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 =
m
ef -2
q~ + U ef (q) = const .
(3.24)
This is the first integral motion. Solving Eq. (24) for q , we get the first-order differential equation,
dq
dt
1/2
= +
m
[H-U e{ (q)]\
(3.25)
ef
which may be readily integrated:
^/« ef A
V ^ J
1/2 q(t)
1
#(/n)
dq '
[H-U tf (q')\
,2 =t ~ t 0 '
(3.26)
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).
Chapter 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:

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
Attractive
Coulomb
potential
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.
(3.49)
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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 )

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. Chapter 3 Page 14 of 20 Essential Graduate Physics CM: Classical Mechanics T = A dA / dt nab L_ / 2m Using Eqs. (60) and (63), this result may be presented in several other forms: 6™V /2 r = np (\-e 2 f n (L z /2m) = na\ r Y /2 m 2\E\ = 2 na 3/2 m yaj (3.64a) (3.64b) 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- specific. 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 Li = E- (3.65) (3.66) 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. Chapter 3 Page 15 of 20 Essential Graduate Physics CM: Classical Mechanics 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

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. Chapter 3 Page 16 of 20 Essential Graduate Physics CM: Classical Mechanics Rutherford scattering formula da a ' 2 1 dd f4 Ej sin 4 (0/ 2) (3.72) 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

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 . r" 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. 6.5. Chapter 3 Page 18 of 20 Essential Graduate Physics CM: Classical Mechanics 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 U(r) 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. Chapter 3 Page 19 of 20 Essential Graduate Physics CM: Classical Mechanics Chapter 3 Page 20 of 20 Essential Graduate Physics CM: Classical Mechanics 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 , (4.1) whose Lagrangian equation of motion, Harmonic oscillator’s equation K mq + Kq = 0, i.e. q + a> 0 q = 0, with of = — > 0 m (4.2) 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: Harmonic oscillator’s motion q(t) = u cos co 0 t + vsin a> 0 t = A cos (a> 0 t -

exp {icot}, with the corresponding sign implications for intermediate formulas, but (of
course) similar final results for real variables.
© 2013-2016 K. Likharev
Essential Graduate Physics
CM: Classical Mechanics
f v = -m »
(4.5)
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 .
(4.6a)
This equation is frequently presented in the form
q + 2 8q + a>lq = 0, with 8 =
1
2m
(4.6b)
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.
(4.7)
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 = —
8
2m
V
(4.10)
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.
Free
oscillator
with
damping
Decaying
free
oscillations
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6) 0 _ m = 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
RWA
solution
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
Duffing
equation
2
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) .
(4.43)
q i0) = A{t)cos[cot-cp{t)\, with A,cp 0 at£— >0.
(4.41)
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

(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. (4.58a) (4.58b) 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 Chapter 4 Page 14 of 34 Essential Graduate Physics CM: Classical Mechanics 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) 8 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 (4.64) 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). Chapter 4 Page 15 of 34 Essential Graduate Physics CM: Classical Mechanics 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 + A 2 co cos cp. (4.67b) 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 ] (4.68) (4.69) Within the range - |A| < <^< + |A|, Eq. (68) has two fixed points on each 2;r-segment of variable cp: cp ± — i arccos + 27m . (4.70) 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. Phase locking equation Chapter 4 Page 16 of 34 Essential Graduate Physics CM: Classical Mechanics 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 result, A =A A , = _LA 2\S\ 2 co sin

0, i.e. oscillation amplitude has to grow if 22
A / 2nd _ n
/ 3a> 0 3 Q
(4.74)
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,
4
Acp = Ad~^-Acos2cp.
(4.77)
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.
Chapter 4
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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
RWA
equations
for
parametric
excitation
u = —8 u — v ,
4
e /UOJ
v = -ov + cu a .
4
(4.78)
(4.79)
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
At
v = c„e
At
(4.80)
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
/LUO
* 4~
„ ua>
■*"T
-8- A
= A 2 +28A + 8 2 +f~
2
V 4 j
= 0,
A ±
- 8 ±
f iiccf 2
J
-Ml
(4.81)
(4.82)
(4.83)
Requiring the fixed point to be unstable, Re/l + > 0 , we get the parametric excitation condition
^->{8 2 +

, 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

2 l .
(5.69)
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 .
(5.70)
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 =

) = £, ^ (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. Chapter 6 Page 19 of 30 Essential Graduate Physics CM: Classical Mechanics 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 =

