Analytical Mechanics Louis N

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INDEX

accelerated systems, Hamiltonian dynamics. See area-preserving map. See map, area-preserving Hamiltonian dynamics, in accelerated systems Aristarchus of Samos, 317 acceleration Aristotle, 317 and relation to mass, 512-513 Arnol'd. See KAM theorem block on inclined plane, 2-3 Arnold diffusion, 451, 470, 472 caused by a constant force, 535-536 asteroid belt, 469 elevator, 20 asteroid, perturbed by Jupiter (prob), 417 transformation of coordinate systems, 266 astrophysical data, table of, 167-169 action functional S, 51, 178, 220 asymmetric top, 288, 331 (prob) action principle. See variation principle kinetic energy, 306 extended. See extended action torque free, 305-312 relativistic. See relativistic action attractive force, hyperbolic orbits. See hyperbolic orbits, variable, 230-231 attractive force action-angle variables, 230-235, 237, 238, attractor, 452 (see also phase space trajectories) 246-247 (prob) periodic, 425 pendulum. See pendulum, action-angle variables strange. See strange attractor adiabatic invariants, 118, 233-235 Atwood machine, double (prob), 31 airplane wings, 397 axial vector. See pseudovector Almagest, 130 axions, 155 almost periodic, 238 amplifiers, parametric. See parametric amplifiers band theory, of solids, 390 analytical mechanics, 23 basins of attraction, 453 angle variable, 230, 232 bead on wire (prob), 30, 34, 114 angular frequency, 526 beam stability, 390 angular momentum, 137-139, 170-172, 283, 245 (prob) beat frequency. See frequency, beat intrinsic. See angular momentum, rotational Bernoulli, Daniel, 45, 86, 344 rotational, 292 Bernoulli, Jacques, 4 total, 291-292 Bernoulli, Jean, 44-46, 344 translational, 292 Bertrand's theorem, 154 angular momentum barrier, 140-141 Bertrand's theorem (prob), 167 angular velocity, 256-257, 264-266, 283, 293, 304, 306, bicycle (prob), 27 276-277 (prob) bifurcation diagrams, 459-460, 484^85, 491 anharmonic oscillator, 398-401 pitchfork, 459, 483-484, 486 driven, 401^11 Big Bang, 325, 456 period of, 401 binding energy, 516 annihilation radiation, 517 Biot-Savart law, 509 antisymmetric four-tensor, electric and magnetic fields, Birkhoff, 439 (see also Poincare-Birkhoff theorem) 530, 539 black body radiation, 325 antisymmetric matrix, 264, 304 black hole M87, 155 aphelion, ellipse. See ellipse, aphelion block on inclined plane, example, 1-7 apparent force, 267 body cone, 297, 298 approximate solutions to non-analytic problems, 383^11 body coordinate system, 259-264, 284, 300 apsides, ellipse. See ellipse, apsides boldface notation, 262 arc length, 138 boost. See Lorentz transformations

566 INDEX

bouncing ball (prob), 475 comets (prob), 163 boundary conditions, matching. See matching boundary complex numbers, 111 conditions solutions, 89 boundary curve for Poincare section of double pendulum, Compton, A. H., 252, 280 431 Compton generator (prob), 280 bowling ball, 38-39, 35 (prob) Compton scattering (prob), 549 box-counting dimension. See dimension, box-counting configuration space, 36-37, 229 brachistochrone, 44-46, 70 (prob) conic section, 144 Brahe, Tycho, 131 conjugate momentum. See canonically conjugate Britain, dimension of coastline. See dimension, of momentum coastline of Britain conservation of angular momentum, 170-172 broken symmetry. See symmetry breaking of charge. See charge conservation brown dwarfs, 155 of energy, relativistic (prob), 547 bug crawling on a phonograph turntable, Hamiltonian and of momentum. See momentum conservation Lagrangian, 190-192 conservative forces, 18, 41-43 butterfly effect, 425 conserved function, 21 quantity, 174 calculus of variations. See variational calculus constant of the motion, 21, 22, 138, 170, 172, 196 (prob) cannonball example, 52, 77 constraint force, 2, 57 canonical transformations, 207-213, 215-218, 237, 250, generalized, 60-61 241-244 (prob) constraints, 12-13 canonically conjugate momentum, 22, 75, 170, 172, 175, holonomic, 13, 57-58, 83, 85, 33 (prob) 181,538, 34-35 (prob) nonholonomic, 36-40, 62-64, 35 (prob) Cantor set, 464-465 nonintegrable, 36 capacitor (prob), 116 rheonomic, 13, 22 capacity dimension. See dimension, box-counting scleronomic, 13, 22, 85 Cassini, 339-340 contact transformation, 208, 211 catenary curve, 62 continued fraction expansion, 436-437 causal Green's function. See Green's function, causal continuity equation, 188, 525 cavity, cylindrical. See gedanken experiment, cylindrical continuous transformations. See transformations, cavity continuous center of mass, 134, 136-137 contravariant components, 529 motion, 283, 285, 292, 298, 160-161 (prob) control parameter, 481 center of momentum. See center of mass coordinate system central force, 24, 133-141, [161-162, 198, 413] (prob) body. See body coordinate system {see also Kepler problem) rotating. See rotating reference frames motion space. See space coordinate system effective potential, 140-141 polar energy diagram, 142 cylindrical, 10, 19, 24-25 Kepler's Laws, 132,139 elliptical, 309-310 orbit equation, 142-143 spherical, 10,51, 138,226 stability of circular orbit, 384-388 coordinates, generalized. See generalized coordinates centrifugal force, 142, 267 Copernicus, Nicolaus, 130-131 centripetal acceleration, 40 Coriolis force, 267 Ceres, 148 cosinelike, 390 cgs units, 507, 540 cosmic rays, 537 chain example, 61 Coulomb potential. See hyperbolic orbits, repulsive chain rule, 6 force Chandler wobble, 299-300 coupled pendulum. See pendulum, two coupled Chandler, S., 300 covariant components, 529 chaos, 237, 475 (prob) procedure, 538-539 in conservative systems, 379, 425, 428^52, 472 Cramer's rule, 380 in dissipative systems, 425, 452-468 critical damping (prob), 117, 121 chaotic dynamics, 423-474 critically damped. See oscillator, damped simple characteristic exponent, 393 harmonic multiplier, 444 cross section (prob), 165 polynomial, 337 cubic polynomials, 315 charge conservation, 188 curvature, of potential energy, 85-86 circular hoop with 3 point masses connected by massless cyclic coordinates. See ignorable coordinates springs, 365-367 cyclotron frequency, relativistic, 540 motion, relativistic. See relativistic circular motion orbits, 144, 385 d'Alembert, 4, 322 classically forbidden, 126 d'Alembert's Principle, 5 clockwork universe, 423 damped driven pendulum, 425, 453^63, 467-468 coastline. See dimension, of coastline of Britain oscillator. See oscillator, damped cofactors, 355, 362, 380-382 dark matter, 154-156

