close to each other that the system 
discreteness may be neglected, so that the particle displacement q may be considered as a continuous 
function of space and time. The subject of this chapter is small deviations from the rigid-body 
approximation discussed in Chapter 6, i.e. small deformations . The deformation smallness allows one to 
consider the displacement vector q as a function of the initial (pre-deformation) position of the particle 
r, and time t- just as was done in the Secs. 5. 3-5. 5 for ID waves. 

The first task of the deformation theory is to exclude from consideration the types of motion 
considered in Chapter 6, namely the translation and rotation unrelated to deformations. This means, first 
of all, the variables describing deformations should not depend on the part of displacement q that does 
not depend on position r (i.e. is common for the whole media), because that part corresponds to a 
translational shift rather than to a deformation (Fig. la). Moreover, even certain non-uniform 
displacements do not contribute to deformation. For example, Eq. (6.7) (with dr replaced with dq to 
comply with our current notation) shows that a small displacement of the type 

^rotation = <Ap Xr , (7. I ) 

where d(p = (odt is an infinitesimal vector common for the whole continuum, corresponds to its rotation 
about the direction of that vector, and has nothing to do with the body deformation (Fig. lb). 

Fig. 7.1. Two types of 
displacement vector 

distributions that are 
unrelated to deformation: 
(a) translation and (b) 

This is why in order to develop an adequate quantitative characterization of deformation, we 
should start with finding suitable appropriate functions of the spatial distribution of displacements, q(r), 
that exist only due to deformations. One of such measures is the change of distance dl= I dr | between 
two close points: 

© 2013-2016 K. Likharev 

Essential Graduate Physics 

CM: Classical Mechanics 

3 3 

(^) 2 | after deformation ' (^) 2 | before deformation = Y^'j + ^7 ^ _ Z ^7 ^ ’ ( 7 - 2 ) 

7= 1 7=1 

where dqj is the / h Cartesian component of the difference dq between the displacements q of the two 
points. If the deformation is small in the sense I f/q I « I dr \ = dl, we may keep in Eq. (2) only the 
terms proportional to the first power of the infinitesimal vector <iq: 

(dl) 2 \ after deformation ' (dl) 2 | before deform a,ion = Y [ 2 <M? 7 + (^ jf ] ~ 2 Y 7 ' ( 7 ’ 3 ) 

7=1 7=1 

Since qj is a function of 3 independent scalar arguments r y , its differential may be presented as 

3 Sq j 

dc ij = Y 

-dr ,. 
1 8r r J 


7=1 — j 

Coefficients dqjldry may be considered as elements of a tensor 1 providing a linear relation between 
vectors dr and dq. Plugging Eq. (4) into Eq. (2), we get 

(dl) 2 

after deformation 

I before deformation 

2 £ Ai dr dr , . 

7^1 5r y 

J J 


A convenience of tensor dqjl dr y for characterizing deformations is that it automatically excludes 
the translation displacement (Fig. la) that is independent of ry Its drawback is that its particular 
components are still affected by the rotation of the body (though the sum (5) is not). Indeed, according 
to the vector product definition, Eq. (1) may be presented in Cartesian coordinates as 

dqj I rotation = [ d <P j* y ~ dtp^j, jyy . , (7.6) 

where £yy is the Levi-Civita symbol 2 equal to (+1) if all indices j,j’, and j” are different and run in a 
“right” order - {1,2, 3}, etc., and (-1) otherwise, so that for any order of non-equal indices, Sjjy = -£jjy. 
Differentiating Eq. (6) over a particular Cartesian coordinate of vector r, and taking into account that 
this partial differentiation (3) is independent of (and hence may be swapped with) the differentiation id) 
over the rotation angle (p , we get the amounts, 


r dqY 

dr ■, 

V J J rotation 

= -e m .d<p r , d 

r dqY 


V J J rotation 

= -£ nr d(p r =S,, r d(p r , 


which may differ from 0. However, notice that the sum of these two differentials equals zero for any t/cp, 
which is possible only if 

r dq r dq ^ 
+ - 

V dr j dr yj 

= 0, for j * j' , 


1 Since both c/q and dr are legitimate physical vectors (whose Cartesian components are properly transformed as 
the transfer between reference frames), the 3x3 matrix with elements dqjdrj’ is indeed a legitimate physical tensor 
- see the discussion in Sec. 6.2. 

2 See, e.g., MA Eq. (13.2). 

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so that the full sum (5), that includes 3 such partial sums, is not affected by rotation - as we already 
know. This is why it is convenient to rewrite Eq. (5) in a mathematically equivalent form 

(. dlf 

after deformation 


[before deformation 

2 Tj s jr dr j dr j” 

j,r = i 


where .sy ■ arc the elements of the so-called symmetrized strain tensor defined as 




(Note that this modification does not affect the diagonal elements: Sjj = dqfdrj .). The advantage of 
symmetrized tensor (9b) over the initial tensor dqfdrj ■ is that according to Eq. (8), at pure rotation all 
elements of the symmetrized strain tensor vanish. 

Now let us discuss the physical meaning of this tensor. At was already mentioned in Sec. 6.2, 
any symmetric tensor may be diagonalized by an appropriate selection of the reference frame axes. In 
such principal axes, .sy ■ = SjjSjy, so that Eq. (4) takes the simple form 

dq, = 

dq j 


dr I =s jJ dr r 


We may use this expression to calculate the change of each side of an infinitesimal cuboid 
(parallelepiped) with sides dqj parallel to the principal axes: 

d rj | after deformation | before deformation dq j Sjj dr. , (7.11) 

and of cuboid’s volume dV= dr\dr 2 dry. 

dV\ -dV 

u y after deformation u v 

before deformation 

= [ [ {dr } 5 jjdrj ) - f] dr j = dV 






Since all our analysis is only valid in the linear approximation in small sy, Eq. (12) is reduced to 


dV\ after deformation ~ dV | be fore deformation ~ dV ^ ,S „ = dV Tr(s), (2.13) 


where Tr {trace) 3 of any matrix (in particular, tensor) is the sum of its diagonal elements; in our case 4 

Tr(s)-Z^- (7-14) 


So, the diagonal components of the tensor characterize medium’s compression/extension; then 
what is the meaning of the off-diagonal components of the tensor? It may be illustrated on the simplest 
example of a purely shear deformation, shown in Fig. 2 (the geometry is assumed to be uniform along 
axis z). In this case, all displacements (assumed small) have just one Cartesian component, in Fig. 2 

3 The traditional European notation for Tr is Sp (from German Spur meaning “trace” or “track”). 

4 Actually, the tensor theory shows that the trace does not depend on the particular choice of the coordinate axes. 

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along axis x: q = n x ay (with a « 1), so that the only nonvanishing component of the initial strain tensor 
dqj/drj ■ is dqJSy = a , and the symmetrized tensor (9b) is 

s = 

A 0 



all 0" 

0 0 . 
0 o y 


Evidently, the change (13) of volume vanishes in this case. Thus, off-diagonal elements of tensor s 
characterize shear deformations. 

Fig. 7.2. Example of a pure shear. 

To conclude this section, let me note that Eq. (9) is only valid in Cartesian coordinates. For the 
solution of some important problems, especially those with a spherical or axial symmetry, it is 
frequently convenient to express six different components of the symmetric strain tensor via three 
components of the displacement vector q in either spherical or cylindrical coordinates. A 
straightforward differentiation, using the definition of such coordinates, 5 yields, in particular, the 
following formulas for the diagonal elements of the tensor in the local mutually orthogonal coordinates 
that are directed along unit vectors - either { n,., n& } or {n p, n (/h n_- } - at the given point: 

(i) in the spherical coordinates: 

„ _ dc lr „ _ q r , 1 dc le 

■v _ a ’ s ee - + 

or r r ov 

s _ q r | q e cos# | \_dq 

<p<p r r sin 0 rsin6* dcp 

<p . 

(ii) in the cylindrical coordinates: 

Sq p 

i dq« 

c h 

s w - — + ■ P 
p p dcp 

s„ = 

oq : 




These expressions, that will be used below for solution of some problems for symmetrical 
geometries, may be a bit counter-intuitive. Indeed, Eq. (16) shows that even for a purely radial, 
spherically-symmetric defonnation, q = n,q(r), diagonal angular components of strain do not vanish: sge 
= s (p( p = q/r. (According to Eq. (17), in cylindrical coordinates, the same effect is exhibited by the only 
angular component of the tensor.) Note, however, that these relations describe a very simple geometric 
effect: the change of the lateral distance rdy « r between two close points with the same distance r 
from a central point, at a small change of r that keeps the angle dy between their radius-vectors r 

5 See, e.g., MA Eqs. (10.1) and (10.7). 

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7.2. Stress 



Now let us discuss the forces that cause deformations. Internal forces acting inside (i.e. between 
arbitrarily defined parts of) a continuous media may be also characterized by a tensor. This stress 
tensor, 6 with elements q;y, relates components of the elementary vector dF of the force acting on an 
elementary area dA of an (possibly, imaginary) interface between two parts of a continuous media with 
elementary vector dA = n dA normal to the area (Fig. 3): 


The usual sign convention here is to take the outer normal <7n, i.e. to direct dA out of “our” part of the 
continuum, i.e. the part on which the calculated force dF is exerted. 

Fig. 7.3. Definition of vectors dA and dF. 


In some cases the stress tensor’s structure is very simple. For example, as will be discussed in 
detail in the next chapter, static or frictionless fluids may only provide a force normal to any surface and 
usually directed toward “our” part of the body, so that 

dF = -Pd A, i.e. <j jr = -Pd jr , 


where scalar P (in most cases positive) is called pressure, and generally depends on both the spatial 
position and time. This type of stress, with P > 0, is frequently called the hydrostatic compression - even 
if it takes place in solids. 

However, in the general case the stress tensor also has off-diagonal terms, which characterize 
shear stress. For example, if the shear strain shown in Fig. 2 is caused by a pair of forces ±F, they create 
internal forces F x n x , with F x > 0 if we speak about the force acting upon a part of the sample below the 
imaginary horizontal interface we are discussing. In order to avoid horizontal acceleration of each 
horizontal slice of the sample, the forces should not depend on y, i.e. F x = const = F. Superficially, it 
may look that this is the only nonvanishing component of the stress tensor is dF x /dA y = FI A = const, so 
that tensor is asymmetric, in contrast to the strain tensor (15) of the same system. Note, however, that 
the pair of forces ±F creates not only the shear stress, but also a nonvanishing rotating torque x = -Fhn z 
= -(dF x !dAy)Ahn : = -( dF x /dA y )Vn z , where V = Ah is sample’s volume. So, if we want to perform a static 
stress experiment, i.e. avoid sample’s rotation, we need to apply some other forces, e.g., a pair of 
vertical forces creating an equal and opposite torque x ’ = (dF y /dA x )Vn z , implying that dF y /dA x = dF x /dA y 
= FI A. As a result, the stress tensor becomes symmetric, and similar in structure to the symmetrized 
strain tensor (15): 

6 It is frequently called the Cauchy stress tensor, partly to honor A.-L. Cauchy (1789-1857) who introduced it, 
and partly to distinguish it from and other possible definitions of the stress tensor, including the 1 st and 2 nd Piola- 
Kirchhoff tensors. (For the infinitesimal deformations discussed in this course, all these notions coincide.) 

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c = 

FJA 0 




In many situations, the body may be stressed not only by forces applied to their surfaces, but also 
by some volume-distributed (bulk) forces dF = fdV, whose certain effective bulk density f. (The most 
evident example of such forces is gravity. If its field is uniform as described by Eq. (1.16b), then f = pg, 
where p is the mass density.) Let us derive the key formula describing the correct summation of the 
surface and bulk forces. For that, consider again an infinitesimal cuboid with sides dr~j parallel to the 
corresponding coordinates axes (Fig. 4) - now not necessarily the principal axes of the stress tensor. 



dF u) + d(dF <j) ) 

Fig. 7.4. Deriving Eq. (23). 

If elements of the tensor do not depend on position, the force dF {J 1 acting on j ’-th face of the 
cuboid is exactly balanced by the equal and opposite force acting on the opposite face, because vectors 
<7A (/ } of these faces are equal and opposite. However, if a u ■ is a function of r, then the net force d(dF {J ') 
does not vanish. Using the expression for to the j ’ th contribution to sum (18), in the first order in dr, the 
/ h components of this vector is 

d(dF] r) ) = d(a ir dA,)=^^dr.,dA, = ^^dV, (7.21) 

J J 1 dr., J 7 dr., 

where cuboid’s volume dV= drjdAj- does not depend on j’. The addition these force components for all 
three pairs of cuboid faces, i.e. the summation of Eqs. (21) for all 3 values of the upper index j\ yields 
the following relation for the / h component of the net force exerted on the cuboid: 

d(dF j ) = Y j d(dF { j i J = Y j 

da , 

/= i 

/=! dr f 

-dV . 


Since any volume may be broken into such infinitesimal cuboids, Eq. (22) shows that the space-varying 
stress is equivalent to a volume-distributed force </F e f = f e f dV, whose effective (not real!) bulk density f e f 
has the following Cartesian components 

(/4=i a<v 

1 ^ dr 

i ’= i or f 


so that in the presence of genuinely bulk forces dF = fdV, densities f e f and f just add up. 

Let us use this addition rule to spell out the 2 nd Newton law for a unit volume of a continuous 

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S q 

P W = f 


+ f . 

Using Eq. (23), the j th Cartesian component of Eq. (24) may be presented as 






This is the key equation of medium’s dynamics, which will be repeatedly used below. 

For solution of some problems, it is also convenient to have a general expression for work SW of 
the stress forces at a virtual defonnation Sq - understood in the same variational sense as the virtual 
displacement Sr in Sec. 2.1. Using the equivalence between the stress forces and the effective bulk 
forces with density f e f, for any volume V of the media we may write 

sw=it c ,wv=Y,l<.f„)j&ijdv= £ J— < 7 - 28 ) 

v j= l v j,j - 1 v ® r j' 

Let us take this integral by parts for a volume so large that deformations Sqj on its surface are negligible. 
Then, swapping the operations of variations and spatial differentiation (just like it was done with the 
time derivative in Sec. 2.1), we get 

SfV = -Y fa,,S^Ld 3 r. 

j,j '= 1 v 



Assuming that tensor oft - is symmetric, we may rewrite this expression as 

SW = — 

j,j '= 1 V 

x dc lj , A 
<j, r 8 — L + (j rj S — - 

V * dr f " 8r rj 

dr r. 


Now, swapping indices j and j ’ in the second expression, we finally get 

jj '= 1 v 

r dq. 5q r A 

— -a H —a 

8r , 11 dr 11 

V J 

d-r = -Y M \n jj .Ss jj .d 3 r. 


j y 

j,j '= i v 

Work of 

where sjj ■ are the components of strain tensor (9b). It is natural to rewrite this important formula as 


and interpret the locally-defined scalar function fivtfr) as the work of stress forces per unit volume, due 
to the small variation of the defonnation. 

7.3. Hooke’s law 

In order to form a complete system of equations describing media dynamics, one needs to 
complement Eq. (25) with an appropriate material equation describing the relation between the stress 
tensor ajj- and the deformation q described (in the small deformation limit) by the strain tensor Sjj\ This 

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relation depends on the medium, and generally may be rather complex. Even leaving alone various 
anisotropic solids (e.g., crystals) and macroscopically-inhomogeneous materials (like ceramics or sand), 
strain typically depends not only on the current value of stress (possibly in a nonlinear way), but also on 
the previous history of stress application. Indeed, if strain exceeds a certain plasticity threshold, atoms 
(or nanocrystals) may slip to their new positions and never come back even if the strain is reduced. As a 
result, deformations become irreversible - see Fig. 5. 

Fig. 7.5. Typical relation between stress 
and strain in solids (schematically). 

Only below the thresholds of nonlinearity and plasticity (which are typically close to each other), 
strain is nearly proportional to stress, i.e. obeys the famous Hooke’s law. 1 However, even in this elastic 
range the law is not quite simple, and even for an isotropic medium is described not by one but by two 
constants, called elastic moduli. The reason for that is that most elastic materials resist the strain 
accompanied by the volume change (say, the hydrostatic compression) differently from how they resist 
the shear deformation. In order to describe this difference, let us first present the symmetrized strain 
tensor (9b) in the mathematically equivalent form 


According to Eq. (13), the traceless tensor in the first parentheses of Eq. (33) does not give any 
contribution to the volume change, e.g., may be used to characterize purely shear deformation, while the 
second one describes the hydrostatic compression alone. Hence we may expect that the stress tensor 
may be presented (again, in the elastic deformation range only!) as 




where K and // are some constants. 7 8 Indeed, experiments show that Hooke’s law in this form is followed, 
at small strain, by all isotropic elastic materials. In accordance with the above discussion, constant // (in 
some texts, denoted as G ) is called the shear modulus, while constant K (sometimes called B ), the bulk 
modulus. Two columns of Table 1 below show the approximate values of these moduli for typical 
representatives of several major classes of materials. 9 

7 Named after R. Hooke (1635-1703) who was first to describe the law in its simplest, ID version. 

8 The inclusion of coefficients 2 and 3 into Eq. (34) is justified by the simplicity of some of its corollaries - see, 
e.g., Eqs. (38) and (43) below. 

9 Since the strain tensor elements, defined by Eq. (5), are dimensionless, while the strain defined by Eq. (18) has 
the dimensionality of pressure (force by unit area), so do the elastic moduli K and p. 

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To better appreciate these values, let us first discuss the physical meaning of K and //, using two 
simple examples of elastic deformation. For that it is convenient first to solve the set of 9 (or rather 6 
different) linear equations (34) for sy. This is easy to do, due to the simple structure of these equations: 
they relate components <J n • and sy with the same indices, besides the involvement of the tensor trace. 
This slight complication may be readily overcome by noticing that according to Eq. (34), 

Tr ( 0 )=Z°> =3 ^ Tr (4 i-e. Tr(s) = ^^ T r(cr). (7.35) 

Plugging this result into Eq. (34) and solving it for sy, we readily get the reciprocal relation, which may 
be presented in a similar form: 

s jf = 

2 M 

<7 , 


Table 7.1. Elastic moduli, density, and sound velocities of a few representative materials (approximate values) 


K (GPa) 

ju (GPa) 

E (GPa) 


p (kg/m 3 ) 

v/( m/s) 

v t (m/s) 

Diamond (a) 








Hardened steel 








Water <b) 








Air (b) 









Averages over crystallographic directions (~10% anisotropy). 

<b) At the so-called ambient conditions (T= 20°C, P= 1 bar = 10‘ 5 Pa). 

Now let us apply Flooke’s law, in the form of Eqs. (34) or (36), to two simple situations in which 
the strain and stress tensors may be found without formulating the exact differential equations of the 
elasticity theory and boundary conditions for them. (That will be the subject of the next section.) The 
first experiment is the hydrostatic compression when the stress tensor is diagonal, and all its diagonal 
components are equal - see Eq. (19). 10 For this case Eq. (36) yields 

JJ 3 K 11 


which means that regardless of the shear modulus, the strain tensor is also diagonal, with all diagonal 
components equal. According to Eqs. (11) and (13), this means that all linear dimensions of the body are 
reduced by a similar fraction, so that its shape is preserved, while the volume is reduced by 

10 It may be proved that such situation may be implemented not only in a fluid with pressure P, but also by 
placing a solid sample of an arbitrary > shape into a compressed fluid. 

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j = 1 




This equation clearly shows the physical sense of the bulk modulus K as the reciprocal compressibility. 

As Table 1 shows, the values of K may be dramatically different for various materials, and that 
even for such “soft stuff’ as water this modulus in actually rather high. For example, even at the bottom 
of the deepest, 10-km ocean well (P ~ 1(T bar « 0.1 GPa), water density increases by just about 5%. As 
a result, in most human-scale experiments, water may be treated as incompressible - a condition that 
will be widely used in the next chapter. Many solids are even much less compressible - see the first two 
rows of Table 1. 

The most compressible media are gases. For a gas, certain background pressure P is necessary 
just for containing it within certain volume V, so that Eq. (38) is only valid for small increments of 
pressure, A P: 

AV _ AP 
V K ' 


Moreover, gas compression also depends on thermodynamic conditions. (For most condensed media, the 
temperature effects are very small.) For example, at ambient conditions most gases are reasonably well 
described by the equation of state for the model called the ideal classical gas: 

~\T1 rj 1 

PV = Nk B T, i.e.P = — . (7.40) 

where N is the number of molecules in volume V, and ks ~ 1.38x10' J/K is the Boltzmann constant. 11 
For a small volume change A Fat constant temperature, this equation gives 


T =const 

Nk B T 


AV = 

. AV 



T =const 

P ' 


Comparing this expression with Eq. (37), we get a remarkably simple result for the isothermal 
compression of gases, 

^Uco„st=^ (7.42) 

which means in particular that the bulk modulus listed in Table 1 is actually valid, at the ambient 
conditions, for almost any gas. Note, however, that the change of thermodynamic conditions (say, from 
isothermal to adiabatic 12 ) may affect gas’ compressibility. 

Now let us consider the second, rather different, fundamental experiment: a pure shear 
deformation shown in Fig. 2. Since the traces of matrices (15) and (20), which describe this experiment, 
are equal to 0, for their off-diagonal elements Eq. (34) gives simply ajp = 2psjf, so that the deformation 
angle a (see Fig. 2) is just 

1 F 

a = . 

p A 


11 For the derivation and detailed discussion ofEq. (40) see, e.g., SM Sec. 3.1 

12 See, e.g., SM Sec. 1.3. 

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Notice that the angle does not depend on thickness h of the sample, though of course the maximal linear 
deformation q x = ah is proportional to the thickness. Naturally, as Table 1 shows, for all fluids (liquids 
and gases) p = 0, because they cannot resist static shear stress. 

However, not all experiments, even the apparently simple ones, involve just either K or //. Let us 
consider stretching a long elastic rod of a small and unifonn cross-section of area A - the so-called 
tensile stress experiment shown in Fig. 6. 13 


F * , \ F 

^~ f) A $ !'•) —► 

^ Fig. 7.6. Tensile stress experiment. 


Though the deformation of the rod near its clamped ends depends on the exact way forces F are applied 
(we will discuss this issue later on), we may expect that over most of its length the tension forces are 
directed virtually along the rod, dF = F z n z , and hence, with the coordinate choice shown in Fig. 6, cr v/ = 
Oyj = 0 for all j, including the diagonal elements a xx and a yy . Moreover, due to the open lateral surfaces, 
on which, evidently, dF x = dF v = 0, there cannot be an internal stress force of any direction, acting on 
any elementary internal boundary parallel to these surfaces. This means that <j zx = a zy = 0. So, of all 
components of the stress tensor only one, <j zz , is not equal to zero, and for a uniform sample, o zz = const 
= FI A. For this case, Eq. (36) shows that the strain tensor is also diagonal, but with different diagonal 



1 1 ^ 

y 9K + 3/U; 



s = s = 

xx yy 



9 K 6/u 



Since the tensile stress is most common in engineering (and physical experiment) practice, both 
combinations of the elastic moduli participating in these two relations have deserved their own names. 
In particular, the constant in Eq. (44) is usually denoted as HE (but in many texts, as 1/7), where E is 
called the Young 's modulus : 





1 1 

9A + 3/? 

i.e. E = 

9 K/u 
3K + p 


As Fig. 6 shows, in the tensile stress geometry s zz = dqjdz = AL/L, so that the Young’s modulus scales 
the linear relation between the relative extension of the rod and the force applied per unit area: 14 

AL _ 1 F 
L ~ E A' 


13 Though the analysis of compression in this situation gives similar results, in practical experiments a strong 
compression may lead to the loss of horizontal stability - the so-called buckling - of the rod. 

14 According to Eq. (47), E may be thought of as the force per unit area, which would double sample’s length, if 
only our theory was valid for deformations that large. 

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The third column of Table 1 shows the values of this modulus for two well-known solids: diamond (with 
the highest kn own value of E of all bulk materials 15 ) and the steel (physically, a solid solution of ~10% 
of carbon in iron) used in construction. Again, for fluids the Young’s modulus vanishes - as it follows 
from Eq. (46) with // = 0. 

I am confident that the reader of these notes has been familiar with Eq. (44), in the form of Eq. 
(47), from his or her undergraduate studies. However, most probably this cannot be said about its 
counterpart, Eq. (45), which shows that at the tensile stress, rod’s cross-section dimension also change. 
This effect is usually characterized by the following dimensionless Poisson ’s ratio : 16 

19 K 





1 J_ 
9 K 3// 



1 3K-2/U 

2 3 K + fi 


According to this formula, for realistic materials with K > 0, /u > 0, values of a may vary from 
(-1) to (+!4) , but for the vast majority of materials, 17 they are between 0 and Vi - see Table 1. The lower 
limit is reached in porous materials like cork whose lateral dimensions almost do not change at the 
tensile stress. Some soft materials like rubber present the opposite case: a ~ Vi. Since according to Eqs. 
(13), (44) and (45), the volume change is 



= s. 

+ 5 , 

+ s. 

