Physics for Mathematicians — Index (A–H)

A

Abel, 288

Abelian integrals, 73

Abel's integral equation, 318

Abraham and Marsden, 404, 590, 596

absolute motion, 12

absolute space, 12, 275, 393

abuse of notation, 88

acceleration corrections, 378

acceleration of gravity, 14

Achilles, 400

action, 449, 464, 529, 634

• quantity of, 497

action-angle terminology, 640

action-angle variables, 630, 632, 637

• frequencies of, 635

action integral, 497, 527

action variables, 631

addition of angular momentum vectors, 344

additivity of mass, 16, 35

adhesion, molecular, 410

adiabatic, 644

adiabatic invariance of, 651

• J

adiabatic invariant, 433, 641, 644, 651

adiabatic limit, 655

Agamemnon, 400

Aharonov-Bohm effect, 654

air trough, 11

Airy equation, 648

Airy functions, 648

Alhazen, 487, 289

allergy, 513

alternating (skew-symmetric) map associated with skew-symmetric matrix

Amontons, 409

Amontons-Coulombfriction, 411

Amontons' first law of friction, 409

Amontons' second law of friction, 409

amplitude, 287

• infinite, 299
• Analysis ad Refractiones, 490
• analytical mechanics, 439
• specialized considerations for, 440
• analytically defined cross-product,82
• Aneas, 400
• angle(s)
• apsidal, 128, 132, 135
• cut, 423
• dynamical, 654
• Euler, 341ff.
• variants of, 342
• geometric, 655
• Hannay, 653, 655, 659, 664
• of friction, 434
• of incidence, 249, 487
• of reflection, 249, 487
• scattering, 114, 147
• laboratory, 115
• angular coordinates, 623
• angular frequency, 18, 42
• angular momentum, 83
• conservation of, 81, 470, 475
• as vector equation, 87
• elementary illustration of, 86
• using to change directions, 87
• law, 84, 186
• elementary example of, 197
• of system, 83
• per unit mass, 121
• rotational, 84, 187
• vectors can be added, 344
• angular velocity, 83
• angular wave number, 313
• Anh, 417
• anomaly
• eccentric, 163, 643
• true, 163, 643
• Anschütz-gyroscope,391
• Anschütz-Kaempfe, 391
• anti-sync, 479
• oscillations in, 303
• aphelion, 128
• apoapsis, 128

apocenter, 128

apogee, 128

Appell, 172, 371

applied mechanics, 110, 206, 417

approximate solutions, 532

approximation, seat-of-the-pants, for

• Tippe Top, 428

approximations to small oscillations

apsidal angle, 128, 132, 135

apsides, 128

• motion of, 129ff.

apsis, 128

Aquarius, 361

aragonite, 509, 526

Archimedean solid, 204

Archimedes, 3

area in polar coordinates, 56, 630, 631

areas, law of, 85

argument of perihelion, 565

Arnold, 73, 151, 154, 283, 384, 468, 561, 573, 590, 592, 603, 621, 640

Arnold and Avez, 640

Arnold tori, 62

Ars Magna, 356

artillery shells, 387

ascending node, 565

asteroid companions of Earth, 401

asteroid, Trojan, 396

Atwood, 242

Atwood machine, 241, 242

• usual analysis of, 245
• with massive wheel, 246

Atzema, 499

Audin, 640

Auerbach, 110

averaged equation, 647

averaging principle, 646, 648

averaging theorem, 648ff.

axis

• parallel axis theorem, 191,196
• perpendicular axis theorem, 202
• principal, of inertia, 189
• rotation about, 191, 193
• rotation about, rigid body, 334
• stable and unstable, 335

α-particles, 148

B

balance, 54

• torsion, 39

Barbour and Pfister, 395

Barger and Olsson, 251, 428

base curve, 683

basis, symplectic, 601

bath-tubs, 392

baton, 81

beats, 299, 309, 390

Belorizky and Sivardière, 386

Beltrami's identity, 271

bending of solid bodies, 206

bending, thin plank, 256

Benedetti, 19

Benzenberg, 384

Bernoulli, 257

• Euler-Bernoulli equation, 256

Bernoulli, Daniel, 279

Bernoulli, Johann, 282

Berra (Yogi), 461

Berry, 654

Bertrand, 151, 172

Bessel, 38, 259

Beutler, 404

bi-axial crystal, 526

bilinear covariant, 590

billiard balls, collision of, 423

billiards

• draw shot, 419
• follow shot, 419
• game of, 417
• role of friction in, 417

billiards.colostate.edu, 420

billiard shot

• high, 418
• low, 419

billiard tables, cushions of, height of

blackboard, chalk on, 412, 414

Boccaletti and Pucacco, 404, 559, 640

bodycone, 341

body coordinates, free symmetric top

• in, 339

bodyderivative, 333

body,rigid, 491

body,rotating axes within, 424

Bohlin, 154

Bohnenberger, 356

Bohr, 427, 558, 640, 641, 646

Bolsinov and Fomenko, 641

Boltzmann, 646

Born, 640, 641, 645

Born and Wolf, 501 502, 563

Boundary Term Corollary, 462, 468, 539, 572

Bou-Rabee, Marsden, and Romero

Bracket Formula, Cartan's, 576, 595

bracket, Poisson, 605

• invariant definition of, 606
• well-defined, 605

braking, self-, 414

bridge, suspension, 269

Brogliato, 417

Bruce, 288

Bruns, 555

Buchwald, 492, 494, 498

bucket, rotating, 393

buckling, column, 257

bulging of earth near equator, 20{21, effect on plumb bob, 381

Burgers, 646

C

Cabannes, 374, 452

cable, hanging, 268{271

calculus, fractional, 319

Calkin, 642, 656

Calkin and March, 106

caloric, 498

camera obscura, 487

camera, pin-hole, 487

cannonball, 276

• deflection of, 386
• shot upwards, 386

canonical, as symplectic, 603

canonical coordinates, 571

canonical equations, Hamilton's, 540, 546, 547

canonical form, in linearization, 324

canonically conjugate, 571

canonical transformation, 567, 569, generalized, 571, 593

• homogeneous, 596
• 1-parameter family of, 575, 576
• time-dependent, 581ff.
• real usefulness of, 586

capstan, 435

Carathéodory, 603

Cardan, Jerome (Cardano, Girolamo)

Cardan suspension, 356

Carlyle, 466

Carnot energy loss, 98

• temperature change of, 98

Carnot, Lazarus, 98

Carnot, Sadi, 98

Cartan formula, 576

Cartan's Bracket Formula, 576, 595

Cartan, see Poincaré{Cartan

Cartan's Magic Formula, 576, 595

Casey, 257

Cauchy problem, 672

Cavendish, 39

Cayley, 102,109

Ceanga and Hurmuzlu, 110

center, fall into, 160

center of gravity, 5, 81, 204

center of mass, 80, 190

• coordinates, 94

central force, 85, 552, 553

• not radially symmetric, 116
• radially symmetric, 74, 88
• towards origin, 71
• Newton's analysis of, 55ff.

centrifugal force, 38, 377, 378, 396, and rocket launches, 117

centripetal force, 365, 378

Chadwick, 113

chain, 100ff.

• folded, falling, 103
• link of, 102{105
• yanked, 102, 105

chain and sprocket wheel, 246

chalk on blackboard, 412, 414

chandelier, 307

Chandler, 340

Chandler wobble, 340

Chandrasekhar, 7, 21, 73, 128, 138, 142, 143, 466

change of momentum, 12

Chapman, 110

characteristic curve, 590, 669, 674, 681, 689

characteristic for, 677, 699

characteristic function

• for mechanics, 527
• Hamilton's, 554
• Hamilton-Jacobi equation for, 554
• in optics, 524, 527

characteristic strip, 683, 690

characteristic subspace, 590

characteristic surface, 562

characteristic vector field, 669, 681, 689, 674

charged particle, moving, 52

Chicone and Jacobs, 133

China, 356

circle, osculating, 62

circuit, electrical, 293

• damping for, 295

circular frequencies of invariant torus

circular frequency, 287

• of driving force, 298

circular orbits, 67, 121

• not stable for inverse cube force, 135

circulation, 573

circumference of earth, 67{68

Clairaut equation, 518,543

cn, elliptic function, 372

coefficient of friction, 225

coefficient of kinetic friction, 411

coefficient of restitution, 94

Cohen, 430

Cohen and Whitman, 7, 55, 67, 68, 73, 142,143, 283, 361

college ring, 426, 430

collinear forces, 53

collisions

• of rigid bodies, 206
• conservation of energy in, 94, 206
• \head-on", 97
• more general, 97

color, 495

column buckling, 257

complete integral, 540, 541

completely elastic, 94, 96

completely inelastic, 94, 96, 98, 102

composition of forces, 28, 30

compoundpendulum, 211

condensed notation, 443, 529, 531

conditional critical points, Lagrange

• multipliers for, 471
• Euler's Rule for, 474

conditionally periodic, 620

cone

• body, 341
• Monge, 679
• polhode, 366
• space, 341

configuration space, 179

• for pendulum, 209
• coordinate system on, 209
• for rolling disc, 231

congruence of matrices, 568

conical refraction, 526

• external, 526
• internal, 526

conic section orbit, centered at focus

conjugate, 525

• canonically, 571
• hyperbola, 159
• momentum, 445, 634

connection, 463

conservation laws, 79

conservation of angular momentum, 81, 470, 475

• as vector equation, 87
• elementary illustration of, 86
• using to change direction, 87

conservation of energy, 87, 89, 102, in collisions, 94

• in general, 97
• in Principia, 141

conservation of energy equation, 92

conservation of kinetic energy and of

• momentum, 113, 206

conservation of momentum, 22, 32, 80, first statement of, 273

• Newton's statement of, 273

conservation of momentum and conservation of kinetic energy, 113,206

conservative force, 47, 90, 92

• Lagrange's equations for, 475
• rigid bodyin field of, 194

constant energy paths, 464

• Euler for, 464

constraint, 205ff.

• d'Alembert's principle for, 210
• differential, 230ff.
• d'Alembert's principle for, 233
• forces, finding with Lagrange multipliers, 238, 239
• holonomic, 230ff.
• ideal, 210
• meaning of, 210
• problems, Lagrange's equations with
• time-dependent, 227, 449

contact, 110

contact curves, 543

contact point, velocity in rolling, 221

continuousbody, 195

continuousforce, 13

continuousstring, 314

continuum mechanics, 468

contour integration, 636, 663

contraction, 512

convolution, 319

coordinates

• angular, 623
• canonical, 571
• center of mass, 94
• cyclic, 445, 468
• elliptic, 559
• for three dimensions, 561
• ignorable, 445
• laboratory, 115
• multiple-valued, 623
• spherical, 290

coordinate system

• non-inertial, 332
• rotating, 332
• symplectic, 604

Cordani, 592

Coriolis, 420

Coriolis force, 377, 381, 396, 420

• amusing movie about, 381
• on rotating platform, 381

corrections, acceleration, 378

cotangent bundle, 511

• invariant definition of 1-form ω on
• special features of, 511
• 2-form on, 512
• !

Cotes' spiral, 127

Coulomb, 410

• friction, 411

Coulomb-Morin friction, 411

couple, 201

coupledoscillators, 302, 479

coupling, 94, 98

• weak, 304

Courant and Hilbert, 562

covariant, bilinear, 590

covariant derivative, 482

coy physicists, 117

critically damped, 297

critical point

• in calculusof variations, 461
• conditional, Lagrange multipliers for
• Euler's Rule for, 474

cross-product, 55, 81, 186

• analytically defined, 82
• geometrically defined, 82

crystal, 494

• bi-axial, 526

crystal ball, viii

curvature, 62

• formula for, 374

curve traced by axis of top in cuspidal

• case, perpendicularity of, 349

cushions of billiard tables, height of

cuspidal case, for top, 349ff., 356

cuspidal polar top, 355

cuspin envelope, 321

cut angle, 423

cyclic coordinates, 445, 468

cycloid, 221, 236, 288, 327

• geometric proofs for, 289
• involute of, 327

cycloidal pendulum, 288

D

d'Alembert's formula, 502, 505

d'Alembert's principle, 184ff.

• for constraints, 210
• for differential constraints, 233
• with Lagrange's equations, 441

damped, critically, 297

• second solution for, 297

damped forced oscillations, 300

• underdamped case, 301

damped oscillations, 295, 390

damping force, 295

Darboux, 172, 604

• equation, 504

da Vinci, 409

de Broglie, 555, 557

decay, exponential, 296

decompose forces, 30

decrease in weight approaching equator, 380

defined operationally, 8

deflection

• of cannonball, 386
• of falling body, 382
• naive calculation for, 383
• of hanging body,379
• southward, 380
• southward, 386, 405

degeneracy, 636, 637

Delaunay, 640

• variables, 640

densities, spherically symmetrical, 137

density, of continuousbody, 195

dependence, domain of, 502, 506, 507

derivative

• body, 333
• covariant, 482
• fibre, 517
• fibre-wise, 517
• rotating observer's, 333

Desaguliers, 410, 411

Descartes, 273, 275, 488

Descartes' Law, 488

descending node, 565

descent, method of, 506

descent, time of, 289, 385

developable surfaces, 519

deviousness of Leibnizian notation, 122

diagonalization of matrices, simultaneous, 478

diffeomorphism, 608

differential constraints, 230ff.

• d'Alembert's principle for, 233

differential equations, stability of solutions for, 324

diffraction, 500, 563

Dijksterhuis, 6

direction of down, 381

direction of force, 9

discrete subgroup of see subgroup

• Rn
• of discrete

disc, rolling, 230, 362

• along circle, 365, 454
• along straight line, 364, 454
• stability of, 364
• compared to rolling sphere, 230, 240
• down inclined plane, 235
• Lagrange's equations for, 452
• non-upright, 231
• Rn

disembodied ghost, 637, 639

displacements, virtual infinitesimal, 178ff.

distinction between mass and weight

distribution, not integrable, 232

div, 503

divergence, 138

divergence theorem, 138, 503

dn, elliptic function, 372

domain of dependence, 502, 506, 507

doubleimage, 494

doublependulum, 229, 307, 480

• tension on string for, 480

downward direction, 381

draw shot, billiards, 419

driving force, 298

• circular frequency of, 298
• oscillating, 298

duality and power force laws, 154

Dublin, University of, 522

Duck and Sudarshan, 640

Dugas, 5, 466, 490, 499

dynamical angle, 654

“dynamics reduces to statics”, 184

E

earth

• bulge near equator, 20{21, 405
• circumference of, 67{68
• radius of, 16
• rotating, 376, 377

Earth, asteroid companions of, 401

earth-moon system, 25

Earth Trojans, 401

eccentric anomaly, 163, 643

eccentricity, 166

eclipses of Jupiter's satellites, 491

ecliptic, plane of, 565

effective mass, 248

effective potential energy, 144

eggs, hard boiled, 426, 432

Ehrenfest, 646

eikonal, 555

Einstein, 17, 39, 108, 394, 640, 646

• letter to Mach, 394

Einstein's relativity principle, 275

elastic

• body
• completely, 94, 96
• perfectly, 206, 249
• limit, 292

elasticity, modulusof, 253

electrical circuit, 293

• damping for, 295

electrical examples, for harmonic motion, 294

electric charge, 24

• opposite, 24

electric field, 138, 199

electricity, 52

electromagnetic waves, 501

electromagnetism, 200

ellipse, compared to rotating elliptical path, focal point property of, 72

ellipsoid, inertia, 189

• rolling, 337

ellipsoid of revolution, 494

elliptic

• coordinates, 559
• for three dimensions, 561
• function, 75, 372, 347
• functions cn, dn, sn, 372
• integral, 51, 73, 101

elliptical

• harmonic motion, 71
• orbit, 16
• of planets, 37
• centered at focus, 124
• orbit, centered at origin, 123
• orbit, revolving, 131,166
• under central force towards origin

energy, 529

• conservation of, 87, 89, 92, 102, 420
• in collisions, 94
• in general, 97
• in Principia, 141
• in Lagrangian, 448
• kinetic, 87, 88
• factor of 1=2 in, 420
• loss of, 97
• of rigid body,193
• rotational part, 194, 204
• for continuous body,196, 204
• translational part, 194
• loss, Carnot, 98, 102
• of curve, 463
• paths of constant, 464
• Euler for, 464
• per unit mass, 121
• potential, 87, 89, 90
• effective, 144
• in Principia, 142
• only determined up to a constant
• transfer, 305, 309
• not complete, 306

engineering

• mechanical, 110, 206, 417
• problems, 257

english, side, 421

envelope, 288, 321, 491

• cusp in, 321
• of family of surfaces, 323
• of normals to an ellipse, 321
• of secondary waves, 491
• of solutions, 542

equal forces, 14

“equal” solutions for Lagrangians, 459

equator

• bulging of earth near, 20{21, 405
• effect on plumb bob, 381
• decrease in weight approaching, 380

equiangular spiral, 126

equilibrium, 213ff.

• and potential function, 214
• of rigid body,175
• rigid, 176
• stable, 215, 476
• unstable, 477

equilibrium of planes, 5

equilibrium points for Lagrange equations, 476

equilibrium point, small oscillations

• about, 478

equinox(es)

• precession of, 340, 361
• by d'Alembert, 361
• exercise in Goldstein [1], 361
• Newton's calculations, 361
• spring, 361
• vernal, 565

equipotential surfaces, 169

equivalent forces, 201

escape velocity, 117

ether, particles of, 491, 563

Eötvös, 38, 379

Euler, 257, 332, 381, 398, 410, 466, 497, 559

• angles, 341ff.
• variants of, 341
• equation, 461, 462
• Boundary Term Corollary, 462
• equations, 318, 333, 334
• dynamic, 333, 334
• geometric, 344
• for rotating principal vectors, 362
• force, 377, 381
• precession, 339, 381

Euler-Bernoulli equation, 256

Euler-Lagrange equations, 463

Euler-Poisson-Darboux equation, 504

Euler's dynamic equations, 344

Euler's geometric equations, 344

Euler's Rule, 474

Euler's theorem for homogeneous quadratic functions, 448

evolute, 288

evolution, 648

exhaustion, method of, 4

exponential decay, 296

extended Hamilton's principle, 534, 583, 592

• paradoxical fact about, 535

external conical refraction, 526

external force, 29, 79

extremal, in calculusof variations, 461

eye-glasses (spectacles), 488

F

falling body, deflection of, 382

• naive calculation for, 383

falling chain, folded, 103

• falling faster than freely falling
• chain, 105
• infinite speed, 106

fall into center, 160

family of surfaces, envelope of, 323

Fasano and Marmi, 561, 640

fast precession, 353

fast top, 349ff.

Fenchel, 514

Fermat, 489

Fermat's principle, 489, 522, 524, 527

Feynman, 26, 97ff., 216,394, 411, 490

fibre derivative, 517

fibre-wise derivative, 517

fibre-wise Legendre transform, 517

fictitious forces, 376, 377

fifth power, inverse, 70

Fignewton, Dr., 43

filament, 241

• wrapped around object, 245, 267

Fine Hall, 108

finite propagation speed, 502

Finsler metric, 495

first integral, 93

first law, Newton's, 10

first order PDE, 540, 542, 668, 673, linear, 668

• most general, 677
• quasi-linear, 673

fixed masses, two, 559

fixed stars, 11

Flaöchensatz, 85

flflip over, gyroscope, 357, 358

flow of Hamiltonian vector field, 575, 576

flux, 139

focal point property of ellipse, 72

focusedimage of lens, 488

fogeys, old, 412

folded chain, falling, 103

foliation, 323

follow shot, billiard, 419

football, spinning, 432

forced oscillation, 298

• damped, 300
• underdamped case, 301
• resolution of, 30

force, moment of, 85

• in friction, 409
• translational, 377

force(s), 8, 9

• 1st power, 123
• nth
• power, 122ff., 135
• azimuthal, 377, 381
• central, towards origin, 71
• centrifugal, 38, 377, 378, 396
• centripetal, 365, 378
• collinear, 53
• composition of, 28, 30
• conservative, 90, 92, 475
• Lagrange's equations for, 475
• rigid bodyin, 194
• constraint
• finding with Lagrange multipliers
• finding with Lagrange multipliers
• in holonomic case, 239
• Coriolis, 377, 381, 396, 420
• amusing move about, 381
• on rotating platform, 381
• damping, 295
• proportional to velocity, 295
• decompose, 30
• direction of, 9
• driving, 298
• oscillating, 298
• equal, 14
• equivalent, 201
• Euler, 377, 381
• external, 29, 79
• ficticious, 376, 377
• frictional, 409
• generalized, 443
• gravitation, 14
• independent of velocity, 17
• normal component of, 225
• impulsive, 12, 57, 278, 279
• intermolecular, 216
• internal, 29, 176
• for pendulum (tension), 210
• for rigid body, work done by, 195
• non-unique for rigid body, 178
• work done by, for rigid body,195
• inverse cube, 126
• circular orbits not stable for, 126
• inverse square, 58, 60, 123
• exactness of, 135

fork, tuning, 295

• frequency of, 295
• standard, 295

Foucault, 356

Foucault's gyroscope, 381

Foucault's pendulum, 387, 661, 664

• intuitive, geometric, explanation for
• path of, 389
• cuspsin, 390
• loops in, 390
• popular explanation for, 387
• small oscillations of, 389

Fourier series, 310,315

fractional calculus, 319

free curve, 672

free for initial condition, 690

free for initial data, 688, 691

freely falling chain, falling faster than, infinite speed, 106

free symmetric top, 339

• in bodycoordinates, 339
• in inertial coordinates, 340

French, 405

frequencies of action-angle variables

frequencies of invariant torus, 621

frequency, 287

• angular, 18, 42
• circular, 287
• of driving force, 298

Fresnel, 501,524, 526, 563

friction, 216,352, 409

• Amontons-Coulomb, 411
• and rolling, 221
• angle of, 434
• coefficient of, 225
• Coulomb, 411
• Coulomb-Morin, 411
• in opposite direction of motion, 412
• increased, 411
• kinetic, 410
• coefficient of, 411
• laws of, 409
• normal component of force in, 409
• proportional to normal component
• of gravitational force, 225
• role in billiards, 417
• rolling, 411
• sliding, 410
• sliding without, 216
• static 410, 411
• torque coming from, 427
• viscous, 411

frictional force, 409

Frobenius integrability, 232

fulcrum, 183

functions, homogeneous quadratic, 447

• homogeneous quadratic, Euler's theorem on, 448

fundamental Poisson brackets, 607

G

Galilean invariance, 273

Galilean relativity principle, 275

Galileo, 10, 13, 14, 19, 87, 220, 224, 275, 276, 279, 382, 383, 488

Gamma function, 319

Garwin, 251

gauge transformation, 459

Gauss, 384

Gauss' law, 139

Gauss' lemma, 525

Gauss' theorem, 138

generalized canonical transformation, 571, 593

generalized forces, 443

general relativity, 39, 394

general relativity, