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528 References

Index

Aaccelerated observer, 108acceleration, 89, 91, 109, 112acceleration, 104action, 113action angle variables, 430action variable, 431, 434, 439, 447action variables, 426addition, 9air resistance, 128algebraic equation, 164algorithms, 31a-particles, 283amplitude, 138, 157, 159, 179amplitude resonance, 161, 163analytic solution, 511analytical calculation, 1analytical solution, 518angle variable, 434, 439, 447

angular frequencies, 232angular moment, 494angular momentum, 37, 122, 216,223, 230, 233, 270, 366, 392, 478angular velocity, 481, 501anharmonic oscillator, 525animation, 24antisymmetry, 401aphelion, 213, 246approximation, mathematical, 36

physical, 36area conserving, 457area velocity, 227Arnold, 442Arnold diffusion, 441arrow, 64astronomical unit, 213asymptotic behavior, 520asymptotic behavior , 519asymptotic motion, 189atoms, 269, 474

attracting set, 189attracting sets, 189average, 162axial vector, 72azimutal angle, 225

Bbackward scattering, 261balance, 110baseball, 95beam, 269beam intensity, 269Bernoulli, 244, 291, 324bi-soliton, 529bifurcation, 149, 463bifurcation diagram, 469body centered coordinate, 478body centered coordinates, 474Boltzmann, 534boundary conditions, 318brachystochrone, 302brachystochrone problem, 291Brahe, 212

calculus of variation, 334calculus of variations, 289canonical equations, 428, 434canonical momentum, 428canonical transformation, 419, 424canonical variables, 421cartesian, 328

Ccartesian coordinates, 332Cartesian coordinates, 42, 68, 83Cauchy, 386, 492cenit angle, 225center of mass, 220, 222, 256, 476center of mass system, 256, 273, 486center of mass velocity, 263central field, 211central field motion, 219central force, 216, 223, 227, 269central force problem, 219central forces, 113, 211, 221centrifugal force, 235, 238centrifugal potential, 235cgs system, 61chaos, 189, 466, 511chaotic, 115, 197chaotic behavior, 194chaotic dynamic, 460chaotic entanglement, 195chaotic motion, 189Chaotic systems, 446characteristic data, 519characteristic frequency, 431circular motion, 90circular torus, 453classical mechanics, 2, 34, 36clock, 107closed orbits, 232cofactor, 48

530 Index

collision, 255column matrix, 45complementary solution, 156complete integrability, 435completely integrable, 436completely integrable equation, 520complex behavior, 511component, 41, 63computer algebra, 4configuration space, 331conic sections, 213, 244conical sections, 244conjungate momentum, 430conservation law, 120, 264, 534

derivation, 534conservation laws, 361conservation of angular momentum, 362conservation of energy, 534conservation of mass, 534, 536conservation of momentum, 362, 536conservative, 127conservative force field, 127conserved quantity, 392, 402, 427constraint, 382constraint of non slip, 342constraints, 316, 333continuity equation, 534–535continuous models, 511continuous spectrum, 520contour integral, 421

contravariant, 68contravariant vector, 67–68convex function, 376coordinate, cyclic, 361

ignorable, 361coordinate change, 419coordinate system, 44coordinate transformation, 76coordinate transformations, 44coordinates, 41Coulomb scattering, 280coupled pendulum, 347Crank-Nicolson procedure, 539critical damping, 149critical phenomena, 470critical point, 469critically damped motion, 149cross product, 72curl, 80current, 522, 534cycle frequency , 434cyclic, 361, 420cyclic coordinate, 362cyclic variable, 424cyclic variables, 361cycloid, 291cylindrical coordinates, 419

D[], 11damped harmonic oscillator, 144, 169

Index 531

Ddamping constant, 190damping factor, 160, 167damping force, 144, 189damping medium, 147damping parameter, 144, 150degrees of freedom, 189density, 293, 298, 397derivative, 11derivatives, 40, 76deviation moments, 477, 479deVries, 511difference method, 539differentiable manifold, 407differential equation, 13differential scattering cross section, 269differentiation rule, 401diffusion, 314Dirac Lagrangian, 311Dirac's delta function, 515direction, 63direction cosine, 45discrete eigenvalues, 521discretization procedure, 540dispersion, 517dispersion relation, 514dispersive, 514distance, 104division, 9dot product, 72double pendulum, 416

drag force, 132driven damped oscillator, 166driven nonlinear oscillator, 188driven oscillations, 155driving force, 158, 189driving frequency, 158–159DSolve[], 129DSolve[], 13duration of oscillation, 175dynamic, 189dynamical principle, 327dynamics, 83, 111

EEarth, 217eccentricity, 244, 247effective potential, 233, 235–236, 245eigenfunction, 522eigenvalue, 520–522Einstein, 34Einstein summation convention, 86elastic collision, 255electric field, 114electromagnetic force, 117electromagnetic forces, 252Elements, 409elevation, 99ellipse, 142ellipses, 213elliptic fixpoints, 454elliptic function, 180

532 Index

elliptic integral, 180elliptic integrals, 174, 231EllipticK[], 180elongation, 149energy, 123, 142energy loss, 148energy of rotation, 235energy resonance, 161, 163equation of motion, 155, 228, 425equilibrium position, 152ergodic, 441Euclidean plane, 294Euler, 289, 376, 475, 489Euler angles, 474, 487Euler derivative, 289, 297, 310Euler equation, 334Euler Lagrange equations, 370Euler method, 539Euler operator, 299, 309, 312Euler operator, 299Euler procedure, 540Euler theorem, 339Euler-Lagrange equation, 345, 361, 375Euler-Lagrange equations, 289, 334, 350, 384Euler-Lagrange operator, 340Euler's equation, 312Euler's equations of motion, 487Euler’s equation, 297event, 107evolution, 385

experimental facts, 104exponentiation, 9external driving force, 155external force, 108external source, 155

Ffalling particle, 128Feigenbaum, 468Feigenbaum constant, 469Ferma's principle, 324Fermat, 324Fermi, 511field equation, 312fields, 511first integral, 332first-order differential equations, 189fixed interatomic distance, 474fixed stars, 108fixed system, 83fixpoint, 453flip chart movie, 360flow, 436flow field, 437force, 111–112, 126, 331

attractive, 113repulsive, 113

force center, 237, 273force free symmetrical top, 492force free top, 491force moment, 491

Index 533

forces, 63forces in nature, 115forward scattering, 261Fourier transform, 514, 518fractals, 2fractional, 470frame of reference, 107free body, 112free oscillations, 155free particle, 112frequency, 137, 145, 181, 447frequency of revolution, 235friction, 155frontend, 5functional, 292–293, 298, 308, 333–334functional program, 30fundamental Poisson brackets, 402fundamental units, 61

GGalilean invariance, 536Galilean transformation, 529Galilei, 34Galileo, 111Gardner transformation, 537Gauss, 326general density, 534general minimum principle, 325generalized velocities, 375generalized coordinates, 86, 89, 189, 332, 375generalized coordinates, 43, 328

generalized momenta, 375, 434generalized velocities, 328, 332generating function, 422, 426, 429,432, 439generating functional, 292generating functions, 421Get[], 14Giorgi system, 61gold atoms, 283golf play, 95gradient, 78gradient operator produc, 78graphics, 16gravitation, 211gravitational constant, 110gravitational field, 174, 219gravitational force, 110, 132, 250gravitational force, 115gravitational mass, 111gravitational masses, 110gravity, 110, 115Green's function, 164, 169–170Green's method, 168

Hhadronic force, 118Hamilton, 34, 292, 327Hamilton dynamics, 375Hamilton equations, 439Hamilton formulation, 375Hamilton function, 378Hamilton manifold, 414

534 Index

Hamilton system, 442Hamilton-Jacobi equation, 427, 430, 433Hamilton-Jacobi theory, 428Hamilton-Poisson manifold, 415Hamiltonian, 382, 385, 387, 412, 416, 420, 423, 428–429, 431, 448, 450Hamiltonian dynamics, 395Hamiltonian formulation, 321Hamiltonian phase space, 395Hamilton's equation, 384, 403Hamilton's equations, 386, 399Hamilton's principle, 323, 327, 333, 339, 384, 388Hamilton's principle, 332HamiltonsEquation[], 386hard spheres scattering, 278harmonic oscillator, 136, 138, 140, 340, 431heat, 147Heisenberg's uncertainty, 34Helmholtz, 127help, 10Henó, 450Henó map, 450Henon, 443Hertz, 326history, 107homogeneity of space, 323homogeneity of time, 323, 330homogeneity relation, 363homogeneous force field, 306homogenous function, 338homogenous functions, 339

Hooke's law, 137Huberman, 470Huygens, 235hyperbolas, 213hyperbolic fixpoint, 453hyperlink, 10hyperon, 36

Iidentity matrix, 48–49impact parameter, 270, 273, 280inclined plane, 341incommensurable, 232inelastic collision, 255inertia, 66inertia moments, 477inertia tensor, 475, 477, 479, 489inertial coordinates, 474inertial frame, 108inertial mass, 111inertial reference frame, 108infinite degree of freedom, 511infinitesimal parameter, 364infinitesimal rotation, 366infinitesimal transformation, 364inhomogeneous differential equation,172initial condition, 518initial conditions, 140input, 8input form, 12input notation, 12

Index 535

integrability, 375integrable, 450integral of motion, 428, 435integral relation, 40integrals, 80integrals of motion, 435, 446integration, 11integro-differential equation, 514, 520intensity, 269interaction, 251interaction laws, 252interaction potential, 224, 235, 252, 521interaction time, 255interactive use, 8invariant, 72invariants, 363, 419, 534, 536Inverse[], 48inverse matrix, 48inverse scattering method, 514, 525inverse scattering theory, 518inverse scattering transform, 524inversion, 167involution, 435isotropy of space, 330iteration, 28iterative mapping, 449

JJacobi determinant, 457Jacobi determinant , 449Jacobi identity, 402

Jacobi matrix, 450Jacobian, 379Jacobian elliptic function, 186Jacobi's identity, 411JacobiSN[], 186Josephson junction, 189Joule, 127Jupiter, 216

KKAM theorem, 442, 454KdV, 511KdV equation, 515Kepler, 20, 212, 227Kepler's laws, 213kernel, 5, 10keyboard short cuts, 9kinematics, 83kinetic energy, 123, 175, 178, 225,348Kolmogorov, 442Korteweg, 511Korteweg-de Vries, 511Kronecker delta symbol, 51Kronecker's symbol, 477Kruskal, 514, 540

Llab system, 266label, 8laboratory system, 256, 261, 273Lagrange, 34, 289, 318, 325Lagrange function, 329

536 Index

Lagrange density, 310, 335, 338–341, 357Lagrange dynamics, 321, 375Lagrange equations, 330–331, 344Lagrange function, 307, 488Lagrange multiplier, 318–319, 344–345Lagrange's equation, 329Lagrangian, 329–330, 363, 384, 419, 487, 489Lagrangian formulation, 321Lagrangien density, 350l-calculus, 31Landua, 330Laplace, 330Laplace equation, 314Laplace transform, 13, 164, 169Laplacian, 79large wavelength, 515latus rectum, 244law of cosines, 267laws of motion, 36leap frog, 539least action, 329Legendre polynomial, 526Legendre transform, 376LegendreTransform[], 380Leibniz, 324, 376Leibniz's rule, 401, 406length, 60leptons, 119Levi-Civita density, 73Levi-Civita tensor, 489lex prima, 111

lex secunda, 111lex tertia, 111libration, 175Lie's symmetry analysis, 520Lifshitz, 330linear differential equations, 164linear differential operator, 168linear integral equation, 521linear models, 511linear momentum, 121linear ordinary differential equation,168linear stability, 541linearity, 401Liouville, 400, 421Liouville's theorem, 395, 400, 449location of a particle, 83log-log plot, 21logistic function, 462logistic map, 462, 468Los Alamos, 514Lyapunov exponent, 460, 466

MMach, 105magnetometers, 189magnitude, 63MANIAC, 514manifolds and classes, 407mapping, 449mapping area, 449mappings and Hamiltonians, 456

Index 537

Marchenko equation, 514, 520, 524, 526–527, 539Marchenko's integral equation, 524mass, 60, 62, 104, 109–110, 112mass center, 474mass point, 83material system, 37Mathematica, 5mathematical approximation, 36mathematical calculation, 1mathematical structure, 36mathematical tools, 40MathSource, 5, 7matrix, 45, 481

column, 45inverse, 48multiplication, 46orthogonal, 51square, 45transposition, 47

Maupertius, 325Maxwell’s equations, 312mean distance, 216mean distances, 245measuring unit, 61mechanics, 35meson, 36minimal principles, 323minimum action, 325minimum principle, 292minor, 48

Miura, 514Miura transformation, 536mks system, 61modulo, 191modulus, 180molecules, 114, 474momentum, 112Moser, 442motion, 83, 109motion of a ball, 96motion of planets, 211motion on a cylinder, 389moving beat on a string, 381moving coordinate, 515moving frame, 43multi-soliton, 520multiplication, 9

NN- particle system, 336natural boundary conditions, 518NDSolve[], 191Neptune, 217Newton, 34, 105, 213, 324Newtonian mechanics, 104Newtonian theory, 104Newton's equation, 133, 334Newton's equations, 323, 331Newton's first law, 221Newton's laws, 104, 111Newton's second law, 221

538 Index

Noether, 368Noether theorem, 369non integrability, 375non-integrable, 450nonholonomic, 333nonlinear coupled chain, 514nonlinear differential equations, 518nonlinear dynamics, 511nonlinear field equation, 511nonlinear initial value problem, 520nonlinear oscillation, 174nonlinear partial differential equation, 519nonlinearity, 517Normal[], 182normalization constant, 522nucleon, 36numerical calculation, 15numerical integration, 15, 190numerical solution, 190, 194

Oobject oriented programs, 31observer, 107–108operating system, 5optics, 323options, 17orbit, 231, 238orbit potential, 234orbits, 244origin of time, 107orthogonal matrix , 51

oscillatory motion, 136output, 8overdamped motion, 150

Ppalettes, 9parabolas, 213parabolic orbit, 96parallelogram law, 114parametric plot, 16parametric representation, 19, 142partial solution , 157particle density, 534particular solution, 156Pasta, 511path, 83, 306pendula, 111pendulum, 174, 179, 196pendulum motion, 176perihelia, 113perihelion, 213, 246period, 179, 181period doubling, 468periodic, 441, 446, 468periodic regime, 470periodic solution, 535periodicity, 430phase, 529phase diagram, 140, 148phase factor, 159, 161phase plane, 140

Index 539

phase portrait, 140phase space, 177–178, 192, 195, 375, 400, 403, 419, 431, 435, 446, 451phase space, 140phase space volume, 422phase transition, 470phase velocity, 514–515philosophy of mechanics, 107physical approximation, 36physical effect, 36physical law, 104physical laws, 104physical theories, 36pivot point, 189planar pendulum, 188planet motion, 238planet movement, 211planetary laws, 213planetary motion, 233platonic body, 214plot, 16Poincaré plane, 449, 452, 454Poincaré section, 189, 193, 196–197, 458Poincaré technique, 189Poincaré-Hopf theorem, 436point mass, 83Poisson, 386Poisson bracket, 400, 412, 414, 435Poisson brackets, 400Poisson manifold, 409, 412PoissonBracket[], 404

polar axis, 225polar coordinates, 42, 86polynomial, 27Pöschel, 525Pöschel-Teller problem, 525position, 83position variable, 140potential energy, 123, 126, 175, 331,348potential reconstruction, 521power law, 231power-law, 132precession, 113principal axes, 479principal axis , 248Principia, 111principle of equivalence, 111principle of least action, 329principle of least constraint, 326procedural function, 29programming, 27projectile, 95

Qquadratic equation, 10quadrature, 175, 430quantum mechanics, 2, 37, 520quasi-periodic, 441, 446Quit[], 8