1 F 

(1 - 2a), 


such materials virtually do not change their volume at the tensile stress. The ultimate limit of this trend, 
AVIV = 0, is provided by fluids and gases, because their Poisson ratio <j is exactly equal to Vi. (This 
follows from Eq. (48) with p = 0.) However, for most practicable construction materials such as steel 
(see Table 1) the change (49) of volume is as high as -40% of that of the length. 

Due to the clear physical sense of coefficients E and a, they are frequently used as a pair of 
independent elastic moduli, instead of K and // . Solving Eqs. (46) and (48) for K and /u , we get 

K = 


3(1 -2a)’ 


2(l + o-)' 


Using these formulas, the two (equivalent) formulations of Hooke’s law, expressed by Eqs. (34) and 
(36), may be rewritten as 


15 It is probably somewhat higher (up to 2,000 GPa) in such nanostructures as carbon nanotubes and monoatomic 
sheets {graphene), though there is still a substantial uncertainty in experimental values of elastic moduli of these 
structures - see, e.g., C. Lee et al. Science 321 , 5887 (2008) and J.-U. Lee et al.,Nano Lett. 12 , 4444 (2012). 

16 Unfortunately, the dominating tradition is to use for the Poisson ratio the same letter {a) as for the stress tensor 
components, but they may be always distinguished by the presence or absence of component indices. 

17 The only kn own exceptions are certain exotic media with very specific internal microstructure - see, e.g., R. 
Lakes, Science 235 , 1038 (1987) and references therein. 



law with 
E and a 

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of elastic 

The linear relation between the strain and stress tensor allows one to calculate the potential 
energy U of an elastic medium due to its elastic deformation. Indeed, to each infinitesimal part of this 
strain increase, we may apply Eq. (32), with the work SW of the surface forces equal to -SU. Let us 
slowly increase the defonnation from a completely unstrained state (in which we may take U = 0) to a 
certain strained state, in the absence of bulk forces f, keeping the deformation type, i.e. the relation 
between the elements of the stress tensor intact. In this case, all elements of tensor ajj ■ are proportional 
to the same single parameter characterizing the stress (say, the total applied force), and according to 
Hooke’s law, all elements of tensor syj are proportional to that parameter as well. In this case, 
integration over the variation yields the final value 18 

U = ^u{r)d i r, u{r) 




<j .. ,s . 
L*t jj jj 
j,f = i 


Evidently, u(r) may be interpreted as the volume density of the potential energy of the elastic 

7.4, Equilibrium 

Now we are fully equipped to discuss dynamics of elastic deformations, but let us start with 
statics. The static (equilibrium) state may be described by requiring the right-hand part of Eq. (25) to 
vanish. In order to find the elastic defonnation, we need to plug from the Hooke’s law (51a), and 
then express elements Sjy via the displacement distribution - see Eq. (9). For a unifonn material, the 
result is 19 

= o 

2(1 + cr) yTq 2(1 + cr)(l - 2cr) dr f dr 


Taking into account that the first sum in Eq. (53) is just the j th component of V 2 q, while the second sum 
is the / h component of V(V-q), we see that all three equations (53) for three Cartesian components (j = 
1 , 2 and 3) of the deformation vector q, may be conveniently merged into one vector equation 


2(l + o-) 

V 2 q + 


2(l + o-)(l-2o-) 

V(V • q) + f = 0. 


For some applications, it is more convenient to recast this equation to another form, using vector 
identity 20 V 2 q =V(V-q) - Vx(Vxq). The result is 


(l + o-)(l-2 o-) 


— - — V x (V x q) + f = 0. 
2(l + o-) 


It is interesting that in problems without volume-distributed forces (f = 0), the Young’s modulus 
E cancels! Even more fascinating, in this case the equation may be re-written in a form not involving the 

18 For clarity, let me reproduce a similar integration for the ID motion of a particle on a spring. In this case, 8U = 
-SW = -FSx, and if spring’s force is elastic, F = -kx, the integration yields U= kx 2 / 2 = Fx/2. 

19 As follows from Eqs. (50), the coefficient before the first sum in Eq. (53) is just the shear modulus //, while that 
before the second sum is equal to (K + ju/3). 

20 See, e.g.,MAEq. (11.3). 

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Poisson ratio a either. Indeed, acting by operator V on the remaining terms of Eq. (55), we get a 
surprisingly simple equation 

V 2 (V -q) = 0. (7.56) 

A natural question here is how do the elastic moduli affect the deformation distribution if they do 
not participate in the differential equation describing it. The answer is two-fold. If what is fixed at the 
body boundary are deformations, then the moduli are irrelevant, because the defonnation distribution 
through the body does not depend on them. On the other hand, if the boundary conditions fix stress (or a 
combination of stress and strain), then the elastic constants creep into the solution via the recalculation 
of these conditions into the strain. 

As a simple but representative example, let us find the deformation distribution in a (generally, 
thick) spherical shell under the effect of different pressures fixed inside and outside it (Fig. 7a). 

Due to the spherical symmetry of the problem, the deformation is obviously spherically- 
symmetric and radial, q = q(r) n r , i.e. is completely described by one scalar function q(r). Since the curl 
of such a radial vector field is zero, 21 Eq. (55) is reduced to 

V(V -q) = 0, 


This equation means that the divergence of function q(r ) is constant within the shell. In spherical 
coordinates this means 22 

(r 2 ^) = const. (7.58) 

r dr 

Naming this constant 3 a (with the numerical factor chosen for later notation convenience), and 
integrating Eq. (58), we get its solution, 

q(r) = ar + \, (7.59) 


that also includes another integration constant b . 

To complete the analysis, we have to determine constants a and b from the boundary conditions. 
According to Eq. (19), 

21 If this is not immediately evident, have a look at MA Eq. (10.1 1) with f = /)(/•) n,.. 

22 See, e.g., MA Eq. (10.10) with f = q(r) n,. 

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j-Pi, r = R l , 
\-P 2 , r = R 2 . 


In order to relate this stress to strain, let us use Hooke’s law, but for that, we first need to calculate the 
strain tensor components for the deformation distribution (59). Using Eqs. (16), we get 

dq _ b q b 

s n .=- = a- 2 — , Sgg=S =- = « + —, 
or r r r 

so that Tr (s) = 3 a. Plugging these relations into Eq. (51a) for cr, T , we get 


cr . = 

1 + cr 

f 9 b 

a -2 


r* J 


H 5a 

1 - 2cr 



Now plugging this relation into Eqs. (60), we get a system of two linear equations for coefficients a and 
b. Solving this system, we get: 

a = 

1 2cr P l R l P 2 R 2 l 

E R\ - Rf ’ 

1 + g(P,-P 2 )RfR 3 2 
2 E Rl - Rf 


Formulas (59) and (63) give a complete solution of our problem. It is rich in contents and 
deserves at least some analysis. First of all, note that according to Eq. (50), coefficient (1 - 2<j)/E in the 
expression for a is just 1/3 K, so that the first tenn in Eq. (59) for deformation is just the hydrostatic 
compression. In particular, the second of Eqs. (63) shows that if R\ = 0, then b = 0. Thus for a solid 
sphere we have only the hydrostatic compression that was discussed in the previous section. Perhaps 
less intuitively, making two pressures equal gives the same result (hydrostatic compression) for arbitrary 
R 2 > R{. 

However, in the general case b 0 , so that the second tenn in the defonnation distribution (59), 
which describes the shear deformation, 23 is also substantial. In particular, let us consider the important 
thin-shell limit R 2 - R\ = t « R 1,2 = R - see Fig. 7b. In this case, q(R\) ~ q{Ri) is just the change of the 
shell radius R , for which Eqs. (59) and (63) (with R 2 - R^ « 3 R 2 t) give 

A R = q(R) « aR + — j ■. 


(P x -P 2 )R 7 

3 1 

1 — 2cr 1 + cr 
+ - 

2 E 

= ^~P 2 ) 

R~ 1 — (7 

t 2 E 


Naively, one could think that at least in this limit the problem could be analyzed by elementary 
means. For example, the total force exerted by the pressure difference (Pi - P 2 ) on the diametrical cross- 
section of the shell (see, e.g., the dashed line in Fig. 7b) is F = nR~(P\ - P 2 ), giving the stress, 


a = — 

^R\p x -p 2 ) 

2 nRt 


directed along shell’s walls. One can check that this simple formula may be indeed obtained, in this 
limit, from the strict expressions for aee and cr w following from the general treatment carried out 
above. However, if we try now to continue this approach by using the simple relation (47) to find the 
small change Rs zz of sphere’s radius, we would arrive at a result with the structure of Eq. (64), but 

23 Indeed, according to Eq. (50), the material-dependent factor in the second of Eqs. (63) is just 1/4//. 

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without factor (1 - a) < 1 in the nominator. The reason for this error (which may be as significant as 
-30% for typical construction materials - see Table 1) is that Eq. (47), while being valid for thin rods of 
arbitrary cross-section, is invalid for thin broad sheets, and in particular the thin shell in our problem. 
Indeed, while at the tensile stress both lateral dimensions of a thin rod may contract freely, in our 
problem all dimensions of the shell are under stress - actually, under much more tangential stress than 
the radial one. 24 

7.5. Rod bending 

The general approach to the static deformation analysis, outlined in the beginning of previous 
section, may be simplified not only for symmetric geometries, but also for the uniform thin structures 
such as thin plates (“membranes” or “sheets”) and thin rods. Due to the shortage of time, in this course I 
will demonstrate typical approaches to such systems only on the example of thin rods. (The theory of 
membrane deformation is very much similar.) Besides the tensile stress analyzed in Sec. 3, two other 
major deformations of rods are bending and torsion. Let us start from a “local” analysis of bending 
caused by a pair of equal and opposite external torques x = ±n y z y perpendicular to the rod axis z (Fig. 8), 
assuming that the rod is “quasi-uniform”, i.e. that on the scale of this analysis (comparable with linear 
scale a of the cross-section) its material parameters and cross-section A do not change substantially. 



Fig. 7.8. This rod bending, in a local reference frame (specific for each cross-section). 

Just as in the tensile stress experiment (Fig. 6), at bending the components of the stress forces 
dF, normal to the rod length, have to equal zero on the surface of the rod. Repeating the arguments made 
for the tensile stress discussion, we arrive at the conclusion that only one diagonal component of the 
tensor (in Fig. 8, <j ~ ) may differ from zero: 

°> =< V 7 =- ( 7 - 66 ) 

However, in contrast to the tensile stress, at pure static bending the net force along the rod has to vanish: 

F z = | cr z _d 2 r = 0, (7.67) 


so that a :z has to change sign at some point of axis x (in Fig. 8, selected to lay in the plane of the bent 
rod). Thus, the bending deformation may be viewed as a combination of stretching some layers of the 
rod (bottom layers in Fig. 8) with compression of other (top) layers. 

Since it is hard to find more about the stress distribution from these general considerations, let us 
turn over to strain, assuming that the rod’s cross-section is virtually constant on the length of the order 

24 Strictly speaking, this is only true if the pressure difference is not too small, namely, if (Pi - » P\^t!R. 

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of its cross-section size. From the above presentation of bending as a combination of stretching and 
compression, it evident that the longitudinal deformation q z has to vanish along some neutral line on the 
rod’s cross-section - in Fig. 8, represented by the dashed line. 25 Selecting the origin of coordinate x on 
this line, and expanding the relative deformation in the Taylor series in x, due to the cross-section 
smallness, we may limit ourselves to the linear term: 



dq z _ x 
dz R 


Flere constant R has the sense of the curvature radius of the bent rod. Indeed, on a small segment dz the 
cross-section turns by a small angle dq\ = - dqjx (Fig. 8b). Using Eq. (68), we get dcp y = dz/R, which is 
the usual definition of the curvature radius R in the differential geometry, for our special choice of the 
coordinate axes. 26 

Expressions for other components of the strain tensor are harder to guess (like at the tensile 
stress, not all of them are equal to zero!), but what we already know about o z: and s zz is already 
sufficient to start fonnal calculations. Indeed, plugging Eq. (66) into the Hooke’s law in the fonn (51b), 
and comparing the result for s zz with Eq. (68), we find 

=~E (7.69) 

From the same Eq. (51b), we could also find the transverse components of the strain tensor, and see that 
they are related to s zz exactly as at the tensile stress: 

s xx =s yy =~crs zz , (7.70) 

and then, integrating these relations along the cross-section of the rod, find the deformation of the cross- 
section shape. More important for us, however, is the calculation of the relation between rod’s curvature 
and the net torque acting on a given cross-section (of area A and orientation dA z > 0): 

Ty = J(r x dF) y =-^ X (J zz d 2 r = ^x 2 d 2 r = — -, (7.71) 


where I v is a geometric constant defined as 

I v =^x 2 dxdy. (7.72) 


Note that this factor, defining the bending rigidity of the rod, grows as fast as a 4 with the linear scale a 
of the cross-section. 27 

In these expressions, x has to be counted from the neutral line. Let us see where exactly does this 
line pass through rod’s cross-section. Plugging result (69) into Eq. (67), we get the condition defining 
the neutral line: 

25 Strictly speaking, that dashed line is the intersection of the neutral surface (the continuous set of such neutral 
lines for all cross-sections of the rod) with the plane of drawing. 

26 Indeed, for ( dx/dz ) 2 « 1, the general formula MA Eq. (4.3) for curvature (with the appropriate replacements / 

— > x and -V — > z) is reduced to MR = d 2 x/dz 2 = d( dx/dz)/dz = d(\w\(p } )ldz « d(p y Jdz. 

27 In particular, this is the reason why the usual electric wires are made not of a solid copper core, but rather a 
twisted set of thinner sub-wires, which may slip relative to each other, increasing the wire flexibility. 

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| xdxdy = 0. (7.73) 


This condition allows a simple interpretation. Imagine a thin sheet of some material, with a constant 
mass density cr per unit area, cut in the form of rod’s cross-section. If we place a reference frame into its 
center of mass, then, by its definition, 

crj rdxdy = 0. (7.74) 


Comparing this condition with Eq. (73), we see that one of neutral lines has to pass through the center of 
mass of the sheet, which may be called the “center of mass of the cross-section”. Using the same 
analogy, we see that integral I y (72) may be interpreted as the moment of inertia of the same imaginary 
sheet of material, with cr formally equal to 1, for its rotation about the neutral line - see Eq. (6.24). This 
analogy is so convenient that the integral is usually called the moment of inertia of the cross-section and 
denoted similarly - just as has been done above. So, our basic result (71) may be re-written as 

1 _ r v 

R Ef, 




vs. torque 

This relation is only valid if the deformation is small in the sense R » a. Still, since the 
deviations of the rod from its unstrained shape may accumulate along its length, Eq. (75) may be used 
for calculations of global deviations arbitrary on the scale of a. In order to describe such deformations, 
this equation has to be complemented by conditions of balance of the bending forces and torques. 
Unfortunately, this requires a bit more of differential geometry than I have time for, and I will only 
discuss this procedure for the simplest case of relatively small deviations q = q x of the rod from its initial 
straight shape, which will be used for axis z (Fig. 9a), by some bulk-distributed force f = n f x {z). (The 
simplest example is a uniform gravity field, for which f x = -pg = const.) Note that in the forthcoming 
discussion the reference frame will be global, i.e. common for the whole rod, rather than local 
(pertaining to each cross-section) as in the previous analysis - cf. Fig. 8. 


q = 0p 


F n 

q = 0 
T = 0 ^ 

F = 0 
T — 0 

F = n X F 0 
T = 0 

Fig. 7.9. Global picture of rod bending: (a) forces acting on a small fragment of a rod and 
(b) two bending problem examples, each with two typical, different boundary conditions. 

First of all, we may write an evident differential equation for the average vertical force F = 
n x F x (z) acting of the part of the rod located to the left of its cross-section located at point z. This 
equation expresses the balance of vertical forces acting on a small fragment dz of the rod (Fig. 9a), 
necessary for the absence of its linear acceleration: F x (z + dz) - F x (z) + f x {z)Adz = 0, giving 

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= ~f* A • (7-76) 


Note that this vertical component of the internal forces has been neglected at our derivation of Eq. (75), 
and hence our final results will be valid only if the ratio FJA is much less than the magnitude of a zz 
described by Eq. (69). However, these lateral forces create the very torque x = n v r v that causes the 
bending, and thus have to be taken into account at the analysis of the global picture. This re-calculation 
is expressed by the balance of torque components acting on the same rod fragment of length dz, 
necessary for the absence of its angular acceleration: 


-T = ~ F x- (7-77) 


These two equations of dynamics (or rather statics) should be complemented by two geometric 
relations. The first of them is dcpfdz = MR, which has already been discussed. We may immediately 
combine it with the basic result (75) of the local analysis, getting: 

d<P y _ T y 
dz EI v 



The final equation is the geometric relation evident from Fig. 9a: 

dq x 


= <Py 


which is (as all expressions of our simple analysis) only valid for small bending angles, I cp y I « 1 . 

Four differential equations (76)-(79) are sufficient for the full solution of the weak bending 
problem, if complemented by appropriate boundary conditions. Figure 9b shows four most frequently 
met conditions. Fet us solve, for example, the problem shown on the top panel of Fig. 9b: bending of a 
rod, clamped in a wall on one end, under its own weight. Considering, for the sake of simplicity, a 
uniform rod, 28 we may integrate equations (70), (72)-(74) one by one, each time using the appropriate 
boundary conditions. To start, Eq. (76), with f x = - pg, yields 

F x = pgAz + const = pgA(z - L ), (7.80) 

where the integration constant has been selected to satisfy the right-end boundary condition: F x = 0 at z 
= L. As a sanity check, at the left wall (z = 0), F x = -pgAL = - mg, meaning that the whole weight of the 
rod is exerted on the wall - fine. 

Next, plugging Eq. (80) into Eq. (77) 

o o? 

r„ = -^(z 2 -2Zz)+ const = -^(z 2 -2 Lz + L 2 )= -^-{z - L) 2 , 


where the integration constant’s choice ensures the second right-boundary condition: z y = 0 at z = L. 
Proceeding in the same fashion to Eq. (78), we get 

28 As clear from their derivation, Eqs. (76)-(79) are valid for any distribution of parameters A, E, I, and p over the 
rod’s length, provided that the rod is quasi-uniform, i.e. its parameters’ changes are so slow that the local relation 
(78) is still valid at any point. 

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EzA. + const = _ mL [ (z _ L f + e l 

2 El 3 6 El L * 

y y 


where the integration constant is selected to satisfy the clamping condition at the left end of the rod: (p y 
= 0 at z = 0. (Note that this is different from the support condition, illustrated on the lower panel of Fig. 
9b, which allows the angle at z = 0 to be finite but requires the torque to vanish.) Finally, integrating Eq. 
(79) with <p y given by Eq. (82), we get rod’s global deformation law, 

dA z ) 


6EI y 

(z-L ) 4 


+ Z 3 z + const 

pgA (z-L) 4 , , 3 _ L 4 

l-LZ , 

6 EI v [4 4 _ 


where the integration constant is zero again to satisfy the second left-boundary condition q = 0 atx = 0. 
So, the bending law is sort of complex even in this very simple problem. It is also remarkable how fast 
does the end’s displacement grow with the increase of rod’s length: 

?,( £ ) 

PgAL 4 
8 El 



To conclude the solution, let us discuss the validity of this result. First, the geometric relation 
(79) is only valid if | <p y (L)\ « 1, and hence if \q x (L)\ « L. Next, the local formula Eq. (78) is valid if 
HR = z(L)/EI y « Ha ~ A~ . Using results (81) and (84), we see that the latter condition is equivalent to 
\q x (L)\ « L la, i.e. is weaker, because all our analysis has been based on the assumption that L » a. 

Another point of concern may be that the off-diagonal stress component <j xz ~ FJA, that is 
created by the vertical gravity forces, has been ignored in our local analysis. For that approximation to 
be, this component must be much smaller than the diagonal component <j zz ~ aE/R = a zll v taken into 
account in that analysis. Using Eqs. (80) and (81), we are getting the following estimates: a xz ~ pgL, a zz 
~ apgAL U y ~ a pgL H y . According to its definition (72), I y may be crudely estimated as a , so that we 
finally get the following simple condition: a « L, which has been assumed from the very beginning. 

7.6. Rod torsion 

One more class of analytically solvable elasticity problems is torsion of quasi-uniform, straight 
rods by a couple of axially-oriented torques x = n z t z (Fig. 10). 

Here the main goal of the local analysis is to relate torque z z to parameter k in the relation 

dcp : 


- K. 


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rigidity C 

C for axially 

If the deformation is elastic and small (in the sense /ca « 1 , where a is again the characteristic size of 
rod’s cross-section), k is proportional to r z , and their ratio, 

dcp. / dz 


is called the torsional rigidity of the rod. Our task is to calculate the rigidity. 

As the first guess (as we will see below, of a limited validity), one may assume that the torsion 
does not change the shape or size of the cross-section, but leads just to the mutual rotation of cross- 
sections about certain central line. Using a reference frame with the origin on that line, this assumption 
immediately allows the calculation of components of the displacement vector dq, by using Eq. (6) with 
r/cp = n z d(p z \ 

dq x = -ydcp, = -Kydz, dq Y = xd <p z = Kxdz, dq. = 0. 


From here, we can calculate all Cartesian components (9) of the strain tensor: 

U* =*»,= Uz = °> = 5 VV = 0, s„ = s a = -—y, s yz = s ^ = —X . (7.88) 

The first of these equalities means that volume does not change, i.e. we are dealing with a pure shear 
deformation. As a result, all nonvanishing components of the stress tensor, calculated from Eqs. (34), 29 
are proportional to the shear modulus alone: 

cr = (7 = cr =0, <J = cr =0, cr = cr = — jukv , cr = cr = ukx. (7.89) 

Now it is straightforward to use this result to calculate the full torque as an integral over the 
cross-section areaz4: 

r z = | (r x dF ) z = J (xdF y - ydF x ) = J (xa yz - ya xz )dxdy. (7.90) 

Using Eq. (89), we get r z = ///U z , i.e. 

C = jul., where I z = j (x 2 + v 2 )dxdv . 



Again, just as in the case of thin rod bending, we have got an integral similar to a moment of 
inertia, this time for rotation about axis z passing through a certain point of the cross-section. For any 
axially-symmetric cross-section, this evidently should be the central point. Then, for example, for the 
practically important case of a round pipe with internal radius Ri and external radius /C, Eq. (91) yields 


c = A 2n J p 3 dp =y p(r* - R [ 4 ). 


In particular, for the solid rod of radius R this gives torsional rigidity C = (7t/2)pR 4 , while for a 
hollow pipe of small thickness t « R, Eq. (92) is reduced to 

29 For this problem, with purely shear deformation, using alternative elastic moduli E and a would be rather 
unnatural. If needed, we may always use the second of Eqs. (50): p = £72(1 + cr). 

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C = 2/rjuR 3 t . (7.93) 

Note that per unit cross-section area A (and hence per unit mass) this rigidity is twice higher that of a 
solid rod: 



thin round pipe 


solid round rod 


This fact is the basis of a broad use of thin pipes in construction. 

However, for rods with axially-asymmetric cross-sections, Eq. (91) gives wrong results. For 
example, for a narrow rectangle of area A = wt with t « w, it yields C = jutw 3 / 12 [WRONG!], even 
functionally different from the correct result - cf. Eq. (106) below. The reason of the failure of the above 
analysis is that does not describe possible bending q z of rod’s cross-section in the direction along the 
rod. (For axially- symmetric rods, such bending is evidently forbidden by the symmetry, so that Eq. (91) 
is valid, and results (92)-(94) are absolutely correct.) Let us describe 30 this, rather counter-intuitive 
effect by taking 

q : =Ki//(x,y), (7.95) 

(where y/ is some function to be determined), but still keeping Eq. (87) for two other components of the 
displacement vector. The addition of y/ does not change the equality to zero of the diagonal components 
of the strain tensor, as well as of s xy = s yx , but contributes to other off-diagonal components: 


s = s = — 

XZ ZX r\ 

Z \ 

■ y h — — 


s = s = — 

vz zy 2 

X + 

dy z 
dy , 


and hence to the corresponding elements of the stress tensor: 

°7z = 07, = 


-y + — 

dx j 

°yz = °7y = 


X + 

dy / 


Now let us find the requirement imposed on function yAx,y) by the fact that the stress force 
component parallel to rod’s axis, 

dF_ = <J zx dA x +<j„ Y dA Y = pncd A 


- V + — 1 — — 1 

dx J dA 

- + 

x + 



dA y 



has to vanish at rod’s surface(s), i.e. at each border of its cross-section. Coordinates {x, y} of points at a 
border may be considered functions of the arc / of that line - see Fig. 11. As this figure shows, the 
elementary area ratios participating in Eq. (98) may be readily expressed via derivatives of functions x(P) 
and v(/): dA JdA = sin a = dy/dl , dA y /dA = cos a = -dx/dl, so that we may write 



y + 



Adi j 


x + 

dy/ Y dx s 
dy dl J 


= 0 . 


Introducing, instead of y/, a new function %(x,y), defined by its derivatives as 

30 I would not be terribly shocked if the reader skipped the balance of this section at the first reading. Though the 
following calculation is very elegant and instructive, its results will not be used in other parts of these notes. 

