# Index

Any inaccuracies in this index may be explained by the fact that it has been prepared with the help of a computer.

Donald E. Knuth, Fundamental Algorithms

(Volume 1 of The Art of Computer Programming)

Page numbers for Scheme procedure definitions are in italics.

Page numbers followed by n indicate footnotes.

∘ (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)

δη (variation operator), 26

λ-calculus, 509

λ-expression, 498499

λ-notation, 498 n

ΠL[q] (Hamiltonian state path), 203

χ (coordinate function), 7

σ (phase-space path), 218

ω matrix, 124

$\stackrel{\to }{\omega }$ (angular velocity), 124

ω (symplectic 2-form), 359

C (local-tuple transformation), 44

CH (canonical phase-space transformation), 337 n

D. See Derivative

Dt (total time derivative), 64

∂. See Partial derivative

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

(energy state function), 82

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

$\stackrel{˜}{J}$ (shuffle function), 350

J, Jn (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

$\stackrel{˙}{Q}$ (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, 913

computing, 1423

coordinate-independence of, 17

free particle, 1420

generating functions and, 421425

Hamilton–Jacobi equation and, 421425

Lagrangian, 12

minimizing, 1823

parametric, 21

principles (see Principle of stationary action)

S, 10

time evolution and, 423425, 435437

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

conservation of, 43, 80, 86, 142143

equilibrium points for, 149

Euler's equations and, 151153

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, 146150, 151153

of rigid body, 135137

sphere of, 148

z component of, 85

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

Euler's equations for, 151153

kinetic energy in terms of, 131, 134

representation of, 123126

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, 434435

of surfaces of section, 272, 434435

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, 513516

Assignment in Scheme, 506508

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

surfaces of section for, 248263

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, 161165, 231232

conserved quantities for, 160

degrees of freedom of, 5 (ex. 1.1)

Euler angles for, 159

Hamiltonian treatment of, 228233

kinetic energy of, 159

Lagrangian treatment of, 157165

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

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

CH (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, 342352

Poisson brackets and, 352353

Canonical equations. See Hamilton's equations

Canonical heliocentric coordinates, 409 (ex. 5.21)

Canonical perturbation theory. See Perturbation theory

Canonical plane, 362 n

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

conditions for, 342357

for driven pendulum, 392

general, 342357

group properties of, 346 (ex. 5.4)

for harmonic oscillator, 344

invariance of antisymmetric bilinear form under, 359362

invariance of phase volume under, 358359

invariance of Poisson brackets under, 358

as Lie series, 448

Lie transforms (see Lie transforms)

point transformations (see Point transformations)

polar-canonical (see Polar-canonical transformation)

to rotating coordinates, 348349, 377378

time evolution as, 426437

total time derivative and, 390393

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, 170171

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, 381389

gravitational, 31

in 2 dimensions, 40, 227228, 381389

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, 405407

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)

for variations, 27 (eq. 1.26)

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, 170176

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

islands and, 309

of pendulum period with drive, 289, 290

periodic orbits and, 309, 316

rational rotation number and, 316

small denominators and, 475

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, 502504

cond, 500

Conditionals in Scheme, 500501

Configuration, 4

Configuration manifold, 7 n

Configuration path. See Path

Configuration space, 45

Conjugate momentum, 79

non-uniqueness of, 239

cons, 503

Consequent in conditional, 500

angular momentum, 43, 80, 86, 142143

coordinate choice and, 7981

cyclic coordinates and, 80

energy, 8183, 142, 211

Jacobi constant, 89 n, 383, 400

Lyapunov exponents and, 267

momentum, 7981

Noether's theorem, 9091

phase space reduction and, 224226

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, 5963

explicit, 99103

in extended bodies, 4

holonomic, 4 n, 109

integrable, 4 n, 109

linear in velocities, 112

nonholonomic (non-integrable), 112

on coordinates, 101

rigid, 4963

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

action-angle (see Action-angle coordinates)

conserved quantities and choice of, 7981

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 selector (Q), 220

Coordinate singularity, 144

Coordinate transformations, 4447

constraints as, 5963

Coriolis force, 47, 49

Correction fluid, 150

Cotangent space, bundle, 203 n

Coupling, spin-orbit. See Spin-orbit coupling

Coupling systems, 105106

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

Dt (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, 499500

Degrees of freedom, 45

Delta function, 454 (ex. 6.12)

