Bibliography In this bibliography, we deliberately restricted ourselves to the textbooks or papers that we effectively consulted and used practically. The subject of this book being very broad, it is obvious that the following list is far from exhaustive and there certainly exist many other excellent books. A more complete bibliography can be found as references in some of the textbooks mentioned below. Textbooks that are recommended for undergraduate students which provide a good overview L. Landau and E. Lifchitz, Mechanics, Pergamon Press, 1969. Unavoidable basic textbook. V. Arnold, Mathematical Methods of Classical Mechanics, Springer, 2nd edition, 1989. Reference textbook which, and this is an additional quality for specialists, introduces and uses the tools of differential geometry. H. Goldstein, Classical Mechanics, Addison-Wesley, Londres, 2nd edition, 1980. Basic textbook for point mechanics and continuous medium mechanics. It is very complete, especially for relativitic aspects. The discussion concerning chaos is absent. H. Goldstein, C. Poole and J. Fasko, Classical Mechanics, Addison-Wesley, Londres, 3rd edition, 2002. This third edition of the famous Goldstein book takes advantage of the experience of two new authors for renewing some obsolete aspects of the previous editions, in particular concerning relativistic topics. This edition addresses the notion of chaos.
458 Bibliography I. Percival and D. Richards, Introduction to Dynamics, Cambridge University Press, 1982. Excellent textbook that we recommend for its clarity and its pedagogical emphasis, written by specialists in this domain. J.V. José and E.J. Saletan, Classical Dynamics, Cambridge University Press, 1998. This very pedagogical textbook takes time to explain and illustrate every new notion. Its contains many explicit diagrams and figures and is a source of many interesting and miscellaneous exercices. S. Hildebrandt and A. Tromba, Mathématiques et formes optimales, Belin, Pour la Science, 1986 (in French). A real masterpiece of clarity which explains the various aspects of the least action principle. G.L. Kotkin and U. Serbo, Collection of Problems in Classical Mechanics, Pergamon Press, 1971. An excellent source of very beautiful original problems on various aspects of classical mechanics. I. Stewart, Does God Play Dice?, Penguin, new edition, 1990. Remarkable popular work concerning chaos, understandable to all. It addresses all interesting aspects, including those which go beyond Hamiltonian systems. Poincaré, collection Les génies de la science, Pour la Science, n 4, November 2000 (in french). Useful for all. Textbooks for undergraduate and graduate students which we used as reference works concerning the summaries of this book C. Lanczos, The Variational Principles of Mechanics, University of Toronto Press, 1970. Very marginal work, which deals with subjects generally forgotten by other authors and which is, consequently, an invaluable complement. H.G. Schuster, Deterministic Chaos: An Introduction, VCH Verlagsgellschaft Germany, 1987. This book easy to read, illustrated with many figures, is an introduction to chaos; it is not restricted to Hamiltonian systems.
Bibliography 459 M. Tabor, Chaos and Integrability in Nonlinear Dynamics, John Wiley & Sons, New York, 1988. Very concise and complete, this textbook has the advantage of avoiding sophisticated mathematical tools. A.J. Lichtenberg and M.A. Lieberman, Regular and Stochastic Motion, Springer-Verlag, 1983. This book is addressed to specialists. E. Ott, Chaos in Dynamical Systems, Cambridge University Press, 1993. This clear and complete textbook is concerned with general dynamical systems and thus goes beyond the ambition of the present work. M.C. Gutzwiller, Chaos in Classical and Quantum Mechanics, Springer- Verlag, New York Inc., 1990. A book for students interested in quantum mechanics. Nevertheless, the part devoted to classical mechanics is well presented. E.N. Lorentz, Essence of Chaos, University of Washington Press, 1995. Easily readable, this book addresses all topics concerning chaos. F. Scheck, Mechanics: From Newton s Law to Deterministic Chaos, Springer- Verlag, 3rd edition, 1999. A book close to ours in spirit, but which employs more abstract mathematical notations. W. Yourgrau and S. Mandelstam, Variational Principles in Dynamics and Quantum Theory, Pitman, 3rd edition, 1968. Original and epistemological analysis of Hamilton s theory. Many historical references and connections with other domains in physics. N. Rasband, Dynamics, John Wiley & Sons, 1983. Many aspects are addressed in this book, but the mathematical notations are abstract. K.T. Alligood, T.D. Sauer and J.A. Yorke, Chaos, an Introduction to Dynamical Systems, Springer-Verlag, 1997. This book is clear and complete, but reserved for students with a good mathematical training.
