Index

Page numbers for Scheme procedure definitions are in italics.

Page numbers followed by n indicate footnotes.

0, for all practical purposes, 20 n. See also Zero-based indexing

∘ (composition), 7 n, 510

Γ[q]

for local tuple, 11

Lagrangian state path, 203

γ (configuration-path function), 7

δ function, 454 (ex. 6.12)

δ_η (variation operator), 26

λ-calculus, 509

λ-expression, 498–499

λ-notation, 498 n

Π_L[q] (Hamiltonian state path), 203

χ (coordinate function), 7

σ (phase-space path), 218

ω matrix, 124

$\overrightarrow{\omega }$ (angular velocity), 124

ω (symplectic 2-form), 359

C (local-tuple transformation), 44

C_H (canonical phase-space transformation), 337 n

D. See Derivative

D_t (total time derivative), 64

∂. See Partial derivative

$$ \mathcal{E}$$ (Euler–Lagrange operator), 98

ℰ (energy state function), 82

F₁–F₄. See also Generating functions

F₁(t, q, q′), 373

F₂(t, q, p′), 373

F₃(t, p, q′), 374

F₄(t, p, p′), 374

H (Hamiltonian), 199

I (identity operator), 517

I with subscript (selector), 64 n, 513

$\tilde{J}$ (shuffle function), 350

J, J_n (symplectic unit), 301, 355

L (Lagrangian), 11

L (Lie derivative), 447

P (momentum selector), 199, 220

$\mathcal{P}$ (momentum state function), 79

Q (coordinate selector), 220

$\dot{Q}$ (velocity selector), 64

q (coordinate path), 7

S (action), 10

Lagrangian, 12

’ (quote in Scheme), 505

, in tuple, 520

:, names starting with, 21 n

; in tuple, 31 n, 520

# in Scheme, 504

{ } for Poisson brackets, 218

[ ] for down tuples, 512

[ ] for functional arguments, 10 n

( ) for up tuples, 512

() in Scheme, 497, 498 n, 503

Action, 9–13

computing, 14–23

coordinate-independence of, 17

free particle, 14–20

generating functions and, 421–425

Hamilton–Jacobi equation and, 421–425

Lagrangian, 12

minimizing, 18–23

parametric, 21

principles (see Principle of stationary action)

S, 10

time evolution and, 423–425, 435–437

variation of, 28

Action-angle coordinates, 311

Hamiltonian in, 311

Hamilton–Jacobi equation and, 413

Hamilton’s equations in, 311

harmonic oscillator in, 346 (eq. 5.31)

perturbation of Hamiltonian, 316, 458

surfaces of section in, 313

Action principle. See Principle of stationary action

Alphabet, insufficient size of, 15 n

Alternative in conditional, 501

angle-axis->rotation-matrix, 184

Angles, Euler. See Euler angles

Angular momentum. See also Vector angular momentum

conservation of, 43, 80, 86, 142–143

equilibrium points for, 149

Euler’s equations and, 151–153

in terms of principal moments and angular velocity, 136

kinetic energy in terms of, 148

Lie commutation relations for, 452 (ex. 6.10)

as Lie generator of rotations, 440

of free rigid body, 146–150, 151–153

of rigid body, 135–137

sphere of, 148

z component of, 85

Angular velocity vector ($\overrightarrow{\omega }$), 124, 139

Euler’s equations for, 151–153

kinetic energy in terms of, 131, 134

representation of, 123–126

Anomaly, true, 171 n

antisymmetric->column-matrix, 126

Antisymmetry of Poisson bracket, 220

Area preservation

by maps, 278

Liouville’s theorem and, 272

Poincaré–Cartan integral invariant and, 434–435

of surfaces of section, 272, 434–435

Arguments. See also Function(s); Functional arguments

active vs. passive in Legendre transformation, 208

in Scheme, 497

Arithmetic

generic, 16 n, 509

on functions, 18 n, 511

on operators, 34 n, 517

on procedures, 19 n

on symbolic values, 511

on tuples, 509, 513–516

Arnold, V. I., xiii, xv n, 113. See also Kolmogorov–Arnold–Moser theorem

Assignment in Scheme, 506–508

Associativity and non-associativity of tuple multiplication, 515, 516

Asteroids, rotational alignment of, 151

Astronomy. See Celestial objects

Asymptotic trajectories, 223, 287, 302

Atomic scale, 8 n

Attractor, 274

Autonomous systems, 82. See also Extended phase space

surfaces of section for, 248–263

Awake top, 231

Axes, principal, 133

of this dense book, 135 (ex. 2.7), 150

Axisymmetric potential of galaxy, 250

Axisymmetric top

awake, 231

behavior of, 161–165, 231–232

conserved quantities for, 160

degrees of freedom of, 5 (ex. 1.1)

Euler angles for, 159

Hamiltonian treatment of, 228–233

kinetic energy of, 159

Lagrangian treatment of, 157–165

nutation of, 162 (fig. 2.5), 164 (ex. 2.15)

potential energy of, 160

precession of, 119, 162 (fig. 2.6), 164 (ex. 2.16)

rotation of, 119

sleeping, 231

symmetries of, 228

Baker, Henry. See Baker–Campbell–Hausdorff formula

Baker–Campbell–Hausdorff formula, 453 (ex. 6.11)

Banana. See Book

Barrow-Green, June, 457

Basin of attraction, 274

Bicycle wheel, 156 (ex. 2.13)

Birkhoff, George David. See Poincaré–Birkhoff theorem

bisect (bisection search), 321, 326

Body components of vector, 134

Boltzmann, Ludwig, 12 n, 203 n, 274 n

Book

banana-like behavior of, 128

rotation of, 119, 150

Brackets. See also Poisson brackets

for down tuples, 512

for functional arguments, 10 n

bulirsch-stoer, 145

Bulirsch–Stoer integration method, 74 n

Butterfly effect, 241 n

C (local-tuple transformation), 44

C_H (canonical phase-space transformation), 337 n

Campbell, John. See Baker–Campbell–Hausdorff formula

canonical?, 344

Canonical-H?, 348

Canonical-K?, 348

canonical-transform?, 351

Canonical condition, 342–352

Poisson brackets and, 352–353

Canonical equations. See Hamilton’s equations

Canonical heliocentric coordinates, 409 (ex. 5.21)