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dZ_ s J_ 

dx 2 


dy/ | dy _ 1 
dy / dy 2 

^ dy/ 

-y + 



we may rewrite condition (99) as 


r ox dy + ox dx ' 

v ' 

= 2 — 


= 0 , 

dy dl dx dl J bordcr dl 

so that function x should be constant at each border of the cross-section. 



In particular, for a singly-connected cross-section, limited by just one continuous border line, the 
constant is arbitrary, because according to Eqs. (100), its choice does not affect the longitudinal 
deformation function yAx,y) and hence the deformation as the whole. Now let use the definition (100) of 
function x to calculate the 2D Laplace operator of this function: 

_ d 2 X 

d 2 ; 


d 2 


]_ 8 _ 

2 dx 

— x — 




2 dy 

■y + 



= -l. 


This a 2D Poisson equation (frequently met, for example, in electrostatics), but with a very simple, 
constant right-hand part. Plugging Eqs. (100) into Eqs. (97), and those into Eq. (90), we may express 
torque r, and hence the torsional rigidity C, via the same function: 






Sometimes, it is easier to use this result in one of its two different forms. The first of them may 
be readily obtained from Eq. (103a) using integration by parts: 

C = -2//(j dy\ xdx + J dx^ydy) = -2//[[ dy(xx bo rdei ~ J xdx)+ j dx(yx hovdcr - { xd) \ 

= 4 M 

J xdxdy - X border l dxdy 


while the proof of one more form, 

C = 4//J (v x v x) dxdy, 



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is left for reader’s exercise. 

Thus, if we need to know rod’s rigidity alone, it is sufficient to calculate function )ix,y) from Eq. 
(102) with boundary condition (101), and plug it into any of Eqs. (103). Only if we are also curious 
about the longitudinal deformation (95) of the cross-section, we may continue by using Eq. (100) to find 
function i/4x,y). Let us see how does this general result work for the two examples discussed above. For 
the round cross-section of radius R , both the Poisson equation (102) and the boundary condition, % = 
const at x + y = R , are evidently satisfied by the axially-symmetric function 

-^-(x 2 +y 2 ) + const. 


For this case, either of Eqs. (103) yields 

c= M 








i.e. the same result (91) that we had for y/ = 0. Indeed, plugging Eq. (104) into Eqs. (100), we see that in 
this case dy/ldx = dy/ldy = 0, so that yAx,y) = const, i.e. the cross-section is not bent. (As we have 
discussed in Sec. 1, a uniform translation dq z = /a// = const does not give any defonnation.) 

Now, turning to a rod with a narrow rectangular cross-section wt with t « vv, we may use this 
strong inequality to solve the Poisson equation (102) approximately, neglecting the derivative along the 
wider dimension (say, v). The remaining ID differential equation d~%ld~x = -1, with boundary conditions 
y\ x = +tn = X\x = -ta has an evident solution % = -x / 2 + const. Plugging this expression into any fonn of Eq. 
(103), we get the correct result for the torsional rigidity: 

C = i//n7 3 . (7.106) 

Now let us have a look at the cross-section bending law (95) for this particular case. Using Eqs. (100), 
we get 

^ = -x-2— = x. 



dx dy 


Integrating these differential equations over the cross-section, and taking the integration constant (again, 
not contributing to the defonnation) for zero, we get a beautifully simple result: 

V = xy, i.e. q z = Kxy . 


It means that the longitudinal deformation of the rod has a “propeller bending” form: while the regions 
near the opposite corners (sitting on the same diagonal) of the cross-section bend toward one direction 
of axis z, corners on the other diagonal bend in the opposite direction. (This qualitative conclusion 
remains valid for rectangular cross-sections with any aspect ratio t/w.) 

For rods with several surfaces, i.e. with cross-sections limited by several boundaries (say, hollow 
pipes), the boundary conditions for function %(x, y) require a bit more care, and Eq. (103b) has to be 
modified, because the function may be equal to a different constant at each boundary. Let me leave the 
calculation of the torsional rigidity for this case for reader’s exercise. 

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3D plane, 

7.7. 3D acoustic waves 

Now moving to elastic dynamics, we may start with Eq. (24) that may be transformed into the 
vector form exactly as this was done for the static case in the beginning of Sec. 4. Comparing Eqs. (24) 
and (54), we immediately see that the result may be presented as 


d 2 q 


2(1 + a) 

V 2 q + 

2(1 + cr)(l - 2cr) 

V(V-q) + f(r,f). 


Let us use this general equation for analysis of probably the most important type of time- 
dependent deformations: elastic waves. First, let us address the simplest case of a virtually infinite, 
uniform elastic medium, without any external forces f. In this case, due to the linearity and homogeneity 
of the resulting equation of motion, and in clear analogy with the ID case (see Sec. 5.3), we may look 
for a particular time-dependent solution in the form of a sinusoidal, linearly-polarized, plane wave 

q(r,0 = Re 


i ( k ■ r - cat j 


where a is the constant complex amplitude of a wave (now a vector!), and k is the wave vector whose 
magnitude is equal to the wave number k. The direction of these two vectors should be clearly 
distinguished: while a determined wave’s polarization, i.e. the direction of the particle displacements, 
vector k is directed along the spatial gradient of the full phase of the wave 

fskr - cot + arg a , 


i.e. along the direction of the wave front propagation. 

The importance of the angle between these two vectors may be readily seen from the following 
simple calculation. Let us point axis z of an (inertial) reference frame along the direction of vector k, 
and axis x in such direction that vector q, and hence a he within the {x, zj plane. In this case, all 
variables may change only along that axis, i.e. V = n z (8/dz), while the amplitude vector may be 
presented as the sum of just two Cartesian components: 

a = a x n x +a : n z . (7.112) 

Let us first consider a longitudinal wave, 31 with the particle motion along the wave direction: a x 

= 0, a z = a. Then vector q in Eq. (109), describing that wave, has only one (z) component, so that V q = 

2 2 2 2 2 

dqjdz and V(V-q) = n r (S q Idz ), and the Laplace operator gives the same expression: V q = n z {8 q/fiz ). 

As a result, Eq. (109), with f = 0, yields 

d 2 q z _ f E E 

P 8t 2 [2(1 + cr) + 2(1 + cr)(l - 2cr) 

Plugging the plane -wave solution (110) into this equation, we see that it is indeed satisfied if the wave 
number and wave frequency are related as 

d 2 q z _ E(\ - a) d 2 q _ 

dz 2 (1 + cr)(l - 2cr) dz 2 


31 In geophysics, the longitudinal waves are known as P-waves (with letter P standing for “primary”), because 
due to their higher velocity (see below) they arrive at the detection site (from a distant earthquake or explosion) 
before waves of other types. 

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co = v,k. 

E{\ - cr) _ K + (4/3)// 
(1 + o-)(l - 2 a)p p 


This expression allows a simple interpretation. Let us consider a static experiment, similar to the 
tensile test experiment shown in Fig. 6, but with a sample much wider than L in both directions 
perpendicular to the force. Then the lateral contraction is impossible, and we can calculate the only 
finite stress component, a zz , directly from Eq. (34) with Tr (s) = s zz : 

f 1 > 

W ~-s B 

fl ^ 

~ S ZZ 


+ 3 K 

l 3 ; 

13 ) 

K + — u 


We see that the nominator in Eq. (114) is nothing more than the static elastic modulus for such a 
uniaxial deformation, and it is recalculated into the velocity exactly as the spring constant in the ID 
waves considered in Sec. 5.3 - cf. Eq. (5. 32). 32 Thus, the longitudinal acoustic waves are just simple 
waves of uniaxial extension/compression along the propagation axis. Formula (114) becomes especially 
simple in fluids, where p = 0, and the wave velocity is described by well-known expression 


Note, however, that for gases, with their high compressibility and temperature sensitivity, the value of K 
participating in this formula may differ, at high frequencies, from that given by Eq. (42), because the 
fast compressions/extensions of gas are nearly adiabatic rather than isothermal. This difference is 
noticeable in Table 1 which, in particular, lists the values of v/ for some representative materials. 

Now let us consider an opposite case of transverse waves with a x = a, a z = 0. In such a wave, the 
displacement vector is perpendicular to z, so that V q = 0, and the second term in the right-hand part of 
Eq. (109) vanishes. On the contrary, the Laplace operator acting on such vector still gives the same non- 
zero contribution, V 2 q = n : (8 2 q/dz 2 ), to Eq. (109), so that the equation yields 

2 q x E d 2 q x 
dt 2 2(l + a) 8z 2 ’ 


and instead of Eq. (1 14) we now get 

co = v t k. 

2(1 + 0-)/? 



We see that the speed of transverse waves depends exclusively from the shear modulus // of the 
medium. 33 This is also very natural: in such waves, the particle displacements q = n x q are perpendicular 
to the elastic forces c/F = n z dF, so that the only one component a xz of the stress tensor is involved. Also, 

32 Actually, we can identify these results even qualitatively, if we consider a medium consisting of n parallel, 
independent ID chains per unit area. Extension of each chain fragment, of length d, by A d « d gives force F = 
kAd, so that the total longitudinal stress, a :z = Fn, is related to strain s zz = Ad/d, as <7 :z /s zz = kn/d. Multiplying both 
parts of Eq. (5.33a) by n/d, and noticing that ( mn/d) is nothing more than the average mass density p, we make 
that equation absolutely similar to Eq. (113), just with a different notation for the longitudinal rigidity <J : Js zx . 

33 Because of that, one can frequently meet term shear waves. In geophysics, they are also known as S- waves, S 
standing for “secondary”, again in the sense of arrival time. 




in fluids 




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the strain tensor sjj- has no diagonal components, Tr (s) = 0, so that p is the only elastic modulus actively 
participating in the Hooke’s law (34). 

In particular, fluids cannot carry transverse waves at all (formally, their velocity (118) vanishes), 
because they do not resist shear deformations. For all other materials, longitudinal waves are faster than 
the transverse ones. Indeed, for all kn own materials the Poisson ratio is positive, so that the velocity 
ratio that follows from Eqs. (114) and (118), 

v , 

( 2 - 2cr V /2 

v 1 - 2cr J 


is above V 2 « 1.4. For the most popular construction materials, with cr« 0.3, the ratio is about 2 - see 
Table 1. 

Let me emphasize again that for both longitudinal and transverse waves the relation between the 
wave number and frequency is linear: co= vk. As has already been discussed in Sec. 5.3, in this case of 
acoustic waves (or just “sound”) there is no dispersion, i.e. a transverse or longitudinal wave of more 
complex form, consisting of several (or many) Fourier components of the type (110), preserves its form 
during propagation: 34 

q(z,t) = q(z-vf,0). (7.120) 

As one may infer from the analysis in Sec. 5.3, the dispersion would be back at very high (< hypersound) 
frequencies where the wave number k becomes of the order of the reciprocal distance between the 
particles of the medium (e.g., atoms or molecules), and hence the approximation of the medium as a 
continuum, used through this chapter, became invalid. 

As we already kn ow from Sec. 5.3, besides the velocity, an important parameter characterizing 
waves of each type is the wave impedance Z of the medium, for acoustic waves frequently called the 
acoustic impedance. Generalizing Eq. (5.44) to the 3D case, we may define the impedance as the ratio of 
the force per unit area (i.e. the corresponding component of the stress tensor) exerted by the wave, to 
particles’ velocity. For example, for the longitudinal waves, propagating in the positive/negative 
direction along z axis, 

Z, = + 

a , 

= + - 

a , 

dqjdt s zz dqjdt 

_ _ o~ g oq z !oz 

s zz dqjdt' 


Plugging in Eqs. (1 10), (1 14), and (115), we get 







(K + -ju)p 



in a clear analogy with Eq. (5.45). Similarly, for the transverse wave, the appropriately modified 
definition, Z, = +a xz /(dq x /dz), yields 


z t =M U2 - 


34 However, if the initial wave is an arbitrary mixture (109) of longitudinal and transverse components, these 
components, propagating with different velocities, will “run from each other”. 

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Just like in the ID waves, one role of impedance is to scale the power carried by the wave. For 
plane 3D waves in infinite media, with their infinite wave front area, it is more appropriate to speak it is 
more appropriate to speak about power density, i.e. power ^ = d '/9dA per unit area of the front, and 
characterize it by not only its magnitude, 

dF <9q 

dA dt ’ 


but also the direction of the energy propagation, that (for a plane wave in an isotropic medium) 
coincides with the direction of the wave vector k: p = ^n k . Using definition (18) of the stress tensor, we 
may present the Cartesian components of this Umov vector 35 as 



dq r 

ajr dt 


Returning to plane waves propagating along axis z, and acting exactly like in Sec. 5.3, for both the 
longitudinal and transverse waves we arrive at the following 3D analog Eq. (5.46), 

2 7 

co Z * 

p-, = aa 

rz 2 


with Z being the corresponding impedance - either Z/ or Z,. 

Just as in ID case, one more important effect in which the notion of impedance is crucial is wave 
reflection from at an interface between two media. The two boundary conditions, necessary for the 
analysis of these processes, may be obtained from the continuity of vectors q and dF. (The former 
condition is evident, while the latter one may be obtained by applying the 2 nd Newton law to the 
infinitesimal volume dV = dAdz, where segment dz straddles the boundary.) Let us start from the 
simplest case of the normal incidence on a plane interface between two uniform media with different 
elastic moduli and mass densities. Due to the symmetry, it is evident that the incident 
longitudinal/transverse wave may only excite longitudinal/trans verse reflected and transferred waves, 
but not the counterpart wave type. Thus we can literally repeat all the calculations of Sec. 5.4, again 
arriving at the fundamental relations (5.53) and (5.54), with the only replacement of Z and Z’ with the 
corresponding values of either Z/ (121) or Z, (123). Thus, at the normal incidence the wave reflection is 
determined solely by the acoustic impedances of the media, while the sound velocities are not involved. 

The situation, however, becomes more involved at a nonvanishing incidence angle d ,] (Fig. 12), 
where the transmitted wave is generally also refracted, i.e. propagates under a different angle, ^ 
to the interface. Moreover, at & ] ■£ 0 the directions of particle motion (vector q) and of the stress forces 
(vector r/F) in the incident wave are neither exactly parallel nor exactly perpendicular to the interface, 
and thus this wave serves as an actuator for reflected and refracted waves of both types - see Fig. 12. (It 
shows the particular case when the incident wave is transverse.) The corresponding four angles, d, ir \ 
0 t (1 \ dp d t ’ , may be readily related to d l) by the “kinematic” condition that the incident wave, as well as 

35 Named after N. Umov who introduced this concept in 1874. Ten years later, a similar concept for 
electromagnetic waves (see, e.g., EM Sec. 6.4) was suggested by J. Poynting, so that some textbooks use the term 
“Umov-Poynting vector”. In a dissipation-free, elastic medium, the Umov vector obeys the following continuity 

equation, d{pv 2 / 2 + uj/dt + V • ff = 0, with u given by Eq. (52), which expresses the conservation of the total 
(kinetic plus potential) energy of elastic deformation. 

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the reflected and refracted waves of both types should have the same spatial distribution along the 
interface plane, i.e. for the material particles participating in all five waves. According to Eq. (110), the 
necessary boundary condition is the equality of the tangential components (in Fig. 12, k x ), of all five 
wave vectors: 

k t sin #/ r * = k, sin #/'* = k] sin 0, = k' t sin#, = k x = k, sin#/'*. (7.127) 

Since the acoustic wave vectors, at fixed frequency, are inversely proportional to the corresponding 
wave velocities, we immediately get the following relations: 


so that generally all 4 angles are different. (In optics, the latter relation, reduced to just one equality for 
the only possible, transverse waves, is known as the Snell law.) These relations show that, just like in 
optics, the direction of a wave propagating into a medium with lower velocity is closer to the normal 
(axis z). In particular, this means that if v ’ > v, the acoustic waves, at larger angles of incidence, may 
exhibit the effect of total internal reflection, so well known from optics 36 , when the refracted wave 
vanishes. In addition, Eqs. (128) show that in acoustics, a reflected longitudinal wave, with velocity v/ > 
v t , may vanish at sufficiently large angles of transverse wave incidence. 


Fig. 7.12. “Kinematic” condition of 
acoustic wave reflection and refraction. 

All these fact automatically follow from general expressions for amplitudes of the reflected and 
refracted waves via the amplitude of the incident wave. These relations are straightforward to derive 
(again, from the continuity of vectors q and dF), but since they are much more bulky then those in the 
electromagnetic wave theory (where they are called the Fresnel formulas 37 ), I would not have 
time/space for spelling them up. Let me only note that, in contrast to the case of normal incidence, these 
relations involve 8 media parameters: the values of impedances Z, Z\ and velocities v, v ’ on both sides 
of the interface, and for both the longitudinal and transverse waves. 

There is another factor that makes boundary acoustic effects more complex. Within certain 
frequency ranges, interfaces (and in particular surfaces) of elastic solids may sustain so-called surface 

36 See, e.g., EM Sec. 7.5. 

37 Their discussion may be also found in EM Sec. 7.5. 

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acoustic waves (SAW), in particular, the Rayleigh waves and Love waves , 38 The main feature that 
distinguishes such waves from their bulk (longitudinal and transverse) counterparts is that the particle 
displacement amplitude is maximal at the interface and decays exponentially into the bulk of both 
adjacent media. The characteristic depth of this penetration is of the order of, though not exactly equal 
to the wavelength. 

In the Rayleigh waves, the particle displacement vector q has two components: one longitudinal 
(and hence parallel to the interface along which the wave propagates) and another transverse 
(perpendicular to the interface). In contrast to the bulk waves discussed above, the components are 
coupled (via their interaction with the interface) and as a result propagate with a single velocity v R . As a 
result, the trajectory of each particle in the Rayleigh wave is an ellipse in the plane perpendicular to the 
interface. A straightforward analysis 39 of the Rayleigh waves on the surface of an elastic solid (i.e. its 
interface with vacuum) yields the following equation for v R : 

( V 2 'j 


( V 2S 


f ,. 2 ) 


= 16 


l u J 

l u J 

l V ! J 


According to this formula, and Eqs. (114) and (118), for realistic materials with 0 < a < Vi, the Rayleigh 
waves are slightly (by 4 to 13%) slower than the bulk transverse waves - and hence substantially slower 
than the bulk longitudinal waves. 

In contrast, the Love waves are purely transverse, with vector q oriented parallel to the interface. 
However, the interaction of these waves with the interface reduces their velocity vl in comparison with 
that (v t ) of the bulk transverse waves, keeping it in the narrow interval between v t and v R : 

v R <v L <v, <v 7 . (7.130) 

The practical importance of surface acoustic waves is that their amplitude decays very slowly 
with distance r from their point-like source: a oc Mr , while any bulk waves decay much faster, as a oc 
Mr. (Indeed, in the latter case power T 3 cc a", emitted by such source, is distributed over a spherical 
surface area proportional to r , while in the former case all the power goes into a thin surface circle 
whose length scales as r.) At least two areas of applications of the surface acoustic waves have to be 
mentioned: in geophysics (for earthquake detection and Earth crust seismology), and electronics (for 
signal processing, with a focus on frequency filtering). Unfortunately, I cannot dwell on these 
interesting topics and I have to refer the reader to special literature. 40 

7.8. Elastic waves in restricted geometries 

From what we have discussed in the end of the last section, it should be pretty clear that 
generally the propagation of acoustic waves in elastic bodies of finite size may be very complicated. 
There is, however, one important limit in which several important results may be readily obtained. This 
is the limit of (relatively) low frequencies, where the wavelength is much larger than at least one 

38 Named, respectively, after Lord Rayleigh (bom J. Strutt, 1842-1919) who has theoretically predicted the very 
existence of surface acoustic waves, and A. Love (1863-1940). 

39 See, e.g., Sec. 24 in L. Landau and E. Lifshitz, Theory of Elasticity, 3 rd ed., Butterworth-Heinemann, 1986. 

40 See, for example, K. Aki and P. G. Richards, Quantitative Seismology /, 2 nd ed., University Science Books, 2002, 
and D. Morgan, Surface Acoustic Waves, 2 nd ed., Academic Press, 2007. 

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dimension of a system. Let us consider, for example, various waves that may propagate along thin rods, 
in this case “thin” meaning that the characteristic size a of rod’s cross-section is much smaller than not 
only the length of the rod, but also the wavelength A = Irdk. In this case there is a considerable range of 
distances z along the rod, 

a«Az«A, (7.131) 

in which we can neglect the dynamic effects due to medium inertia, and apply results of our earlier static 

For example, for a longitudinal wave of stress, which is essentially a wave of periodic tensile 
extensions and compressions of the rod, within range (131) we can use the static relation (44): 

^ = Es zz . (7.132) 

For what follows, it is easier to use the general equation of elastic dynamics not in its vector form (109), 
but rather in the precursor, Cartesian-component form (25), with fj= 0. For plane waves propagating 
along axis z, only one component (with j ’ — » z) of the sum in the right-hand part of this equation is non- 
vanishing, and it is reduced to 

d 2 q da 


In our current case of longitudinal waves, all components of the stress tensor but a zz are equal to zero. 
With a zz from Eq. (132), and using the definition s zz = dqjdz = dqjdz, Eq. (133) is reduced to a very 
simple wave equation, 


d 2 q : 

d 2 r 

which shows that the velocity of such tensile waves is 


f E 'l 



V = 






Comparing this result with Eq. (1 14), we see that the tensile wave velocity, for any medium with a > 0, 
is lower than the velocity v/ of longitudinal waves in the bulk of the same material. The reason for this 
difference in simple: in thin rods, the cross-section is free to oscillate (e.g., shrink in the longitudinal 
extension phase of the passing wave), 41 so that the effective force resisting the longitudinal deformation 
is smaller than in a border-free space. Since (as clearly visible from the wave equation), the scale of the 
force gives the scale of v , this difference translates into slower waves in rods. Of course as wave 
frequency is increased, at ka ~ 1 there is a (rather complex and cross-section-depending) crossover from 
Eq. (135)toEq. (114). 

Proceeding to transverse waves in rods, let us first have a look at long bending waves, with 
vector q = n x q x (with axis x along the bending direction - see Fig. 8) being approximately constant in the 
whole cross-section. In this case, the only component of the stress tensor contributing to the net 
transverse force F x is a xz , so that the integral of Eq. (133) over the cross-section is 

41 Due to this reason, the tensile waves can be called longitudinal only in a limited sense: while the stress wave is 
purely longitudinal a xx = a vy = 0, the strain wave is not: s xx = Syy = -<js :z ^ 0, i.e. q(r, t) ^ n : q z ,. 

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8F r 



J a x: dA - 


Now, if Eq. (131) is satisfied, we again may use static local relations (77)-(79), with all derivatives dldz 
duly replaced with their partial form dldz, to express force F x via the bending deformation q x . Plugging 
these relations into each other one by one, we arrive at a very unusual differential equation 




Looking for its solution in the form of a sinusoidal wave (110), we get a nonlinear dispersion relation: 42 


co = 

\P A J 



Such relation means that the bending waves are not acoustic at any frequency, and cannot be 
characterized by a single velocity that would be valid for all wave numbers k, i.e. for all spatial Fourier 
components of a waveform. According to our discussion in Sec. 5.3, such strongly dispersive systems 
cannot pass non-sinusoidal waveforms too far without changing their wavefonn very considerably. 

This situation changes, however, if the rod has an initial uniform longitudinal stress <%, = T/A 
(where force T is usually called tension), on whose background the transverse waves propagate. To 
analyze its effect, let us redraw Fig. 6, for a minute neglecting the bending stress - see Fig. 13. 

(p Y (z + dz) 

Fig. 7.13. Additional forces in a thin rod 
(“string”), due to background tension T. 

Still sticking to the limit of small angles (p, the additional vertical component dT x of the net force 
acting on a small rod fragment of length dz is T x {z - dz) - T x (z) = T cp y (z + dz) - T (p v (z) « T(dcp y ldz)dz, so 
that dFJdz = T(d(p y !dz). With the geometric relation (79) in its partial-derivative form dqjdz = cp y , this 
additional term becomes T (d qjdz ). Adding it to the right-hand part of into Eq. (137), we get the 
following dispersion relation 

— (Elk 4 +7k 2 ) 
P A ' ' ' 


42 Note that since the “moment of inertia” I v , defined by Eq. (72), may depend on the bending direction (unless 
the cross-section is sufficiently symmetric), the dispersion relation (138) may give different results for different 
directions of the bending wave polarization. 





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At low k (and hence low frequencies), it describes acoustic waves with the “guitar string” velocity that 
should be well known to the reader from undergraduate courses: 

on a string: 

where the denominator is nothing else than the linear mass density. However, as the frequency grows, 
Eq. (139) describes a crossover to highly-dispersive bending waves (138). 

Now let us consider the so-called torsional waves that are essentially the dynamic propagation of 
the torsional deformation discussed in Sec. 6. The easiest way to describe these waves, again within the 
limits given by Eq. (131), is to write the equation of rotation of a small segment dz of the rod about axis 
z, passing through the “center of mass” of its cross-section, under the difference of torques t = n-r z 
applied on its ends - see Fig. 10: 

pl ; dz^ = dr ; , (7.141) 



where I z is the “moment of inertia” defined by Eq. (91), which now, after its multiplication by pdz, i.e. 
by the mass per unit area, has turned into the real moment of inertia of a dz - thick slice of the rod. 
Dividing both parts by dz, using the static local relation (86), r r = Ctc = C(d(pJdz), we get the following 
differential equation 

pi z 


Just as Eqs. (114), (118), (135) and (140), this equation describes an acoustic (dispersion-free) wave that 
propagates with frequency-independent velocity 


As we have seen in Sec. 6, for rods with axially-symmetric cross-sections, the torsional rigidity C is 
described by the simple equation (91), C = pl z , so that expression (143) is reduced to Eq. (118) for the 
transverse waves in infinite media. The reason for this similarity is simple: in a torsional wave, particles 
oscillate along small arcs (Fig. 14a), so that if the rod’s cross-section is round, the stress-free surface 
does not perturb or modify the motion in any way, and hence does not affect the transverse velocity. 








Fig. 7.14. Particle trajectories in two 
different transverse waves with the same 
velocity: (a) torsional waves in a thin 
round rod and (b) circularly-polarized 
waves in an infinite (or very broad) 

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This fact raises an interesting issue of the relation between the torsional and circularly-polarized 
waves. Indeed, in Sec. 7, I have not emphasized enough that Eq. (118) is valid for a transverse wave 
polarized in any direction perpendicular to vector k (in our notation, directed along axis z). In particular, 
this means that such waves are doubly-degenerate: any isotropic elastic medium can carry 
simultaneously two non-interacting transverse waves propagating in the same direction with the same 
velocity (118), with mutually perpendicular linear polarizations (directions of vector a), for example, 
directed along axes x and y. If both waves are sinusoidal (110), with the same frequency, each point of 
the medium participates in two simultaneous sinusoidal motions within the \x,v] plane: 

i(kz - cot) 
a e 


= A x cos'?. 

a y e 

i{kz - cot) 

Ay cos(V + cp), 


where T* = hz - cot + (p x , and cp= <p y - cp x . Trigonometry tells us that the trajectory of such motion on the 
[x, y] plane is an ellipse (Fig. 15), so that such waves are called elliptically-polarized. The most 
important particular cases of such polarization are: 

(i) cp = 0 or tv. a linearly-polarized wave, with vector a turned by angle 0 = Arctan (Ay/A, x ) from 
axis x; and 

(ii) cp = ± nil and A x = Ay. circularly polarized waves, with the right or left polarization, 

The circularly polarized waves play an important role in quantum mechanics, where such waves 
may be most naturally quantized, with elementary excitations (in the case of mechanical waves we are 
discussing, called phonons) having either positive or negative angular momentum L z = ±h. 

Now comparing the trajectories of particles in the torsional wave in a thin round rod (or pipe) 
and the circularly-polarized wave in a broad sample (Fig. 14), we see that, despite the same wave 
propagation velocity, these transverse waves are rather different. In the former case (Fig. 14a) each 
particle moves back and forth along an arc, with the arc length different for different particles (and 
vanishing at rod’s center). On the other hand, in a circularly-polarized, plane wave all particles move 
along similar, circular trajectories. 

Fig. 7.15. Trajectory of a particle of an infinite 
medium with elliptically-polarized transverse 
wave, within the plane perpendicular to the 
direction of wave propagation. 

In conclusion, let me briefly mention the opposite limit, when the size of the body, from whose 
boundary are completely reflected, 43 is much larger than the wavelength. In this case, the waves 

43 For acoustic waves, such condition is easy to implement. Indeed, from Sec. 7 we already know that the strong 
inequality of wave impedances Z is sufficient for such reflection. The numbers of Table 1 show that, for example, 

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volume per 
wave vector 

propagate almost as in an infinite 3D medium (Sec. 7), and the most important new effect is the finite 
numbers of wave modes in the body. Repeating ID analysis of Sec. 5.4 for each dimension of a 3D 
cuboid of volume V = Z 1 E 2 T 3 (for example, using the Born-Karman boundary conditions in each 
dimension), we obtain Eq. (5.59) for the spectrum of components of wave vector k along each side. This 
means that all possible wave vectors are located in nodes of a rectangular 3D mesh with steps InlLj in 
each direction, and hence with the A-spacc (“reciprocal space”) volume 

In 2 n 2 n _ ( 2n ) 3 

T x T 2 T,~~T 


per each vector. It is possible (though not quite as straightforward as it is sometimes assumed) to prove 
that this relation is valid regardless of the shape of volume V. Hence the number of different wave 
vectors within the reciprocal space volume cfk » f) is 

dN = 





d^k » 1 . 


of states 

In quantum mechanics, this relation takes the form of the density of quantum states in k-space: 


8k ~ S d 3 k 


(2n) 3 


where g is the number of possible different quantum states with the same de Broglie wave vector k. In 
this form, Eq. (146) is ubiquitous in physics. 44 For phonons, formed from quantization of one 
longitudinal mode, and two transverse modes with different polarizations, g = 3. 

7.9. Exercise problems 

7.1 . A uniform thin sheet of an isotropic, elastic material is 
compressed, along its thickness t, by two plane, parallel, broad (of area 
A » t ) rigid surfaces - see Fig. on the right. Assuming no slippage 
between the sheet and the surfaces, calculate the relative compression (- 
A tit) as a function of the compressing force. Compare the result with 
that for the tensile stress, given by Eq. (47). 

7.2 . A thin, wide sheet of an isotropic, elastic material is clamped in two rigid, plane, parallel 
surfaces that are pulled apart with force F. Find the relative extension AL/L of the sheet in the direction 
of the force, and its relative compression A t/t in the perpendicular direction, and compare the results 
with Eqs. (47)-(48) for the tensile stress, and the solution of Problem 1. 

< > 


the impedance of a longitudinal wave in a typical metal (say, steel) is almost two orders of magnitude higher than 
that in air, ensuring their virtually full reflection from the surface. 

44 See, e.g., EM Secs. 7.7 and 7.9, and QM Sec. 1.5. 

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7.3 . Calculate the radial extension A R of a thin, long, round 
cylindrical pipe under the effect of its rotation with a constant angular 
velocity <x> about its symmetry axis (see Fig. on the right), in terms of the 
elastic moduli E and o, assuming that pressure both inside and outside the 
pipe is negligible. 

7.4 . A long, uniform rail with the cross-section shown in Fig. on 
the right, is being bent with the same (small) torque twice: first within 
plane xz and then within plane yz. Assuming that t « L, find the ratio of 
rail defonnations in these two cases. 

7.5 . Two thin rods of the same length and mass have 
been made of the same elastic, isotropic material. The cross- 
section of one of them is a circle, while another one is an 
equilateral triangle - see Fig. on the right. Which of the rods 
is more stiff for bending along its length? Quantify the 
relation. Does the result depend on the bending plane 

1.6. A thin, elastic, unifonn, initially straight beam is placed on 
two point supports at the same height - see Fig. on the right. What 
support point placement minimizes the largest deviation of the beam 
from the horizontal line, under its own weight? 

7.7 . Calculate the largest compression force T that may be 
withstood by a thin, straight, elastic rod without bucking (see Figs, 
on the right) for two shown cases: 







(i) rod’s ends are clamped, and 

(ii) the rod it free to rotate about the support points. 

7.8 . Calculate the potential energy of a small and slowly changing, but otherwise arbitrary 
bending deformation of a uniform, elastic, initially straight rod. Can the result be used to derive the 
dispersion relation (7.138)? 

7.9 .* Calculate the spring constant dF/dL of a coil 
spring made of a uniform, elastic wire, with circular cross- 
section of diameter d, wound as a dense round spiral of N 
» 1 turns of diameter D » d - see Fig. on the right. 
Comment on the type of material’s deformation. 

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a F 

7.10. The coil discussed in Problem 9 is now used as what is 
sometimes called the torsion spring - see Fig. on the right. Find the 
corresponding spring constant dzldcp, where r is the torque of external forces F 
relative the center of the coil (point O). 

7.1 1 . Use Eqs. (101) and (102) to recast Eq. (103b) for the torsional 
given by Eq. (103c). 

7.12 . * Generalize Eq. (103b) to the case of rods with more than one cross-section boundary. Use 
the result to calculate the torsional rigidity of a thin round pipe, and compare it with Eq. (93). 

7.13 . Calculate the potential energy of a small but otherwise arbitrary torsional deformation (p z (z) 
of a uniform, straight, elastic rod. 

7.14 . A steel wire with the circular cross-section of a 3 -mm diameter is stretched with a constant 
force of 10 N and excited at frequency 1 kHz by an actuator that excites all modes of longitudinal and 
transverse waves. Which wave has the highest group velocity? Accept the following parameters for steel 
(see Table 7.1): E = 170 GPa, o = 0.30 , p = 7.8 g/cnr. 

7.15 . Define and calculate appropriate wave impedances for (i) tensile 

and (ii) torsional waves in a thin rod. Use the results to calculate what fraction ^ 

of each wave’s power is reflected from the connection of a long rod with 
round cross-section to a similar rod, but with twice larger diameter - see Fig. 
on the right. 


rigidity C into the form 

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Chapter 8. Fluid Mechanics 

This chapter describes the basic notions of mechanics of fluids, discusses a few core problems of statics 
and dynamics of ideal and viscous fluids, and gives a very brief review of such a complex phenomenon 
as turbulence. Also, the viscous fluid flow is used to give an elementary introduction to numerical 
methods of partial differential equation solution - whose importance extends well beyond this particular 

8.1. Hydrostatics 

The mechanics of fluids (the class of materials that includes both liquids and gases) is both more 
simple and more complex than that of the elastic solids, with the simplicity falling squarely to the 
domain of statics - often called hydrostatics, because water has always been the main fluid for the 
human race and hence for science and engineering. Indeed, fluids are, by definition, the media that 
cannot resist static shear deformations. There are two ways to express this fact. First, we can formally 
take the shear modulus ju, describing this resistance, to be equal zero. Then the Hooke’s law (7.34) 
shows that the stress tensor is diagonal: 

( 8 . 1 ) 

Alternatively, the same conclusion may be reached by looking at the stress tensor definition (7.19) and 
saying that in the absence of shear stress, the elementary interface dF has to be perpendicular to the area 
element dA, i.e. parallel to vector dA. 

Moreover, in fluids at equilibrium, all three diagonal components ojj of the stress tensor have to 
be equal. To prove that, it is sufficient to single out (mentally rather than physically) from a fluid a small 
volume in the shape of a right prism, with mutually perpendicular faces normal to the two directions we 
are interested in (Fig. 1, along axes x and v). 

XX X XX ' 

Fig. 8.1. Proving the pressure isotropy. 


The prism is in equilibrium if each Cartesian component of the total force acting on all its faces 
nets to zero. For the x-component this balance is a xx dA x - (a U(/ dA)cosa = 0. However, from the geometry 
(Fig. 1), dA x = dAcosa, and the above balance condition yields a aa = o xx . A similar argument for the 
vertical forces gives o aa = <J yy , so that a xx = a yy .Since such equality holds for any pair of diagonal 
components of the stress tensor, ojj, all three of them have to be equal. This common component is 
usually represented as (-P), because in the vast majority of cases, parameter P, called pressure, is 
positive. Thus we arrive at the key relation (which has already been mentioned in Ch. 7): 

a = -PS , 

( 8 . 2 ) 

© 2013-2016 K. Likharev 

Essential Graduate Physics 

CM: Classical Mechanics 

In the absence of bulk forces, pressure should be constant through the volume of fluid, due to 
symmetry. Let us see how this result is affected by bulk forces. With the simple stress tensor (2), the 
general condition of equilibrium of a continuous medium, expressed by Eq. (7.25) with zero left-hand 
part, becomes just 


- — + /,■ =0, (8.3) 

and may be re-written in a convenient vector form: 

— VP + f = 0. (8.4) 

In the simplest case of a heavy fluid, with mass density p, in a uniform gravity field, f = pg, and the 
equation of equilibrium becomes, 

-VP + pg = 0, (8.5) 

with only one nonvanishing component (vertical, near the Earth surface). If, in addition, the fluid may 
be considered incompressible, with its density p constant, 1 this equation may be readily integrated to 
give the so-called Pascal equation : 2 

P + pg\> = const, 


where y is the vertical coordinate, with the direction opposite to that of vector g. 

Let me hope that this equation, and its simple applications (including buoyant force calculations 
using the Archimedes principle), are well familiar to the reader from his or her undergraduate physics 
courses, so that I may save time by skipping their discussion. I would only like to note, that the 
integration of Eq. (4) may be more complex in the case if the bulk forces f depend on position, 3 and/or if 
the fluid is substantially compressible. In the latter case, Eq. (4) should be solved together with the 
media-specific equation of state p = pfP) describing the compressibility law - whose example is given 
by Eq. (7.40) for ideal gases: p = mN/V= mPIkfT, where m is the mass of one gas molecule. 

8.2. Surface tension effects 

Besides the bulk (volume-distributed) forces, one more possible source of pressure is surface 
tension. This effect results from the difference between the potential energy of atomic interactions on 
the interface between two different fluids and that in their bulks, and thus may be described by an 
additional potential energy 

U t =yA, 


1 As was discussed in Sec. 7.3 in the context of Table 7.1, this is an excellent approximation, for example, for 
human-scale experiments with water. 

2 The equation, and the SI unit of pressure 1 Pa = lN/m 2 , are named after B. Pascal (1623-1662) who has not only 
pioneered hydrostatics, but also invented the first mechanical calculator and made several other important 
contributions to mathematics - and Christian philosophy! 

3 An example of such a problem is given by fluid equilibrium in coordinate systems rotating with a constant 
angular velocity. Here the real bulk forces should be complemented by the centrifugal “force” - the only inertial 
force which does not vanish at constant © and r - see Eq. (6.92). 






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where A is the interface area, and /is called the surface tension constant (or just the “surface tension”), 
evidently of the dimensionality of J/m , i.e. N/m. For a stable interface of any two fluids, y is always 
positive. 4 In the absence of other forces, the surface tension makes a liquid drop spherical to minimize 
its surface area at fixed volume. 

For the analysis of the surface tension effects in the presence of other forces, it is convenient to 
reduce it to a certain additional effective pressure drop A/fr at the interface. In order to calculate AP e f, 
let us consider the condition of equilibrium of a small part dA of a smooth interface between two fluids 
(Fig. 2), in the absence of bulk forces. 

+ 8{dA) 

Fig. 8.2. Deriving the Young-Laplace 
formula (10). 

If pressures P\g on two sides of the interface are different, the work of stress forces on fluid 1 at 
a small virtual displacement Sr = n Sr of the interface (where n = dAJdA is the unit vector nonnal to the 
interface) equals 5 

SW = dA5r{P l -P 2 ). (8.8) 

For equilibrium, this work has to be compensated by an equal change of the interface energy, SUi = 
ySidA). Differential geometry tells us that in the linear approximation in Sr, the relative change of the 
elementary surface area, corresponding to a fixed solid angle dCl, may be expressed as 

+ (8.9) 

dA R , R 2 

where Rig are the so-called principal radii of the interface curvature. 6 Combining Eqs. (7)-(9), we get 
the Young-Laplace formula 7 

4 If y of the interface of certain two fluids is negative, it self-reconfigures to decrease U s by the interface area, i.e. 
fragments the system into a solution. 

5 Thi s equality readily follows from the general Eq. (7.32), with the stress tensor elements expressed by Eq. (2), 
but in this simple case of the net stress force d¥ = (/ J , - P 2 )d\ parallel to the interface element vector dA, it may 
be even more simply obtained just from the definition of work SW= dF-Sr at the virtual displacement Sr = n Sr. 

6 This general formula may be verified by elementary means for a sphere of radius r (for which R\= R 2 = r and dA 
= r 2 dCl, so that S(dA)/dA = S(r 2 )/r 2 = 2 Sr/r), and a round cylindrical interface of radius R (for which R\ = r, R 2 = 
oo, and dA = rdcpdz, so that S(dA)ldA = Sr/r). 

1 This formula (not to be confused with Eq. (12), called the Young’s equation) was derived in 1806 by P.-S. 
Laplace (of the Laplace operator/equation fame) on the basis of the first analysis of the surface tension effects by 
T. Y oung a year earlier. 

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-p 2 =AP ef = r 


R R 

2 y 

( 8 . 10 ) 




In particular, this formula shows that the additional pressure created by surface tension inside a 
spherical drop of a liquid, of radius R, equals 2 y/R, i.e. decreases with R. In contrast, according to Eqs. 
(5)-(6), the effects of bulk forces, for example gravity, grow as pgR. The comparison of these two 
pressure components shows that if the drop radius (or more generally, the characteristic linear size of a 
fluid sample) is much larger than the so-called capillary length 



a , = 

l PS) 

( 8 . 11 ) 



the surface tension may be safely ignored - as will be done in the following sections of this chapter, 
besides a brief discussion of Eq. (48). For the water surface, or more exactly its interface with air at 


ambient conditions, y~ 0.073 N/m, while p~ 1,000 kg/m , so that a c ~ 4 mm. 

On the other hand, in very narrow tubes, such as blood capillary vessels with radius a ~ 1 pm, 
i.e. a « a c , the surface tension effects are very important. The key notion for the analysis of these 
effects is the equilibrium contact angle 6 C (also called the “wetting angle”) at the edge of a liquid 
wetting a solid - see Fig. 3. 

Fig. 8.3. Contact angles 
for (a) hydrophilic and 
(b) hydrophobic surfaces. 

According to its definition (7), constant y may be interpreted as a force (per unit length of the 
interface boundary) directed along the interface and trying to reduce its area. As a result, the balance of 
horizontal components of the three such forces, shown in Fig. 3, immediately yields 

Ysx+Yxz cos 0 c =y sg , 

( 8 . 12 ) 



where the indices at constants y correspond to three possible interfaces between the liquid, solid and gas. 
For the so-called hydrophilic surfaces that “like to be wet” by this particular liquid (not necessarily 
water), meaning that y & i < y sg , this relation yields cos6f > 0, i.e. 6 C < nil - the situation shown in Fig. 3a. 
On the other hand, for hydrophobic surfaces with y s i > y sg , Young’s equation (12) yields larger contact 
angles, 6 C > nil - see Fig. 3b. 

Let us use this notion to solve the simplest but perhaps the most important problem of this field - 
find the height h of the fluid column in a narrow vertical tube made of a hydrophilic material, lifted by 
the surface tension forces, assuming its internal surface to be a round cylinder of radius a - see Fig. 4. 
Inside an incompressible fluid, pressure drops with height according to the Pascal equation (6), so that 
just below the surface, P ~ Po - pgh, where Po is the background (e.g., atmospheric) pressure. This 

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means that at a « h the pressure variation along the concave surface (called the meniscus ) of the liquid 
is negligible, so that according to the Young-Poisson equation (10) the sum (l/f?i + HRf) has to be 
virtually constant along the surface. Due to the axial symmetry of the problem, this means that the 
surface has to be a part of a sphere. 8 From the contact angle definition, radius R of the sphere is equal to 
a! cos0 c - see Fig. 4. 

Plugging this relation into Eq. (10) with P\-P 2 = pgh, we get the following equation for h: 

. 2rcos6> 

Pgh = 1 




In hindsight, this result might be obtained more directly - by requiring the total weight pgV = pg(m h ) 
of the lifted liquid’s column to be equal to the vertical component Fcos6 c of the full surface tension 
force F = yp acting on the perimeter p = 2 m of the meniscus. Using the definition (11) of the capillary 
length a c , Eq. (13a) may be presented as the so-called Jurin rule: 


according to our initial assumption h » a, Eq. (13) is only valid for narrow tubes, with radius a « a c . 
This capillary rise is the basic mechanism of lifting water with nutrients from roots to the branches and 
leaves of plants, so that the tallest tree height is practically established by the Jurin rule (13), with cos# c 
« 1 and the pore radius a limited from below by a few microns, because of the viscosity effects 
restricting the fluid discharge - see Sec. 5 below and in particular the Poiseuille formula (60). 

8.3. Kinematics 

In contrast to the stress tensor, which is useful and simple - see Eq. (2), the strain tensor is not a 
very useful notion in fluid mechanics. Indeed, besides a very few situations, 9 typical problems of this 
field involve fluid flow, i.e. a state when velocity of fluid particles has some nonzero time average. This 

8 Note that this is not true for tubes with different shapes of their cross-section. 

9 One of them is the sound propagation, where particle displacements q are typically small, so that results of Sec. 
7.7 are applicable. As a reminder, they show that in fluids, with p = 0, the transverse sound cannot propagate 
(formally, has zero velocity and impedance), while the longitudinal sound’s velocity is finite - see Eq. (7. 1 16). 

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means that the trajectory of each individual particle is a long line, and the notion of its displacement q 
becomes impracticable. However, particle’s velocity v = dq/dt is a much more useful notion, especially 
if it is considered as a function the observation point r and (generally) time t. In an important class of 
fluid dynamics problem, the so-called stationary (or “steady”, or “static”) flow, the velocity defined in 
this way does not depend on time, v = v(r). 

There is, however, a price to pay for the convenience of this notion: namely, due to the 
difference between vectors q and r, particle’s acceleration a = d~q/dt (that participates, in particular, in 
the 2 nd Newton law) cannot be calculated just as a time derivative of velocity v(r, t ). This fact is evident, 
for example, for the static flow case, in which the acceleration of individual fluid particles may be very 
significant even if v(r) does not depend on time - just think about the acceleration of a drop of water 
flowing over the Niagara Falls rim, first accelerating fast and then virtually stopping below, while the 
water velocity v at every particular point, as measured from a bank-based reference frame, is nearly 
constant. Thus the main task of fluid kinematics is to express a via v(r,f); let us do this. 

Since each Cartesian component Vj of the velocity has to be considered as a function of four 
independent scalar variables, three Cartesian components rp of vector r and time t, its full time 
derivative may be presented as 



dv : ^ dVjdr r 

— - + > — - — — . 
dt dry dt 


Let us apply this general relation to a specific set of infinitesimal changes {dr\, dn, dr 3 } that follows a 
small displacement dq of a certain particular particle of the fluid, dr = dq = \dt, i.e. 

dr j =v j dt. (8.15) 

In this case dvfdt is the y'-th component aj of the particle’s acceleration a, so that Eq. (14) yields the 
following key relation of fluid kinematics: 


Using operator V, this result may be rewritten in the following compact vector form : 10 


This relation already signals the main technical problem of the fluid dynamics: many equations 
involving particle’s acceleration are nonlinear in velocity, excluding such a powerful tool the linear 
superposition principle from the applicable mathematical arsenal. 




One more basic relation of the fluid kinematics is the so-called continuity equation, which is 
essentially just the differential version of the mass conservation law. Let us mark, inside a fluid flow, an 

10 The operator relation d/dt = 5/dt + (v-V), applicable to an arbitrary (scalar or vector) function, is frequently 
called the convective derivative. (Alternative adjectives, such as “Lagrangian”, “substantial”, or “Stokes”, are 
sometimes used for this derivative as well.) The relation has numerous applications well beyond the fluid 
dynamics - see, e.g., EM Chapter 9 and QM Chapter 1. 

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arbitrary volume V limited by stationary (time-independent) surface S. The total mass of the fluid inside 
the volume may change only due to its flow though the boundary: 

= — [ pd 3 r = -[ pv n d 2 r = -f px-dA, (8.17a) 

dt dt * * • 

where the elementary area vector dA is defined just as in Sec. 7.2 - see Fig. 5. 

Using the same the same divergence theorem that has been used several times in this course, 11 
the surface integral in Eq. (17a) may be transformed into the integral of V(pv) over volume V, so that 
this relation may be rewritten as 

+ V j 

d 3 r = 0 . 


where vector j = pv defined is called either the mass flux density or the mass curren t. Since Eq. (17b) is 
valid for an arbitrary volume, the function under the integral has to vanish at any point: 





+ Vj = 0. 


Note that such continuity equation is valid not only for mass, but for other conserved physics 
quantities (e.g., the electric charge, quantum-mechanical probability, etc.), with the proper re-definition 
of /?and j. 12 

8.4. Dynamics: Ideal fluids 

Let us start our discussion of fluid dynamics from the simplest case when the stress tensor obeys 
the simple expression (2) even at the fluid motion. Physically, this means that fluid viscosity effects, 
including mechanical energy loss, are negligible. (We will discuss the conditions of this assumption in 
the next section.) Then the equation of motion of such an ideal fluid (essentially the 2 nd Newton law for 
its unit volume) may be obtained from Eq. (7.25) using the simplifications of its right-hand part, 
discussed in Sec. 1: 

pa = -VP + f. (8.19) 

11 If the reader still needs a reminder, see MA Eq. (12.1). 

12 See, e.g., EM Sec. 4.1 and QM Sec. 1.4. 

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Now using the basic kinematic relation (16), we arrive at the following Euler equation : 


p h p{\ • V )v = -VP + f . 



Generally this equation has to be solved together with the continuity equation (11) and equation 
of state of the particular fluid, p = p(P). However, as we have already discussed, in many situations the 
compressibility of water and other important fluids is very low and may be ignored, so that p may be 
treated as a given constant. Moreover, in many cases the bulk forces f are conservative and may be 
presented as a gradient of a certain potential function u( r) - the potential energy per unit volume: 

f = -Vw ; 

( 8 . 21 ) 

for example, for a uniform gravity field, u = pgh. In this case the right-hand part of Eq. (20) becomes - 
V(P + u). For these cases, it is beneficial to recast the left-hand of that equation as well, using the 
following well- kn ow identity of vector algebra 14 

( 2 ^l 

(v-V)v = V — -vx(Vxv). 

V 2 ) 

( 8 . 22 ) 

As a result, the Euler equation takes the form 

p /tvx(Vxv) + V 


2 A 

P + u + p — 

= 0 . 


In a stationary flow, the first term of this equation vanishes. If the second term, describing fluid’s 
vorticity, is zero as well, then Eq. (23) has the first integral of motion, 

P + u + ^-v 2 

= const , 


called the Bernoulli equation. Numerous examples of application of Eq. (17) to simple problems of 
stationary flow in pipes, in the Earth gravity field (giving u = pgh), should be well known to the reader, 
so I hope I can skip their discussion without much harm. 

In the general case an ideal fluid may have vorticity, so that Eq. (24) is not always valid. 
Moreover, due to absence of viscosity in an ideal fluid, the vorticity, once created, does not decrease 
along the streamline - the fluid particle’s trajectory, to which the velocity is tangential in every point. 15 
Mathematically, this fact 16 is expressed by the following Kelvin theorem : ( Vxv)-<7A = const along any 
small contiguous group of streamlines crossing an elementary area c/A. 17 

13 It was derived in 1755 by the same L. Euler whose name has already been (reverently) mentioned several times 
in this course. 

14 It readily follows, for example, from MA Eq. (1 1.6) with g = f = v. 

15 Perhaps the most spectacular manifestation of the vorticity conservation are the famous toroidal vortex rings 
(see, e.g., nice photo and movie at ring) , predicted in 1858 by H. von 
Helmholtz, and then demonstrated by P. Tait in a series of spectacular experiments with smoke in air. The 
persistence of such a ring, once created, is only limited by fluid’s viscosity - see the next section. 

16 First formulated verbally by H. von Helmholtz. 

17 Its proof may be found, e.g., in Sec. 8 of L. Landau and E. Lifshitz, Fluid Mechanics, 2 nd ed., Butterworth- 
Heinemann, 1987. 





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In many important cases the vorticity of fluid is negligible. For example, if a solid body of 
arbitrary shape is embedded into an ideal fluid that is uniform (meaning, by definition, that v(r ,t) = Vo = 
const) at large distances, its vorticity is zero everywhere. (Indeed, since Vxv at the uniform flow, the 
vorticity is zero at distant points of any streamline, and according to the Kelvin theorem, should equal 
zero everywhere.) In this case the velocity, as any curl-free vector field, may be presented as a gradient 
of some effective potential function, 

v = -V (/). (8.25) 

Such potential flow may be described by a simple differential equation. Indeed, the continuity equation 
(18) for a steady flow of an incompressible fluid is reduced to V-v = 0. Plugging Eq. (25) into this 
relation, we get the scalar Laplace equation, 

VV = 0, (8.26) 

which should be solved with appropriate boundary conditions. For example, the fluid flow may be 
limited by solid bodies inside which that the fluid cannot penetrate. Then the fluid velocity at these 
boundaries should not have a normal component: 

d(f> _ 


On the other hand, at large distances from the body in question the fluid flow is known, e.g., uniform: 

V^ = -v 0 , atr— >go. (8.28) 

As the reader may already kn ow (for example, from a course of electrodynamics 18 ), the Laplace 
equation (26) is readily solvable analytically in several simple (symmetric) but important situations. Let 
us consider, for example, the case of a round cylinder, with radius R, immersed into a flow with the 
initial velocity Vo perpendicular to the cylinder axis (Fig. 6). 19 

For this problem, it is natural to use cylindrical coordinates with axis z parallel to cylinder’s axis. 
In this case the velocity distribution is evidently independent of z, so that we may simplify the general 
expression of the Laplace operator in cylindrical coordinates 20 by taking d/dz = 0. As a result, Eq. (26) is 
reduced to 21 

_ 1 _ 

P dp 

d ( d(j)^ 



1 d 2 (j) 
p 2 d6 2 

= 0, at p > R. 


The general solution of this equation may be obtained using the variable separation method: 22 

18 See, e.g., EM Secs. 2.3 and 2.4. 

19 Evidently, motion of the cylinder, with constant velocity (-Vo), in the otherwise stationary fluid leads to exactly 
the same problem - in the reference frame bound to the moving body. 

20 See, e.g., MAEq. (10.3). 

21 Let me hope that letter p , used here for the magnitude 2D radius-vector p = {x, y}, will not be confused with 
fluid’s density - which does not participate in this boundary problem. 

22 See, e.g., EM Eq. (2. 1 12). Note that the most general solution of Eq. (29) also includes a term proportional to 
(p, but this term should be zero for such a single-valued function as the velocity potential. 

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00 / \ 
(f> = a Q +b () In P + X( c » cosrup + s n sinmp\a lt p" +b n p 

where coefficients a„ and b„ have to be found from the boundary conditions (27) and (28). Choosing 
axis x = rcoscp to be parallel to vector Vo (Fig. 6a) we may rewrite these the conditions in the form 

|^ = 0, at p = R, 

(j> — > -v 0 pcos(p + (j) 0 , at p»R, 



where <fx) is an arbitrary constant, which does not affect the velocity distribution, and may be taken for 
zero. The latter condition is incompatible with all tenns of Eq. (30) except the term with n = 1 (with si = 
0 and c\a\ = - v 0 ) , so it is reduced to 

f cb \ 

(f)= —v^p -\ — - — — cos cp. 

\ P ) 

Now, plugging this solution into Eq. (31), we get c x b l = -v 0 R 2 , so that, finally, 


0 = -v o p + — cosy?. 

I P J 


Fig. 8.6. Flow of ideal, incompressible fluid around a round cylinder: (a) equipotential surfaces and 
(b) streamlines. 

Figure 6a shows the surfaces of constant velocity potential (j). In order to find the fluid velocity, it 
is easier to rewrite result (34) in the Cartesian coordinates x = /xosr/y y = /isiru/i: 

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(f) = -v 0 x 

f . R 2A 

1 + ~~2 
V P J 

1 + 


2 A 

X +y 


From this equation, we may readily calculate the Cartesian components v x = - 80/dx and v y = - d(f)ldy of 
the fluid velocity. Figure 6b shows particle streamlines. 23 One can see that the largest potential gradient, 
and hence the maximum speed, is achieved at points near the vertical diameter ( p = R , <p = ± tz/ 2), where 

R=r t = 2v o- (8-36) 


Now the pressure distribution may now be found from the Bernoulli equation (24). For w(r) = 0, 
it shows that the pressure reaches maximum at the ends of the longitudinal diameter y = 0, while at the 
ends of the transverse diameter x = 0, where the velocity is largest, it is less by 2 pv 0 ~ (where p is the 
fluid density again - sorry for the notation jitters!) Note that the distributions of both velocity and 
pressure are symmetric about the transverse axis x = 0, so that the fluid flow does not create any net 
drag force in its direction. This result, which stems from the conservation of the mechanical energy of 
an ideal fluid, remains valid for a solid body of arbitrary shape moving inside an infinite volume of such 
ideal fluid - the so-called D ’Alambert paradox. However, if a body moves near ideal fluid’s surface, its 
energy may be transformed into that of surface waves, and the drag becomes possible. 

Speaking about the surface waves in a gravity field 24 , their description is one more classical 
problem of the ideal fluid dynamics. Let us consider an open surface of an ideal fluid of density p in a 
uniform gravity field f = pg = -pgn y ~ see Fig. 7. If the wave amplitude A is sufficiently small, we can 
neglect the nonlinear term (v-V)v oc A in the Euler equation (13) in comparison with the first term, 
dv/dt, that is linear in A. For a wave with frequency co and wavenumber k, particle’s velocity v = dq/dt is 
of the order of coA, so that this approximation is legitimate if of A » k(coA)~, i.e. when 

kA « 1, (8.37) 

i.e. when the wave amplitude is much smaller than its wavelength X = 2nlk. By this assumption, we may 
neglect the fluid vorticity effects, and again use Eq. (25) and (for an incompressible fluid) Eq. (26). 

Fig. 8.7. Small “plane” surface wave on 
a deep fluid. Dashed lines show fluid 
particle trajectories. (For clarity, the 
displacement amplitude A = (A7ft>)O 0 is 
strongly exaggerated.) 

23 They may be found by integration of the evident equation dy/dx = vfx,y)/v x {x,y). For our simple problem this 
integration may be done analytically, giving the relation y[l - R 2 /(x 2 + y 2 )] = const, where the constant is specific 
for each streamline. 

24 The alternative, historic term “gravity waves” for this phenomenon may nowadays lead to a confusion with the 
relativistic effect of gravity waves (which may propagate in vacuum), whose direct detection is a focus of so 
much current experimental effort. 

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Looking for the solution of the Laplace equation (26) in the natural form of a ID sinusoidal 

wave, 25 

(j) = Re ®( y)e i(kz ~ at) , (8.38) 

we get a simple equation 

7 2>K 

-k 2 O = 0, (8.39) 


with an exponential solution (decaying, as it has to, at y — » - oo) ® = ct>oexp{Ay}, so that Eq. (38) 

^ = Re ^ 0 e ky e i(kz ~ OJt) =® 0 e k} ’cos (kz-cot), (8.40) 

where the last form is valid if Oo is real - which may be always arranged by a proper selection of origins 
of z and/or t. Note that the rate of wave decay with depth is exactly equal to the wavenumber of its 
propagation along the surface. Because of that, the trajectories of fluid particles are exactly circular. 
Indeed, using Eqs. (25) and (40) (with amplitude) to calculate velocity components, 

v x = 0, v y = ~~~ = cos(kz - cot), v. = ~~~ = sin (kz - cot), (8.41) 

we see that they have equal real amplitudes, and are phase-shifted by nil. This result becomes even 
more clear if we use the velocity definition v = dq/dt to integrate Eqs. (41) over time to recover the 
particle displacement law q(f). Due to the strong inequality (37), the integration may be done at fixed y 
and z: 

q = — sin {kz - cot), q, = — cos(kz - cot ) . (8.42) 

co ' co 

Note that the phase of oscillations of v = coincides with that of q v . It means, in particular, that at wave’s 
top (“crest”), fluid particles are moving in the direction of wave’s propagation - see arrows in Fig. 7. 

It is remarkable that all this picture follows from the Laplace equation alone! The “only” 
remaining feature to calculate is the dispersion law oik), and for that we need to combine Eq. (40) with 
what remains, in our linear approximation, of the Euler equation (23). In this approximation, and with 
the bulk force potential u = pgy, this equation is reduced to 

( dd 

V -p^ + P + pgy =0. (8.43) 

This equation means that the function in the parentheses is constant in space; at the surface, it should 
equal to pressure P 0 above the surface (say, the atmospheric pressure), that we assume to be constant. 
This means that on the surface, the contributions to P that come from the first and the third tenn in Eq. 
(43), should compensate each other. Let us take the average surface position for y = 0; then the surface 
with waves is described by relation y = q y - see Fig. 7. Due to the strong relation (37), which means k\q v \ 
« 1, we can use Eqs. (40) and (42) with y = 0, so that the above compensation condition yields 

25 Such a wave is “plane” only in direction x (perpendicular to the propagation direction z, see Fig. 4). 

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- pcoft> Q sin {kz - cot) + pg — <& 0 sin(fe - cot) = 0 . 



This condition is identically satisfied on the whole surface (and for any ®o) as soon as 

co 2 = gk , 

this equality giving the dispersion relation we were looking for. 


Looking at this surprisingly simple result (which includes just one constant, g), note, first of all, 
that it does not involved fluid’s density. This is not too much surprising, because due to the weak 
equivalence principle, particle masses always drop out of the results of problems involving gravitational 
forces alone. Second, the dispersion law (45) is strongly nonlinear, and in particular does not have the 
acoustic wave limit. This means that the surface wave propagation is strongly dispersive, with the phase 
velocity colk oc Hco diverging at co — > 0. This divergence is an artifact of our assumption of the infinite 
fluid thickness. A rather straightforward generalization of the above calculations to a layer of finite 
thickness h, using the additional boundary condition v y \ y=.h = 0, yields the following modified dispersion 

co 2 = gk tanh kh . 


It shows that relatively long waves, with A » h, i.e. with kh « 1, propagate without dispersion (i.e. 
have co/k = const = v), with velocity 

v = {gh) V2 . (8.47) 

For the Earth oceans, this velocity is rather high, approaching 300 m/s (!) for h = 10 km. This result 
explains, in particular, the very fast propagation of tsunami waves. 

In the opposite limit of very short waves (large k), Eq. (45) also does not give a good description 
of experimental data, due to the effects of surface tension (see Sec. 2 above). Using Eq. (8.10), it is easy 
(and hence left for the reader :-) to show that their account leads (at kh » 1) to the following 
modification of Eq. (45): 

co 2 = g k + ^. (8.48) 


According to this fonnula, the surface tension is important at wavelengths smaller than the capillary 


constant a s given by Eq. (11). Much shorter waves, for whom Eq. (48) yields co oc k , are called the 
capillary waves - or just “ripples”. 

All these generalizations are still limited to potential forces, and do not allow one to describe 
energy loss, in particular the attenuation of either bulk or surface waves in fluids. For that, as well as for 
the drag force description, we need to proceed to the effects of viscosity. 

8.5. Dynamics: Viscous fluids 

Fluid viscosity of many fluids, at not too high velocities, may be described surprisingly well by 
adding, to the static stress tensor (2), additional components proportional to velocity v = dq/dt: 

<r, r =-PS ir +a l J\). 


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Since the Hooke law (7.34) has taught us about the natural structure of such a tensor in the case of stress 
proportional to displacement q, we may expect a similar expression with replacement q — » v = dq/dt: 

a ir = 111 e„ - ^..,Tr(e)l + 3^f ^ /y ,Tr(e)\ . 


where cy ■ are the elements of the symmetrized strain derivative tensor: 

ds , 



j + dv r 

dr., dr, 


J J 


Experiment confirms that Eq. (50) gives a good description of the viscosity effects in a broad range of 
isotropic fluids. Coefficient r] is called either the shear viscosity, or the dynamic viscosity, or just 
viscosity, while £ is called the second (or bulk ) viscosity. 

In the most frequent case of a virtually incompressible fluid, Tr(u) = d[Tr(s)\/dt = ( dV/dt)/V= 0, 
so that the term proportional to vanishes, and // is the only important viscosity parameter. 26 Table 1 
shows the approximate values of the viscosity, together with the mass density p, for several common 
fluids. One can see that i] may vary in extremely broad limits; the extreme cases are glasses (somewhat 
counter-intuitively, these amorphous materials are not stable solids even at room temperature, but rather 
may “flow”, though extremely slowly, until they eventually crystallize) and liquid helium. 27 

Fluid (all at 300 K, besides the helium) 

ij (mPa-s) 

p (kg/m 3 ) 


10 21 -10 24 


Machine oils (SAE 10W - 40 W) 









Liquid helium 4 (at 4.2K, 10 5 Pa) 



Air (at 10 5 Pa) 



Incorporating the additional components of ■ to the equation (20) of fluid motion, absolutely 
similarly to how it was done at the derivation of Eq. (7.109) of the elasticity theory, with the account of 
Eq. (16) we arrive at the famous Navier-Stokes equation: 28 





26 Probably the most important effect we miss by neglecting f is the attenuation of (longitudinal) acoustic waves, 
into which the second viscosity makes a major (and in some cases, the main) contribution. 

27 Actually, at even lower temperatures (for He 4, T < T A » 2.17 K), helium becomes a superjluid , i.e. looses 
viscosity completely, as result of the Bose-Einstein condensation - see, e.g., SM Sec. 3.4. 

28 Named after C.-L. Navier (1785-1836) who had suggested the equation, and G. Stokes (1819-1903) who has 
demonstrated its relevance by solving it for several key situations. 

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The apparent simplicity of this equation should not mask an enormous range of phenomena, 
notably including turbulence (see the next section), that are described by it, and the complexity of its 
solutions even for some simple geometries. In most problems interesting for practice the only option is 
to use numerical methods, but due to the large number of parameters ( p , //, £ plus geometrical 
parameters of the involved bodies, plus the distribution of bulk forces f, plus boundary conditions), this 
way is strongly plagued by the “curse of dimensionality” that was discussed in Sec. 4.8. 

Let us see how does the Navier-Stokes equation work, on several simple examples. As the 
simplest case, let us consider the so-called Couette flow caused in an incompressible fluid layer between 
two wide, horizontal plates (Fig. 8) by mutual sliding of the plates with a constant relative velocity v 0 . 

Fig. 8.8. The simplest problem of 
the viscous fluid flow. 

Let us assume a laminar (vorticity-free) fluid flow. (As will be discussed in the next section, this 
assumption is only valid within certain limits.) Then we may use the evident symmetry of the problem, 
to take, in the reference frame shown in Fig. 8, v = n z v(y). Let the bulk forces be vertical, f = n f , so 
they do not give an additional drive to fluid flow. Then for the stationary flow (dw/d t = 0), the vertical, 
v-component of the Navier-Stokes equation is reduced to the static Pascal equation (3), showing that the 
pressure distribution is not affected by the plate (and fluid) motion. In the horizontal, z-component of the 
equation only one term, V v, survives, so that for the only Cartesian component of velocity we get the 
ID Laplace equation 


In contract to the ideal fluid (see, e.g., Fig. 6b), the relative velocity of a viscous fluid and a solid 
wall it flows by should approach zero at the wall, 29 so that Eq. (52) should be solved with boundary 

0, at y = 0, 

v = \ 

[v 0 , at y = d. 


Using the evident solution of this boundary problem, v(y) = (y/d)v 0 , illustrated by arrows in Fig. 8, we 
can now calculate the horizontal drag force acting on a unit area of each plate. For the bottom plate, 

F. | dv\ v 0 


29 This is essentially an additional experimental fact, but may be readily understood as follows. A solid may be 
considered as an ultimate case of a fluid (with infinite viscosity), and the tangential component of velocity should 
be a continuous an interface between two fluids, in order to avoid infinite stress - see Eq. (50). 

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(For the top plate, the derivative dv/dy has the same value, but the sign of dA y has to be changed to 
reflect the direction of the outer nonnal to the solid surface, so that we get a similar force but with the 
negative sign.) The well-known result (54) is often used, in undergraduate courses, for a definition of 
the dynamic viscosity 77 , and indeed shows its physical meaning very well. 

As the next, slightly less trivial example let us consider the so-called Poiseuille problem 30 of the 
relation between the constant external pressure gradient % = -dP/dz applied along a round pipe with 
internal radius R (Fig. 9) and the so-called discharge Q - defined as the mass of fluid flowing through 
pipe’s cross-section per unit time. 



P i 



-* v(p) 




Fig. 8.9. The Poiseuille problem. 

Again assuming a laminar flow, we can involve the problem uniformity along the z axis and its 
axial symmetry to infer that v = n z v(p), and P = -%z + J{p, cp) + const (where p = {p, cp} is the 2D radius- 
vector rather than fluid density), so that the Navier-Stokes equation (44) for an incompressible fluid 
(with V-v = 0) is reduced to a 2D Poisson equation 

pV;v = - X . (8.55) 

After spelling out the 2D Laplace operator in polar coordinates for our axially-symmetric case d/dcp = 0, 
Eq. (55) becomes a simple ordinary differential equation, 

1 d dv 

d — — (p—) = -X, (8-56) 

p dp dp 

that has to be solved at the segment 0 < p<R, with the following boundary conditions: 

at p = R, 

v = 0 , 

— = 0 , at p = 0 . 


(The latter condition is required by the axial symmetry.) A straightforward double integration yields: 

v = j-{R 2 -p 2 ), 


so that the integration of the mass flow density over the cross-section of the pipe, 


Q = \pvd 1 r = 2np^-\{R 1 - p ,2 )p'dp', 

A 4 d 0 



immediately gives us the so-called Poiseuille (or “Hagen-Poiseuille”) law for the fluid discharge: 

30 It was solved theoretically by G. Stokes in 1845 in order to explain Eq. (60) that had been formulated by J. 
Poiseuille in 1840 on the basis of his experimental results. 

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Q = 





where (sorry!) p is the mass density again. 


Fig. 8.10. Application of the finite- 
difference method with a very coarse 
mesh (with step h = a/2) to the 
problem of viscous fluid flow in a 
pipe with a square cross-section. 

Of course, not for each cross-section shape the 2D Poisson equation (55) is so readily solvable. 
For example, consider a very simple, square-shape cross-section with side a (Fig. 10). For it, it is natural 
to use the Cartesian coordinates, so that Eq. (55) becomes 

<3 2 v 8~v y 

— j = = const, for0<x,jp<a, (8.61) 

dx dy ~ r] 

and has to be solved with boundary conditions 

v = 0, atx,y = 0,T. (8.62) 

For this boundary problem, analytical methods 31 give answers in the form of an infinite series 
that ultimately require computers for their plotting and comprehension. Let me use this pretext to 
discuss how explicitly numerical methods may be used for such problems - or any partial differential 
equations involving the Laplace operator. The simplest of them is the finite-difference method 32 in 
which the function to be calculated, f[r\,r 2 ,...), is represented by its values in discrete points of a 
rectangular grid (frequently called mesh ) of the corresponding dimensionality (Fig. 1 1). 








Fig. 8.11. Idea of the finite- 
difference method in (a) one and 

h ' 

(b) two dimensions. 

31 For example, the Green’s function method (see, e.g., EM Sec. 2.7). 

32 For more details see, e.g., R. J. Leveque, Finite Difference Methods for Ordinary > and Partial Differential 
Equations, SIAM, 2007. 

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In Sec. 4.7, we have already discussed how to use such a grid to approximate the first derivative 
- see Eq. (4.98). Its extension to the second derivative is straightforward - see Fig. 11a: 

d 2 f 


~ 1 

_p_ 1 

~ 1 

|7W /-/«-] 

, dr o 

~ h 

1%, Sr ij) 

~ ~h 

h h 

/-,+/«- - 2 / 
h 1 


The relative error of this approximation is of the order of h 2 ^/dr/, quite acceptable in many cases. As a 
result, the left-hand part of Eq. (61), treated on a square mesh with step h (Fig. 1 lb), may be presented 
as the so-called 5-point scheme : 

d 2 v d 2 v 
~dx 2+ ~d/ 

+ - 2v V T + V i - 2v 

v_> +iy +V T +v ; -4v 
h 2 


(The generalization to the 7 -point scheme, appropriate for 3D problems, is straightforward.) 

Let us apply this scheme to the pipe with the square cross-section, using an extremely coarse 
mesh with step h = all (Fig. 10). In this case the fluid velocity v should equal zero on the walls, i.e. in 
all points of the five-point scheme (Fig. 1 lb) except for the central point (in which velocity is evidently 
the largest), so that Eqs. (61) and (64) yield 33 

0 + 0 + 0 + 0 - 4v max ^ x 

{all) 2 ~ r) ’ 


1 %a 2 

16 77 


The resulting expression for the maximal velocity is only ~20% different from the exact value. 
Using a slightly finer mesh with h = a! A, which gives a readily solvable system of 3 linear equations for 
3 different velocity values (the exercise highly recommended to the reader), brings us within a couple 
percent from the exact result. This shows that such “numerical” methods may be more efficient 
practically than the “analytical” ones, even if the only available tool is a calculator app on your 
smartphone rather than an advanced computer. 

Of course, many practical problems of fluid dynamics do require high-performance computing, 
especially in conditions of turbulence (see the next section) with its complex, irregular spatial-temporal 
structure. In these conditions, the finite-difference approach may become unsatisfactory, because is 
implies the same accuracy of derivative approximation through the whole volume. A more powerful (but 
also much more complex for implementation) approach is the finite-element method in which the 
discrete point mesh is based on triangles with uneven sides, and is (in most cases, automatically) 
generated in accordance with the system geometry - see Fig. 12. Unfortunately I do not have time for 
going into the details of that method, so the reader is referred to the special literature on this subject . 34 

Before proceeding to our next topic, let me note one more important problem that is analytically 
solvable using the Navier-Stokes equation (51): a slow motion of a solid sphere of radius R, with a 
constant velocity v 0 , through an incompressible viscous fluid - or equivalently, a slow flow of the fluid 

33 Note that value (65) is exactly the same as given for v max = v\p=o by the analytical formula (58) for the round 
cross-section with radius R = all. This is not an occasional coincidence. The velocity distribution given by (58) is 
a quadratic function of both x and y. For such functions, with all derivatives higher than d 2 /dr 2 being equal to 
zero, equation (64) is exact rather than approximate. 

34 See, e.g., C. Johnson, Numerical Solution of Partial Differential Equations by the Finite Element Method, 
Dover, 2009, or T. J. R. Hughes, The Finite Element Method, Dover, 2000. 

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(uniform at large distances) around an immobile sphere. Indeed, in the limit v — > 0, the second term in 
the left-hand part of this equation is negligible (just as at the surface wave analysis in Sec. 3), and the 
equation takes the form 

-VP + rjV 2 y = 0, (8.66) 

which should be complemented with the incompressibility condition V-v = 0 and boundary conditions 

v = 0, at r - R, 
v — » v 0 , at r — » oo. 