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, 518521

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, 45

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, 329330, 486

Distribution functions, 276

Divided phase space, 244, 258, 286290

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

(energy state function), 82

Earth

precession of, 176 (ex. 2.18)

rotational alignment of, 151

Effective Hamiltonian, 230

Effects in Scheme, 505508

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, 8183, 142, 211

dissipation of (see Dissipation of energy)

Energy state function (), 82

Hamiltonian and, 200

Epicyclic motion, 381389

eq?, 505

for angular momentum, 149

inverted, for pendulum, 246, 282 (ex. 3.14), 491494, 496 (ex. 7.4)

linear stability of, 291295

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, 137141

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, 151157

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, 451453

homoclinic tangle and, 307

Expressions in Scheme, 497

Extended phase space, 394402

generating functions in, 407

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

elliptic, 299, 320

equilibria or periodic motion and, 290, 295

for Hamiltonian systems, 298

hyperbolic, 299, 320

linear stability of, 295297

manifolds for, 303

parabolic, 299

Poincaré–Birkhoff fixed points, 320

Poincaré–Birkhoff theorem, 316321

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, 1420

Lagrange equations for, 33

Lagrangian for, 1415

Free rigid body. See Rigid body (free)

Freudenthal, Hans, xiv n

Friction

internal, 150

tidal, 170

Function(s), 510511

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, 248252

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 momentum, 79

transformation of, 337 (eq. 5.5)

Generalized velocity, 8

transformation of, 45

Generating functions, 364394

in extended phase space, 407

F1F4, 373374

F1, 364368

F2, 371373

F2 and point transformations, 375376

F2 for polar coordinate transformation, 376377

F2 for rotating coordinates, 377378

integral invariants and, 368373

Lagrangian action and, 421425

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, 165169

rigid-body, 166

Group properties

of canonical transformations, 346 (ex. 5.4)

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, 202203

Hamiltonian state derivative, 202, 204

Hamiltonian state path ΠL[q], 203

Hamilton–Jacobi equation, 411413

action-angle coordinates and, 413

action at endpoints and, 425

for harmonic oscillator, 413417

for Kepler problem, 417421

separation in spherical coordinates, 418421

time-independent, 413

Hamilton-equations, 203

Hamilton's equations, 197200

in action-angle coordinates, 311

computation of, 203205

dynamical, 217

for central potential, 227

for damped harmonic oscillator, 275

for harmonic oscillator, 344

from action principle, 215217

from Legendre transformation, 210211

numerical integration of, 236

Poisson bracket form, 220

Hamilton's principle, 38

for systems with rigid constraints, 4950

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, 413417

Hausdorff, Felix. See Baker–Campbell–Hausdorff formula

Heisenberg, Werner, 12 n, 203 n

Heliocentric coordinates, 409 (ex. 5.21)

Hénon, Michel, 195, 241, 248

Hénon–Heiles problem, 248263

computing surfaces of section, 261263

Hamiltonian for, 252

history of, 248252

integrals of motion, 251, 254, 256260

interpretation of model, 256260

model of, 252254

potential energy, 253

surface of section, 254263

Hénon's quadratic map, 280 (ex. 3.13)

Heteroclinic intersection, 305

Higher-order perturbation theory, 468473, 489494

History

Hénon–Heiles problem, 248252

variational principles, 10 n, 13 n, 39 n

Holonomic system, 4 n, 109

Homoclinic intersection, 304

Homoclinic tangle, 302309

chaotic regions and, 307

computing, 307309

exponential divergence and, 307

Homogeneous function, Euler's theorem, 83 n

Huygens, Christiaan, 10 n

Hyperion, chaotic tumbling of, 151, 170176

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 tensor, 127

diagonalization of, 132133

kinetic energy in terms of, 131

principal axes of, 133

transformation of, 130132

Initial conditions. See Sensitivity to initial conditions; State

Inner product of tuples, 515

free-body rotation, 149151

Integers in Scheme, 18 n

Integrable constraints, 4 n, 109

Integrable systems, 285, 309316

periodic orbits of near-integrable systems, 316

perturbation of, 316, 322, 457

surfaces of section for, 313316

Integral, definite, 10 n

Integral invariant

generating functions and, 368373

Poincaré, 362364

Poincaré–Cartan, 402, 431434

Integration. See Numerical integration

Invariant curves, 243, 322330

dissolution of, 329330, 486

finding (computing), 326329

finding (strategy), 322325

irrational rotation number and, 322

Kolmogorov–Arnold–Moser theorem, 322

Irrational number, continued-fraction approximation, 325

for Hénon–Heiles problem, 259

for periodically driven pendulum, 244246, 289290, 483486

for standard map, 279

perturbative vs. actual, 483486

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

$\stackrel{˜}{J}$ (shuffle function), 350

J, Jn (symplectic unit), 301, 355

J-func, 351

J-matrix, 353

Jac (Jacobian of map), 270

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, 378381

solution via Hamilton–Jacobi equation, 417421

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

Kinematics of rotation, 122126

Kinetic energy

ellipsoid of, 148

in Lagrangian, 3839

of axisymmetric top, 159

of free rigid body, 148150

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, 2325

at a moment, 97

computing, 3336

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

derivation of, 2530

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 gravitational potential, 32