Rradial equation, 228radial oscillations, 232

540 Index

radial velocity, 231radial velocity , 233random motion, 123rank, 66, 68rational number, 13reaction, 113recurrence, 232reduced mass, 221reference point, 83reflection, 521reflection coefficient, 521reflection index, 523reflection-less potential, 539refraction, 324regular dynamic, 511regular motion, 189–190relative coordinates, 219relative motion, 107relative velocity, 108resonance, 161resonance frequency, 161rest, 112restoring force, 136revolutions, 178rheonimic, 333rigid body, 474, 478rolling wheel, 318rolling wheel, 341rotating frame, 475rotation, 474rotation matrix, 49, 59

rotation symmetry, 269rotations, 56Rudnick, 470Rudolphine table, 213rule based program, 31ruler, 107Russel, 511Rutherford scattering, 280Rutherford's scattering formula, 282

SSarturn, 217scalar field, 40scalar product, 71scalars, 40, 60scaling, 515scaling exponent, 470scaling law, 218, 470scaling property, 469scattering, 251scattering angle, 260–261, 265, 270, 274scattering cross section, 269, 271, 273, 283scattering data, 520–521scattering data , 519scattering particles, 269scattering potential, 520scattering problem, 269, 520scattering process, 519Schrödinger's equation, 312scleronomic, 333, 362self-similar, 470

Index 541

self-similar structure, 454self-similarity, 454, 470sensitivity, 189separating variables, 179separation, 428separation ansatz, 526separation of Hamiltonians, 433separatrix, 178shallow channels, 514sliding beat, 387sliding mass, 347Snell's law, 324solitary wave, 511solitary waves, 514soliton, 520, 525–526, 529Solve[], 10spectral characteristic, 524spectral method, 539spherical coordinates, 42, 225spherical symmetry, 88, 224spherical top, 490square matrix, 45standard form, 12standard map, 458standard package, 14standard packages, 7StandardForm, 11stationary characteristic, 521stationary coordinate, 43Steiner's theorem, 486–487

Stokes theorem, 421strange attractor, 189, 196strange entangled curve, 196stroboscopic map, 193stroboscopic snapshot, 189strong nuclear force, 118Sturm-Liouville problem, 518,520–521subtraction, 9sum, 12super cyclic, 468surface, 18symbolic calculation, 10symmetrical tensor, 477symmetries, 361symmetry, 123symmetry analysis, 520symmetry group, 149symmetry line, 486symmetry point, 486symplectic matrix, 436syntax, 1, 8

Ttangent map, 461tangent representation, 460target coordinates, 421Taylor series, 12, 136Taylor-Chiricov map, 458Teller, 525temperature, 60, 123temporal change, 86

542 Index

tensor, 66rank, 66

tensors, 40test function, 292–293theoretical analysis, 36theory of scattering, 520thermal energy, 123time, 60, 109time, 104time of revolution, 216time-dependent potential, 521top, spherical , 480

symmetric , 480unsymmetrical, 480

topology, 436tori, 446torque, 122torques, 63torus, 437total differential, 421, 538total energy, 126, 138, 177, 233, 447total kinetic energy, 475total length, 294traditional form, 12trajectory, 430–431, 447transformation matrix, 45transformations, 40, 241translation, 240, 474translations, 121translations in time, 362

transmission, 521transmission coefficient, 521transmission rate , 523transposed matrix, 47, 49transposition, 47triangle addition law, 65triangle law, 64trigonometric function, 27trigonometric functions, 9, 138tunneling junction, 189turning points, 231twist map, 449twist mapping, 450two body problem, 211, 222, 251two particle collision, 251two-body forces, 114two-dimensional oscillator system,310

UUlam, 511underdamped motion, 145uniform motion, 43, 112uniformly accelerated, 43units, 61upper reversal point, 179

Vvacuum, 132variation, 308, 329variational derivative, 314variational principle, 323, 388vector, 63–64, 67, 83

Index 543

vector addition, 64vector analysis, 14, 63vector field, 40vector product, 40, 71–72vectors, 40velocities, 63velocity, 85velocity, 104velocity of sound, 133Venus, 217volume integration, 80

Wwater waves, 514wave, 511wave function, 520weak nuclear force, 119winding number, 448, 450work, 123, 139world-line, 107

ZZabusky, 514, 540

544 Index