Solutions
Problems for Chapter 2
2.1 We obtain directly dr / dz = f(1 + f2 - rr) / (1 + f2)3/2. The equation of
the curve is 1 + f2 - rr = 0, from which the result follows.
Therefore
r(z) = avl + f(z)2.
Setting f(z) = sinh(¢(z)), we obtain
r(z) = acosh(¢(z));
i.e.,
f
= a¢(z) sinh(¢(z)),
and therefore a¢(z) = 1 and the solution r(z) = acosh((z - zo)/a). This is a
particular case of the use of conserved quantities discussed in Chapter 3.
2.2 Lagrange Multipliers
We must minimize
(7.68)
with the constraints
z(O) = zo,
z(a) =
Zl,
and
i. VI
B
+ z(x)2dx = L.
One can transform the problem into
min
V=
loa (p,gz + >')Vl + z(x)2dx,
(7.69)
with z(O) = zo, z(a) = Zl.
The conserved quantity
(p,gz + >.) = C
+ z(x)2
VI
(7.70)
168
Solutions
yields z
= sinh cf>(x) , i.e.,
Z
j.£gZ
+ A = Ccoshcf> with C¢ = j.£g.
The solution is
A + -C cosh (J-Lg
= --(x - xo) ) .
J-Lg
(7.71)
C
J-Lg
The constants xo, C, and A are fixed by the conditions z(O) = zo,
and Joa JI + z(x)2dx = L.
z(a) =
Zl,
2.3 Brachistochrone
Energy conservation gives
-I (dS)2
+ g(z dt
2
a) = O.
(7.72)
We want to minimize
T-
l
b
I
(
+ Z2
2g(a-z)
a
)
dx
(7.73)
with the constraints z(a) = a, z(b) = {3.
The Lagrange function I:- = ylr'cI-+---'-,z2"-j""-2-g-;-(a---z--:-) does not depend on x, and
therefore there is conservation of
(7.74)
where we introduce a positive constant R. Setting
the parametric form
Z-Zo
Rcoscf>
= ---,
2
x - Xo
=
z=
tan(cf>j2), we obtain
R(cf> + sincf»
2
'
(7.75)
which is the equation of a cycloid.
2.4 Win a Slalom
1. With this definition of the variable x, we have (z - zo) = (x - xo) sina
and the potential energy is V = mg(z - zo) = -mgxsina.
2. The total energy is E = ~m(j;2+1?)-mgxsina. Since energy is conserved,
and since it is taken to be zero initially, we have j;2 + iP = 2gx sin a.
3. Therefore dt 2 = (dx 2 + dy2)j(2gx sin a).
4. The total time to get from 0 to A is therefore
T
=
fAo dt = v'2gsina
I
fA JI +x(y')2 dx
0
5. Using the Lagrange-Euler equation, we obtain
o=
-
d
y'
"'t=::;=====;=~
dx ylx(1 + (y')2)
Solutions
169
6. We deduce
where C is a constant. However,
y'
y'x(l + (y')2)
dy
y'x(dx 2 + dy2)
if
=C
xyf2g sin 0:
'
(7.76)
and therefore if = Kx with K = Cyf2g sin 0:.
7. The parametric form x(B) = (1- cos2B)/2C 2 = sin 2 B/C 2, y(B) = (2Bsin2B)/2C 2 satisfies the equation (y')2 = C 2x/(1- C 2x); i.e., (dy/dB)2 =
(dx/dB)2tan 2 B. From if/x = K, we obtain (dy/dB)(dB/dt)/x = K; i.e.,
dB/dt = K/2 and B = Kt/2 since, for t = 0, B = O.
8. The curve is a portion of a cycloid. We have dy / dx = tan B and therefore
y' » 1 for B rv 7r /2. The trajectory starts vertically (dy / dx = 0 for B = 0)
and becomes horizontal if y(A) » x(A), as shown in Figure 7.1.
o
y
A
x
Fig. 7.1. Optimal trajectory from 0 to A.
9. Since point A is fixed, the velocity VA at A is fixed by energy conservation. It is the maximum velocity of the skier. Therefore, the time to get
horizontally from y(A) to y(O) is larger than the time (y(A) - y(O))/VA it
would take to cover this distance at the maximum velocity. On the other
hand, one must start vertically in order to acquire the maximum velocity
as quickly as possible. The ideal trajectory comes from an optimization
between these two effects.
2.5 Strategy of a Regatta
1. We have by definition x = Vx = V cos B, i = Vz = v sin B, and therefore
z' = dz/dx = tanB.
2. We have Vx = vcosB = w/h. This velocity is maximum when h(z') is
minimum; i.e., for z' = 1, namely B = 7r/4. We then have Vx = w/2. In
fact, it is sufficient to multiply h by a constant to be in the appropriate
situation for a given sailboat for which vx,max = )..w.
3. We have dt = dx/v x = h'(z') dx/w(z), and therefore
-l
T-
L
o
dx
h'(z')
().
w z
(7.77)
170
Solutions
4. Setting <fJ = h'(z')/w(z), the Lagrange-Euler equation that optimizes the
total time T is
~: = :X (~:,) .
5. The function <fJ does not depend explicitly on x. Therefore, we have
~<fJ _
, 8<fJ
- z 8z
dx
,,8<fJ
8z'·
+z
Consequently,
~
dx
(<fJ - z' 8<fJ) = 0
8z'
,
which gives (h'(z')z' - h(z'))/w(z) = constant.
6. We have z'h' - h = -2/z'. We therefore obtain the first-order differential
equation for the function x(z), (-2/A)dx/dz = w(z), and hence the result
x = L WoZ - wlzoln(l + (z/zo))
WOZI - WIZO In(l + (zI/zo)) '
(7.78)
where we have incorporated the conditions (x = 0, z = 0) and (x = L, z =
zd·
7. We obtain
+ (zI/zo))
woL - wILzo/(z + zo)
dz - WOZI - wlzoln(l
z , - -- dx -
~~~--~~~~~~~
« Land Zl « Zo, the velocity of the wind does not vary appreciably
over the whole path, and one has z' '" zI/ L « l.
In the second question, we have seen that the optimal velocity for a
constant wind velocity is attained for z' = 1. The present configuration
certainly does not correspond to the best strategy. One must tack at some
point (Xl, Z) with 0 < Xl < Land Z » Zl, as represented in Figure 7.2
in order to benefit fully from the power of the wind (this possibility was
excluded in the text).
If Zl
z
x:::L
Z :::z,
shore
x
Fig. 7.2. Path of the boat with a tacking at x = L/2.
Solutions
171
The trajectory drawn with an angle of fJ = 45 degrees (lz'l = 1) and a
tacking fJ -+ -fJ at x = L/2 has a total length LV2 and a velocity greater
than (wO - wl)/2. The time along this path, Tv = 2LV2/(wO - wI),
is obviously shorter than the time along the path with no tacking, T rv
2L(zl/L)/(wO - wI) = 2zl/(wO - wI) .
In realistic cases, for instance the America's Cup, one can see how
subtle the regatta problem is. Skippers must make quick decisive choices
between very different options.
Problems for Chapter 3
3.1 Moving Pendulum
3.2 Properties of the Action
1. Free particle
s=
m (X2 - xd 2
2
t2 - h
2. Harmonic oscillator
3. Constant force
with Va = (X2 - Xd/(t2 - td - (1/2)(F/m)(t2 - h).
4. One varies the endpoint of arrival in the integration by parts of
5. One varies t2, taking into account that the variation of the time of arrival
yields a variation of the trajectory.
172
Solutions
3.3 Conjugate Momenta in Spherical Coordinates
1. The Lagrangian is C = ~m(f2
2. The conjugate momenta are
Pr =
ac
af
.
= mr,
P9
+ r2 iJ2 + r2 sin2 0 ¢2) - V(r).
ac
2·
= aiJ = mr 0,
P</>
ac
2
2 .