for harmonic oscillator, 30, 72

for periodically driven pendulum, 74

for spin-orbit coupling, 173

from Newton's equations, 3638, 5458

vs. Newton's equations, 39

numerical integration of, 73

off the beaten path, 97

singularities in, 143

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

coordinate transformations of, 44

cyclic in coordinate, 80

energy and, 12

for axisymmetric top, 159

for central potential (polar), 4243, 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 gravitational potential, 31

for harmonic oscillator, 21

for spin-orbit coupling, 173

for systems with rigid constraints, 49

generating functions and, 421423

Hamiltonian and, 200 (eq. 3.19), 210

kinetic energy as, 14, 122, 141

non-uniqueness of, 6366

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, 233236

Lagrangian state. See State tuple

Lagrangian state derivative, 71

Lagrangian state path Γ[q], 203

lambda, 498

Lambda calculus, 509

Lambda expression, 498499

Lanczos, Cornelius, 335

Least action, principle of. See Principle of stationary action

Legendre polynomials, 167

Legendre-transform, 212

Legendre transformation, 205212

active arguments in, 208

passive arguments in, 208209

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 LH, 447

Lie series, 443451

computing, 448451

for central field, 450

for harmonic oscillator, 448

in perturbation theory, 458460

Lie-transform, 448

Lie transforms, 437443

composition of, 451

computing, 448

exponential identities, 451453

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, 297302

nonlinear stability and, 302

of equilibria, 291295

of fixed points, 295297

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, 268272

from canonical transformation, 428

Lipschitz condition (Rudolf Lipschitz), 69 n

Lisp, 503 n

list, 503

list-ref, 503

Lists in Scheme, 502504

literal-function, 15, 512, 521

Literal symbol in Scheme, 504505

Local names in Scheme, 501502

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

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, 303309

Map

area-preserving, 278

Chirikov–Taylor, 278 n

Poincaré, 242

representation in programs, 247

standard, 277280

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

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, 1496

Newtonian vs. variational formulation, 3, 39

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

Minimization

of action, 1823

in Scmutils, 19 n, 21 n

minimize, 19 n

Mixed-variable generating functions, 374

Moment(s) of inertia, 126130

principal, 132135

of top, 159

conjugate to coordinate (see Conjugate momentum)

conservation of, 7981

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

history of, 9 n

libration of, 174, 175

rotation of, 119, 151, 170176, 496 (ex. 7.5)

Moser, Jürgen. See Kolmogorov–Arnold–Moser theorem

Motion

atomic-scale, 8 n

chaotic (see Chaotic motion)

dense, on torus, 312 n

deterministic, 9

epicyclic, 381389

ergodic, 312 n

periodic (see Periodic motion)

quasiperiodic, 243, 312

realizable vs. conceivable, 2

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, 514516

as composition, 516

as contraction, 514

Multiply periodic functions, Poisson series for, 474

Multipole expansion of potential energy, 165169

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

Newton, Sir Isaac, 3

Newtonian formulation of mechanics, 3, 39

Newton's equations

as Lagrange equations, 3638, 5458

vs. Lagrange equations, 39

Noether, Emmy, 81 n

Noether's integral, 91

Noether's theorem, 9091

angular momentum and, 143

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

Non-axisymmetric top, 263

exponential(s) of noncommuting operators, 451453

of some partial derivatives, 427 n, 520

of some tuple multiplication, 516

Nonholonomic system, 112

Nonsingular structure, 368 n

{ } for Poisson brackets, 218

( ) for up tuples, 512

[ ] for down tuples, 512

[ ] for functional arguments, 10 n

ambiguous, xivxv

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: Dt, 64

traditional, xivxv, 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, 451453

function vs., 448 n, 517

generic, 16 n

Operators

derivative (D) (see Derivative)

Euler–Lagrange ($\mathcal{E}$), 98

Lie derivative (LH), 447

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

partial derivative (∂) (see Partial derivative)

variation (δη), 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

in a central potential, 78 (ex. 1.30)

Lagrange equations for, 3132

retrodiction of, 9 n

Euler's equations and, 153154

nonsingular coordinates for, 181191

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

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, 518519, 520

chain rule, 519, 523 (ex. 9.1)

notation: ∂, xv, 24, 520

Particle, free. See Free particle

Path, 2

finding, 2023

momentum path, 80

osculation of, 94

parametric, 20

realizable (see Realizable path)

variation of, 12, 18, 26

Path functions, abstraction of, 94

Peak, 222

behavior of, 223, 286287

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, 460473

phase plane of, 223, 286

phase-volume conservation for, 268

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

width of oscillation region, 466

drive as modification of gravity, 66

Hamiltonian for, 392

Lagrange equations for, 52

Lagrangian for, 51, 66

Pericenter, 171 n

Period doubling, 245

behavior of, 196, 244246

chaotic behavior of, 76, 243

emergence of divided phase space, 286290

Hamiltonian for, 236, 476

inverted equilibrium, 246, 282 (ex. 3.14), 491494, 496 (ex. 7.4)

islands in sections for, 244246, 289290, 483486

Lagrange equations for, 74

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

as perturbed rotor, 476478

phase-space descriptions for, 239

phase space evolution of, 236

resonances for, 481491

spin-orbit coupling and, 173

surface of section for, 242248, 282 (ex. 3.14), 287290, 483494

undriven pendulum vs., 244

with zero-amplitude drive, 286289

Periodically driven systems, surfaces of section, 241248

Periodic motion, 312

fixed points and, 295

integrable systems and, 309, 316

Periodic points, 295

Poincaré–Birkhoff theorem, 316321

rational rotation number and, 316

resonance islands and, 290

Perturbation of action-angle Hamiltonian, 316, 458

Perturbation theory, 457

for many degrees of freedom, 473478

for pendulum, 466468

for periodically driven pendulum, 491494

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

higher-order, 468473, 489494

Lie series in, 458460

nonlinear resonance, 478494

secular-term elimination, 471473

secular terms in, 470

small denominators in, 475, 476

Phase portrait, 231, 248 (ex. 3.10)

chaotic regions, 241

divided, 244, 258, 286290

extended, 394402

non-uniqueness, 238239

of pendulum, 223, 286

qualitative features, 242246, 258260, 285286

reduced, 402407

regular regions, 241

two-dimensional, 222

volume (see Phase-volume conservation)

Phase space reduction, 224226

conserved quantities and, 224226

Lagrangian, 233236

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, 358359

Phobos, rotation of, 171

Pit, 222

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, 316321

computing fixed points, 321322

recursive nature of, 321

Poincaré–Cartan integral invariant, 402

time evolution and, 431434

Poincaré integral invariant, 362364

generating functions and, 368373

Poincaré map, 242

Poincaré recurrence, 272

Poincaré section. See Surface of section

computing, 339341

general canonical transformations vs., 357

generating functions for, 375376

polar-rectangular conversion, 339, 376377

to rotating coordinates, 348349, 377378

time-independent, 338

Poisson, Siméon Denis, 33 (ex. 1.10)

Poisson brackets, 218222

canonical condition and, 352353

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 $\stackrel{˜}{J}$, 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, 4243

point transformation to rectangular, 339, 376377

transformation to rectangular, 46

Potential. See Central force; Gravitational potential

Potential energy

of axisymmetric top, 160

Hénon–Heiles, 253

in Lagrangian, 3839

multipole expansion of, 165169

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, 132135

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), 813

Hamilton's equations and, 215217

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, 497498

Procedures

arithmetic operations on, 19 n

generic, 16 n

Products of inertia, 128

Q (coordinate selector), 220

$\stackrel{˙}{Q}$ (velocity selector), 64

q (coordinate path), 7

qcrk4 (quality-controlled Runge–Kutta), 145

Quadratic functions, Legendre transformation of, 211

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, 181191

Hamilton's discovery of, 39 n

quaternion units, 188

Quotation in Scheme, 504505

qw-state->L-space, 190

qw-sysder, 189

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, 913

uniqueness, 12

Recurrence theorem of Poincaré, 272

Recursive procedures, 501

Reduced mass, 35 (ex. 1.11), 380

Reduced phase space, 402407

Reduction

Lagrangian, 233236

of phase space (see Phase space reduction)

ref, 15 n, 514

Regular motion, 241, 243, 258

linear separation of trajectories, 263

Renormalization, 267 n

center, 480

islands (see Islands in surfaces of section)

nonlinear, 478494

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

overlap criterion, 488489, 496 (ex. 7.5)

for periodically driven pendulum, 481491

spin-orbit, 177181

width, 483 (ex. 7.2), 488

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

Rigid body, 120

free (see Rigid body (free))

kinetic energy, 120122

kinetic energy in terms of inertia tensor and angular velocity, 126129, 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, 135137

Rigid body (free), 141

angular momentum, 151153

angular momentum and kinetic energy, 146150

computing motion of, 143145

Euler's equations and, 151154

(in)stability, 149151

orientation, 153154

Rigid constraints, 4963

as coordinate transformations, 5963

Rotating coordinates

in extended phase space, 400402

generating function for, 377378

point transformation for, 348349, 377378

active, 130

composition of, 123, 187

computing, 93

group property of, 187

(in)stability of, 149151

kinematics of, 122126

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

Lie generator for, 440

matrices for, 138

of celestial objects, 151, 165, 170171

of Hyperion, 170176

of Mercury, 171, 193 (ex. 2.21)

of Moon, 119, 151, 170176, 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, 460473

periodically driven pendulum as perturbation of, 476478

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

Salam, Abdus, 509

Saturn. See Hyperion

for Gnu/Linux, where to get it, xvi

Schrödinger, Erwin, 12 n, 203 n

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, 471473

Selector function, 64 n, 513

coordinate selector (Q), 220

momentum selector (P), 199, 220

velocity selector ($\stackrel{˙}{Q}$), 64

Selectors in Scheme, 503

Semicolon in tuple, 31 n, 520

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

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 $\stackrel{˜}{J}$, 350

Simplification of expressions, 511

Singularities, 202 n

Euler angles and, 143, 154

in Euler's equations, 154

quaternions, 181191

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, 165181

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, 177181

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, 303309

computing, 307309

standard-map, 278

Standard map, 277280

Stars. See Galaxy

State, 6871

evolution of (see Time evolution of state)

Hamiltonian vs. Lagrangian, 202203

in terms of coordinates and momenta (Hamiltonian), 196

in terms of coordinates and velocities (Lagrangian), 69

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)

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, 239248

in action-angle coordinates, 313

area preservation of, 272, 434435

computing (Hénon–Heiles), 261263

computing (stroboscopic), 246

fixed points (see Fixed points)

for autonomous systems, 248263

for Hénon–Heiles problem, 254263

for integrable system, 313316

for non-axisymmetric top, 263

for periodically driven pendulum, 242248, 282 (ex. 3.14), 287290, 483494

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

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

for standard map, 277280

invariant curves (see Invariant curves)

islands (see Islands in surfaces of section)

stroboscopic, 241248

Symbolic values, 511512

Symbols in Scheme, 504505

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), 359362

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), 353357

antisymmetric bilinear form and, 359362

Symplectic unit J, Jn, 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, 8690, 283 (ex. 3.16), 399402

chaotic motion, 283 (ex. 3.16)

surface of section for, 283 (ex. 3.16)

Tidal friction, 170

time, 15 n

Time-dependent transformations, 347349

Time evolution of state, 6878

action and, 423425, 435437

as canonical transformation, 426437

in phase space, 236238

Poincaré–Cartan integral invariant and, 431434

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, 6365

affecting conjugate momentum, 239

canonical transformation and, 390393

commutativity of, 91 n

computing, 97

constraints and, 108

identifying, 68 (ex. 1.28)

notation: Dt, 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, 347349

Transpose, 351 n

transpose, 126, 351 n

True anomaly, 171 n

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

Tuples, 512516

arithmetic on, 509, 513516

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, 514516

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, 378381

Two-trajectory method, 265

Undriven pendulum. See Pendulum

Uniform circle map, 326

Uniqueness

of Lagrangian—not!, 6366

of phase-space description—not!, 238239

of realizable path, 12

of solution to Lagrange equations, 69

unstable-manifold, 308

Unstable manifold, 303309

computing, 307309

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: δη, 26

Variational equations, 266

Variational formulation of mechanics, 23, 39

Variational principle. See Principle of stationary action

Vector

body components of, 134

in Scheme, 502504

square of, 18 n, 21 n

vector, 504

vector?, 504

vector-ref, 504

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 ($\stackrel{˙}{Q}$), 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, 286289

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