460 Bibliography Books which were consulted occasionally to address very special points. The interested reader may thus go further by looking at them L. Landau and E. Lifchitz, The Classical Theory of Fields, Pergamon Press, 1994. L. Landau and E. Lifchitz, Theory of Elasticity, Pergamon Press, 1986. J.D. Jackson, Classical Electrodynamics, John Wiley & Sons, New York, 1962. A. Messiah, Quantum Mechanics, North Holland, 1963. L.E. Ballentine, Quantum Mechanics, World Scientific Publishing Co., 1998. B. Cagnac and J.C. Pebay- Peyroula, Modern Atomic Physics, MacMillan interacting publishing, 1975. J. Bass, Cours de mathématiques, Masson, 1977 (in French). J.W.S. Rayleigh, Theory of Sound, Dover, 1945. R.P. Feynman, Lectures on Physics, Addison-Wesley Publishing Company, 1963. S. Weinberg, Gravitation and Cosmology, John Wiley & Sons, 1972. G. Bruhat, Optique, Masson, 1992 (in French). W.H. Press, S.A. Teukolsky, W.T. Vetterling and B.P. Flannery, Numerical Recipes, Cambridge University Press, 1992. R. Campbell, Théorie générale de l équation de Mathieu et de quelques autres équations, Masson, 1955 (in French). C. Rosensweig and J.B. Krieger, Exact Quantization Conditions, J. Math. Phys., 9, 849, 1968. F.L. Moore, J.C. Robinson, C.F. Barucha, Bala Sundaram and M.G. Raizen, Atom Optics Realisation of the Quantum Delta-kicked Rotor, Phys. Rev. Lett. 75, 4598, 1995. H. Stapelfeldt and T. Seideman, Aligning Molecules with Strong Laser Pulses, Rev. Mod. Phys. 75, 543, 2003.
Index acceleration mode, 401, 425, 437 acoustical frequency, 75, 108 action, 112 function, 233, 239 functional, 112 variable, 284 adiabatic invariant, 341, 346 Aharonov Bohm effect, 246 d Alembert principle, 12, 14 angle action variables, 283 Anosov s mapping, 403, 432 aphelion, 223, 229 areal velocity, 56 Arnold s cat, 403, 432 atomic chain, 75, 107 attractor, 446 aurora borealis, 354, 379 autonomous, 167 axial frequency, 95 axle, 19, 39 bead, 16, 28, 186, 224, 356, 382 bifurcation, 187, 228, 394, 398, 419 billiard, 183, 409, 447 Binet equation, 56, 133, 134, 143, 163, 222, 228 blade, 71, 102 boost, 304 brachistochrone, 148 calculus of variations, 114 canonical perturbation, 342, 348 transformation, 283, 334 caustic, 243, 264 centrifugal force, 23, 50 chaos, 390, 392, 395 chaos ergodicity, 399, 423 Compton, 23, 47 conjugate point, 119, 138 variable, 52 conservative, 167 constant of the motion, 53, 167, 282, 288 constraint equation, 40 continuous fraction, 396, 417 convergent direction, 394 separatrix, 394 Coriolis force, 12, 17, 23, 49 Coulomb problem, 302 Curie principle, 125 cyclic coordinate, 53 cycloid, 31 cyclotron frequency, 96, 352, 376 motion, 379 radius, 379 damped pendulum, 407, 443 declination angle, 69 degree of freedom, 10 divergent separatrix, 394 drift, 66, 67, 91, 354, 381 ecliptic, 68, 98 elastic bar, 64 moment, 71
462 Index electromagnetic potential, 51 electrostatic lens, 243, 265 ellipsoid, 150 elliptic coordinate, 248, 276 integral, 349 point, 168 energy, 53, 167 equinox precession, 68, 97 Fermat path, 236 principle, 122, 144, 181 Fermi accelerator, 405, 438 Fibonacci sequence, 403, 435 field, 112 fine structure, 293 constant, 292, 294, 318 first integral, 281, 288 fixed point, 169, 390, 392 flexion vibration, 71, 102 flow, 167, 204 parameter, 287, 304, 337 Foucault pendulum, 59, 79 free fall, 242, 261 friction force, 12 Galilean transformation, 304, 338 general action, 253 