In spherical coordinates, with the polar axis directed along vector vo, this boundary problem has the 
axial symmetry (so that d\!d(p= 0 and v (ZJ = 0), and allows the following analytical solution: 

v r = v 0 cos 6 

\ 3 R R 3 A 

1 + r 

2 r 2 r 2 j 

v e = _v o s i n & 

\_ 3R _ Ri ^ 

4 r 4 r 2 j 

( 8 . 68 ) 












Fig. 8.12. Typical finite-element mesh generated 
automatically for an object of complex geometry - 
in this case, a plane wing’s cross-section. (Figure 
adapted from .') 

Calculating pressure from Eq. (66), and integrating it over the surface of the sphere it is now 
straightforward to obtain the famous Stokes formula for the drag force acting on the sphere: 



F = 6 7ZTJ RVq . 


Historically, this formula has played an important role in the first precise (with accuracy better than 1%) 
calculation of the fundamental electric charge e by R. Millikan and H. Fletcher from their famous oil 
drop experiments in 1909-1913. 

8.6. Turbulence 

The Stokes formula (69), whose derivation is limited to low velocities at that the nonlinear term 
(v-V)v could be neglected, become invalid if the fluid velocity is increased. For example, Fig. 13 shows 
the drag coefficient defined as 

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pv\At 2 


where A is the cross-section of the body as seen from the fluid flow direction, for a sphere of radius R 
(so that A = nR 2 ), as a function of the so-called Reynolds number? 5 for this particular geometry defined 

Re _ PV o(2g) = pv 0 D 

V V 


Fig. 8.13. The drag coefficient for a sphere and a round thin disk as functions of the Reynolds number. 

Adapted from F. Eisner, Das Wider standsproblem, Proc. 3 rd Int. Cong. On Appl. Mech., Stockholm, 1931. 

In this notation, the Stokes formula (69) reads Cd = 24/Re. One can see this formula is only valid 
at Re « 1, while at larger velocities the drag force becomes substantially higher than that prediction, 
and its dependence on velocity very complicated, so that only its general, semi-quantitative features may 
be readily understood from simple arguments. 36 

The reason for this complexity is a gradual development of very intricate, time-dependent fluid 
patterns, called turbulence, rich with vortices - for an example, see Fig. 14. These vortices are 
especially pronounced in the region behind the moving body (so-called wake), while the region before 

35 This notion was introduced in 1851 by the same G. Stokes, but eventually named after O. Reynolds who 
popularized it three decades later. 

36 For example, Fig. 13 shows that, within a very broad range of Reynolds numbers, from ~10 2 to ~3xl0 5 , Cd for 
sphere is of the order of (and for a flat disk, remarkably close to) unity. This level, i.e., the approximate equality F 
~ pvoA/2, may be understood (in the picture where the object is moved by an external force F with velocity Vo 
through a fluid which is initially at rest) as the equality of force’s power Fvq and fluid’s kinetic energy (pvy/2) V 
created in volume V = vo A in unit time. This relation would be exact if the object gave velocity Vo to each and 
every fluid particle its cross-section runs into, for example by dragging all such particles behind itself. In reality, 
much of this kinetic energy goes into vortices - see Fig. 14 and its discussion below. 

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the body is virtually unperturbed. Figure 14 indicates that turbulence exhibits rather different behaviors 
in an extremely broad range of velocities (i.e. values of Re), and sometimes changes rather abruptly - 
see, for example, the significant drag drop at Re ~ 5x10 s . 



fluid flow 



Fig. 8.14. Snapshot of the turbulent tail (wake) behind a sphere moving in a fluid with a 
high Reynolds number, showing the so-called von Karman vortex street. A nice animation 
of such a pattern may be found at number . 

In order to understand the conditions of this phenomenon, let us estimate the scale of various 
terms in the Navier-Stokes equation (51) for the generic case of a body with characteristic size / moving 
in an otherwise static, incompressible fluid, with velocity v. In this case the characteristic time scale of 
possible non-stationary phenomena is given by the ratio //v, 37 so that we arrive at the following 

Equation term: 
Order of magnitude: 

yC»(v-V)v f pV 2 y 

V 2 V 

P T pg 


(I have skipped term VP, because as we saw in the previous section, in typical fluid flow problems it 
balances the viscosity term, and hence is of the same order of magnitude.) This table shows that relative 
importance of the terms may be characterized by two dimensionless ratios. 38 

The first of them is the so-called Froude number 

37 The time scale of some problems may be different from //v; for example, for forced oscillations of a fluid flow 
it is given by the reciprocal oscillation frequency f. For such problems, ratio S = /(// v) serves as another, 
independent dimensionless constant, commonly called either the Strouhal number or the reduced frequency. 

38 For substantially compressible fluids (e.g., gases), the most important additional dimensionless parameter is the 
Mach number M = v/v/, where v; = (Kip) 112 is the velocity of the longitudinal sound - which is, as we already 
know, the only wave mode possible in an infinite fluid. Especially significant for practice are supersonic effects 
(including the shock wave in the form of the famous Mach cone with half-angle (f i = arcsin M' 1 ) which arise at 
M> 1. For a more thorough discussion of these issues, I have to refer the reader to more specialized texts - e.g., 
Chapter IX of the Landau and Lifshitz volume cited above, or Chapter 15 in I. M. Cohen and P. K. Kundu, Fluid 
Mechanics, 4 th ed., Academic Press, 2007 - which is generally a good book on the subject. Another popular, rather 
simple textbook is R. A. Granger, Fluid Mechanics, Dover, 1995. 

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_ 2 / 7 2 

/2V // _ V 


which characterizes the relative importance of bulk gravity - or, upon an appropriate modification, other 
bulk forces. In most practical problems (with the important exception of surface waves, see Sec. 4 
above) F » 1 , so that the gravity effects may be neglected. 

Much more important is another ratio, the Reynolds number (71), in the general case defined as 


(. ' v number 

which is a measure of the relative importance of the fluid particle’s inertia in comparison with the 
viscosity effects. 39 Thus, it is not quite surprising that for a sphere, the role of the vorticity-creating term 
(v-V)v becomes noticeable already at Re ~ 1 — see Fig. 13. Much more surprising is the onset of 
turbulence in systems where the laminar (turbulence-free) flow is formally an exact solution to the 
Navier-Stokes equation for any Re. For example, at Re > Re t ~ 2,100 (with l = 2R and v = v max ) the 
laminar flow in a round pipe, described by Eq. (58), becomes unstable, and the resulting turbulence 
decreases the fluid discharge Q in comparison with the Poiseuille law (60). Even more strikingly, the 
critical value of Re is rather insensitive to the pipe wall roughness. 

Since Re » 1 in many real-life situations, 40 turbulence is very important for practice. However, 
despite nearly a century of intensive research, there is no general, quantitative analytical theory of this 
phenomenon, 41 and most results are still obtained either by rather approximate analytical treatments, or 
by the numerical solution of the Navier-Stokes equations using the approaches discussed in the previous 
section, or in experiments (e.g., on scaled models 42 in wind tunnels). 

Unfortunately, due to the time/space restrictions, for a more detailed discussion of these results I 
have to refer the reader to more specialized literature, 43 and will conclude the chapter with a brief 
discussion of just one issue: can the turbulence be “explained by a singe mechanism”? (In other words, 
can it be reduced, at least on a semi-quantitative level, to a set of simpler phenomena that are commonly 
considered “well understood”?) Apparently the answer in no, 44 though nonlinear dynamics of simpler 
systems may provide some useful insights. 

Res pdll = p± t 

Tjv/l 2 77 

39 Note that the “dynamic” viscosity 77 participates in this number (and many other problems of fluid dynamics) 
only in the combination ////? that thereby has deserved a special name of kinematic viscosity. 

40 For example, the values of 77 and p for water listed in Table 1 imply that for a few-meter object, Re > 1,000 at 
any speed above just ~1 mm/s. 

41 A rare exception is the relatively recent theoretical result by S. Orszag (1971) for the turbulence threshold in a 
flow of an incompressible fluid through a gap of thickness 7 between two parallel plane walls: Re, « 5,772 (for / = 
7/2, v = Vmax). However, this result does not predict the turbulence patterns at Re > Re,. 

42 The crucial condition of correct modeling is the equality of the Reynolds numbers (74) (and if relevant, also of 
the Froude numbers and/or the Mach numbers) of the object of interest and its model. 

43 See, e.g., P. A. Davidson, Turbulence, Oxford U. Press, 2004. 

44 The following famous quote is attributed to W. Heisenberg on his deathbed: “When I meet God, I will ask him 
two questions: Why relativity? And why turbulence? I think he will have an answer for the first question.” 
Though probably inaccurate, this story reflects rather well the understandable frustration of the fundamental 
physics community, known for their reductionist mentality, with the enormous complexity of phenomena which 
obey simple (e.g., Navier-Stokes) equations. 

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At the middle of the past century, the most popular qualitative explanation of turbulence had 
been the formation of an “energy cascade” that would transfer energy from larger to smaller vortices. 
With our background, it is easier to retell that story in the time-domain language (with velocity v serving 
as the conversion factor), using the fact that in a rotating vortex each component of the particle radius- 
vector oscillates in time, so that to some extent the vortex plays the role of an oscillatory motion mode. 
Let us consider the passage of a solid body between the two, initially close, small parts of fluid. The 
body pushes them apart, but after its passage these partial volumes are free to return to their initial 
positions. However, the domination of inertia effects at motion with Re » 1 means that the volumes 
continue to “oscillate” for a while about those equilibrium positions. (Since elementary volumes of an 
incompressible fluid cannot merge, these oscillations take the form of rotating vortices.) 

Now, from Sec. 4.8 we know that intensive oscillations in a system with quadratic nonlinearity, 
in this case provided by the convective tenn (v-V)v, are equivalent, for small perturbations, to the 
oscillation of the system parameters at the corresponding frequency. On the other hand, the discussion in 
Sec. 5.5 shows that in a system with two oscillatory degrees of freedom, a periodic parameter change 
with frequency co p may lead to non-degenerate parametric excitation of oscillations with frequencies 
a >\2 satisfying relation co\ + a >2 = co p . Moreover, the spectrum of oscillations in such system also has 
higher combinational frequencies such as ( co p + 0 )\ ), thus pushing the oscillation energy up the 
frequency scale. In the presence of other oscillatory modes, these oscillations may in turn produce, via 
the same nonlinearity, even higher frequencies, etc. In a fluid, the spectrum of these “oscillatory modes” 
(actually, vortex structures) is essentially continuous, so that the above arguments make very plausible a 
sequential transfer of energy to a broad spectrum of modes - whose frequency spectrum is limited from 
above by the energy dissipation due to viscosity. When excited, these modes interact (in particular, 
phase-lock) through system’s nonlinearity, creating the complex motion we call turbulence. 

Though not having much quantitative predictive power, such handwaving explanations, which 
are essentially based on the excitation of a large number of effective degrees of freedom, had been 
dominating the fluid dynamics reviews until the mid-1960s. At that point, the discovery (or rather re- 
discovery) of quasi-random motion in classical dynamic systems with just a few degrees of freedom 
altered the discussion substantially. Since this phenomenon, called the deterministic chaos, extends well 
beyond the fluid dynamics, and I will devote to it a separate (albeit short) next chapter, and in its end 
briefly return to the discussion of turbulence. 

8.7. Exercise problems 

8.1 . A solid round cylinder of radius r is let to float in a water 
inside the glass, also of a round cylindrical form with radius R - see 
Fig. on the right, which shows the water levels before and after the 
submersion, and some vertical dimensions of the system. Calculate the 
buoyant force F exerted by water on the floating body. 

Hint'. This is just a fast check whether the reader understands 
the Archimedes principle correctly. 

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8.2 . Pressure P under a free water surface crudely obeys the Pascal law, Eq. (6). Find the first- 
order corrections to this result, due to small compressibility of water. 

8.3 . Find the stationary shape of the open surface of an 
incompressible, heavy fluid rotated about a vertical axis with a constant 
angular velocity co - see Fig. on the right. 

8.4 .* Calculate the shape of the surface of an incompressible fluid of 
density p near a vertical plane wall, in a uniform gravity field - see Fig. on the 
right. In particular, find the height h of liquid’s rise at the wall surface as a 
function of the contact angle Q c . 

8.5 .* A soap film with surface tension y is stretched between two 
similar, coaxial, thin, round rings of radius R , separated by distance d- see Fig. 
on the right. Neglecting gravity, calculate the equilibrium shape of the film, and 
the force needed for keeping it stretched. 

8.6 . A solid sphere of radius R is kept in a steady, vorticity-free flow of an ideal incompressible 
fluid, with velocity v 0 . Find the spatial distribution of velocity and pressure, and in particular their 
extremal values. Compare the results with those obtained in Sec. 4 for a round cylinder. 

8.7 . * A small source, located at distance d from a plane wall of a container 
filled with an ideal, incompressible fluid of density p, injects additional fluid 
isotropically, at a constant mass current (“discharge”) Q = dMIdt - see Fig. on the 
right. Calculate fluid’s velocity distribution, and its pressure on the wall, created 
by the flow. 

Hint. Recall the charge image method in electrostatics, 45 and contemplate 
its possible analog. 

8.8 . Derive Eq. (46) for surface waves on a finite-thickness layer of a heavy liquid. 

8.9 . Derive Eq. (48) for the capillary waves (“ripples”). 

45 See, e.g., EM Secs. 2.6, 3.3, and 4.3. 

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8.10 . * Derive a 2D differential equation describing propagation of waves on the surface of a 
broad layer, of constant thickness h, of an ideal, incompressible fluid, and use it to calculate the longest 
standing wave modes and frequencies in a layer covering a spherical planet of radius R » h. 

Hint : The second assignment requires some familiarity with the basic properties of spherical 
harmonics . 46 

8.11 . Calculate the velocity distribution and dispersion relation of waves propagating along the 
horizontal interface of two ideal, incompressible fluids of different densities. 

8.12 . Calculate the energy of a monochromatic, plane surface wave on an, ideal, incompressible, 
deep fluid, and the power it carries (per unit width of wave’s front). 

8.13 . Use the finite-difference approximation for the Laplace operator, with mesh h = al A, to find 
the maximum velocity and total mass flow Q of a viscous, incompressible fluid through a long pipe with 
a square-shaped cross-section of side a. Compare the results with those described in Sec. 4 for: 

(i) the same problem with mesh h = at. 2 , and 

(ii) a pipe with circular cross-section of the same area. 


8.14 . A layer, of thickness h, of a heavy, viscous, 
incompressible fluid flows down a long and wide incline plane, 
under its own weight - see Fig. on the right. Find the stationary 
velocity distribution profile, and the total fluid discharge (per unit 

8.15. Calculate the torque exerted on a unit length of a solid round cylinder of radius R that 
rotates about its axis, with angular velocity co, inside an incompressible fluid with viscosity rj. 

8.16 . Calculate the tangential force (per unit area) exerted by incompressible fluid, with density 
p and viscosity 77 , on a broad solid plane placed over its surface and forced to oscillate, along the 
surface, with amplitude a and frequency ox 

8.17 . A massive barge, with a flat 
bottom of area A, floats in shallow water, 
with clearance h « A (see Fig. on the 
right). Calculate the time dependence of 
barge’s velocity V(t), and the water velocity 
profile, after the barge’s engine has been 
turned off. Discuss the limits of large and 
small values of the dimensionless parameter MtpAh. 


V(t) = ? 




P> V 

46 See, e.g., EM Sec. 2.5(iv) and/or QM Sec. 3.6. 

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8.18 . * Derive a general expression for mechanical energy loss rate in a viscous incompressible 
fluid that obeys the Navier-Stokes equation, and use this expression to calculate the attenuation 
coefficient of surface waves, assuming that the viscosity is small (quantify this condition). 

8.19 . Use the Navier-Stokes equation to calculate the attenuation coefficient for a plane, 
sinusoidal acoustic wave. 

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Chapter 9. Deterministic Chaos 

This chapter gives a very brief review of chaotic phenomena in deterministic maps and dynamic systems 
with and without dissipation, and an even shorter discussion of the possible role of chaos in fluid 

9.1. Chaos in maps 



Chaotic behavior of dynamic systems 1 (sometimes called the deterministic chaos ) has become 
broadly recognized 2 after the publication of a 1963 paper by E. Lorenz who was examining numerical 
solutions of the following system of three nonlinear, ordinary differential equations, 

f =afq 2 -q x ), 
q 2 = a 2 q x -q 2 -q x q 3 , 
q 3 =q x q 2 - a 2 q 2 . 


as a rudimentary model for heat transfer through a horizontal liquid layer between two solid plates. 
(Experiment shows that if the bottom plate is kept hotter than the top one, the liquid may exhibit 
turbulent convection.) He has found that within a certain range of constants < 21 , 2 , 3 , the solutions of Eq. (1) 
follow complex, unpredictable, non-repeating trajectories in the 3D g-space. Moreover, the resulting 
functions qft) (where j = 1 , 2,3) are so sensitive to initial conditions qfi 0) that at sufficiently large times 
t, solutions corresponding to slightly different initial conditions are completely different. 



Very soon it was realized that such behavior is typical for even simpler mathematical objects 
called maps, so that I will start my discussion of chaos from these objects. AID map is essentially a rule 
for finding the next number q n +\ of a series, in the simplest case using only its last kn own value q n , in a 
discrete series numbered by integer index n. The most famous example is the so-called logistic map : 3 

q n+ 1 =/(?„) = ^(i-?,,)- 


The basic properties of this map may be understood using the (hopefully, self-explanatory) 
graphical presentation shown in Fig. I. 4 One can readily see that at r < 1 (Fig. la) the map rapidly 
converges to the trivial fixed point q {0) = 0, because each next value of q is less than the previous one. 
However, if r is increased above 1 (as in the example shown in Fig. lb), fixed point q {0] becomes 
unstable. Indeed, at q„ « 1, map (2) yields q n + 1 = rq n , so that at r > 1, values q n grow with each 
iteration. Instead of the unstable point q (0) = 0, in the range 1 <r<r\, where r\ = 3, the map has a stable 
fixed point, q (l \ that may be found by plugging this value into both parts of Eq. (2): 

1 In this context, this term is understood as “systems described by deterministic differential equations”. 

2 Actually, the notion of quasi-random dynamics due to the exponential divergence of trajectories may be traced 
back at least to (apparently independent) works by J. Poincare in 1892 and by J. Hadamard in 1898. Citing 
Poincare, “ may happen that small differences in the initial conditions produce very > great ones in the final 
phenomena. [...] Prediction becomes impossible .” 

3 Its chaotic properties were first discussed in 1976 by R. May, though the map itself is one of simple ecological 
models repeatedly discussed earlier, and may be traced back at least to the 1838 work by P. Verhulst. 

4 Since the maximum value of function /(<?), achieved at q = Vi, equals r/4, the mapping may be limited by 
segment x = [0, 1], if parameter r is between 0 and 4. Since all interesting properties of the map, including chaos, 
may be found within these limits, I will focus on this range. 

© 2013 -2016 K. Likharev 

Essential Graduate Physics 

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q m =rq">(l-q">\ 

, (!) 

(1) > 

giving g (1) = (1 - Mr) - see the left branch of the plot shown in Fig. 2. 


(a) (b) 


f = 3 

r 2 M 

0.8 - 

0.6 - 

0.4 - 

0.2 - 


q w =\-- 


~i — i — i — i — i — i — i — i — i — i — i — i — i — r~ 

2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 

Fig. 9.2. Fixed points and 
chaotic regions of the logistic 
map. The plot is adapted from 
gistic map; a very nice live 
simulation of the map is also 
available on this Web site. 

At r > r\ = 3, the plot gets thicker: here the fixed point q (l) also becomes unstable. To prove that, 
let us take q n = <y + q n , assume that deviation q n from the fixed point q 1 1 is small, and linearize map 
(3) in q n , just as we repeatedly did for differential equations earlier in this course. The result is 

< 1 ,, 



q,=r(l-2q m )q,=(2- r )q n . 


It shows that 0 < 2 — r <1, i.e. 1 < r <2, deviations q n decrease monotonically. At -1 < 2 - r < 0, i.e. 

in the range 2 < r <3, the deviation signs alternate but the magnitude still decreases (as in a stable focus 
- see Sec. 4.6). However, at -1 < 2 - r, i.e. r > r\ = 3, the deviations are growing by magnitude, while 
still changing sign, at each step. Since Eq. (2) has no other fixed points, this means that at n — » oo, values 

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q n do not converge to one point; rather, within the range r\ < r < ri, they approach a limit cycle of 
alternation of two points, q+ 2) and q} 2) that satisfy the following system of algebraic equations 

q? =f(q i : ) ),q a) =f{q?)- (9-5) 

(These points are also plotted in Fig. 2, as functions of parameter r.) What has happened at point r\ is 
called the period-doubling bifurcation. The story repeats at r = r 2 = 1 + V6 « 3.45 where the system goes 
from the 2-point limit cycle to a 4-point cycle, then at point r = r^ « 3.54 at that the limit cycle becomes 
consisting of 8 alternating points, etc. Most remarkably, the period-doubling bifurcation points r n , at that 
the number of points in the limit cycle doubles from 2"' 1 points to 2" points, become closer and closer. 
Numerical calculations have shown that these points obey the following asymptotic behavior: 




r n —>r x , where r = 3.5699..., A = 4.6692... 


Parameter 8 is called the Feigenbaum constant ; for other maps, and some dynamic systems (see the 
next section), period-doubling sequences follow a similar law, but with different parameter 8. 

More important for us, however, is what happens at r > r Numerous numerical experiments, 
repeated with increasing precision, 5 have confirmed that here the system is fully disordered, with no 
reproducible limit cycle, though (as Fig. 2 shows) at r « r x , all sequential values q n are still confined to a 
few narrow regions. 6 However, as parameter r is increased well beyond r, r „ these regions broaden and 
merge. This the so-called full, or well-developed chaos, with no apparent order at all. 7 

The most important feature of chaos (in this and any other system) is the exponential divergence 
of trajectories. For a ID map, this means that even if the initial conditions q\ in two map 
implementations differ by a very small amount Ac/i, the difference A q n between the corresponding 
sequences q n is growing (on the average) exponentially with n. Such exponents may be used to 
characterize chaos. Indeed, let us assume that Aq\ is so small that N first values q„ are relatively close to 
each other. Then an evident generalization of the first of Eqs. (4) to an arbitrary point q n is 

kq„ + i= e ,Aq n , e„ 

Using this result iteratively for N steps, we get 


Aq N = A q x n e n , so that In 

n = 1 



A q N 

A q x 

q=q n * 



5 The reader should remember that just as the usual (“nature”) experiments, numerical experiments also have 
limited accuracy, due to unavoidable rounding errors. 

6 The geometry of these regions are essentially fractal, i.e. has a dimensionality intermediate between 0 (which 
any final set of geometric points would have) and 1 (pertinent to a ID continuum). An extensive discussion of 
fractal geometries, and their relation to the deterministic chaos may be found, e.g., in the book by B. B. 
Mandelbrot, The Fractal Geometry of Nature, W. H. Freeman, 1983. 

7 This does not mean that the chaos development is a monotonic function of r. As Fig. 2 shows, within certain 
intervals of this parameter chaos suddenly disappears, being replaced, typically, with a few-point limit cycle, just 
to resume on the other side of the interval. Sometimes (but not always!) the “route to chaos” on the borders of 
these intervals follows the same Feigenbaum sequence of period-doubling bifurcations. 

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Numerical experiments show that in most chaotic regimes, at N — » oo such a sum fluctuates about 
an average, which grows as AN, with parameter 

(9.9) exponent 

called the Lyapunov exponent , 8 being independent on the initial conditions. The bottom panel in Fig. 3 
shows it as a function of the parameter r for the logistic map (2). 

1 N 

^ = lim A ^0 lim A^ooT7Z ln KI’ 

n = 1 

Fig. 9.3. The Lyapunov exponent for 
the logistic map. Adapted from the 
monograph by Schuster and Just (cited 
below). © WileyVCFl Verlag GmbH & 
Co. KGaA. 

Note that at r < r, E , A is negative, indicating trajectory’s stability, besides points r\, r 2 , ... where 
A would become positive if the limit cycle change had not brought it back to the negative territory. 
However, at r > r,,-, A becomes positive, returning the negative values only in limited intervals of stable 
limit cycles. It is evident that in numerical experiments (which dominate the studies of the deterministic 
chaos) the Lyapunov exponent may be used as a good measure of chaos’ “depth”. 9 

Despite all the abundance of results published for particular maps, 10 and several interesting 
general observations (like the existence of the Feigenbaum bifurcation sequences), to the best of my 
knowledge nobody can yet predict the patterns like those shown in Fig. 2 and 3, from just looking at the 
map rule itself, i.e. without carrying out actual numerical experiments with in. Unfortunately the 
situation with chaos in other systems is not much better. 

8 After A. Lyapunov (1857-1918), famous for his studies of stability of dynamic systems. 

9 A'-di mens ions maps, which relate A'-di mens ional vectors rather than scalars, may be characterized by N 
Lyapunov exponents rather than one. In order to have chaotic behavior, it is sufficient for just one of them to 
become positive. For such systems, another measure of chaos, the Kolmogorov entropy, may be more relevant. 
This measure, and its relation with the Lyapunov exponents, are discussed, e.g., in SM Sec. 2.2. 

10 See, e.g., Chapters 2-4 in H. G. Schuster and W. Just, Deterministic Chaos, 4 th ed., Wiley-VCH, 2005, or 
Chapters 8-9 in J. M. T. Thompson and H. B. Stewart, Nonlinear Dynamics and Chaos, 2 nd ed., Wiley, 2002. 

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9.2. Chaos in dynamic systems 

Proceeding to the discussion of chaos in dynamic systems, it is more natural, with our 
background, to illustrate this discussion not with the Lorenz’ system Eqs. (1), but with the system of 
equations describing a dissipative pendulum driven by a sinusoidal external force, which was repeatedly 
discussed in Chapter 4. Introducing two new variables, the nonnalized momentum p = q/ co 0 and the 

external force’s full phase y/= cot, we may rewrite Eq. (4.42) describing the pendulum, 

q + 2 Sq + a>l sin q = f 0 cos cot , (9. 10a) 

in a form similar to Eq. (1), i.e. as a system of three first-order ordinary differential equations: 

q = co 0 p, 

p = -co 0 sing - 2Sp + (/ 0 / co 0 ) cost//, (9.10b) 

i// = co. 

Figure 4 several results of numerical solution of Eq. (10). 11 In all cases, the internal parameters 
8 and coo of the system, and the external force amplitude fo are fixed, while the external frequency co is 
gradually changed. For the case shown on the top panel, the system still tends to a stable periodic 
solution, with low contents of higher harmonics. If the external force frequency is reduced by a just few 
percent, the 3 rd subharmonic may be excited. (This effect has already been discussed in Sec. 4.8 - see, 
e.g., Fig. 4.15.) The next panel shows that just a very small further reduction of frequency leads to a new 
tripling of the period, i.e. the generation of a complex waveform with the 9 th subhannonic. Finally, even 
a minor further change of parameters leads to oscillations without any visible period, e.g., chaos. 

In order to trace this transition, direct observation of the oscillation waveforms q(t) is not very 
convenient, and trajectories on the phase plane [q, /;] also become messy if plotted for many periods of 
the external frequency. In situations like this, the Poincare (or “stroboscopic”) plane, already discussed 
in Sec. 4.6, is much more useful. As a reminder, this is essentially just the phase plane [q, p\, but with 
the points highlighted only once a period, e.g., at y/ = 2 7m, with n = 1,2,... On this plane, periodic 
oscillations of frequency co are presented just as one fixed point - see, e.g. the top panel in the right 
column of Fig. 4. The beginning of the 3 ld subhannonic generation, shown on the next panel, means 
tripling of the oscillation period, and is reflected on the Poincare plane by splitting the fixed point into 
three. It is evident that this transition is similar to the period-doubling bifurcation in the logistic map, 
besides the fact (already discussed in Sec. 4.8) that in systems with an asymmetric nonlinearity, such as 
the pendulum (10), the 3 ld subhannonic is easier to excite. From this point, the 9 th hannonic generation 
(shown on the 3 rd panel of Fig. 4), i.e. one more splitting of the points on the Poincare plane, may be 
understood as one more step on the Feigenbaum-like route to chaos - see the bottom panel of that figure. 

So, the transition to chaos in dynamic systems may be at least qualitatively similar to than in ID 
maps, with the similar law (6) for the critical values of some parameter r of the system (in Fig. 4, 
frequency co), though generally with a different value of exponent 8. Moreover, it is evident that we can 
always consider the first two differential equations of system (10b) as a 2D map that relates the vector 
{q„+i,Pn+i} of the coordinate and velocity, measured at i// = 2 jin + 1), with the previous value {q n , p n ) 

1 1 In the actual simulation, a small term eq, with s « 1 , has been added to the left-hand part of this equation. This 
term slightly somewhat tames the trend of the solution to spread along q axis, and makes the presentation of 
results easier, without affecting the system dynamics too much. 

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of that vector (reached at y/ = 2 m). Unfortunately this similarity also implies that chaos in dynamical 
systems is at least as complex, and it as little understood, as in maps. 

288 290 292 294 296 298 300 0 

0)t !2 k 

Fig. 9.4. Oscillations in a pendulum with weak damping, 81 co o =0.1, driven by a sinusoidal external 
force with a fixed effective amplitude fjcqp = 1, and several close values of the frequency (listed on 
the panels). Left column: oscillation waveforms q(t) recorded after certain initial transient intervals. 
Right column: representations of the same processes on the Poincare plane of variables [p, q]. 

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For example, Fig. 5 shows (a part of) the state diagram of the externally-driven pendulum, with 
the red bar marking the route to chaos traced in Fig. 4, and shading/hatching styles marking different 
regimes. One can that the pattern is at least as complex as that shown in Figs. 2 and 3, and besides a few 
features, 12 is equally unpredictable from the form of the equation. 


0 0.2 0.4 0.6 0.8 1 

CD t CD ^ 

Fig. 9.5. Phase diagram of an externally- driven pendulum with 
S/coq - 0.1. Regions of oscillations with the basic period are not 
shaded. The notation for other regions is as follows. Doted: 
subharmonic generation; cross-hatched: chaos; hatched: chaos 
or basic period (depending on the initial conditions); hatch- 
dotted: basic period or subharmonics. Solid lines show 
boundaries of single-regime regions, while dashed lines are 
boundaries of regions in which several types of motion are 
possible, depending on history. (Figure courtesy V. Komev.) 

Are there any valuable general results concerning chaos in dynamic systems? The most 
important (though an almost evident) result is that this phenomenon is impossible in any system 
described by one or two first-order differential equations with right-hand parts independent of time. 
Indeed, let us start with a single equation 

q = f(q), (9.11) 

where j{q) is any single-valued function. This equation may be directly integrated to give 

t _{_dq_ — + const (9.12) 

J /(<?') 

showing that the relation between q and t is unique and hence does not leave place for chaos. 

Now, let us explore the system of two such equations: 

q.\ ~ f\^si\^qi)i (9 13) 

q 2 =f 2 (q l ,q 2 )- 

Consider its phase plane shown schematically in Fig. 6. In a “usual” system, the trajectories approach 
either some fixed point (Fig. 6a) describing static equilibrium, or a limit cycle (Fig. 6b) describing 
periodic oscillations. (Both notions are united by the term attractor, because they “attract” trajectories 
launched from various initial conditions.) Fiowever, phase plane trajectories of a chaotic system of 

12 In some cases, it is possible to predict a parameter region where chaos cannot happen, due to lack of any 
instability-amplification mechanism. Unfortunately, typically the analytically predicted boundaries of such region 
form a rather loose envelope of the actual (numerically simulated) chaotic regions. 

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equations that describe real physical variables (which cannot tend to infinity), should be confined to a 
limited phase plane area, and simultaneously cannot start repeating each other. (This topology is 
frequently called the strange attractor .) For that, 2D trajectories need to cross - see, e.g., point A in Fig. 

Fig. 9.6. Attractors in dynamical systems: (a) a fixed point, (b) a limit cycle, and (c) a strange attractor. 

However, in the case described by Eqs. (13), this is clearly impossible, because according to 
these equations, the tangent slope on the phase plane is a unique function of point coordinates}*/!, c/ 2 }: 

dq\ = fM\,q 2 ) (9 14 ) 

dq 2 f 2 (q l ,q 2 ) 

Thus, in this case the deterministic chaos is impossible. 13 It becomes, however, readily possible if the 
right-hand parts of a system similar to Eq. (13) depend either on other variables of the system or time. 
For example, if we consider the first two differential equations of system (10b), in the case fo = 0 they 
have the structure of the system (13) and hence chaos is impossible, even at 3 < 0 when (as we know 
from Sec. 4.4) the system allows self-excitation of oscillations - leading to a limit-cycle attractor. 
However, if 0, this argument does not work any longer and (as we have already seen) the system 
may have a strange attractor - which is, for dynamic systems, a synonym for the deterministic chaos. 
Thus, chaos is possible in dynamic systems that may be described by three or more differential 
equations of the first order. 14 

9.3, Chaos in Hamiltonian systems 

The last analysis is of course valid for Hamiltonian systems, which are just a particular type of 
dynamic systems. However, one may wonder whether these systems, that feature at least one first 
integral of motion, H = const, and hence are more “ordered” than the systems discussed above, can 
exhibit chaos at all. The question is yes, because such systems still can have mechanisms for an 
exponential growth of a small initial perturbation. 

13 A mathematically-strict formulation of this statement is called the Poincare-Bendixon theorem, which was 
proved by I. Bendixon as early as in 1901. 

14 Since a typical dynamic system with one degree of freedom is described by two such equations, the number of 
the first-order equations describing a dynamic system is sometimes called the number of half-degrees of freedom. 
This notion is very useful and popular in statistical mechanics - see, e.g., SM Sec. 2.2 and on. 

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As the simplest way to show it, let us consider a so-called mathematical billiard, i.e. a ballistic 
particle (a “ball”) moving freely by inertia on a horizontal plane surface (“table”) limited by rigid 
impenetrable walls. In this idealized model of the usual game of billiards, ball’s velocity v is conserved 
when it moves on the table, and when it runs into a wall, the ball is elastically reflected from it as from a 
mirror, 15 with the reversal of the sign of the normal velocity v n , and conservation of the tangential 
velocity v r , and hence without any loss of its kinetic (and hence the full) energy 

E = H = 

= ™(v 2 
2 V " 

2 +v: 



This model, while being a legitimate 2D dynamic system, 16 allows geometric analyses for several simple 
table shapes. The simplest case is a rectangular billiard of area axb (Fig. 7), whose analysis may be 
readily carried out by the replacement of each ball reflection event with the mirror reflection of the table 
in that wall - see dashes lines in panel (a). 


Fig. 9.7. Ball motion on 
a rectangular billiard at 

(a) a commensurate, and 

(b) an incommensurate 
launch angle. 

Such analysis (left for reader’ pleasure :-) shows that if the tangent of the ball launching angle cp 
is commensurate with the side length ratio, 

tan (p = ±——, (9.16) 

n a 

where n and in are non-negative integers without common integer multipliers, the ball returns exactly to 
the launch point O, after bouncing m times from each wall of length a, and n times from each wall of 
length b. (Red lines in Fig. 7a show an example of such trajectory for n = m = 1, while blue lines, for m 
= 3, n = 1.) Thus the larger is the sum ( m + n ), the more complex is such closed trajectory - “orbit”. 

Finally, if ( n + m) — » oo, i.e. tan cp and b/a are incommensurate (meaning that their ratio is an 
irrational number), the trajectory covers all the table area, and the ball never returns exactly into the 
launch point. Still, this is not the real chaos. Indeed, a small shift of the launch point shifts all the 
trajectory fragments by the same displacement. Moreover, at any time t, each of Cartesian components 
Vj(t) of the baft’s velocity (with coordinate axes parallel to the table sides) may take only two values, 
±Vj( 0), and hence may vary only as much as the initial velocity is being changed. 

15 A more scientific-sounding name for such a reflection is specular (from Latin “speculum” meaning a metallic 

16 Indeed, it is fully described by Lagrangian function L = mv 2 /2 - U( p), with U(p) = 0 for 2D radius-vectors p 
belonging to the table area, and C(p) = +oo outside of the area. 

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In 1963, Ya. Sinai showed that the situation changes completely if an additional wall, in the 
shape of a circle, is inserted into the rectangular billiard (Fig. 8). For most initial conditions, ball’s 
trajectory eventually runs into the circle (see the red line on panel (a) as an example), and the further 
trajectory becomes essentially chaotic. Indeed, let us consider ball’s reflection from the circle-shaped 
wall - Fig. 8b. Due to the conservation of the tangential velocity, and the sign change of the normal 
velocity component, the reflection obeys the mechanical analog of the Snell law (cf. Fig. 7.12 and its 
discussion): 6 r = 6,. Figure 8b shows that as the result, a small difference dcp between the angles of two 
close trajectories (as measured in the lab system), doubles by magnitude at each reflection from the 
curved wall. This means that the small deviation grows along the ball trajectory as 

\S(p(N)\ ~ \8(p{ 0)| x 2^ = \S<p(0)\e Nln2 , (9.17) 

where N is the number of reflections from the convex wall. 17 As we already know, such exponential 
divergence of trajectories, with a positive Lyapunov exponent, is the sign of deterministic chaos. 18 


Fig. 9.8. (a) Motion on a Sinai 
billiard table, and (b) the 
mechanism of the exponential 
divergence of close trajectories. 

The most important new feature of the dynamic chaos in Hamiltonian systems is its dependence 
on initial conditions. (In the systems discussed in the previous two previous sections, that lack the 
integrals of motion, the initial conditions are rapidly “forgotten”, and the chaos is usually characterized 
after cutting out the initial transient period - see, e.g., Fig. 4.) Indeed, even a Sinai billiard allows 
periodic motion, along closed orbits, at certain initial conditions - see the blue and green lines in Fig. 8a 
as examples. Thus the chaos “depth” in such systems may be characterized by the “fraction” 19 of the 
phase space of initial parameters (for a 2D billiard, the 3D space of initial values of x, y, and <p) resulting 
in chaotic trajectories. 

This conclusion is also valid for Hamiltonian systems that are met in experiments more 
frequently than the billiards, for example, coupled nonlinear oscillators without damping. Perhaps, the 

17 Superficially, Eq. (17) is also valid for a plane wall, but as was discussed above, a billiard with such walls 
features a full correlation between sequential reflections, so that angle (p always returns to its initial value. In a 
Sinai billiard, such correlation disappears. Because of that, concave walls may also make a billiard chaotic. A 
famous example is the stadium billiard, suggested by L. Bunimovich, with two straight, parallel walls connecting 
two semi-circular, concave walls. Another example, which allows a straightforward analysis, is the Hadamard 
billiard : an infinite (or rectangular) table with non-horizontal surface of negative curvature. 

18 Billiards are also a convenient platform for a discussion of a conceptually important issue of quantum 
properties of classically chaotic systems (sometimes improperly named “quantum chaos”). 

19 Actually, quantitative characterization of the fraction is not trivial, because it may have fractal dimensionality. 
Unfortunately, due to lack of time I have to refer the reader interested in this issue to special literature, e.g., the 
monograph by B. Mandelbrot (cited above) and references therein. 

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earliest and the most popular example is the so-called Henon-Heiles system, 20 which may be desorbed 
by the following Lagrangian function: 

r m \ ( ■ 2 

2 2 \ . m 2 ( ■ 2 2 2 ) 

( 1 ^ 
2 1 2 

L _ v? 1 

0J \ 9| J ^ — H2 a> 2 q 2 , 

- £ 

q 1 -~q 2 

q 2 • 


l 3 2 


It is straightworward to use Eq. (18) to derive the Lagrangian equations of motion, 

m\q x +(0[q x )=-2sq x q2, 
m 2 ($2 "h ) — _ ^($1 — ^ 2 )’ 


and find its first integral of motion (physically, the energy conservation law): 

+® r «] 2 

2 2 \ 

+ CO 2 C[ 2 j “t" £ 



-q 2 

q 2 = const . 



In the conext of our discussions in Chapter 4 and 5, Eqs. (19) may be readily interpreted as those 
describing two oscillators, with small-oscillation eigenfrequencies co\ and oh, nonlinearly coupled only 
as described by the terms in the right-hand parts of the equations. This means that as the oscillation 
amplitudes A\^, and hence the total energy E of the system, tend to zero, the oscillator subsystems are 
virtually independent, each performing sinusoidal oscillations at its own frequency. This observation 
suggestes a convenient way to depict the system motion. 21 Let us consider a Poincare plane for one of 
the oscillators (say, with coordinate c/ 2 ), similar to that discussed in Sec. 2 above, with the only 
difference is that (because of the absence of an explicit function of time in system’s equations), the 
trajectory on the [ q 2 , q 2 ] plane is highlighted at the moments when c/i = 0. 

Let us start from the limit A \ 2 — > 0, when oscillations of c /2 are virtually sinusoidal. As we 
already know (see Lig. 4.9 and its discussion), if the representation point highlighting was perfectly 
synchronous with frequency ah of the oscillations, there would be only one point on the Poincare plane 
- see, e.g. the right top plane in Pig. 4. However, at the q\ - initiated highlighting, there is not such 
synchronism, so that each period, a different point of the elliptical (at the proper scaling of the velocity, 
circular) trajectory is highlighted, so that the resulting points, for certain initial conditions, reside on a 
circle of radius Aj. If we now vary the initial conditions, i.e. redistribute the initial energy between the 
oscillators, but keep the total energy E constant, on the Poincare plane we get a series of ellipses. 

Now, if the initial energy is increased, nonlinear interaction of the oscillations start to deform 
these ellipses, causing also their crossings - see, e.g., the top left panel of Pig. 9. Still, below a certain 
threshold value of E, all Poincare points belonging to a certain initial condition sit on a single closed 

20 It was first studied in 1964 by M. Henon and C. Heiles as a simple model of star rotation about a gallactic 
center. Most studies of this equation have been carried out for the following particular case: m 2 = 2m 1 , m , oq 2 = 
m 2 a> 2 2 . In this case, introducing new variables x = cq 1 y= sq 2 , and r = oqt, it is possible to rewrite Eqs. (18)-(20) 
in parameter-free forms. All the results shown in Fig. 9 below are for this case. 

21 Generally, it has a trajectory in 4D space, e.g., that of coordinates q 1,2 and their time derivatives, although the 
first integral of motion (20) means that for each fixed energy E, the motion is limited to a 3D sub-space. Still, this 
is too much for convenient representation of the motion. 

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contour. Moreover, these contours may be calculated approximately, but with a pretty good accuracy, 
using a straighforward generalization of the small parameter method discussed in Sec. 4.2. 22 

Fig. 9.9. Poincare planes of the Henon- 
Heiles system (19), in notation y = sq 2 , for three 
values of the dimensionless energy e = E/Eq, 
with E 0 = m\(Q\l£. Adapted from M. 
Henon and C. Heiles, The Astron. J. 69, 73 
(1964). ©AAS. 

Flowever, starting from some value of energy, certain initial conditions lead to series of points 
scattered over final-area parts of the Poincare plane - see the top right panel of Fig. 9. This means that 
the corresponding oscillations q 2 {t) do not repeat from one (quasi-) period to the next one - cf. Fig. 4 for 
the dissipative, forced pendulum. This is chaos. 23 However, some other initial conditions still lead to 
closed contours. This feature is similar to Sinai billiards, and is typical for Hamiltonian systems. As the 
energy is increased, the larger and larger part of the Poincare plane belongs to the chaotic motion, 
signifying deeper and deeper chaos. 

22 See, e.g., M. V. Berry, in: S. Joma (ed.), Topics in Nonlinear Dynamics , AIP Conf. Proc. No. 46, AIP, 1978, 

pp. 16-120. 

23 This fact complies with the necessary condition of chaos, discussed in the end of Sec. 2, because Eqs. (19) may 
be rewritten as a system of four differential equations of the first order. 

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9.4, Chaos and turbulence 

This extremely short section consists of essentially just one statement, extending the discussion 
in Sec. 8.5. The (re-) discovery of the detenninistic chaos in systems with just a few degrees of freedom 
in the 1960s changed the tone of debates concerning origins of turbulence very considerably. At first, an 
extreme point of view that equated the notions of chaos and turbulence, became the debate’s favorite. 24 
However, after an initial excitement, a significant evidence of the Landau-style mechanisms, involving 
many degrees of freedom, has been rediscovered and could not be ignored any longer. To the best 
knowledge of this author, who is a very distant albeit interested observer of that field, most experimental 
and numerical-simulation data carry features of both mechanisms, so that the debate continues. 25 Due to 
the age difference, most readers of these notes have much better chances than the author to see where 
will this discussion end (if it does :-). 26 

9.5. Exercise problems 

9.1 . Generalize the reasoning of Sec. 1 to an arbitrary ID map q n+ \ = with function f{q) 
differentiable at all points of interest. In particular, derive the condition of stability of an A-point limit 
cycle g (1) — » q a) — > ...— > q {N) — > q [l) . 

9.2 . Use the stability condition, derived in Problem 9.1, to analyze chaos excitation in the so- 
called tent map : 

/ Wq, for 9<q<\/2, 

1 \ r(\ - q), for \/2<q< 1, 

with 0 < r < 2. 

9.3 . A dynamic system is described by the following system of ordinary differential equations: 

=~qi +a x ql, 

q 2 =a 2 q 2 -a 3 q\ +a 4 q 2 (\-q;). 

Can it exhibit chaos at some set of constant parameters a\JX\l 

9.4 . A periodic function of time has been added to the right-hand part of the first equation of the 
system considered in the previous problem. Is chaos possible now? 

24 An important milestone on that way was the work by S. Newhouse et al., Comm. Math. Phys. 64, 35 (1978), 
who proved the existence of a strange attractor in a rather abstract model of fluid flow. 

25 See, e.g., U. Frisch, Turbulence: The Legacy of A. N. Kolmogorov, Cambridge U. Press, 1996. 

26 The reader interested in the deterministic chaos as such, may also like to have a look at a very popular book by 
S. Strogatz, Nonlinear Dynamics and Chaos, Westview, 2001. 

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Chapter 10. A Bit More of Analytical Mechanics 

This concluding chapter reviews two alternative approaches to analytical mechanics, whose main 
advantage is a closer parallel to quantum mechanics in general and to its quasiclassical (WKB) 
approximation in particular. One of them, the Hamiltonian formalism, is also used to derive an 
important asymptotic result, the adiabatic invariance, for classical systems with slowly changing 

10.1. Hamilton equations 

Throughout this course we have seen how useful the analytical mechanics, in its Lagrangian 
form, may be invaluable for solving various particular problems of classical mechanics. Now let us 
discuss several alternative formulations 1 that may not be much more useful for this purpose, but shed 
light on possible extensions of classical mechanics, most importantly to quantum mechanics. 

As was already discussed in Sec. 2.3, the partial derivative Pj = 8L/ 5q , participating in the 
Lagrange equations (2.19) 

d dL dL 

dt dqj 8q j 


may be considered as the generalized momentum corresponding to generalized coordinate qj, and the 
full set of this momenta may be used to define the Hamiltonian function (2.32): 

H = YjPifi ~ L - 



Now let us rewrite the full differential of this function 2 in the following form: 

dH = d 

YjPjdj ~L =Y J [ d (P j )d j +P j d(q J )]-dL 

V J 

= Y\ d( <pMj + Pj d (<ij)\- 



dt + y 

r dL dL . ^ 

— d (q j )+— d (q j ) 

\ 8 q j 8c p j 


According to the definition of the generalized momentum, the second terms of each sum over j cancel, 
while according to the Lagrange equation (1), the derivative dL! dq } is just pj , so that 

O T 

dH = -—dt + ^{q jdp j - p jdq j) . (10.4) 

So far, this is just a universal identity. Now comes the main trick of Hamilton’s approach: let us 
consider H a function of the following independent arguments: time t, the generalized coordinates qj, 

1 Due mostly to W. Hamilton (1805-1865) and C. Jacobi (1804-1851). 

2 Actually, this differential has already been used in Sec. 2.3 to derive Eq. (2.35). 

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and the generalized momenta pj (rather than generalized velocities). With this commitment, the general 
rule of differentiation of a function of several arguments gives 

dH = 

dH , x 

dt + / 

dt y 

dH , dH , 
dq . H dp 


dp i 


where dt, dqj, and dpj are independent differentials. Since Eq. (5) should be valid for any choice of these 
argument differentials, it should hold in particular if the differentials correspond to the real law of 
motion, for which Eq. (4) is valid as well. The comparison of Eqs. (4) and (5) gives us three relations: 

dH _ dL 
dt dt 

Comparing the first of them with Eq. (2.35), we see that 

dH _ dH 
dt dt 

( 10 . 6 ) 




( 10 . 8 ) 

meaning that function H{t, qj, pj) can change in time only via its explicit dependence on t. Eqs. (7) are 
even more substantial: provided that such function I I(t, qj, pj) has been calculated, they give us two first- 
order differential equations (called the Hamilton equations) for the time evolution of the generalized 
coordinate and generalized momentum of each degree of freedom of the system. 3 

Let us have a look at these equations for the simplest case of a system with one degree of 
freedom, with the simple Lagrangian function (3.3): 


L = ^-q 2 -U e{ (q,t). (10-9) 

In this case, p = dL / dq = m ef q , and H = pq - L = m ef q 2 / 2 + U e{ (q,t) . In order to honor our new 
commitment, we need to express the Hamiltonian function explicitly via t, q and p (rather than q ): 

H = ~~ ' r U ef {q,t). 
2 m ef 

Now we can spell out Eqs. (7) for this particular case: 

dH p 

q = ^ = ’ 

dp m e{ 

( 10 . 10 ) 

( 10 . 11 ) 

3 Of course, the right-hand part of each equation (7) generally can include coordinates and momenta of other 
degrees of freedom as well, so that the equations of motion for different j are generally coupled. 

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P = 



dU e f 


( 10 . 12 ) 

While the first of these equations just repeats the definition of the generalized momentum 
corresponding to coordinate q, the second one gives the equation of momentum change. Differentiating 
Eq. (11) over time, and plugging Eq. (12) into the result, we get: 

9 = 

1 dU ei 

m e f dq 


So, we have returned to the same equation (3.4) that had been derived from the Lagrangian approach. 