0 this expression may be readily simplified:
U c[ ( 6 ) « const +
f L 2
_zL
SI,
M gl\ 32
(6.84)
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,
Threshold
(6.85) angular
velocity
2
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 }
ii
1/2
\ 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 '
(6.87)
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
Transformation
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)
dt
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),
(6.90)
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 .
(6.92)
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
Transformation
of
acceleration
2 nd Newton
law in non-
inertial
reference
frame
Chapter 6
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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 - 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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FJA O'
c =
FJA 0
0
0
(7.20)
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.
dAj,
'U
dF u) + d(dF ~~ x r) - C/ ef , (6.95)
where the effective potential energy, 20
U ef =U-j{~~** shape into a compressed fluid.
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AV
V
3
j = 1
P
K
(7.38)
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 '
(7.39)
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
AP
T =const
Nk B T
~V^~
AV =
. AV
i.e.
V
T =const
AP
P '
(7.41)
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
(7.43)
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
L
F * , \ F
^~ f) A $ !'•) —►
^ Fig. 7.6. Tensile stress experiment.
z
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 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
AV
V
= s.
+ 5 ,
+ s.
1 F
~E~4
(1 - 2a),
(7.49)
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 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 ■.
R~
(P x -P 2 )R 7
3 1
1 — 2cr 1 + cr
+ -
2 E
= ^~P 2 )
R~ 1 — (7
t 2 E
(7.64)
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,
F
a = —
A
^R\p x -p 2 )
2 nRt
(7.65)
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//.
Chapter 7
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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.
(a)
(b)
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, =< 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)
A
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:
s
zz
dq z _ x
dz R
(7.68)
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)
AAA
where I v is a geometric constant defined as
I v =^x 2 dxdy. (7.72)
A
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)
A
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)
A
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,
y
(7.75)
Rod
bending
curvature
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.
(a)
q = 0p
(b)
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
Chapter 7
Page 18 of 38
Essential Graduate Physics
CM: Classical Mechanics
dF
= ~f* A • (7-76)
dz
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:
dr
-T = ~ F x- (7-77)
dz
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**

= 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, 0 X
2 m ef 2
where of =
k
ef
m
ef
(10.14)
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):
dq
dt
-co 0 q.
(10.15)
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):
df_
dt
df
= — +
dt
I
df

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 E co Q ‘ (10.40) 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). Chapter 10 Page 7 of 14 Essential Graduate Physics CM: Classical Mechanics dJ dt 1 In 1 2 n ( dp dE dp dA \ , — + — — dq . \dE dt dA dt ) (10.41) 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 . (10.42) Adiabatic invariance 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 (10.43) 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) dJ 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 (10.45) is valid for an arbitrary oscillator described by Eq. (10). Thus, Eq. (44) becomes 0 = 2n — + const . T (10.46) 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}. Chapter 10 Page 8 of 14 Essential Graduate Physics CM: Classical Mechanics Action Hamilton principle 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 = (10.47) 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 principle. 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 , (10.48) 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 Sq,+—Sq, dq 8q i =z — / a?. dL d + / 5q,- j dq j dt (10.49) 14 See, e.g., Chapter 6 in J. Jose and E. Saletan, Classical Dynamics, Cambridge U. Press, 1998. Chapter 10 Page 9 of 14 Essential Graduate Physics CM: Classical Mechanics After plugging the last expression into Eq. (48), we can integrate the second term by parts: dL 'fin ar 'fin dL d ‘fin = J xl^^/'+X t.jdqj j dL —8q, ~\U dq \_ 1 J At- fin ‘fin -X j f dL A j t = 0 . (10.50) 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 d_ 7 dt r dL A dq ‘fin dt = - J x v oy d_ dt r dL A dq v j dL 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. Chapter 10 Page 10 of 14 Essential Graduate Physics CM: Classical Mechanics 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: (10.59) 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 Hamilton- Jacobi action .-N'l j ‘ ‘/ / t . . j d_ dt r 5L A v a?, v dL dq. 5q, (10.60) 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. Chapter 10 Page 11 of 14 Essential Graduate Physics CM: Classical Mechanics 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 (10.61) dS dq, = Pi (10.62) (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 (10.63) Now, by the very definition (59), the full derivative dS/dt is nothing more that the Lagrange function L, so that Eq. (63) yields as dt L-'ZPjVj- j (10.64) 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 dt (10.65) 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) = 2m ef 2m ef 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 ef ydqj (10.67) 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) Hamilton- Jacobi equation Chapter 10 Page 12 of 14 Essential Graduate Physics CM: Classical Mechanics 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 ef dq + U e f (<7) = 0. (10.69) Integrating it, we get S 0 = J {2 m e{ [E - U e{ ( 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. Chapter 10 Page 14 of 14