generalized acceleration, 10 coordinate, 9 force, 12 momentum, 52, 165 potential, 51 velocity, 10 generating function, 289, 297, 298, 301, 302, 324, 330, 332, 353, 375 generator, 284, 287, 303, 336 geodesic, 241, 261 group speed, 239 velocity, 252, 279 gyroscope, 21, 44, 46 Hamilton equation, 167 function, 166 principle, 111 Hamilton Jacobi, 233, 235 Hamiltonian, 166 Hannay s phase, 356, 382 harmonic oscillator, 295, 298, 302 Heiles, 62 Hénon, 62 Hénon and Heiles potential, 84 heterocline point, 395 holonomic, 13, 40 constraint, 18, 116, 123 homocline point, 395 hoop, 16, 28, 186, 224 Huygens construction, 238, 243 pendulum, 17, 31 hyperbolic point, 168 index, 236 inertial force, 23, 48 integrable system, 281 integral constraint, 115, 159 involution, 282, 323 isochronous, 17 Jacobi theorem, 236 Jupiter s Greeks, 412, 450 Trojans, 412, 450 KAM curve, 390 theorem, 391 Kepler problem, 174, 292, 295, 314 kicked rotor, 385 kinetic energy, 10 Koenig theorem, 18, 76, 99 Lagrange
Index 463 equation, 10, 52 function, 52 multiplier, 13, 19, 41, 115, 150, 159 Lagrangian, 52 point, 412, 413, 450, 452 system, 52 Landau levels, 312 Laplace law, 127, 158 least action principle, 111, 118, 135 Legendre transform, 166 libration, 227 Liouville theorem, 167 Lorentz force, 116, 131, 200 transformation, 304, 339 magnetic flux, 246 forces, 12 moment, 378 magnetron frequency, 96 Mathieu, 211 equation, 180 Maupertuis principle, 121, 141, 235, 245, 268 Mercury, 128, 162 Minkowski metric, 176 mixture, 395 Noether theorem, 54, 58, 78 non-resonant torus, 389 optical frequency, 75, 108 path, 214 Painlevé integral, 58, 77 parabolic coordinate, 247, 271 parametric resonance, 170 Paul trap, 180, 211 Penning trap, 67, 94 perihelion, 197, 223, 229 perturbation theory, 341, 342 phase portrait, 37, 168, 190, 220, 227, 229 space, 165 speed, 239 velocity, 252, 279 Poincaré section, 183, 388 Poincaré Birkhoff theorem, 393 Poisson bracket, 281, 327, 329 precession of perihelia, 129, 164 prolate shape, 249 propagator, 169, 208, 212 proper mode, 57 quadrupolar approximation, 249 quadrupole interaction, 97 quantum, 112 quasi-integrable system, 341 reaction force, 16, 19, 28 reduced action, 121, 234, 239, 253 mass, 55 resonant torus, 286, 390, 396, 415 reverse pendulum, 178, 207 revisiting theorem, 168 rope, 16, 27 rotating frame, 65, 89, 172, 192 rule EBK, 285 Runge Lenz vector, 173, 195 Rydberg constant, 294, 318 saddle point, 120 sawtooth mapping, 402, 428 scalar potential, 51 scale invariance, 395 Schwarzschild metric, 129, 162 radius, 129, 164 secular term, 348, 362 self-similarity, 395 separation of variables, 236 separatrice, 168 shearing modulus, 64
464 Index sidereal day, 69 sine-gordon equation, 73, 105 sling, 15, 26 small denominators, 345 Snell Descartes law, 144, 215 soap film, 125, 154 soft mode, 63, 86 solitary wave, 73, 105 Sommerfeld atom, 293, 316 spiral point, 446 square well, 351 stability islet, 392 stable elliptic fixed point, 394 node, 168 standard mapping, 387, 398, 401, 402, 418, 425, 428 Stark effect, 247, 271 surface tension, 127, 158 Toda net, 62 torus, 282 trajectory, 11 transversal wave, 64, 88 turn indicator, 21, 43 turning point, 168, 274, 278 Ulam approximation, 407 mapping, 443 unstable hyperbolic point, 394 node, 168 vector potential, 51 virial theorem, 186, 223, 230 virtual displacement, 12 work, 12 wave front, 242, 243, 263, 264 wheel jack, 14, 24 Young modulus, 72