= a¢ = mr sin O¢.
3. Taking the derivative of (3.73) with respect to time, and taking into account that in Cartesian coordinates p = mv, one obtains directly the
result L z = mr2 sin2 O¢ = P</>o
4. The conservation of P</>' or L z , corresponds to the invariance under translation in ¢; i.e., rotation invariance around the z axis.
5. If a charged particle is in a magnetic field B parallel to Oz, there is
rotational invariance around the z axis and the component L z is conserved.
Problems for Chapter 4
4.1 Coupled Oscillators
1. One obtains directly
{X,P} = 1
{X,Q} = 0
p2
H = 2m
+
mw 2X 2
2
{Y,P} = 0
Q2
+ 2m +
{Y,Q} = 1
m(w2 + ,n2)y2
2
2. The eigenfrequencies of the system are therefore
Jw 2 +,n2 .
.
WI
= wand
3. The general form of the motion follows from
4.2 Three Coupled Oscillators
We obtain with no difficulty
m
2
2
2
H = 2(PI +P2 +P3)
2
mw
2
2
2)
3m,n2 (2
2)
+ -2-(XI
+X2 +X3 + - 2 - Xl +X2 .
4.3 Forced Oscillations
1. We obtain with no difficulty
{X,P} = 1.
W2
Solutions
173
2. In these variables, which are the same as those used by Dirac in the
quantum harmonic oscillator,
H = w(a*a).
3. We obtain {a, a*} = -i.
4. The evolution equation in time of a is
a = {a,H} = -iwa,
which is a first-order differential equation. The general solution is
a(t) = ao exp (-iwt),
where ao is a complex constant. The energy of the oscillator is E = wlaol 2 .
5. For t ::::; 0, we have ao = O. In the presence of Hpoh the Hamiltonian
becomes
H = w(a*a) + b(a + a*) sin fit.
Therefore, we have
a = {a, H} = -iwa -
ib sin fit.
This is solved by standard techniques. With the condition E(t
one obtains
e-i(D-w)T _
E(t > T) = wb 21 2i(D _ w)
1
+
e-i(D+w)T -
2i(D + w)
< 0) = 0,
1
12.
6. This is a resonance phenomenon at D = w (or at D = -w, which is
equivalent). In the vicinity of D = w, the energy acquired by the oscillator
is of the form
E(
T) = b2sin2(D - w)T/2
t>
w
(D-W)2
'
which has a peak of height wb 2T2/4 at D = w.
4.4 Closed Chain of Coupled Oscillators.
1. a) In the definition, we see that
Yk = y'N-k,
b) We have
The summation over k gives onn' and the result
174
Solutions
N
N
L,qkqk = L,p;.
k=l
n=l
(7.80)
Similarly
t t (~ t
k=l
qkqk =
VN n=l
k=l
e-2ikmr/N pn) (
t
~
VN n'=l
e2ikn'7r/N
p~) .
(7.81)
The summation over k gives bnn" and hence the result.
c) On the other hand, we have
~(xn
- X n +,)'
~ ~~
x
(t,
(t/
e-2ikn'IN (1-
ik' n·IN
(1 - e
e- 2ihIN
2ik' . IN)
)Yk)
Yk) . (7.82)
The summation over n gives bkk' and the result.
2. Equations of motion and their solution.
a) We have
with
b) We have
{Yj, qd = bjk' {Yj, qk} = bjk, {Yj, qF,r -d = bjk, {yj, qN -d = bjk.
(7.83)
c) We obtain
Yk = {Yk, H} = ; (qk + qN-k) = mqk'
Yk = {Yk' H} = ; (qk
. _{
qk -
qk,
H} -
- -
.* _ { * H} -
qk -
qkl
- -
+ qF,r-k) =
mqk,
mfl'%(Yk + YN-k) _ fl,2 *
2
- m kYk,
m
fl,2 ( + * )
k Yk YN-k _
2
- m
fl,2
kYk·
d) We therefore have {Yk(t)} = ak cos(fl\t + ¢k), and hence {xn(t)}.
