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