INDEX 567

day, length of in age of dinosaurs (prob), 282 eccentric anomaly, 148-150 degeneracy, 339, 343, 353 eccentricity, 323, 470-473 modes, 365-367 ellipse, 144 degrees of freedom, 2, 8, 10-12, 36, 86, 27 (prob) ecliptic, 317 rotational, 283 effective potential, 140-142, 385 delta function, 99-100 eigenvalues, 302-303, 336-339, 392, 444, derivative 449 directional. See directional derivative generalized, 353, 359 Lie. See Lie derivative eigenvectors, 302-303, 337-339, 392, 444, determinant, 203, 352 449 deterministic chaos, 424 generalized, 353 diagonalization, 392 Einstein, A., 154, 493-495, 508, 514, 516, 519, of symmetric matrices, 336-339 533 of t and v matrices, 364-365 Einstein summation convention, 253, 529 diatomic molecule (prob), 377 electric and magnetic field, antisymmetric four-tensor. diffeomorphism, 55 See antisymmetric four-tensor, electric and differential equation magnetic field linear, inhomogeneous, 95 electric field, 493, 508-510, 522 ordinary, second-order, and homogeneous, 88-89 charged particle. See particle in electric and magnetic superposition, 88-89 fields second order, 24 transformation law for light pulse. See transformation differential equations of motion. See equations of motion law, electric field for light pulse dimension, 463-468 electric force, transformation law. See transformation law, box counting, 463-466 electric and magnetic forces fractional, 463-464 {see also fractal) electromagnetic field (prob), 199-200, 550, 551 Hausdorff, 463 electromagnetic field, relativistic Lagrangian. See Lyapunov, 467 relativistic Lagrangian, electromagnetic field magnification, 466 electron capture, 185 of coastline of Britain, 465^466 electron-positron annihilation, 517 of strange attractor of damped driven pendulum, 468 electrostatics (prob), 67 Dirac delta function. See delta function elevator, 20, 200 (prob) Dirac, P. A. M., 44 ellipse, 144-145 directional derivative, 236 aphelion, 145 discontinuity, in Green's function, 101-102 apsides, 145 discrete transformations. See transformations, discrete eccentricity, 144 displaced axis theorem, 289 perihelion, 145 displacement ratio. See mode displacement ratio semimajor axis, 145-146 dissipative flow, 189 semiminor axis, 145-146 divergence, 192 ellipsoid exponential of trajectories. See characteristic exponent equation of, 306 Doppler shift, 505, 507, 515, 546 (prob) inertia, 306 dot cancellation, 14 inertia tensor (prob), 327 dot product. See scalar product momental, 306 double pendulum oblate, 289 exact, 425, 426-428, 430-433, 435, 440-442, 451-452 prolate, 289 Hamiltonian, 426^27 elliptic fixed points. See fixed points, elliptic Lagrangian, 349 orbits, 145,385, 162 (prob) small angle approximation, 348-355, 358-359, elliptical motion 377 (prob) period of, 145-147 double-well potential (prob), 157 time dependence, 147-150 driven oscillator. See oscillator, driven energy, 21 {see also Hamiltonian) driven support of pendulum. See pendulum, with driven of light wave, transformation law. See transformation support law, energy of light wave DSHO. See oscillator, damped simple harmonic binding. See binding energy Duffing oscillator, 384, 398-401, [158, 4191 (prob) relativistic, 515-517, 533 driven, 401^11 stored in oscillator. See oscillator, stored energy period of, 401 energy-momentum four-vector. See four-vector, dumbbell, inertia tensor, 290-291 energy-momentum dynamical equilibrium. See equilibrium, dynamical entrainment (prob), 480 instability. See instability, dynamical EOM. See equations of motion stability. See stability, dynamical epicyles, 130-131 variable, 7 equant, 130-131 equations of motion, 17-19, 23-24, 51, 54 Earth. See solar system, stability equator, 339-340 as noninertial reference frame, 267-275 equatorial bulge equatorial bulge. See equatorial bulge Earth's, 339-342, 332 (prob)