Solutions
175
3. If, at time t = 0, we have YN(O) = 1, YN(O) = 0 and {Yn(O) = 0, Yn(O) =
O}, 'Vn =1= N, then YN(t) = cos(wt) and Yn(t) = 0, 'Vn =1= N. Therefore
xn(t) = (l/VN) cos(wt). Oscillators of the same amplitude at a given
time are always in phase, and only the global motion with respect to the
plane x = 0 with frequency w appears.
4. Wave propagation.
If w = 0, the eigenfrequencies are !?~ = 2!?sin(k1r/N) rv 2!?(k1r/N)
for k « N. The boundary conditions give Y1 = cos 2!?7rt / N, YN -1 =
cos2!?7rt/N, and Yn = 0 otherwise.
a) Therefore, we obtain
lXn = XN-n =
1
-_ VN
[
2 cos (2!?7rt)
VN
~
cos (2!?7rtN+ 2n7r)
cos2n7rN
+ cos (2!?7rtN-
(7.84)
2n7r)] . (7.85 )
b) We observe a propagation phenomenon in both directions since
in the notation above. The point x n +m has the same amplitude at
time t + m/!? as the point Xn at time t.
c) If we write xn(t) = f(t, Y = na), the function f is
1_ [
(2!?7rt + 2Y7r/a)
f( t,y ) -__
VN
cos
N
+ cos (2!?7rt -N 2Y7r/a)]
and satisfies the wave equation
1 82 f
!?2a 2 8t 2
82 f
8x2
------=0.
In this chain of coupled oscillators, a progressive wave of velocity !?a
propagates.
4.5 Virial Theorem
1. One obtains
p2
{A, H} = - - r . V'V.
m
The time evolution of A is simply
dA
dt = {A, H} =
p2
m - r . V'v.
2. We have (.,4)
(A(T) - A(O))/T = O. Therefore, inserting this in the
result above, we obtain
2 ( : : ) = (r· V'V).
176
Solutions
3. If V = gr n , we have
8V
r· V'V = ra;: = nV.
We therefore obtain 2(Ec) = n(V).
4. The total energy is E = Ec + V. We therefore obtain
a) For a harmonic oscillator, E = 2(Ec) = 2(V).
b) For a Newtonian potential, E = -(Ec) = (1/2)(V), which is obvious
on a circular trajectory, but holds for any elliptic trajectory.
5. In general, for an arbitrary potential, the orbits of bound states are not
closed. However, they remain confined in a given region of space at any
time. The generalization of the averaging (4.107) is
(I) = lim (T---+oo)
r
T
T1 Jo f(t) dt.
With this definition, we have
(A) = lim (T---+oo) (A(T) - A(O))/T = 0
since A(t) is bounded for any t. With this definition, the result remains
true.
4.6
{Lx,Ly} = L z
4.7 We obtain
and cyclic permutations.
Problems for Chapter 5
5.1 Telegraph Equation
The Lagrangian density is
(7.86)
where 'lj;* is the "mirror" density which concentrates instead of diffusing. This
leads to the propagation equation
3 8 2'lj;
28'lj;
2~ -i1'lj;+a ~ =0.
v ut
ut
(7.87)
This equation can be solved by Fourier transformation if the coefficients v
and a are constants. (This is not the case if the medium is inhomogeneous or
discontinuous. )
Solutions
177
Problems of Chapter 6
6.2 Geodesics
Solutions exist only for p 2: R (which is explained by equation (6.136)).
The energy is
(7.88)
The calculation is similar to previous cases such as (2). We define the
parameters wand , as before:
2
2E
w = mR2'
(7.89)
We obtain
(7.90)
and
tanh(¢(t) - ¢o) = ,tanhw(t - to).
(7.91)
Problems for Chapter 7
7.1 Propagator of a Harmonic Oscillator
The classical action for a harmonic one-dimensional oscillator is
The calculation of the propagator involves only Gaussian integrals, and the
result follows directly. One recovers (7.61).