Canonical perturbation theory. See Perturbation theory

Canonical plane, 362 n

Canonical transformations, 335–336. See also Generating functions; Symplectic transformations

composition of, 346 (ex. 5.4), 381, 393 (ex. 5.12)

conditions for, 342–357

for driven pendulum, 392

general, 342–357

group properties of, 346 (ex. 5.4)

for harmonic oscillator, 344

invariance of antisymmetric bilinear form under, 359–362

invariance of phase volume under, 358–359

invariance of Poisson brackets under, 358

invariants of, 357–364 (see also Integral invariants)

as Lie series, 448

Lie transforms (see Lie transforms)

point transformations (see Point transformations)

polar-canonical (see Polar-canonical transformation)

to rotating coordinates, 348–349, 377–378

time evolution as, 426–437

total time derivative and, 390–393

Cantorus, cantori, 244 n, 330

car, 503

Cartan, Élie. See Poincaré–Cartan integral invariant

Cauchy, Augustin Louis, 39 n

cdr, 503

Celestial objects. See also Asteroids; Comets; Earth; Galaxy; Hyperion; Jupiter; Mercury; Moon; Phobos; Planets

rotation of, 151, 165, 170–171

Center of mass, 121

in two-body problem, 381

Jacobi coordinates and, 409 (ex. 5.21)

kinetic energy and, 121

vector angular momentum and, 135

Central force

collapsing orbits, 389 (ex. 5.11)

epicyclic motion, 381–389

gravitational, 31

in 2 dimensions, 40, 227–228, 381–389

in 3 dimensions, 47 (ex. 1.16), 84

Lie series for motion in, 450

orbits, 78 (ex. 1.30)

reduced phase space for motion in, 405–407

Central potential. See Central force

Centrifugal force, 47, 49

Chain rule

for derivatives, 517, 523 (ex. 9.1)

for partial derivatives, 519, 523 (ex. 9.1)

for total time derivatives, 64 (ex. 1.26)

in traditional notation, xiv n

for variations, 27 (eq. 1.26)

Chaotic motion, 241. See also Exponential divergence

homoclinic tangle and, 307

in Hénon–Heiles problem, 259

in restricted three-body problem, 283 (ex. 3.16)

in spin-orbit coupling, 282 (ex. 3.15), 496 (ex. 7.5)

near separatrices, 290, 484, 486

of Hyperion, 151, 170–176

of non-axisymmetric top, 263

of periodically driven pendulum, 76, 243

overlapping resonances and, 488

Characteristic exponent, 293

Characteristic multiplier, 296

Chirikov, Boris V., 278 n

Chirikov–Taylor map, 278 n

Church, Alonzo, 498 n

Colon, names starting with, 21 n

Comets, rotation of, 151

Comma in tuple, 520

Commensurability, 312. See also Resonance

islands and, 309

of pendulum period with drive, 289, 290

periodic orbits and, 309, 316

rational rotation number and, 316

small denominators and, 475

Commutativity. See also Non-commutativity

of some tuple multiplication, 515

of variation (δ) with differentiation and integration, 27

Commutator, 451

of angular-momentum Lie operators, 452 (ex. 6.10)

Jacobi identity for, 451

of Lie derivative, 452 (ex. 6.10)

Poisson brackets and, 452 (ex. 6.10)

compatible-shape, 351 n

Compatible shape, 351 n

component, 15 n, 514

compose, 500

Composition

of canonical transformations, 346 (ex. 5.4), 381, 393 (ex. 5.12)

of functions, 7 n, 510, 523 (ex. 9.2)

of Lie transforms, 451

of linear transformations, 516

of operators, 517

of rotations, 123, 187

Compound data in Scheme, 502–504

cond, 500

Conditionals in Scheme, 500–501

Configuration, 4

Configuration manifold, 7 n

Configuration path. See Path

Configuration space, 4–5

Conjugate momentum, 79

non-uniqueness of, 239

cons, 503

Consequent in conditional, 500

Conserved quantities, 78, 195. See also Hénon–Heiles problem, integrals of motion

angular momentum, 43, 80, 86, 142–143

coordinate choice and, 79–81

cyclic coordinates and, 80

energy, 81–83, 142, 211

Jacobi constant, 89 n, 383, 400

Lyapunov exponents and, 267

momentum, 79–81

Noether’s theorem, 90–91

phase space reduction and, 224–226

phase volume (see Phase-volume conservation)

Poisson brackets of, 221

symmetry and, 79, 90

for top, 160

Constant of motion (integral of motion), 78. See also Conserved quantities; Hénon–Heiles problem

Constraint(s)

augmented Lagrangian and, 102, 109

configuration space and, 4

as coordinate transformations, 59–63

explicit, 99–103

in extended bodies, 4

holonomic, 4 n, 109

integrable, 4 n, 109

linear in velocities, 112

nonholonomic (non-integrable), 112

on coordinates, 101

rigid, 49–63

as subsystem couplers, 105

total time derivative and, 108

velocity-dependent, 108

velocity-independent, 101

Constraint force, 104

Constructors in Scheme, 503

Contact transformation. See Canonical transformations

Continuation procedure, 247

Continued-fraction

approximation of irrational number, 325

Contraction of tuples, 514

coordinate, 15 n

Coordinate(s). See also Generalized coordinates

action-angle (see Action-angle coordinates)

conserved quantities and choice of, 79–81

constraints on, 101

cyclic, 80, 224 n

heliocentric, 409 (ex. 5.21)

ignorable (cyclic), 80

Jacobi, 409 (ex. 5.21)

polar (see Polar coordinates)

redundant, and initial conditions, 69 n

rotating (see Rotating coordinates)

spherical, 84

Coordinate function (χ), 7

Coordinate-independence

of action, 17

of Lagrange equations, 30, 43 (ex. 1.14)

of variational formulation, 3, 39

Coordinate path (q), 7. See also Local tuple

Coordinate selector (Q), 220

Coordinate singularity, 144

Coordinate transformations, 44–47

constraints as, 59–63

Coriolis force, 47, 49

Correction fluid, 150

Cotangent space, bundle, 203 n

Coupling, spin-orbit. See Spin-orbit coupling

Coupling systems, 105–106

Curves, invariant. See Invariant curves

Cyclic coordinate, 80, 224 n

D. See Derivative

D (Scheme procedure for derivative), 16 n, 516

D-as-matrix, 355 n

D-phase-space, 347

∂. See Partial derivative

D_t (total time derivative), 64

d’Alembert–Lagrange principle (Jean leRond d’Alembert), 113

Damped harmonic oscillator, 274

define, 499

definite-integral, 17

Definite integral, 10 n

Definitions in Scheme, 499–500

Degrees of freedom, 4–5

Delta function, 454 (ex. 6.12)

Derivative, 8 n, 516–521. See also Total time derivative

as operator, 517

as Poisson bracket, 446

chain rule, 517, 523 (ex. 9.1)

in Scheme programs: D, 16 n, 516

notation: D, 8 n, 516

of function of multiple arguments, 29 n, 518–521

of function with structured arguments, 24 n

of function with structured inputs and outputs, 522

of state, 71

partial (see Partial derivative)

precedence of, 8 n, 516

with respect to a tuple, 29 n

determinant, 144

Differentiable manifold, 7 n

Dimension of configuration space, 4–5

Dirac, Paul Adrien Maurice, 12 n

Dissipation of energy

in free-body rotation, 150

tidal friction, 170

Dissipative system, phase-volume conservation, 274

Dissolution of invariant curves, 329–330, 486

Distribution functions, 276

Divided phase space, 244, 258, 286–290

Dot notation, 32 n

Double pendulum. See Pendulum, double

down, 15 n, 513

Down tuples, 512

Driven harmonic oscillator, 430 (ex. 6.6)

Driven pendulum. See Pendulum (driven)

Driven rotor, 317, 321

Dt (total time derivative), 97

Dynamical state. See State

$\mathcal{E}$ (Euler–Lagrange operator), 98

ℰ (energy state function), 82

Earth

precession of, 176 (ex. 2.18)

rotational alignment of, 151

Effective Hamiltonian, 230

Effects in Scheme, 505–508

Eigenvalues and eigenvectors

for equilibria, 293

for fixed points, 296

for Hamiltonian systems, 298

of inertia tensor, 132

for unstable fixed point, 303

Einstein, Albert, 1

Einstein summation convention, 367 n

else, 500

Empty list, 503

Energy, 81

as sum of kinetic and potential energies, 82

conservation of, 81–83, 142, 211

dissipation of (see Dissipation of energy)

Energy state function (ℰ), 82

Hamiltonian and, 200

Epicyclic motion, 381–389

eq?, 505

Equilibria, 222–223, 291–295. See also Fixed points

for angular momentum, 149

inverted, for pendulum, 246, 282 (ex. 3.14), 491–494, 496 (ex. 7.4)

linear stability of, 291–295

relative, 149

stable and unstable, 287

Equinox, precession of, 176 (ex. 2.18)

Ergodic motion, 312 n

Ergodic theorem, 251

Euler, Leonhard, 13 n

Euler->M, 139

Euler-state->omega-body, 140

Euler angles, 137–141

for axisymmetric top, 159

kinetic energy in terms of, 141

singularities and, 143, 154

Euler–Lagrange equations. See Lagrange equations

Euler-Lagrange-operator ($\mathcal{E}$), 98

Euler–Lagrange operator ($\mathcal{E}$), 98

Euler’s equations, 151–157

singularities in, 154

Euler’s theorem on homogeneous functions, 83 n

Euler’s theorem on rotations, 123

Euler angles and, 182

Evolution. See Time evolution of state

evolve, 75, 145, 238

explore-map, 248

Exponential(s)

of differential operator, 443

of Lie derivative, 447 (eq. 6.147)

of noncommuting operators, 451–453

Exponential divergence, 241, 243, 263–267. See also Chaotic motion; Lyapunov exponent

homoclinic tangle and, 307

Expressions in Scheme, 497

Extended phase space, 394–402

generating functions in, 407

F₁–F₄. See also Generating functions

F₁(t, q, q′), 373

F₂(t, q, p′), 373

F₃(t, p, q′), 374

F₄(t, p, p′), 374

F->C, 46, 96

F->CH, 339

F->K, 340

Fermat, Pierre, 13 (ex. 1.3)

Fermat’s principle (optics), 13 (ex. 1.3), 13 n

Fermi, Enrico, 251

Feynman, Richard P., 12 n

find-path, 21

First amendment. See Degrees of freedom

First integral, 78

Fixed points, 295. See also Equilibria

elliptic, 299, 320

equilibria or periodic motion and, 290, 295

for Hamiltonian systems, 298

hyperbolic, 299, 320

linear stability of, 295–297

manifolds for, 303

parabolic, 299

Poincaré–Birkhoff fixed points, 320

Poincaré–Birkhoff theorem, 316–321

rational rotation number and, 316

Floating-point numbers in Scheme, 18 n

Floquet multiplier, 296 n

Flow, defined by vector field, 447 n

Force

central (see Central force)

exerted by constraint, 104

Forced libration of the Moon, 175

Forced rigid body. See Rigid body, forced

Formal parameters

of a function, 14 n

of a procedure, 499

Foucault pendulum, 62 (ex. 1.25), 78 (ex. 1.31)

frame, 76 n

Free libration of the Moon, 175

Free particle

action, 14–20

Lagrange equations for, 33

Lagrangian for, 14–15

Free rigid body. See Rigid body (free)

Freudenthal, Hans, xiv n

Friction

internal, 150

tidal, 170

Function(s), 510–511

arithmetic operations on, 18 n, 511

composition of, 7 n, 510, 523 (ex. 9.2)

homogeneous, 83 n

operator vs., 448 n, 517

orthogonal, tuple-valued, 101 n

parallel, tuple-valued, 101 n

selector (see Selector function)

tuple of, 7 n, 521

vs. value when applied, 509, 510

with multiple arguments, 518, 519, 523 (ex. 9.2)

with structured arguments, 24 n, 519, 523 (ex. 9.2)

with structured output, 521, 523 (ex. 9.2)

Functional arguments, 10 n

Functional mathematical notation, xiv, 509

Function definition, 14 n

Fundamental Poisson brackets, 352

Γ[q]

for local tuple, 11

Lagrangian state path, 203

Galaxy, 248–252

axisymmetric potential of, 250

Galilean invariance, 68 (ex. 1.29), 341 (ex. 5.1)

Gamma (Scheme procedure for Γ), 16

optional argument, 36 (ex. 1.13)

Gamma-bar, 95

Gas in corner of room, 273

Generalized coordinates, 6–8, 39. See also Coordinate(s)

Euler angles as, 138 (see also Euler angles)

Generalized momentum, 79

transformation of, 337 (eq. 5.5)

Generalized velocity, 8

transformation of, 45

Generating functions, 364–394

in extended phase space, 407

F₁–F₄, 373–374

F₁, 364–368

F₂, 371–373

F₂ and point transformations, 375–376

F₂ for polar coordinate transformation, 376–377

F₂ for rotating coordinates, 377–378

integral invariants and, 368–373

Lagrangian action and, 421–425

Legendre transformation between F₁ and F₂, 373

mixed-variable, 374

Generic arithmetic, 16 n, 509

Gibbs, Josiah Willard, 12 n, 203 n

Golden number, 325

Golden ratio, a most irrational number, 325

Golden rotation number, 328

Goldstein, Herbert, 119

Goldstein’s hoop, 110

Golf ball, tiny, 108 (ex. 1.41)

Grand Old Duke of York. See neither up nor down

Graphing, 23 (ex. 1.5), 75, 248

Gravitational potential

central, 31

of galaxy, 250

multipole expansion of, 165–169

rigid-body, 166

Group properties

of canonical transformations, 346 (ex. 5.4)

of rotations, 187 (see also Euler’s theorem on rotations)

H (Hamiltonian), 199

H-central, 339

H-harmonic, 448

H-pend-sysder, 237

Hamilton, Sir William Rowan, 39 n, 183

Hamiltonian, 199

in action-angle coordinates, 311

computing (see H-…)

cyclic in coordinate, 224 n

energy state function and, 200

for axisymmetric potential, 250

for central potential, 227, 339, 381, 382

for damped harmonic oscillator, 275

for driven pendulum, 392

for driven rotor, 317

for harmonic oscillator, 344

for harmonic oscillator, in action-angle coordinates, 346 (eq. 5.31)

for Kepler problem, 418

for pendulum, 460

for periodically driven pendulum, 236, 476

for restricted three-body problem, 399, 400

for spin-orbit coupling, 496 (ex. 7.5)

for top, 230

for two-body problem, 378

Hénon–Heiles, 252, 455 (ex. 6.12)

Lagrangian and, 200 (eq. 3.19), 210

perturbation of action-angle, 316, 458

time-dependent, and dissipation, 276

Hamiltonian->Lagrangian, 213

Hamiltonian->state-derivative, 204

Hamiltonian flow, 447 n

Hamiltonian formulation, 195

Lagrangian formulation and, 217

Hamiltonian state, 202–203

Hamiltonian state derivative, 202, 204

Hamiltonian state path Π_L[q], 203

Hamilton–Jacobi equation, 411–413

action-angle coordinates and, 413

action at endpoints and, 425

for harmonic oscillator, 413–417

for Kepler problem, 417–421

separation in spherical coordinates, 418–421

time-independent, 413

Hamilton-equations, 203

Hamilton’s equations, 197–200

in action-angle coordinates, 311

computation of, 203–205

dynamical, 217

for central potential, 227

for damped harmonic oscillator, 275

for harmonic oscillator, 344

from action principle, 215–217

from Legendre transformation, 210–211

numerical integration of, 236

Poisson bracket form, 220

Hamilton’s principle, 38

for systems with rigid constraints, 49–50

Harmonic oscillator

coupled, 105

damped, 274

decoupling via Lie transform, 442

driven, 430 (ex. 6.6)

first-order equations for, 72

Hamiltonian for, 344

Hamiltonian in action-angle coordinates, 346 (eq. 5.31)

Hamilton’s equations for, 344

Lagrange equations for, 30, 72

Lagrangian for, 21

Lie series for, 448

solution of, 34, 344

solution via canonical transformation, 344

solution via Hamilton–Jacobi, 413–417

Hausdorff, Felix. See Baker–Campbell–Hausdorff formula

Heiles, Carl, 241, 248. See also Hénon

Heisenberg, Werner, 12 n, 203 n

Heliocentric coordinates, 409 (ex. 5.21)

Hénon, Michel, 195, 241, 248

Hénon–Heiles problem, 248–263

computing surfaces of section, 261–263

Hamiltonian for, 252

history of, 248–252

integrals of motion, 251, 254, 256–260

interpretation of model, 256–260

model of, 252–254

potential energy, 253

surface of section, 254–263

Hénon’s quadratic map, 280 (ex. 3.13)

Heteroclinic intersection, 305

Higher-order perturbation theory, 468–473, 489–494

History

Hénon–Heiles problem, 248–252

variational principles, 10 n, 13 n, 39 n

Holonomic system, 4 n, 109

Homoclinic intersection, 304

Homoclinic tangle, 302–309

chaotic regions and, 307

computing, 307–309

exponential divergence and, 307

Homogeneous function, Euler’s theorem, 83 n

Huygens, Christiaan, 10 n

Hyperion, chaotic tumbling of, 151, 170–176

I (identity operator), 517

I with subscript (selector), 64 n, 513

if, 501

Ignorable coordinate. See Cyclic coordinate

Indexing, zero-based. See Zero-based indexing

Inertia, moments of. See Moment(s) of inertia

Inertia matrix, 128. See also Inertia tensor

Inertia tensor, 127

diagonalization of, 132–133

kinetic energy in terms of, 131

principal axes of, 133

transformation of, 130–132

Initial conditions. See Sensitivity to initial conditions; State

Inner product of tuples, 515

Instability. See also Equilibria; Linear stability

free-body rotation, 149–151

Integers in Scheme, 18 n

Integrable constraints, 4 n, 109

Integrable systems, 285, 309–316

periodic orbits of near-integrable systems, 316

perturbation of, 316, 322, 457

reduction to quadrature and, 311 (see also Quadrature)

surfaces of section for, 313–316

Integral, definite, 10 n

Integral invariant

generating functions and, 368–373

Poincaré, 362–364

Poincaré–Cartan, 402, 431–434

Integral of motion, 78. See also Conserved quantities; Hénon–Heiles problem

Integration. See Numerical integration

Invariant curves, 243, 322–330

dissolution of, 329–330, 486

finding (computing), 326–329

finding (strategy), 322–325

irrational rotation number and, 322

Kolmogorov–Arnold–Moser theorem, 322

Invariants of canonical transformations, 357–364. See also Integral invariants

Irrational number, continued-fraction approximation, 325

Islands in surfaces of section. See also Resonance

for Hénon–Heiles problem, 259

for periodically driven pendulum, 244–246, 289–290, 483–486

for standard map, 279

perturbative vs. actual, 483–486

in Poincaré–Birkhoff construction, 321

Poisson series and, 488

secondary, 260, 290

size of, 322, 488

small denominators and, 322, 488

iterated-map, 308 n

Iteration in Scheme, 502

$\tilde{J}$ (shuffle function), 350

J, J_n (symplectic unit), 301, 355

J-func, 351

J-matrix, 353

Jac (Jacobian of map), 270

Jacobi, Carl Gustav Jacob, 39 n. See also Hamilton–Jacobi equation

Jacobian, 270

Jacobi constant, 89 n, 383, 400

Jacobi coordinates, 409 (ex. 5.21)

Jacobi identity

for commutators, 451

for Poisson brackets, 221

Jeans, Sir James, “theorem” of, 251

Jupiter, 129 (ex. 2.4)

KAM theorem. See Kolmogorov–Arnold–Moser theorem

Kepler, Johannes. See Kepler…

Kepler problem, 31, 35 (ex. 1.11)

in reduced phase space, 406

reduction to, 378–381

solution via Hamilton–Jacobi equation, 417–421

Kepler’s third law, 35 (ex. 1.11), 173

Kinematics of rotation, 122–126

Kinetic energy

ellipsoid of, 148

in Lagrangian, 38–39

as Lagrangian for free body, 122, 141

as Lagrangian for free particle, 14

of axisymmetric top, 159

of free rigid body, 148–150

of rigid body, 120–122 (see also Rigid body, kinetic energy…)

rotational and translational, 122

in spherical coordinates, 84

Knuth, Donald E., 531

Kolmogorov, A. N.. See Kolmogorov–Arnold–Moser theorem

Kolmogorov–Arnold–Moser theorem, 302, 322

L (Lagrangian), 11

L (Lie derivative), 447

L-axisymmetric-top, 229

L-body, 137

L-body-Euler, 141

L-central-polar, 43, 47

L-central-rectangular, 41

L-free-particle, 14

L-harmonic, 22

L-pend, 52

L-periodically-driven-pendulum, 74

L-rectangular, 213

L-space, 137

L-space-Euler, 141

L-uniform-acceleration, 40, 61

Lagrange, Joseph Louis, 13 n, 39 n

Lagrange-equations, 33

Lagrange equations, 23–25

at a moment, 97

computing, 33–36

coordinate-independence of, 30, 43 (ex. 1.14)

derivation of, 25–30

as first-order system, 72

for central potential (polar), 43

for central potential (rectangular), 41

for damped harmonic oscillator, 275

for driven pendulum, 52

for free particle, 33

for free rigid body, 141

for gravitational potential, 32

for harmonic oscillator, 30, 72

for periodically driven pendulum, 74

for spin-orbit coupling, 173

from Newton’s equations, 36–38, 54–58

vs. Newton’s equations, 39

numerical integration of, 73

off the beaten path, 97

singularities in, 143

traditional notation for, xiv, 24

uniqueness of solution, 69

Lagrange-interpolation-function, 20 n

Lagrange interpolation polynomial, 20

Lagrange multiplier. See Lagrangian, augmented

Lagrangian, 12

adding total time derivatives to, 65

augmented, 102, 109

computing, 14–15 (see also L-…)

coordinate transformations of, 44

cyclic in coordinate, 80

energy and, 12

for axisymmetric top, 159

for central potential (polar), 42–43, 227

for central potential (rectangular), 41

for central potential (spherical), 84

for constant acceleration, 40

for damped harmonic oscillator, 275

for driven pendulum, 51, 66

for free particle, 14–15

for free rigid body, 122, 141

for gravitational potential, 31

for harmonic oscillator, 21

for spin-orbit coupling, 173

for systems with rigid constraints, 49

generating functions and, 421–423

Hamiltonian and, 200 (eq. 3.19), 210

kinetic energy as, 14, 122, 141

kinetic minus potential energy as, 38–39 (see also Hamilton’s principle)

non-uniqueness of, 63–66

parameter names in, 14 n

rotational and translational, 141

symmetry of, 90

Lagrangian-action, 17

Lagrangian->energy, 82

Lagrangian->Hamiltonian, 213

Lagrangian->state-derivative, 71

Lagrangian action, 12

Lagrangian formulation, 195

Hamiltonian formulation and, 217

Lagrangian reduction, 233–236

Lagrangian state. See State tuple

Lagrangian state derivative, 71

Lagrangian state path Γ[q], 203

lambda, 498

Lambda calculus, 509

Lambda expression, 498–499

Lanczos, Cornelius, 335

Least action, principle of. See Principle of stationary action

Legendre, Adrien Marie. See Legendre…

Legendre polynomials, 167

Legendre-transform, 212

Legendre transformation, 205–212

active arguments in, 208

passive arguments in, 208–209

of quadratic functions, 211

Leibniz, Gottfried, 10 n

let, 501

let_*, 502

Libration of the Moon, 174, 175

Lie, Sophus. See Lie…

Lie-derivative, 448, 448 n

Lie derivative, 447 n

commutator for, 452 (ex. 6.10)

Lie transform and, 447 (eq. 6.147)

operator L_H, 447

Lie series, 443–451

computing, 448–451

for central field, 450

for harmonic oscillator, 448

in perturbation theory, 458–460

Lie-transform, 448

Lie transforms, 437–443

advantage of, 441

composition of, 451

computing, 448

exponential identities, 451–453

for finding normal modes, 442

Lie derivative and, 447 (eq. 6.147)

in perturbation theory, 458

Lindstedt, A., 471

linear-interpolants, 20 n

Linear momentum, 80

Linear separation of regular trajectories, 263

Linear stability, 290

equilibria and fixed points, 297–302

nonlinear stability and, 302

of equilibria, 291–295

of fixed points, 295–297

of inverted equilibrium of pendulum, 492, 496 (ex. 7.4)

Linear transformations

as tuples, 515

composition of, 516

Liouville, Joseph. See Liouville…

Liouville equation, 276

Liouville’s theorem, 268–272

from canonical transformation, 428

Lipschitz condition (Rudolf Lipschitz), 69 n

Lisp, 503 n

list, 503

list-ref, 503

Lists in Scheme, 502–504

literal-function, 15, 512, 521

Literal symbol in Scheme, 504–505

Local names in Scheme, 501–502

Local state tuple, 71

Local tuple, 11

component names, 14 n

functions of, 14 n

in Scheme programs, 15 n

transformation of (C), 44

Log, falling off, 84 (ex. 1.33)

Loops in Scheme, 502

Lorentz, Hendrik Antoon. See Lorentz transformations

Lorentz transformations as point transformations, 399 (ex. 5.18)

Lorenz, Edward, 241 n

Lyapunov, Alexey M.. See Lyapunov exponent

Lyapunov exponent, 267. See also Chaotic motion

conserved quantities and, 267

exponential divergence and, 267

Hamiltonian constraints, 302

linear stability and, 297

M-of-q->omega-body-of-t, 126

M-of-q->omega-of-t, 126

M->omega, 126

M->omega-body, 126, 185

MacCullagh’s formula, 168 n

make-path, 20, 20 n

Manifold

differentiable, 7 n

stable and unstable, 303–309

Map

area-preserving, 278

Chirikov–Taylor, 278 n

fixed points of, 295–297 (see also Fixed points)

Hénon’s quadratic, 280 (ex. 3.13)

Poincaré, 242

representation in programs, 247

standard, 277–280

symplectic, 301

twist, 315

Mars. See Phobos

Mass point. See Point mass

Mathematical notation. See Notation

Mather, John N. (discoverer of sets named cantori by Ian Percival), 244 n

Matrix

inertia, 128 (see also Inertia tensor)

orthogonal, 124, 130 n

symplectic, 301, 355, 356 (ex. 5.6)

as tuple, 515

Maupertuis, Pierre-Louis Moreau de, 13 n

Mean motion, 175 n

Mechanics, 1–496

Newtonian vs. variational formulation, 3, 39

Mercury, resonant rotation of, 171, 193 (ex. 2.21)

Minimization

of action, 18–23

in Scmutils, 19 n, 21 n

minimize, 19 n

Mixed-variable generating functions, 374

Moment(s) of inertia, 126–130

about a line, 128

about a pivot point, 159

principal, 132–135

of top, 159

Momentum. See also Angular momentum

conjugate to coordinate (see Conjugate momentum)

conservation of, 79–81

generalized (see Generalized momentum)

variation of, 216 n

momentum, 204

Momentum path, 80

Momentum selector (P), 199, 220

Momentum state function ($\mathcal{P}$), 79

Moon

head-shaking, 174

history of, 9 n

libration of, 174, 175

rotation of, 119, 151, 170–176, 496 (ex. 7.5)

Moser, Jürgen. See Kolmogorov–Arnold–Moser theorem

Motion

atomic-scale, 8 n

chaotic (see Chaotic motion)

constrained, 99–103 (see also Constraint(s))

dense, on torus, 312 n

deterministic, 9

epicyclic, 381–389

ergodic, 312 n

periodic (see Periodic motion)

quasiperiodic, 243, 312

realizable vs. conceivable, 2

regular vs. chaotic, 241 (see also Regular motion)

smoothness of, 8

tumbling (see Chaotic motion, of Hyperion; Rotation(s), (in)stability of)

multidimensional-minimize, 21, 21 n

Multiplication of operators as composition, 517

Multiplication of tuples, 514–516

as composition, 516

as contraction, 514

Multiply periodic functions, Poisson series for, 474

Multipole expansion of potential energy, 165–169

n-body problem, 408 (ex. 5.21). See also Three-body problem, restricted; Two-body problem

Nelder–Mead minimization method, 21 n

Newton, Sir Isaac, 3

Newtonian formulation of mechanics, 3, 39

Newton’s equations

as Lagrange equations, 36–38, 54–58

vs. Lagrange equations, 39

Noether, Emmy, 81 n

Noether’s integral, 91

Noether’s theorem, 90–91

angular momentum and, 143

Non-associativity and associativity of tuple multiplication, 515, 516

Non-axisymmetric top, 263

Non-commutativity. See also Commutativity

exponential(s) of noncommuting operators, 451–453

of some partial derivatives, 427 n, 520

of some tuple multiplication, 516

Nonholonomic system, 112

Nonsingular structure, 368 n

Notation, 509–523. See also Subscripts; Superscripts; Tuples

{ } for Poisson brackets, 218

( ) for up tuples, 512

[ ] for down tuples, 512

[ ] for functional arguments, 10 n

ambiguous, xiv–xv

composition of functions, 7 n

definite integral, 10 n

derivative, partial: ∂, xv, 24, 520

derivative: D, 8 n, 516

functional, xiv, 509

functional arguments, 10 n

function of local tuple, 14 n

selector function: I with subscript, 64 n, 513

total time derivative: D_t, 64

traditional, xiv–xv, 24, 200 n, 218 n, 509

Numbers in Scheme, 18 n

Numerical integration

of Hamilton’s equations, 236

of Lagrange equations, 73

in Scmutils, 17 n, 74 n, 145

symplectic, 453 (ex. 6.12)

Numerical minimization in Scmutils, 19 n, 21 n

Nutation of top, 162 (fig. 2.5), 164 (ex. 2.15)

Oblateness, 170

omega (symplectic 2-form), 361

omega-cross, 126

Operator, 517

arithmetic operations on, 34 n, 517

composition of, 517

exponential identities, 451–453

function vs., 448 n, 517

generic, 16 n

Operators

derivative (D) (see Derivative)

Euler–Lagrange ($$ \mathcal{E}$$), 98

Lie derivative (L_H), 447

Lie transform ($E_{\mathit{ϵ},W}^{\prime }$), 439

partial derivative (∂) (see Partial derivative)

variation (δ_η), 26

Optical libration of the Moon, 174

Optics

Fermat, 13 (ex. 1.3)

Snell’s law, 13 n

Orbit. See Orbital motion; Phase-space trajectory

Orbital elements, 421

Orbital motion. See also Epicyclic motion; Kepler problem

in a central potential, 78 (ex. 1.30)

Lagrange equations for, 31–32

retrodiction of, 9 n

Orientation. See also Rotation(s)

Euler’s equations and, 153–154

nonsingular coordinates for, 181–191

specified by Euler angles, 138

specified by rotations, 123

Orientation vector, 182

Orthogonal matrix, 124, 130 n

Orthogonal transformation. See Orthogonal matrix

Orthogonal tuple-valued functions, 101 n

Oscillator. See Harmonic oscillator

osculating-path, 96

Osculation of paths, 94

Ostrogradsky, M. V., 39 n

Out-of-roundness parameter, 173

P (momentum selector), 199, 220

$\mathcal{P}$ (momentum state function), 79

p->r (polar-to-rectangular), 46

pair?, 504

Pairs in Scheme, 503

Parallel tuple-valued functions, 101 n

Parameters, formal. See Formal parameters

Parametric path, 20

parametric-path-action, 21

with graph, 23 (ex. 1.5)

Parentheses

in Scheme, 497, 498 n

for up tuples, 512

partial, 33 n

Partial derivative, 24, 518–519, 520

chain rule, 519, 523 (ex. 9.1)

notation: ∂, xv, 24, 520

Particle, free. See Free particle

Path, 2

coordinate path (q), 7 (see also Local tuple)

finding, 20–23

momentum path, 80

osculation of, 94

parametric, 20

realizable (see Realizable path)

variation of, 12, 18, 26

Path-distinguishing function, 2, 8. See also Action

Path functions, abstraction of, 94

Peak, 222

Pendulum. See also Pendulum (driven); Periodically driven pendulum

behavior of, 223, 286–287

constraints and, 103

degrees of freedom of, 5 (ex. 1.1)

double (planar), 6, 117 (ex. 1.44)

double (spherical), 5 (ex. 1.1)

equilibria, stable and unstable, 287

Foucault, 62 (ex. 1.25), 78 (ex. 1.31)

Hamiltonian for, 460

Lagrangian for, 32 (ex. 1.9)

periodically driven pendulum vs., 244

as perturbed rotor, 460–473

phase plane of, 223, 286

phase-volume conservation for, 268

spherical, 5 (ex. 1.1), 86 (ex. 1.34)

width of oscillation region, 466

Pendulum (driven), 50–52. See also Pendulum; Periodically driven pendulum

drive as modification of gravity, 66

Hamiltonian for, 392

Lagrange equations for, 52

Lagrangian for, 51, 66

Pericenter, 171 n

Period doubling, 245

Periodically driven pendulum. See also Pendulum (driven); Pendulum

behavior of, 196, 244–246

chaotic behavior of, 76, 243

emergence of divided phase space, 286–290

Hamiltonian for, 236, 476

inverted equilibrium, 246, 282 (ex. 3.14), 491–494, 496 (ex. 7.4)

islands in sections for, 244–246, 289–290, 483–486

Lagrange equations for, 74

linear stability analysis, 492, 496 (ex. 7.4)

as perturbed rotor, 476–478

phase-space descriptions for, 239

phase space evolution of, 236

resonances for, 481–491

spin-orbit coupling and, 173

surface of section for, 242–248, 282 (ex. 3.14), 287–290, 483–494

undriven pendulum vs., 244

with zero-amplitude drive, 286–289

Periodically driven systems, surfaces of section, 241–248

Periodic motion, 312

fixed points and, 295

integrable systems and, 309, 316

Periodic points, 295

Poincaré–Birkhoff theorem, 316–321

rational rotation number and, 316

resonance islands and, 290

Perturbation of action-angle Hamiltonian, 316, 458

Perturbation theory, 457

for many degrees of freedom, 473–478

for pendulum, 466–468

for periodically driven pendulum, 491–494

for spin-orbit coupling, 496 (ex. 7.5)

higher-order, 468–473, 489–494

Lie series in, 458–460

nonlinear resonance, 478–494

secular-term elimination, 471–473

secular terms in, 470

small denominators in, 475, 476

Phase portrait, 231, 248 (ex. 3.10)

Phase space, 203. See also Surface of section

chaotic regions, 241

divided, 244, 258, 286–290

evolution in, 236–238 (see also Time evolution of state)

extended, 394–402

non-uniqueness, 238–239

of pendulum, 223, 286

qualitative features, 242–246, 258–260, 285–286

reduced, 402–407

regular regions, 241

two-dimensional, 222

volume (see Phase-volume conservation)

Phase space reduction, 224–226

conserved quantities and, 224–226

Lagrangian, 233–236

Phase-space state function, 519

in Scheme, 521

Phase-space trajectory (orbit)

asymptotic, 223, 287, 302

chaotic, 243, 259

periodic, 309, 312, 316

quasiperiodic, 243, 312

regular, 243, 258

regular vs. chaotic, 241

Phase-volume conservation, 268, 428

for damped harmonic oscillator, 274

for pendulum, 268

under canonical transformations, 358–359

Phobos, rotation of, 171

Pit, 222

Planets. See also Earth; Jupiter; Mercury

moment of inertia of, 129 (ex. 2.4)

rotational alignment of, 151

rotation of, 165

plot-parametric-fill, 308

plot-point, 76 n

Plotting, 23 (ex. 1.5), 75, 248

Poe, Edgar Allan. See Pit; Pendulum

Poincaré, Henri, 239 n, 251, 285, 302, 471

Poincaré–Birkhoff theorem, 316–321

computing fixed points, 321–322

recursive nature of, 321

Poincaré–Cartan integral invariant, 402

time evolution and, 431–434

Poincaré integral invariant, 362–364

generating functions and, 368–373

Poincaré map, 242

Poincaré recurrence, 272

Poincaré section. See Surface of section

Point mass, 4 n. See also Golf ball, tiny

Point transformations, 336–341. See also Canonical transformations

computing, 339–341

general canonical transformations vs., 357

generating functions for, 375–376

polar-rectangular conversion, 339, 376–377

to rotating coordinates, 348–349, 377–378

time-independent, 338

Poisson, Siméon Denis, 33 (ex. 1.10)

Poisson brackets, 218–222

canonical condition and, 352–353

commutator and, 452 (ex. 6.10)

of conserved quantities, 221

as derivations, 446

fundamental, 352

Hamilton’s equations in terms of, 220

in terms of $\tilde{J}$, 352

in terms of symplectic 2-form, ω, 360

invariance under canonical transformations, 358

Jacobi identity for, 221

Lie derivative and, 447

Poisson series

for multiply periodic function, 474

resonance islands and, 488

polar-canonical, 345

Polar-canonical transformation, 344

generating function for, 365

harmonic oscillator and, 346

Polar coordinates

Lagrangian in, 42–43

point transformation to rectangular, 339, 376–377

transformation to rectangular, 46

Potential. See Central force; Gravitational potential

Potential energy

of axisymmetric top, 160

Hénon–Heiles, 253

in Lagrangian, 38–39

multipole expansion of, 165–169

Precession

of equinox, 176 (ex. 2.18)

of top, 119, 162 (fig. 2.6), 164 (ex. 2.16)

Predicate in conditional, 500

Predicting the past, 9 n

principal-value, 76 n

Principal axes, 133

of this dense book, 135 (ex. 2.7), 150

Principal moments of inertia, 132–135

kinetic energy in terms of, 134, 141, 148

Principle of d’Alembert–Lagrange, 113

Principle of least action. See Principle of stationary action

Principle of stationary action (action principle), 8–13

Hamilton’s equations and, 215–217

principle of least action, 10 n, 13 n, 39 n

statement of, 12

used to find paths, 20

print-expression, 444, 511 n

Probability density in phase space, 276

Procedure calls, 497–498

Procedures

arithmetic operations on, 19 n

generic, 16 n

Products of inertia, 128

Q (coordinate selector), 220

$\dot{Q}$ (velocity selector), 64

q (coordinate path), 7

qcrk4 (quality-controlled Runge–Kutta), 145

Quadratic functions, Legendre transformation of, 211

Quadrature, 161 n, 222. See also Integrable systems

integrable systems and, 311

reduction to, 224 n

Quartet, 300 (fig. 4.5)

Quasiperiodic motion, 243, 312

quaternion->angle-axis, 184

quaternion->RM, 184

quaternion->rotation-matrix, 185

quaternion-state->omega-body, 186

Quaternions, 181–191

Hamilton’s discovery of, 39 n

quaternion units, 188

Quotation in Scheme, 504–505

qw-state->L-space, 190

qw-sysder, 189

Radial momentum, 80

Reaction force. See Constraint force

Realizable path, 9

conserved quantities and, 78

as solution of Hamilton’s equations, 202

as solution of Lagrange equations, 23

stationary action and, 9–13

uniqueness, 12

Recurrence theorem of Poincaré, 272

Recursive procedures, 501

Reduced mass, 35 (ex. 1.11), 380

Reduced phase space, 402–407

Reduction

Lagrangian, 233–236

of phase space (see Phase space reduction)

to quadrature, 224 n

ref, 15 n, 514

Regular motion, 241, 243, 258

linear separation of trajectories, 263

Renormalization, 267 n

Resonance. See also Commensurability

center, 480

islands (see Islands in surfaces of section)

nonlinear, 478–494

of Mercury’s rotation, 171, 193 (ex. 2.21)

overlap criterion, 488–489, 496 (ex. 7.5)

for periodically driven pendulum, 481–491

spin-orbit, 177–181

width, 483 (ex. 7.2), 488

Restricted three-body problem. See Three-body problem, restricted

Rigid body, 120

forced, 154–157 (see also Spin-orbit coupling; Top)

free (see Rigid body (free))

kinetic energy, 120–122

kinetic energy in terms of inertia tensor and angular velocity, 126–129, 131

kinetic energy in terms of principal moments and angular momentum, 148

kinetic energy in terms of principal moments and angular velocity, 134

kinetic energy in terms of principal moments and Euler angles, 141

vector angular momentum, 135–137

Rigid body (free), 141

angular momentum, 151–153

angular momentum and kinetic energy, 146–150

computing motion of, 143–145

Euler’s equations and, 151–154

(in)stability, 149–151

orientation, 153–154

Rigid constraints, 49–63

as coordinate transformations, 59–63

Rotating coordinates

in extended phase space, 400–402

generating function for, 377–378

point transformation for, 348–349, 377–378

Rotation(s). See also Orientation

active, 130

composition of, 123, 187

computing, 93

group property of, 187

(in)stability of, 149–151

kinematics of, 122–126

kinetic energy of (see Rigid body, kinetic energy…)

Lie generator for, 440

matrices for, 138

of celestial objects, 151, 165, 170–171

of Hyperion, 170–176

of Mercury, 171, 193 (ex. 2.21)

of Moon, 119, 151, 170–176, 496 (ex. 7.5)

of Phobos, 171

of top, book, and Moon, 119

orientation as, 123

orientation vector and, 182

passive, 130

as tuples, 515

Rotation number, 315

golden, 328

irrational, and invariant curves, 322

rational, and commensurability, 316

rational, and fixed and periodic points, 316

Rotor

driven, 317, 321

pendulum as perturbation of, 460–473

periodically driven pendulum as perturbation of, 476–478

Routh, Edward John

Routhian, 234

Routhian equations, 236 (ex. 3.9)

Runge–Kutta integration method, 74 n

qcrk4, 145

Rx, 63 (ex. 1.25), 93 n

Rx-matrix, 139

Ry, 63 (ex. 1.25), 93 n

Rz, 63 (ex. 1.25), 93 n

Rz-matrix, 139

S (action), 10

Lagrangian, 12

s->m (structure to matrix), 353

s->r (spherical-to-rectangular), 85

Saddle point, 222

Salam, Abdus, 509

Saturn. See Hyperion

Scheme, xvi, 497–508, 509. See also Scmutils

for Gnu/Linux, where to get it, xvi

Schrödinger, Erwin, 12 n, 203 n

Scmutils, xvi, 509–523. See also Scheme

generic arithmetic, 16 n, 509

minimization, 19 n, 21 n

numerical integration, 17 n, 74 n, 145

operations on operators, 34 n

simplification of expressions, 511

where to get it, xvi

Second law of thermodynamics, 274

Section, surface of. See Surface of section

Secular terms in perturbation theory, 470

elimination of, 471–473

Selector function, 64 n, 513

coordinate selector (Q), 220

momentum selector (P), 199, 220

velocity selector ($\dot{Q}$), 64

Selectors in Scheme, 503

Semicolon in tuple, 31 n, 520

Sensitivity to initial conditions, 241 n, 243, 263. See also Chaotic motion

Separatrix, 147, 222. See also Asymptotic trajectories

chaos near, 290, 484, 486

motion near, 302

series, 462

series:for-each, 444

series:sum, 463

set-ode-integration-method!, 145

show-expression, 16, 46 n

Shuffle function $\tilde{J}$, 350

Simplification of expressions, 511

Singularities, 202 n

Euler angles and, 143, 154

in Euler’s equations, 154

quaternions, 181–191

Sleeping top, 231

Small denominators

for periodically driven pendulum, 477

in perturbation theory, 475, 476

resonance islands and, 322, 488

Small divisors. See Small denominators

Snell’s law, 13 n

Solvable systems. See Integrable systems

solve-linear-left, 71 n

solve-linear-right, 339 n

Spherical coordinates

kinetic energy in, 84

Lagrangian in, 84

Spin-orbit coupling, 165–181

chaotic motion, 282 (ex. 3.15), 496 (ex. 7.5)

Hamiltonian for, 496 (ex. 7.5)

Lagrange equations for, 173

Lagrangian for, 173

periodically driven pendulum and, 173

perturbation theory for, 496 (ex. 7.5)

resonances, 177–181

surface of section for, 282 (ex. 3.15)

Spivak, Michael, xiv n, 509

Spring–mass system. See Harmonic oscillator

square, 21 n, 499

for tuples, 40 n, 499 n

Stability. See Equilibria; Instability; Linear stability

Stable manifold, 303–309

computing, 307–309

standard-map, 278

Standard map, 277–280

Stars. See Galaxy

State, 68–71

evolution of (see Time evolution of state)

Hamiltonian vs. Lagrangian, 202–203

in terms of coordinates and momenta (Hamiltonian), 196

in terms of coordinates and velocities (Lagrangian), 69

state-advancer, 74

State derivative

Hamiltonian, 204

Hamiltonian vs. Lagrangian, 202

Lagrangian, 71

State path

Hamiltonian, 203

Lagrangian, 203

State tuple, 71

Stationarity condition, 28

Stationary action. See Principle of stationary action

Stationary point, 2 n

Steiner’s theorem, 129 (ex. 2.2)

String theory, 119 n, 150. See also Quartet

Stroboscopic surface of section, 241–248. See also Surface of section

computing, 246

Subscripts

down and, 15 n

for down-tuple components, 513

for momentum components, 79 n, 338 n

for selectors, 513

Summation convention, 367 n

Superscripts

for coordinate components, 7 n, 15 n, 79 n

for up-tuple components, 513

for velocity components, 15 n, 338 n

up and, 15 n

Surface of section, 239–248

in action-angle coordinates, 313

area preservation of, 272, 434–435

computing (Hénon–Heiles), 261–263

computing (stroboscopic), 246

fixed points (see Fixed points)

for autonomous systems, 248–263

for Hénon–Heiles problem, 254–263

for integrable system, 313–316

for non-axisymmetric top, 263

for periodically driven pendulum, 242–248, 282 (ex. 3.14), 287–290, 483–494

for restricted three-body problem, 283 (ex. 3.16)

for spin-orbit coupling, 282 (ex. 3.15)

for standard map, 277–280

invariant curves (see Invariant curves)

islands (see Islands in surfaces of section)

stroboscopic, 241–248

Symbolic values, 511–512

Symbols in Scheme, 504–505

Symmetry

conserved quantities and, 79, 90

continuous, 195

of Lagrangian, 90

of top, 228

symplectic-matrix?, 355

symplectic-transform?, 355

symplectic-unit, 355

Symplectic bilinear form (2-form), 359–362

invariance under canonical transformations, 359

Symplectic condition. See Symplectic transformations

Symplectic integration, 453 (ex. 6.12)

Symplectic map, 301

Symplectic matrix, 301, 356 (ex. 5.6), 353–357

Symplectic transformations, 355. See also Canonical transformations

antisymmetric bilinear form and, 359–362

Symplectic unit J, J_n, 301, 355

Syntactic sugar, 499

System derivative. See State derivative

T-body, 134

T-body-Euler, 141

T-func, 347

Taylor, J. B., 278 n

Tensor. See Inertia tensor

Tensor arithmetic

notation and, 79 n, 338 n

summation convention, 367 n

tuple arithmetic vs., 509, 513

Theology and principle of least action, 13 n

Thermodynamics, second law, 274

Three-body problem, restricted, 86–90, 283 (ex. 3.16), 399–402

chaotic motion, 283 (ex. 3.16)

surface of section for, 283 (ex. 3.16)

Tidal friction, 170

time, 15 n

Time-dependent transformations, 347–349

Time evolution of state, 68–78

action and, 423–425, 435–437

as canonical transformation, 426–437

in phase space, 236–238

Poincaré–Cartan integral invariant and, 431–434

Time-independence. See also Extended phase space

energy conservation and, 81

Top

axisymmetric (see Axisymmetric top)

non-axisymmetric, 263

Top banana. See Non-axisymmetric top

Torque, 165 (ex. 2.16)

in Euler’s equations, 154

in spin-orbit coupling, 173

Total time derivative, 63–65

adding to Lagrangians, 65

affecting conjugate momentum, 239

canonical transformation and, 390–393

commutativity of, 91 n

computing, 97

constraints and, 108

identifying, 68 (ex. 1.28)

notation: D_t, 64

properties, 67

Trajectory. See Path; Phase-space trajectory

Transformation

canonical (see Canonical transformations)

coordinate (see Coordinate transformations)

Legendre (see Legendre transformation)

Lie (see Lie transforms)

orthogonal (see Orthogonal matrix)

point (see Point transformations)

symplectic (see Symplectic transformations)

time-dependent, 347–349

Transpose, 351 n

transpose, 126, 351 n

True anomaly, 171 n

Tumbling. See Chaotic motion, of Hyperion; Rotation(s), (in)stability of

Tuples, 512–516

arithmetic on, 509, 513–516

commas and semicolons in, 520

component selector: I with subscript, 64 n, 513

composition and, 516

contraction, 514

of coordinates, 7

down and up, 512

of functions, 7 n, 521

inner product, 515

linear transformations as, 515

local (see Local tuple)

matrices as, 515

multiplication of, 514–516

rotations as, 515

semicolons and commas in, 520

squaring, 499 n, 514

state tuple, 71

up and down, 512

Twist map, 315

Two-body problem, 378–381

Two-trajectory method, 265

Undriven pendulum. See Pendulum

Uniform circle map, 326

Uniqueness

of Lagrangian—not!, 63–66

of phase-space description—not!, 238–239

of realizable path, 12

of solution to Lagrange equations, 69

unstable-manifold, 308

Unstable manifold, 303–309

computing, 307–309

up, 15 n, 513

Up tuples, 512

Vakonomic mechanics, 114 n

Variation

chain rule, 27 (eq. 1.26)

of action, 28

of a function, 26

of a path, 12, 18, 26

operator: δ_η, 26

Variational equations, 266

Variational formulation of mechanics, 2–3, 39

Variational principle. See Principle of stationary action

Vector

body components of, 134

in Scheme, 502–504

square of, 18 n, 21 n

vector, 504

vector?, 504

vector-ref, 504

Vector angular momentum, 135–137. See also Angular momentum

center-of-mass decomposition, 135

in terms of angular velocity and inertia tensor, 136

in terms of principal moments and Euler angles, 141

Vector space of tuples, 514

Vector torque. See Torque

Velocity. See Angular velocity vector; Generalized velocity

velocity, 15 n

Velocity dispersion in galaxy, 248

Velocity selector ($\dot{Q}$), 64

Web site for this book, xvi

Wheel, 156 (ex. 2.13)

Whittaker transform (Sir Edmund Whittaker), 357 (ex. 5.9)

Width of oscillation region, 466 n

write-line, 505 n

Zero-amplitude drive for pendulum, 286–289

Zero-based indexing, 7 n, 15 n, 503, 513, 514, 531