568 INDEX

equilibrium, 82-87, 114 (prob) cyclotron. See cyclotron frequency dynamical, 384 mode. See mode frequency neutral, 85 of oscillator, 88 pendulum, 82-83 resonant, 108 stable, 82, 125 frequency of light wave, transformation law. See unstable, 82, 125 transformation law, frequency of light wave equinoxes friction, 38, 90 nutation of (prob), 333 full width at half max, 109 precession of, 317-323 functional, 44, 46 ether, 493 functional calculus. See variational calculus Euler angles, 284, 300-302 fundamental theorem of symmetric real matrices, Euler equation, 46^9, 65 336 Euler force, 267 Euler's equations, 283, 292-293, 297-299, Galilean Invariance Principle, 54 329-330 (prob) Galilean transformation laws, 504 Euler, Leonard, 23, 46, 299 Galileo, G., 87, 105, 131-132, 494 Euler-Lagrange equations, 19, 23-24, 51, 54 Gauss, 148 matrix form, 352 Gauss-Jordan elimination, 337 normal coordinates, 359 Gaussian units. See cgs units exotic particle decay (prob). See particle decay (prob) gedanken experiment, cylindrical cavity, 519-521 exponent, characteristic. See characteristic general relativity, 44, 154, 494, 530, exponent 420-422 (prob) extended action, 76 generalized coordinates, 10, 350 extremum, 46, 49 equations of motion, 17-18 forces, 14-15, 18-19,82 Falkland Islands battle (prob), 279 central force, 24-25 Feigenbaum constant, 458^459, 487, 490 kinetic energy, 16-17 Feigenbaum, M. J., 457, 485, 486 momentum. See canonical momentum Fermat, 55 velocities, 14 Fermat's principle, 55, 75, 229, 68 (prob) generating function, 207, 210-217, 240-244 (prob) fictitious force, 25, 253, 267, [279-282, 290] (prob) geodesic curve, 51, 66-67 (prob) fig tree, 485, 486 Gibbs, J. W., 253 filamentation, 186 global chaos, 442, 451 fixed point, 481-482 golden mean, 437 elliptic, 431, 434-435, 440-441, 445 governor (prob), 32 hyperbolic, 435, 440-442, 446-448 gradient vector, 307 neutral, 428^83, 485 grandfather's clock. See pendulum, nonlinear stable, 482, 485 gravitational constant, 339-342 superstable, 482-486 gravitational field, particle in, Hamiltonian-Jacobi unstable, 482-382, 485 equation, 220-222, 224-228 Fizeau, M, 493 potential energy of, 318 flip hyperbolic fixed point, 446 gravity, 134, 141-142, 159 (prob) Floquet matrix, 390-392 Gravity Probe B, 317 Floquet Theory, 390-395, 397, 445 Green's function, 96, 102, 399, 120 (prob) fluid, incompressible. See incompressible fluid causal, 102 forbidden, classically. See classically forbidden drive with arbitrary force, 103-105 force, causing constant acceleration, 535-536 simple harmonic oscillator, 99-102, 103, 105 forced oscillator. See oscillator, driven Gutzwiller, M. C, 237, 442 Foucault pendulum, 253, 272-275, 324 gyroscope, 323, 330 (prob) Foucault pendulum (prob), 281 four-acceleration, 555, 543 (prob) hairy ball theorem. See Poincare-Hopf theorem four-momentum, 527 Hale-Bopp, 163 four-vector, 522 Halley, Edmond, 132-133, 147-148 energy-momentum, 502, 526-528 Hamilton, 48 four-velocity, 527 Hamilton's Characteristic Function, 222, 230 Fourier analysis, 86, 403-404 equations of motion, 180-181, 207, 218, 248-249 decomposition, 390 Principal Function, 219 series, 402 Principle, 46, 51, 53-55, 75, 178-186, 220, 72 (prob) fractal, 425, 453,463^68 cannonball example. See cannonball example frame-dragging, 324-325 Hamilton-Jacobi equation, 207, 218-230, frames. See reference frames 246-247 (prob) free particle, relativistic Lagrangian. See relativistic particle in a gravitational field, 220-222, 224-228 Lagrangian, free particle Hamiltonian, 21-22, 175, 178, 180, 211, 234, [29-30, free solution. See oscillator, transient solution 551] (prob) frequency bug crawling on a phonograph turntable. See bug beat, 348 crawling on a phonograph turntable