References
1. L. Landau and E. Lifshitz, The Classical Theory of Fields, Pergamon
Press, Oxford (1965).
2. Arthur Koestler, The Act of Creation, Hutchinson & Co., London
(1964).
3. R.P. Feynman, R.B. Leighton, and M. Sands, The Feynman Lectures
on Physics, Addison-Wesley, Reading MA (1964).
4. Wolfgang Yourgenau and Stanley Mandelstam, Variational Principles
in Dynamics and Quantum Theory, Dover Publications, New York
(1979).
5. Izrail Moiseevich Gelfand and Sergei Vasilevich Fomin, Calculus of Variations, Rev. English ed. Prentice-Hall, Englewood Cliffs, NJ, (1963).
Andrew Russell Forsyth, Calculus of Variations, Dover, New York
(1960). Jean-Pierre Bourguignon, Calcul Variationnel, Ecole Polytechnique, Palaiseau (1990).
6. Erwin Schr6dinger, Statistical Thermodynamics, Dover Publications,
New York (1989).
7. J.-L. Basdevant and Jean Dalibard, Quantum Mechanics, Springer Verlag, Heidelberg (2005).
8. L. Landau and E. Lifshitz, Mechanics, Pergamon Press, Oxford (1965).
9. Herbert Goldstein, Charles Poole and John Safko, Classical Mechanics,
Addison Wesley, Boston (2002).
10. Philip M. Morse and Herman Feshbach, Methods of Theoretical Physics,
Mc Graw-Hill, New York (1953).
11. Ian Percival and Derek Richards, Introduction to Dynamics, Cambridge
University Press, Cambridge (1982).
12. Max Born and Emil Wolf, Principles of Optics, Pergamon Press, Oxford
(1964).
13. Albert Messiah, Quantum Mechanics, North-Holland, Amsterdam
(1962).
14. J.L. Basdevant, J. Rich, and M. Spiro, Fundamentals in Nuclear
Physics, Springer, New York (2005).
15. Hans Stefani, General Relativity, Cambridge University Press, Cambridge (1982).
180
References
16. Steven Weinberg, Gravitation and Cosmology, John Wiley & Sons, New
York (1972).
17. P. A. M. Dirac, General Theory of Relativity, John Wiley & Sons, New
York (1975).
18. Charles W. Misner, Kip S. Thorne, and John Archibald Wheeler, Gravitation, W.H. Freemann and Company, New York (1973).
19. James Rich, Fundamentals of Cosmology, Springer-Verlag, Heidelberg
(2001).
20. R.P. Feynman and A.R. Hibbs, Quantum Mechanics and Path Integrals,
McGraw-Hill, New York (1965).
21. Lawrence S. Schulman, Techniques and Applications of Path Integration, John Wiley & Sons, New York (1981).
22. Julian Schwinger, Selected Papers on Quantum Electrodynamics, Dover,
New York, (1958).