Thus, the Hamiltonian formalism does not give much new for the solution of most problems of 
classical mechanics. (This is why I have postponed its discussion until the very end of this course.) 
Moreover, since the Hamiltonian function H(t, qj, pj) does nor include generalized velocities explicitly, 
the phenomenological introduction of dissipation in this approach is less straightforward than that in the 
Lagrangian equations whose precursor form (2.17) is valid for dissipative forces as well. However, the 
Hamilton equations (7), which treat the generalized coordinates and momenta in a manifestly symmetric 
way, are aesthetically appealing and heuristically fruitful. This is especially true in the cases where these 
arguments participate in H in a similar way. For example, for the very important case of a dissipation- 
free harmonic oscillator, for which U e f = fc et q / 2, Eq. (10) gives the famous symmetric form 

H = 

v 2 


P ,n c\ (t - > 0 X 
2 m ef 2 

where of = 






The Hamilton equations (7) for this system preserve the symmetry, especially evident if we introduce 
the normalized momentum /? = p/m e fCOo (already used in Secs. 4.3 and 9.2): 



-co 0 q. 


More practically, the Hamilton approach gives additional tools for the search for the integrals of 
motion. In order to see that, let us consider the full time derivative of an arbitrary function /(/, qj,pf): 




= — + 



<h + 


dq, dp : 



Plugging in q . and p j from the Hamilton equations (7), we get 

of arbitrary 





' dH df 

y dpj dq j 

dH df 
dq j dp j y 


= — + 





where the last tenn in the right-hand part is the so-called Poisson bracket 4 that is defined, for two 
arbitrary functions fit, qj, pf and g(t, qj, pf, as 

fe»/} = Z 


dg df 

y dp j dq j 

df dg 

dpj dq j y 


4 Named after S. Poisson - of the Poisson equation and the Poisson statistical distribution fame. 

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From this definition, one can readily verify that besides evident relations {/,/} = 0 and {/,' g} = - {g, /}, 
the Poisson brackets obey the following important Jacobi identity. 

{/.fe,d}+fe ,{*./}}+{*.{/, g}} = 0- (1019) 

Now let us use these relations for a search for integrals of motion. First, equation (17) shows that 
if a function / does not depend on time explicitly, and 

{H,f}= 0, (10.20) 

then df/dt = 0, i.e. function/ is an integral of motion. 

Moreover, if we already know two integrals of motion, say/ and g, then function 

F = {f,g) (10.21) 

is also an integral of motion - the so-called Poisson theorem. In order to prove it, we may use the Jacobi 
identity (19) with h = H. Now using Eq. (17) to express the Poisson brackets {g, H}, {H, g}, and 
g} } = {//, F\ via the full and partial time derivatives of functions /, g, and F, we get 

L dg dg\ ( df df\ dF 8F _ 
Y’dt dt\ + { 8 ’dt dt)dt 8t 

( 10 . 22 ) 

so that if / and g are indeed integrals of motion, i.e., df/dt = dg/dt = 0, then 

dF 8F 


Sg) 8F 

— = — + - 



dt 8t 


dt dt 

y- ,g} + if,y L 




Plugging Eq. (21) into the first term of the right-hand part of this equation, and differentiating it by 
parts, we get dF/dt = 0, i.e. F is indeed an integral of motion as well. 

Finally, one more important role of the Hamilton formalism is that it allows one to trace the 
close connection between the classical and quantum mechanics. Indeed, using Eq. (18) to calculate the 
Poisson brackets of the generalized coordinates and momenta, we readily get 

l</,9/} =0 , \pj>Pr\=Q> \9j ’ Pf } = -djy ■ (10.24) 

In quantum mechanics, 5 operators of these quantities (“observables”) obey commutation relations 

[/,/']= °, \pj,Pj]= °> [dj,Pj] = i^jf, (10.25) 

where the definition of the commutator, 

\gj\ = gf- 

■ f g , is to a certain extent 6 similar to that (18) of 

the Poisson bracket. We see that the classical relations (24) are similar to quantum-mechanical relations 
(25) if we following parallel has been made: 




5 See, e.g., QM Sec. 2.1. 

6 There is of course a conceptual difference between the “usual” products of function derivatives participating in 
the Poisson brackets, and the operator “products” (meaning their sequential action on a state vector - see, e.g., 
QM Sec. 4.1) forming the commutator. 

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This analogy extends well beyond Eqs. (24)-(25). For example, making replacement (26) in Eq. 
(17), we get 



df i 
— + — 
dt h 


■ df .. df 
i.e. ih — = ih — + 
dt dt 


which is the correct equation of operator evolution in the Heisenberg picture of quantum mechanics. 7 

This analogy implies, in particular, that the quantum-mechanical operators (and the matrices 
used for their representation in a particular basis) should satisfy the same identities including Eq. (17). 

10.2. Adiabatic invariance 

One more application of the Hamiltonian formalism in classical mechanics is the solution of the 
following problem. 8 Earlier in the course, we already studied some effects of time variation of 
parameters of a single oscillator (Sec. 4.5) and coupled oscillators (Sec. 5.5). However, those 
discussions were focused on the case when the parameter variation frequency is comparable with the 
initial oscillation frequency (or frequencies) of the system. Another practically important case is when 
some system’s parameter (let us call it A) is changed much more slowly ( adiabatically 9 ), 




where T is a typical time period of oscillations in the system. Let us consider a ID system whose 

Hamiltonian H(q, p. A) depends on time only via the slow (28) evolution of parameter A = Aft), and 
whose initial energy restricts system’s motion to a finite coordinate interval - see Fig. 3.2c. 

Then, as we know from Sec. 3.3, if parameter A is constant, the system performs a periodic 
(though not necessarily sinusoidal) motion back and forth axis q, or, in a different language, along a 
closed trajectory on the phase plane [q,p] - see Fig. I. 10 According to Eq. (8), in this case H is constant 
on the trajectory. (In order to distinguish this particular value from the Hamiltonian function as such, I 
will assume that this constant coincides with the full mechanical energy E, like is does for Hamiltonian 
(10), though this assumption is not necessary for the calculation made below.) 

The oscillation period T may be calculated as a contour integral along this closed trajectory: 

T = 


Using the first of the Hamilton equations (7), we may now present this integral as 

7 See, e.g., QM Sec. 4.6. 

8 Various aspects of this problem and its quantum-mechanical extension were first discussed by L. Le Cornu 
(1895), Lord Rayleigh (1902), H. Lorentz (1911), P. Ehrenfest (1916), and M. Born and V. Fock (1928). 

9 This term has come from thermodynamics and statistical mechanics, where it implies not only a slow parameter 
variation, but also the thermal insulation of the system - see, e.g., SM Sec. 1.3. Evidently, the latter condition is 
irrelevant in our current context. 

10 In Sec. 4.6, we discussed this plane for the particular case of sinusoidal oscillations - see Fig. 9 

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dH / dp 

-dq . 


At each given point q, H = E is a function of p alone, so that we may flip the partial derivative in the 
denominator just as a full derivative, and rewrite Eq. (30) as 


For the particular Hamiltonian (10), this relation is immediately reduced to Eq. (3.27) in the form of a 
contour integral: 

T = 


. 1/2 



[E-U ef (q)\ 


clq . 


Fig. 10.1. Phase-plane representation of periodic 
oscillations of a ID Hamiltonian system, for two 
values of energy (schematically). 

Superficially, it looks that these formulas may be also used to find the motion period change 
when parameter X is being changed adiabatically, for example, by plugging known functions m c \(X) and 
U e f{q, X) into Eq. (32). However, there is no guarantee that energy E in that integral would stay constant 
as the parameter change, and indeed we will see below that this is not necessarily the case. Even more 
interestingly, in the most important case of the harmonic oscillator (U e f = ic c p 2 !2), whose oscillation 
period T does not depend on E (see Eq. (3.29) and its discussion), its variation in the adiabatic limit (28) 

may be readily predicted: T{X) = In/opiX) = 2si[m e f{X)//c e t{A)\ , but the dependence of the oscillation 
energy E (and hence the oscillation amplitude) on X is not immediately obvious. 

In order to address this issue, let us use Eq. (8) (with E = H) to present the energy change with 
X(t), i.e. in time, as 

dE _ dH _ dH dX 
dt dt dX dt 


Since we are interested in a very slow (adiabatic) time evolution of energy, we can average Eq. (33) 
over fast oscillations in the system, for example over one oscillation period T , treating dXIdt as a 
constant during this averaging. 11 The averaging yields 

11 This is the most critical point of this proof, because at any finite rate of parameter change the oscillations are, 
strictly speaking, non-periodic. Because of the approximate nature of this conjecture (which is very close to the 
assumptions made at the derivation of the RWA equations in Sec. 4.3), new, more strict (but also much more 

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dE dA dH dA I "rd/I , 

— « = dt 

dt dt 8 A dt T ' 8a 


Transforming the time integral to the contour one, just as we did at the transition from Eq. (29) to Eq. 
(30), and using Eq. (3 1) for 7] we get 

r dH ISA 

— <p dq 

dE _dA^ dH / dp 


At each point q of the contour, H is a function of not only A, but also of p, which may be also A- 
dependent, so that if E is fixed, the partial differentiation of relation E = H over A yields 

dH dH dp _ . dH / dA dp 

+ — = 0, i.e. = — — . 

dA dp dA dH I dp dA 

Plugging the last relation into Eq.(35), we get 

Aw A 2 § — dq 

dE _ dA 1 qx 

Tr ~ Y, W88 q 

J dE 



Since the left-hand part of Eq. (37), and the derivative dA/dt do not depend on q, we may move them 
into the integrals over q as constants, and rewrite that relation as 

dp dE dp dA ' 
v dE dt dA dt y 

dq = 0. 




Now let us consider the following integral over the same phase-plane contour, 


called the action variable. Just to understand its physical sense, let us calculate J for a harmonic 
oscillator (14). As we know very well from Chapter 4, for such oscillator, q = A cos'P, p = -m e f<A>4sin l P 
(with 'P = coot + const), so that Jmay be easily expressed either via oscillations’ amplitude A, or their 
energy E = H = A 2 !2\ 

J = 

| (- m cf a) 0 A sin T* )d{A cos T*) = A 2 


co Q ‘ 


Returning to the general oscillator with adiabatically changed parameter A, let us use the 
definition of J, Eq. (39), to calculate its time derivative, again taking into account that at each point q of 
the trajectory,/? is a function of E and A: 

cumbersome) proofs of Eq. (42) are still being offered in literature - see, e.g., C. Wells and S. Siklos, Eur. J. 
Phys. 28, 105 (2007) and/or A. Lobo et al., Eur. J. Phys. 33, 1063 (2012). 

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2 n 

( dp dE dp dA \ , 

— + — — dq . 

\dE dt dA dt ) 


Within the accuracy of our approximation, in which the contour integrals (38) and (41) are calculated 
along a closed trajectory, factor dE/dt is indistinguishable from its time average, and these integrals 
coincide, so that result (38) is applicable to Eq. (41) as well. Hence, we have finally arrived at a very 
important result: at a slow parameter variation, dJIdt = 0, i.e. the action variable remains constant: 

J = const . 




This is the famous adiabatic invariance , n In particular, according to Eq. (40), in a harmonic oscillator, 
energy of oscillation changes proportionately to the (slowly changed) eigenfrequency. 

Before moving on, let me briefly note that the adiabatic invariance is not the only application of 
the action variable J. Since the initial choice of generalized coordinates and velocities (and hence the 
generalized momenta) in analytical mechanics is arbitrary (see Sec. 2.1), it is almost evident that J may 
be taken for a new generalized momentum corresponding to a certain new generalized coordinate 0, 13 
and that pair {J, 0} should satisfy the Hamilton equations (7), in particular, 

dS _ dH 
dt dJ 


Following the commitment of Sec. 1 (made there for the “old” arguments qj, pf), before the 
differentiation in the right-hand part in Eq. (43), El should be expressed as a function of t, J, and 0. For 
time-independent Hamiltonian systems, H is uniquely defined by J - see, e.g., Eq. (40). Hence the right- 
hand part of Eq. (43) does not depend on either t or 0, so that according to that equation, 0 (called the 
angle variable) is a linear function of time: 

0 = ^-f + const. (10.44) 


For a harmonic oscillator, according to Eq. (40), derivative dH/dJ= dE/dJ= op = 2nlT, so that 0 
opt + const. It may be shown that a more general form of this relation, 

dH _ 2 n 


is valid for an arbitrary oscillator described by Eq. (10). Thus, Eq. (44) becomes 

0 = 2n — + const . 



12 For certain particular oscillators, e.g., a mathematical pendulum, Eq. (42) may be also proved directly - an 
exercise highly recommended to the reader. 

13 This, again, is a plausible argument but not a strict proof. Indeed, though, according to its definition (39), J is 
nothing more than a sum of several (formally, infinite number of) values of momentum p, they are not 
independent, but have to be selected on the same closed trajectory on the phase plane. For more mathematical 
vigor, the reader is referred to Sec. 45 of Mechanics by Landau and Lifshitz (which was repeatedly cited above), 
which discusses the general rules of the so-called canonical transformations from one set of Hamiltonian 
arguments to another one - say from {p, q} to {J, 0}. 

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To summarize, for a harmonic oscillator, the angle variable 0 is just the full phase 'F that we 
used so much in Ch. 4, while for an arbitrary (nonlinear) ID oscillator, this is a convenient 
generalization of that notion. Due to this reason, variables J and 0 present a convenient tool for 
discussion of certain fine points of dynamics strongly nonlinear oscillators - for whose discussion I, 
unfortunately, do not have time. 14 

10.3. The Hamilton principle 

Now let me show that the Lagrangian equations of motion, that have been derived in Sec. 2.1 
from the Newton laws, may be also obtained from the so-called Hamilton principle, namely the 
condition of a minimum (or rather an extremum) of the integral called action : 

S = 


where t- m and bin are, respectively, the initial and final moments of time, at which moments all 
generalized coordinates and velocities are considered fixed (not varied) - see Fig. 2. 

Fig. 10.2. Deriving the Hamilton 

The proof of that statement is rather simple. Considering, similarly to Sec. 2.1, a possible virtual 
variation of the motion, described by infinitesimal deviations {Sq j (t) , Sq . ( t ) } from the real motion, the 

necessary condition for S to be minimal is 

SS = J SL dt = 0 , 


where SS and SL are the variations of the action and the Lagrange function, corresponding to the set 
{5q (t) , Sq jit ) }. As has been already discussed in Sec. 2.1, we can use the operation of variation just 

as the usual differentiation (but at fixed time, see Fig. 2.1), swapping these two operations if needed - 
see Fig. 2.3 and its discussion. Thus, we may write 

dL dL 



8q i 

=z — 

/ a?. 

dL d 

+ / 5q,- 

j dq j dt 


14 See, e.g., Chapter 6 in J. Jose and E. Saletan, Classical Dynamics, Cambridge U. Press, 1998. 

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After plugging the last expression into Eq. (48), we can integrate the second term by parts: 


'fin ar 'fin dL d 


= J xl^^/'+X 

t.jdqj j 





\_ 1 J At- 



-X j 

f dL A 

j t 

= 0 . 


Since the generalized coordinates in the initial and final points are considered fixed (not affected 
by the variation), all 5qj{t x n i) = 8qj(k m ) = 0, the second term in the right-hand part of Eq. (50) vanishes. 
Multiplying and dividing the last tenn of that part by dt, we finally get 

'fin p j 

t ■ j U( i j j t ■ • 

ini ini 

7 dt 

r dL A 



dt = - J x 

v oy 



r dL A 


v j 

dq , 

Sq :dt = 0. (10.51) 

This relation should hold for an arbitrary set of functions Sq/J), and for any time interval, so that it is 
only possible if the expressions in square brackets equal zero for all j, giving us the set of Lagrange 
equations (2.19). So, the Hamilton principle indeed gives the Lagrange equations of motion. 

It is very useful to make the notion of action S, defined by Eq. (47), more transparent by 
calculating it for the simple case of a single particle moving in a potential field that conserves its energy 
E = T + U. In this case the Lagrangian function L = T -U may be presented as 

L = T-U = 2T -(T + U) = 2T - E = mv 2 -E, (10.52) 

with E = const, so that 

5 = 1 Ldt = J mv 2 dt -Et + const. (10.53) 

Presenting the expression under the remaining integral as my-vdt = p-(dr/dt)dt = p dr, we finally get 

S = J p • dr - Et + const = S 0 - Et + const , (10.54) 

where the time-independent integral 

S 0 = Jp -dr (10.55) 

is frequently called the abbreviated action. 15 

This expression may be used to establish one more connection between the classical and 
quantum mechanics, now in its Schrodinger picture. Indeed, in the quasiclassical (WKB) approximation 
of that picture 16 a particle of fixed energy is described by a De Broglie wave 

v P(r,r) x expj/(j k- dr-cot + const)], (10.56) 

15 Please note that despite a close relation between the abbreviated action S 0 and the action variable J defined by 
Eq. (39), these notions are not identical. Most importantly, J is an integral over a closed trajectory, while S 0 in 
defined for an arbitrary point of a trajectory. 

16 See, e.g., QM Sec. 2.3. 

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where wavevector k is proportional to the particle’s momentum, while frequency co, to its energy: 

k = ^, ® = (10.57) 

n n 

Plugging these expressions into Eq. (56) and comparing the result with Eq. (54), we see that the WKB 
wavefunction may be presented as 

'Pocexp{/S/4 (10.58) 

Hence the Hamilton’s principle (48) means that the total phase of the quasiclassical 
wavefunction should be minimal along particle’s real trajectory. But this is exactly the so-called eikonal 
minimum principle well known from the optics (though valid for any other waves as well), where it 
serves to define the ray paths in the geometric optics limit - similar to the WKB approximation 
condition. Thus, the ratio S/ti may be considered just as the eikonal, i.e. the total phase accumulation, of 
the de Broigle waves. 17 

Now, comparing Eq. (55) with Eq. (33), we see that the action variable J is just the change of the 
abbreviated action So along a single phase-plane contour (divided by 2 tv). This means that in the WKB 
approximation, J is the number of de Broglie waves along the classical trajectory of a particle, i.e. an 
integer value of the corresponding quantum number. If system’s parameters are changed slowly, the 
quantum number has to stay integer, and hence J cannot change, giving a quantum-mechanical 
interpretation of the adiabatic invariance. It is really fascinating that a fact of classical mechanics may 
be “derived” (or at least understood) more easily from the quantum mechanics’ standpoint. 18 

10.4, The Hamilton-Jacobi equation 

Action S, defined by Eq. (47), may be used for one more formulation of classical mechanics. For 
that, we need one more, different commitment: S to be considered a function of the following 
independent arguments: the final time point tf m (which I will, for brevity, denote as t in this section), and 
the set of generalized coordinates (but not of the generalized velocities!) at that point: 


Let us calculate a variation of this (essentially, new!) function, resulting from an arbitrary combination 
of variations of final values qj(f) of the coordinates, while keeping t fixed. Formally this may be done by 
repeating the variation calculations described by Eqs. (49)-(52), besides that now variations dqj at the 
finite point (t) do not necessarily equal zero. As a result, we get 





j ‘ ‘/ / t . . j 



r 5L A 

v a?, v 





17 Eq. (58) was the starting point for R. Feynman’s development of his path-integral formulation of quantum 
mechanics - see, e.g., QM Sec. 5.3. 

18 As a reminder, we have run into a similar situation at our discussion of the non-degenerate parametric 
excitation in Sec. 5.5. 

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For the motion along the real trajectory, i.e. satisfying the Lagrange equations of motion, the second 
term of this expression equals zero. Hence Eq. (60) shows that, for (any) fixed time t, 

dS _ dL 

dq j dqj ' 

But the last derivative is nothing else than the generalized momentum pj - see Eq. (2.3 1), so that 




= Pi 


(As a reminder, both parts of this relation refer to the final moment t of the trajectory.) As a result, the 
full derivative of action S[t, q,(t)] over time takes the form 

dS dS v dS . r . 

— = — + > q : = — + > p Mr 

dt dt j dq 6t j 


Now, by the very definition (59), the full derivative dS/dt is nothing more that the Lagrange 
function L, so that Eq. (63) yields 






However, according to the definition (2) of the Hamiltonian function H, the right-hand part of Eq. (63) 
is just (-11), so that we get an extremely simply-looking Hamilton- Jacobi equation 



This simplicity is, however, rather deceiving, because in order to use this equation for the 
calculation of function S(t, q j) for any particular problem, the Hamiltonian function has to be first 
expressed as a function of time t, generalized coordinates qj, and the generalized momenta p, (which 
may be, according to Eq. (62), presented just as derivatives dSIdqj). Let us see how does this procedure 
work for the simplest case of a ID system with the Hamiltonian function given by Eq. (10). In this case, 
the only generalized momentum is p = dS/dq, so that 

P 1 

H = — vU e{ (q,t) = 





r ds} 

\dq j 

+ U ef (q,t), 

( 10 . 66 ) 

and the Hamilton-Jacobi equation (65) is reduced to a partial differential equation, 

+ U ef (q,t) = 0. 

dS 1 
- + - 

r ds v 

dt 2 m 




Its solution may be readily found in the particular case of time-independent potential energy U e f 
= U e f ( q ). In this case, Eq. (67) is evidently satisfied by a variable-separated solution 

S(t, q) = Sq (q) + const x t . (10.68) 




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Plugging this solution into Eq. (67), we see that since the sum of two last terms in the left-hand part of 
that equation presents the full mechanical energy E, the constant in Eq. (68) is nothing but ( -E ). Thus for 
function So we get an ordinary differential equation 

-E + - 

r ds ^ 2 

2 m 



+ U e f (<7) = 0. 


Integrating it, we get 

S 0 = J {2 m e{ [E - U e{ (<?)]} 1/2 dq + const, (10.70) 

so that, finally, the action is equal to 

S = ^ {2m ef [E -U ef (q)ty l 2 dq - Et + const (10.71) 

For the case of ID motion of a single ID particle, i.e. for q = x, m e f = m, U e ^q) = U(x ), this 
solution is just the ID case of the more general Eqs. (54)-(55), which were obtained by a much more 
simple way. (In particular, S 0 is just the abbreviated action.) 

This particular case illustrates that the Hamilton-Jacobi equation is not the most efficient way for 
solution of most practical problems. However, it may be rather useful for studies of certain mathematical 
aspects of dynamics. 19 Moreover, in the 1940s this approach was extended to a completely different 
field - the optimal control theory, in which the role of action S is played by the so-called cost function - 
a certain functional of a dynamic system, that should be minimized by an optical choice of a control 
signal - a function of time that affects system’s dynamics. From the point of view of this mathematical 
theory, Eq. (65) is a particular case of a more general Hamilton-Jacobi-Bel/man equation. 20 

19 See, e.g., Chapters 6-9 in I. C. Percival and D. Richards, Introduction to Dynamics, Cambridge U. Press, 1983. 

20 See, e.g., T. P. Bertsekas, Dynamic Programming and Optimal Control, vols. 1 and 2, Aetna Scientific, 2005 
and 2007. The reader should not be deceived by the unnatural term “dynamic programming” that was invented by 
the founding father of this field, R. Bellman, to lure government bureaucrats into funding his research, which had 
been deemed too theoretical at that time, but now has a broad range of important applications. 

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10.3. Perform the same tasks as in Problems 1 and 2, for 
the system already considered in Problem 2.5 - a block of mass m 
that can slide, without friction, along the inclined surface of a 
heavy wedge (mass in ’). The wedge is free to move, also without 
friction, along a horizontal surface - see Fig. on the right. (Both 
motions are within the vertical plane containing the steepest slope 

10.4 . Find and solve equations of motion of a particle with the following Hamiltonian function: 

where a is a constant scalar. 

H = -^-(p + or) 2 , 

10.5 . Let L be the Lagrange function, and H the Hamilton function, of the same system. What 
three of the following four statements, 

(0f = 0, 


(U)f = °, 

..... dH 

(m) — = 0, 


(l v)f = 0. 

are equivalent? Give an example when those three equalities hold, but the forth one does not. 

10.6 . Calculate the Poisson brackets of the Cartesian components of the angular momentum L of 
a particle moving in a central force field and its Hamiltonian function H, and discuss the most important 
implication of the result. 

10.7 . After small oscillations had been initiated in a simple pendulum (Fig. on the 
right), the thread is being pulled up slowly, so that the pendulum length / is being 
reduced. Neglecting dissipation, 

(i) prove by a direct calculation that the oscillation energy is indeed changing 
proportionately to the oscillation frequency, as it follows from the constancy of the 
corresponding adiabatic invariant (40), and 

(ii) find the /-dependence of amplitudes of the angular and linear deviations from 
the equilibrium. 

10.8 . The mass m of a small body that performs ID oscillations in potential U(x) = ax 2 ' 1 , with n > 
0, is being changed slowly. Calculate the oscillation energy A as a function of m. 

10.9 . A stiff ball is bouncing vertically from the floor of an elevator whose upward acceleration 
changes very slowly. Neglecting energy dissipation, calculate how much does the bounce height h 
change during acceleration’s increase from 0 to g. 

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