INDEX 569

chaos. See chaos, in conservative systems infinity, orders of, 465 charged particle in electric and magnetic fields, inhomogenous differential equation. See differential 192 equation double pendulum, exact. See double pendulum, exact initial conditions, 7-8, 89 Hamiltonian instability, 392 dynamics, 175-184, 197-198 (prob) dynamical, 383, 386 in accelerated systems, 190-195 instantaneous angular velocity. See angular velocity invariance. See invariance, Hamiltonian axis of rotation. See rotation axis particle on a parabolic wire. See particle on a parabolic rest frame, 526-527, 535-536 wire, Hamiltonian integrable, 425, 447-448, 451^52 relativistic. See relativistic Hamiltonian integrable systems, 207, 235-237 simple harmonic oscillator, 88, 182 intermittency, 457 spherical pendulum. See spherical pendulum, invariable line, 308-309 Hamiltonian invariable plane, 308 handedness, 456 invariance, 28-29 (prob) harmonic analysis, 402-404 Hamiltonian, 211 harmonic oscillations, 383 Lagrangian, 171, 173,209 Hausdorff dimension. See dimension, Hausdorff rotational, 170-172 heavy symmetric top. See symmetric top, heavy translational, 135-136 heliocentric, 130-132, 317 theorems. See Lorentz invariant, invariance theorems Henon-Heiles Hamiltonian (prob), 378, 478 transformation. See transformation, invariance Hermitian matrices, 337 invariant tori, 237-239, 424, 431^33, 435 herpolhode, 308, 311-312 invariants, adiabatic. See adiabatic invariants heteroclinic point, 447 Lorentz. See Lorentz invariant tangle, 448 inverse square force, 134 Hill equation, 383, 389-395, 409, 411 invertible mapping, 55 Hill, G. W., 389 involution, 235-236 Hipparchus of Rhodes, 317 irrational winding number. See winding number, irrational holonomic constraints. See constraints, holonomic irregular motion, 425, 447, 451 homoclinic point, 447 iteration methods, 384, 405 tangle, 446-448 homogenous differential equation. See differential Jacobian, 203, 249-251, 432, 443^44, 452, 454, equation 241 (prob) Hooke, Robert, 132 of Lorentz transformation, 501 Hooke'sLaw, 154 Jupiter, perturbation effects. See asteroid, perturbed by Hubble telescope, 155 Jupiter (prob) hurricanes, 253, 271-272, 281 (prob) See solar system, stability Huygens, C, 87, 125, 480 Hyakutake, 163 KAM barriers, 451 hyperbola, 144, 150 theorem, 435-439 hyperbolic fixed points. See fixed points, hyperbolic tori, 433-439, 448, 451, 469 orbits, 145, 150-154, 164 (prob) Kepler ellipses, precession of (prob), 420-422 attractive force, 150-151, 153 Kepler, J., 131-132 repulsive force, 150-154 Kepler orbit (prob), 413 hyperplane, three-dimensional, 429 problem, 130-133, 141-150, 196 (prob), 226, 228 {see hysteresis, 402, 408^09 also central force problem) perturbation of, 384-388 ice ages, 323 Kepler's equation, eccentric anamoly, 149 ice cream cone rolling on a table, 257-259 Laws, 131-132, 139-140, 147 identity matrix, 262 kinematics, 252 identity transformation. See transformations, identity kinetic energy, [27, 328] (prob) ignorable coordinates, 22-23, 138, 226 asymmetric top. See asymmetric top, kinetic energy impact parameter (prob), 165 block on inclined plane, 4 impulse force, 99 function of both coordinates and velocities, 16-17 arbitrary force as superposition of, 104-105 matrix, Taylor series near equilibrium, 350 response of oscillator to, 100-101 moving constraints, bead on rotating wire example, inclination, 472-473 7-9 inclined plane, example, 1-7 relativistic change, 515-516 incompressible fluid, 188 rigid body, 284-286 index of refraction (prob). See refractive index (prob) rotational, 285-287 inertia ellipsoid. See ellipsoid, inertia symmetric top. See symmetric top, rotational kinetic inertia tensor, 283, 286-291, 339, 327-328 (prob) energy dumbbell, 290-291 sliding ladder example, 16 inertial reference frame, 20, 252, 323-324, 494, 502, total, 285 535 translational, 285 infinitesimal rotation. See rotation, infinitesimal kinetic momentum, 538-539

570 INDEX

Kirkwood, 469 linear differential equation. See differential equation Kirkwood gap, 469-470 oscillator. See oscillator, simple harmonic Kohlrausch, 493 pendulum. See pendulum, linear Kolmogorov. See KAM theorem system, 343 Kovalevskaya, S. V.,451 linearizing, 383, 385-386 Kronecker delta, 286, 307 linearly accelerated reference frames. See reference frames, linearly accelerated Lagrange multipliers, 56-65, 73 (prob) Liouville's theorem, 184-189, 202-204, 250, 391, 444, chain example, 61 449, 452 for problems with explicit holonomic constraints, local chaos, 442, 451 57-58 locomotive (prob), 267 linear pendulum example, 58-59 logistic map, 425, 481-492, 474 (prob) nonintegrable nonholonomic constraints, penny Lorentz, 494 example, 62-64 contraction, 500 Lagrange, Joseph Louis, 1, 23, 86 force, 199, 508, 537-539 Lagrangian, 1, 19, 51, 54-55, 83-86, 209, [28, 30-34, invariant, 521-526, 538 199-200] (prob) action, 530-532 bug crawling on a phonograph turntable. See bug charge, 508, 534 crawling on a phonograph turntable invariance theorems, 524 cannonball. See cannonball mass, 519, 527 central force. See central force Planck's constant, 512 central force, spherical polar coordinates, 138 proper time, 527 charged particle in electric and magnetic fields, 192 transformation, 498-501, 503-504, 555-556, equation of motion, 19 544 (prob) Foucault pendulum. See Foucault pendulum Lorentzian line shape, 109 heavy symmetric top. See symmetric top, heavy, Lorenz equations (prob), 477 Lagrangian Lyapunov dimension. See dimension, Lyapunov in center of mass frame, 137 exponents invariance. See invariance, Lagrangian conservative systems, 449-450 linear pendulum. See pendulum, linear dissipative systems, 458-459, 467, MX-M2 matrix form, 351 logistic map, 483, 485-486 mechanics, relativistic. See relativistic Lagrangian Lyapunov time, 471 mechanics momentum space. See momentum space Lagrangian Mach,E. 324, 513,528 nonlinear pendulum. See pendulum, nonlinear MACHOS, 155 normal coordinates, 359 Mach's principle, 323-325 penny. See penny Maclaurin, 342 simple harmonic oscillator, 88 magnetic field, 493, 509-510, 522, [30, 74, 277, Lao-Tze, 437 550-551] (prob) Laplace, 423, 468 charged particle. See particle in electric and magnetic Laplace-Runge-Lenz vector (prob), 196 fields Larmor frequency, 193 relativistic particle. See particle in a magnetic field, Larmor's Theorem, 192-193 relativistic Laskar,J., 471^74 magnetic force, transformation law. See transformation least action. See Hamilton's Principle law, electric and magnetic forces Legendre transformation, 175-180, 211, 213, magnification. See dimension, magnification 197 (prob) manifold Leibnitz, 45, 75 stable, 440, 446 length contraction. See Lorentz contraction unstable, 440, 446 Lenz vector, 228 map LHC, 452 area-preserving, 431-432 libration, of pendulum. See pendulum, nonlinear, libration logistic. See logistic map Lie derivative, 236 Poincare section, 434, 439-440 light cone, 522 standard (prob). See standard map (prob) light pulse, electric field transformation law. See tangent. See stability matrix transformation law, electric field for light pulse marginal stability. See stability, marginal light wave energy, transformation law. See transformation Mars. See solar system, stability law, energy of light wave mass light definition of, 512-513 bending (prob), 68-69 inertial, 324 emitted by atom, 514-517 reduced, 136-137 minimal optical path, 55 rest, 513-517 speed of. See speed of light matching boundary conditions, 96-98 Lighthill, Sir James, 474 Mathematical Principles of Natural Philosophy, 133 Lindstedt-Poincare perturbation theory, 384, 398^01, Mathieu equation, 383, 395-398 419-420 (prob) functions, 397

INDEX 571

matrix Newton's Second Law, 2, 5, 17, 512-513 antisymmetric. See antisymmetric matrix Third Law, 3 Hermitian. See Hermitian matrix Noether, Emmy, 172 identity. See identity matrix Noether's theorem, 172-175, 245 mondronomy. See stability matrix nonconservative force, 90 notation, 262 example of a rubber band, 42-43 orthogonal. 263 nonconstraint force, 5 stability. See stability matrix nondissipative chaos. See chaos, in conservative transformation, 265 systems unitary. See unitary matrix nonholonomic constraints. See constraints, nonholonomic Maupertuis, 55, 339 noninertial reference frames. See reference frames, cannonball trajectory, 77 noninertial Maupertuis' Principle, 75-80, [74, 162] (prob) nonintegrable, 36, 63, 207, 237, 305, 425, 442, 447-448, Maxwell, J., 493 451^52 Mayer, 514 nonlinear oscillator. See oscillator, nonlinear m -cycle, logistic map, 481 pendulum. See pendulum, nonlinear Mechanique Analytique, 23, 86 resonance, 425 Mercury. See solar system, stability double pendulum, 440-442 precession of, 154 normal coordinates, 343, 359 method of matching boundary conditions, 96-98 mode, 343, 352-355 of successive approximations. See successive mode frequency. See mode frequency approximations, method of, 356 normalization, 338, 356-357 metric tensor, 51, 529 North Pole, 339-340 Michelson-Morley experiment (prob), 551 N-torus. See torus Milankovitch, M, 323 nuclear magnetic resonance, 277 minimal action principle of Maupertuis. See Maupertuis' nutation, 316, 333 (prob) Principle minimal optical path for light, 55 oblate ellipsoid. See ellipsoid, oblate Minkowski, 495, 501 obliquity, 317 mixer, 402 chaotic of Earth, 474 mks units. See SI units ODE. See differential equation, ordinary mode. See normal mode one-dimensional systems, 123-125, 158 (prob) amplitude ratio. See mode displacement ratio orbital period, 147 displacement ratio, 343, 346, 352, 353-355, 357, 361, orbits, of Kepler problem, 143-151 363-367 orders of infinity. See infinity, orders of frequency, 343, 353, 357, 362-366, 374 ordinary differential equation. See differential equation, vector, 352, 355, 362-367 ordinary mode-locking. See phase locking orthogonal matrix, 263, 211-21% (prob) molecule modes, 355-356 diatomic (prob). See diatomic molecule (prob) transformation, 288, 302, 336 triatomic linear. See triatomic molecule, linear oscillations moment of inertia tensor. See inertia tensor harmonic. See harmonic oscillations momental ellipsoid. See ellipsoid, momental small. See small oscillations moments, principal. See principal moments subharmonic. See subharmonic oscillations momentum oscillator canonically conjugate. See canonically conjugate anharmonic. See anharmonic oscillator momentum damped, 81 relativistic, 517-519, 547 (prob) damped simple harmonic, 90-94, 103, 106-110, transverse, relativistic, 517, 532, 548 (prob) 116-120 (prob) momentum conservation, 519 complex driving force, steady state, 108 space, 189-190, 201-202 (prob) critically damped, 92 Lagrangian, 190 overdamped, 92 mondronomy matrix. See stability matrix response to sinusoidal driving force, relative phase, Moon. See solar system, stability 110-113 Moser. See KAM theorem underdamped, 91 multiplicative ergodic theorem. See Oseledic's driven, 81,94-113 multiplicative ergodic theorem damped simple harmonic (prob), 120-121 mylar, aluminized, 517 simple harmonic (prob), 119 Duffing. See Duffing oscillator neutral equilibrium. See equilibrium, neutral linear. See oscillator, simple harmonic neutral fixed point. See fixed point, neutral nonlinear, 125-127, 121-122 (prob) neutrinos, 155, 528 simple harmonic, 81, 88-90, 124, 212-213, 218, Newton, Isaac, 45, 132-133, 284, 299, 324, 339-342, 237-239, [115, 202, 240, 246, 551] (prob) 423, 468, 493 general solution with Green's function, 99-102, 105 Newtonian mechanics, review with sliding block example, Hamiltonian. See Hamiltonian, oscillator, simple 1-3 harmonic