Index
action, 9, 50, 82, 146
amplitude, 147
angle-action variables, 77
angular momentum, 57
and rotations, 57
attractor, 71
baryonic dark matter, 139
black holes, 139
Boltzmann entropy, 41
Boltzmann factor, 38
brachistochrone, 43
Buridan, Jean de, 11
B6lyai, J., 112
canonical
commutation relations, 80
conjugate variables, 77, 83
equations, 69
formalism, 68
formulation, 16, 67
transformation, 75-77, 79
catenoid, 35
chaos, 71
Christoffel symbols, 114
classical limit, 161
commutator, 80
configuration, 36
conjugate momentum, 53, 61
conservation laws, 53
conservative systems, 87
conserved quantities, 43
constant of the motion, 55, 74, 113
curvature of space-time, 122
curved rays, 27
curved space, 53, 108, 112
cyclic variable, 54, 77, 78
d'Alembert, 15
Descartes, R., 22
diffusion equation, 104
Dirac, P.A.M., 81
disorder, 41
dissipative systems, 58
distribution, 36
dynamical symmetries, 57, 77
dynamical systems, 70
economic models, 15, 41
Ehrenfest theorem, 80
eikonal, 89
eikonal approximation, 90
eikonal equation, 90
Einstein ring, 138
Einstein, A., 17, 53, 107
electromagnetic field, 102
energy, 54
energy-momentum, 62
entropy, 41
Boltzmann, 41
equation of the geodesics, 118
equivalence principle, 108
Eratosthenes, 109
Euclid, 108
Euler, L., 12, 26
Eotvos, R., 18, 107
Fermat principle, 8, 21, 90
182
Index
Fermat, P. de, 8, 21, 50
Feynman principle, 145
Feynman, R.P., 145
field equations, 99
field theory, 17, 97
flow, 17, 68, 70, 88
flow of a vector field, 79
Fourier equation, 104
Legendre transformation, 69
Leibniz, G. W., 9
Liouville theorem, 78, 79
Lobatchevsky, N.I., 109, 112
Lorentz force, 48, 59, 60, 63
Lorentz invariance, 49, 61
Lorentz invariant, 62, 63
Lorenz attractor, 71
Galileo, G., 47
gauge invariance, 60
gauge transformations, 60
Gauss, C.F., 109, 112
general relativity, 1, 17, 107
generalized momentum, 53
geodesics, 117
geometrical optics
and wave optics, 89
gravitation and the curvature of
space-time, 122
gravitational
deflection, 130
lens, 138, 139
microlensing, 140
gravitational lensing, 130, 133, 135
by a cluster of galaxies, 134, 137
time delay, 134
machos, 139
Magellanic clouds, 140
Maupertuis principle, 9, 22, 30, 87, 88,
121
Maupertuis, P.L. de, 9, 15, 22, 24, 30,
47,50
Maxwell distribution, 41
Mercury's perihelion, 125
metric, 110
metric tensor, 110
minimal interaction, 63
mirage, 22, 28
inferior, 28
superior, 28
mirages in the Abell cluster, 139
mirror system, 58
momentum, 56
Hamilton, W.R., 12, 50, 69
Hamilton-Jacobi equation, 82, 85
Hamiltonian, 69, 81
heat, 42
Hero of Alexandria, 10
Huygens principle, 91
interfering alternatives, 147
Jacobi identity, 74
Jacobi theorem, 86
Klein, Felix, 112
Lagrange function, 26
Lagrange multipliers, 37, 43
Lagrange, J.-L., 12, 15, 26, 48, 49
Lagrange-Euler equations, 27, 50
Lagrangian, 50
Laplace, P.S. de, 67
least action principle, 48, 49
least time principle, 21
neutron stars, 139
Newton, I., 47
Newtonian gravitation, 122
optimisation under constraints, 10
partition function, 39
path integrals, 105, 148
phase, 163
phase space, 73, 75, 77, 78
Philoponus, John, 11
photon, 130
Poincare, 71
point transformation, 75
Poisson brackets, 73, 75, 76, 80
Poisson law, 33
Poisson theorem, 75
precession of the perihelion, 125
principle
of maximal disorder, 35
of equal probability of states, 35
of least action, 48, 49
Index
of least time, 9, 24
of natural economy, 9, 21, 30
of the Best, 9
propagator, 152
proper time, 125
Pythagorean music scale, 2
reduced action, 87
refraction, 23
relativistic particle, 61
rescuing, 25
Riemann, B., 110
scalar field, 101
Schri:idinger equation, 104, 154, 160
Schwarzschild metric, 124
Schwarzschild, K., 124
Schwinger variational principle, 163
semiclassical approximation, 91
Shapiro, 1.1., 109
soap bubble, 34
183
state, 36
superposition principle, 147
telegraph equation, 106
temperature, 39, 41
Thales, 110
thermodynamic equilibrium, 36
thermostat, 41
Titius Bode law, 7
translation in time, 54
translations in space, 56
twin paradox, 62
variational calculus, 21, 26
variational principle, 52
verifications of general relativity, 125
vibrating string, 98
white dwarfs, 139
WKB approximation, 91
work,42