572 INDEX

simple harmonic response to sinusoidal driving force, phase space, 127, 203-204, 248 106 density, 186-189 steady-state solution, 95, 97-98, 107-108 phase trajectories, 127-128, 424 step function drive force, 96-99 of damped drive pendulum, 454-457, stored energy, 94, 109 461-462 transient solution, 95, 97-98, 107-108 phase velocity, 526 Oseledic's multiplicative ergodic theorem, 449 phase-locking, 425, 461 overdamped. See oscillator, damped simple harmonic phenomenological, 453 photon disintegration (prob), 546 parabola, 144 physical pendulum (prob), 32, 34, 157, 332 parabolic orbits, 145, 164 (prob) pitchfork bifurcation. See bifurcation, pitchfork parallel axis theorem, 289 Planck energy-frequency relation, 512 parallel plate capacitor, 508-510 planet, perturbed by Jupiter (prob). See asteroid, parametric amplification, 384 perturbed by Jupiter (prob) amplifiers, 390 planetary orbits (prob), 162-164 resonance, 384, 388-398, 414-419 (prob) Pluto. See solar system, stability unstable, 397 Poincare, 423-424, 426, 428, 433, 439, 447-448, 468, particle decay, 528, 546 (prob) 494 in a gravitational field, Hamiltonian-Jacobi equation, Recurrence Theorem, 204-206 220-222, 224-228 section, 393, 428-436, 442-443, 445, 451, in a magnetic field, [30, 74] (prob) 459 relativistic, 539-540 damped driven pendulum, 454-457, in electric and magnetic fields 462^63 Lagrangian and Hamiltonian, 192 double pendulum, 430^33, 435, 441, 451 on a parabolic wire section map. See map, Poincare section Hamiltonian, 182-183 Poincare-Birkhoff theorem, 439-442 particular solution. See oscillator, steady-state solution Poincare-Hopf theorem, 235 pendulum. See also oscillator, [33-34, 116-117, 373, Poinsot construction, 283, 307-312 415, 419] (prob) Poinsot, L., 306 action-angle variables, 231-232 point mass sliding on a bowling ball, 38-39 damped driven. See damped driven pendulum masses double. See double pendulum N connected by a string, 367-371, 379 (prob) equilibrium, 82-83 transformation, 208, 216 linear, 58-59 Poisson brackets, 207, 217-218, 236-237, 251, nonlinear, 125-130 243-245 (prob) libration, 126 Polaris, 317 period, 128-130 polhode, 308, 311-312 phase portrait, 127-128 positive definite, 288, 337, 351 rotation, 126 positronium acceleration (prob), 550 spherical. See spherical pendulum postulates, of special relativity. See special relativity, two spring. See spring pendulum postulates upside-down. See upside-down pendulum potential energy, 18-19, 42, 85-86. (See also with driven support, 388-398, 414 (prob) conservative forces) two coupled, 344-348, 376-377 (prob) for the earth in a gravitational field, 318 pendulum (prob), physical. See physical pendulum (prob) matrix, Taylor series near equilibrium, 350 pendulum (prob), triple. See triple pendulum (prob) potential, effective. See effective potential penny example, 62-64, 73 (prob) Poynting vector, 507, 517 perihelion, ellipse. See ellipse, perihelion precessing orbits, 154 period precession, 273, 295-298, 315-316, 321 anharmonic oscillator. See anharmonic oscillator, of equinoxes. See equinoxes, precession of period of of Foucault pendulum. See Foucault pendulum Duffing oscillator. See Duffing oscillator, period of of Kepler ellipses (prob). See Kepler ellipses, nonlinear pendulum. See pendulum, nonlinear precession of (prob) of elliptical motion. See elliptical motion, period of of Mercury, 154 orbital. See orbital period of torque free top. See symmetric top, torque free, period doubling, 456-458, 483^85 asymmetric top, torque free cascade, 457 of symmetric heavy top, 313-316 perpetual motion machine, 514 retrograde, 317 perturbation, 383, 384-388, 433, 439, 442 pressure, radiation. See radiation pressure perturbation theory, 398, 404-408 primed frame, 260 Lindstedt-Poincare. See Lindstedt-Poincare principal axis, 283, 292, 306, 339 perturbation theory transformation, 288 phase advance, 390, 393 principal moments, 288 phase portrait (prob), 156-158 Principia, 133,340,474 phase portrait, pendulum. See pendulum, nonlinear, phase Principle of Least Action. See Hamilton's Principle portrait of Relativity, 493-494

INDEX 573

prolate ellipsoid, See ellipsoid, prolate restricted three-body problem (prob). See three- body proper time, 527,531 problem, restricted (prob) for a particle in a constant force, 536-537 retrograde precession. See precession, retrograde proton-proton annihilation, 517-518 rheonomic constraints. See constraints, rheonomic pseudovector, 257, 277 (prob) rigid body, 254-255 Ptolemy, 130,317 as collection of mass points 11 pumping a swing (prob), 416 kinetic energy, 284-286 Pythagorean theorem, 50 motion, 283-325 torque free motion. See Euler's equations Q, See quality factor rotated reference frames, 259-263 quadratic form, 22, 351, 29 (prob) rotating reference frames, 190-192, 263-264, 266 quadrature, 22, 125 rotating turntable (prob), 243 quality factor, 90, 93-94, 453 rotation matrix, 259-263 quantum mechanics, 44, 229-230, 67 (prob) axis, 255, 257-259 quasiperiodic, 238 infinitesimal, 171, 254-259 route to chaos, 428 number. See winding number of hurricanes. See hurricanes radiation pressure, 517 of pendulum. See pendulum, nonlinear radiation reaction force, 533 rotational angular momentum. See angular momentum, radiofrequency resonant cavity, 185 rotational rank, of tensor, 333 invariance. See invariance, rotational rapidity (prob), 544 kinetic energy. See kinetic energy, rotational rational winding number. See winding number, rational rotations, 261-263, 302 reduced echelon form, 337 simultaneous. See simultaneous rotations reduced mass, 137 three-dimensional, 239, 261 reference frames, 495 rotator, 289 body, 260 (see also body coordinate system) Routhian, 23, [35, 197, 331, 377] (prob) inertial. See inertial reference frames row elimination, 347, 355, 380 linearly accelerated, 193-195 reduction. See Gauss-Jordan elimination noninertial, 20 rubber band, 42-43 noninertial (prob), 200 Rutherford, 151 rotated. See rotated reference frames scattering, 151-152, 165-167, 165 (prob) rotating. See rotating reference frames space, 260. See also space coordinate system saddle hyperbolic fixed point, 446 reflection, 257 (see also transformations, discrete) saddle point, 53, 87 hyperbolic fixed point, 446 satellite (prob), 200 refractive index (prob), 69 scalar, 333-334 regular motion, 425, 447, 451 potential, 525, 534 relative phase, 110-113 product, 254, 334, 523 relativistic action, 530-532 scaling function, universal. See universal scaling circular motion, 539-540 function energy. See energy, relativistic scaling transformation, 487 Hamiltonian, 533 scattering angle, 152 Lagrangian, 550-551 (prob) Schrodinger equation, 229, 384, 67 (prob) electromagnetic field, 534-536 scleronomic constraints. See constraints, scleronomic free particle, 532 second-order differential equation. See differential Lagrangian mechanics, 530-540 equation, second-order mechanics, 494 (see also special relativity) secular terms, 400, 404^05 momentum. See momentum, relativistic self-similarity, 487-488 velocity. See velocity transformation rules, relativistic semi-axes, 306 relativity semimajor axes, 472 general. See general relativity ellipse, 145-146 special. See special relativity semiminor axis, ellipse, 145-146 of simultaneity. See simultaneity separability, 208, 225-226 renormalization, 486^90 separation constants, 221 repulsive force, hyperbolic orbits. See hyperbolic orbits, of variables, 221,228 repulsive force separatrix, 128, 185, 232, 435, 442 resonance, 105-110 sgrad, 236 nonlinear. See nonlinear resonance SHO. See oscillator, simple harmonic parametric. See parametric resonance SI units, 507, 540 resonant cavity, radiofrequency. See radiofrequency similarity transformation, 487 resonant cavity simultaneity, 494, 500, 501-503 frequency. See frequency, resonant simultaneous rotations, 258 terms, 400 (see also secular terms) sinelike, 390 rest frame, instantaneous. See instantaneous rest frame skew gradient, 236 rest mass, 513-517 SLAC, 536

574 INDEX

sleeping top, 316 tachyons (prob), 544 small oscillations, theory of, 343-371, tangent map. See stability matrix 373-374 (prob) Taylor series, 482 vibrations. See small oscillations expansion, 24, 47^8, 83-84, 128 small-angle approximation, 349 tensor, 286, 333-336 Snell's Law (prob), 71 first-rank, 333-334 solar system, stability, 424, 468^74 inertia. See inertia tensor solenoid lens for electrons (prob), 199 second-rank, 286, 319, 333-336 Southern cross, 317 zero-rank, 333-334 space cone, 297, 298 tether ball (prob), 162 coordinate system, 259-264, 284, 300 three-body problem, 226-235 spacelike separation, 522 restricted (prob), 478 space-time areas three-dimensional rotations. See rotations, diagrams, 495^98, 501-503 three-dimensional event, 521 tides, 322, 163 (prob) transformation of, 501 time derivative special relativity, 493-540, [542, 546-548, 550, transformation of coordinate systems, 553] (prob) 266 two postulates, 494 vector. See vector, time derivative speed of light, 494 time dilation, 499-500, 543 (prob) spherical pendulum, 27 (prob) timelike separation, 521-522 Hamiltonian, 183-184 tippe top (prob), 331 spin, 293,296-297, 313 top. See symmetric top, asymmetric top of electron in magnetic field (prob), 277 topology, 235 spring pendulum (prob), 27 tori, KAM. See KAM tori square pulse, 100-101, 104 torus, 235, 429. See also invariant tori stability, 392, 413^414 (prob) tower of Pisa problem, 253, 267-271 matrix, 443^46, 449, 458, 485 trace, 204, 303, 392, 445 dynamical, 383, 386 transformation law marginal, 474 electric and magnetic forces 508-510 of many-dimensional linear systems near electric field for light pulse, 510-511 equilibrium, 359-360 energy of light wave, 507-512 solar system. See solar system, stability frequency of light wave, 505-507 stable equilibrium. See equilibrium, stable Galilean. See Galilean transformation laws fixed point. See fixed point, stable volume, 511-512 manifold, See manifold, stable transformation Staeckel conditions, 226 canonical. See canonical transformations standard map (prob), 476 contact. See contact transformation standing waves on a string, 371 continuous, 171 state vector, 350, 355, 357 discrete, 171 stationary action, path for, 53 identity, 171,215 steady-state solution. See oscillator, steady-state solution invariance, 174 stellar aberration (prob), 553 Lorentz. See Lorentz transformation step function, 96-97 matrix. See matrix transformation Stoke's theorem, 231 of coordinates, 173 strange attractor, 425, 453, 461-463 orthogonal. See orthogonal transformation subharmonic oscillations, 383, 409-411 point. See point transformation successive approximations, method of, 270-271 principal axis. See principal axis, transformation summation convention for indices, 253 scaling. See scaling transformation supercycle, 484, 486-487 similarity. See similarity transformation superposition, 89, 344, 120 (prob) velocity. See velocity transformation rules superstable fixed point. See fixed point, superstable transient solution. See oscillator, transient solution surface of section. See Poincare section translation invariance. See invariance, translation Sussman, G. J., MX-A12 translational angular momentum. See angular momentum, swing. See pumping a swing (prob) translational symmetric matrices kinetic energy. See kinetic energy, translational diagonalization, 336-339 transpose, 249-250, 262 top, 289 transverse momentum, relativistic. See momentum, heavy, 284, 313-317 (see also gyroscope), transverse, relativistic 331 (prob) Tremaine, S., 471 Lagrangian, 313 triatomic molecule, linear, 360-363, 377 (prob) rotational kinetic energy, 305 triple pendulum (prob), 373 torque free, 293-300, 374 (prob) true anomoly, 148 symmetry breaking, 455^56 tune, 445 symplectic, 248-251, 431^432 turning point, 125

INDEX 575

turntable (prob), 279 velocity twin paradox, 554-557 generalized. See generalized velocity two-body problem, 134 relativistic, 504-505, 542 (prob) transformation of coordinate systems, 264-265 uncountable, 465 transformation rules underdamped. See oscillator, damped simple Venus. See solar system, stability harmonic vibrational state, complex. See state vector unitary matrix, 263 vibrations, small. See small oscillations universal scaling function, 488-490 virtual displacement, 4-5, 14-15 unprimed frame, 260 work, 4-5, 9, 14-15, 27 (prob) unstable equilibrium. See equilibrium, stable Voltaire, 75, 340 fixed point. See fixed point, unstable volume, transformation law. See transformation law, manifold. See manifold, unstable volume motion. See instability voyager, 163 upside-down pendulum, 397-398, 415 (prob) water molecule (prob), 375 wave equation, 371 Van der Pol equation (prob), 420 vector, 526 variation 8y(x), 47 Weber, 493 variation in the integral /, 48 Whittaker, E. T., 283, 308-309 principle, 54-55 width, full at half max, 109 variational calculus, 44, 51 (see also Euler equation), Wigner, E. P., 44 66-71 (prob) winding number, 237-239, 425, 427^32, 434, 461 shortest distance between two points, 49-50 irrational, 238-239, 436^37 derivative, 53 rational, 238-239 principle, 532-533 Wisdom, J., 469-472 varied curve, 47 WKB limit, 230 vector, 253-254, 333-334 work, definition of, 4-5 field, 235 world line, 496, 531 potential, 525, 534, 538 Wren, Sir Christopher, 132 time derivative, 253 WWI, Falkland Islands battle (prob). See Falkland Islands vectorial mechanics, 23 battle (prob)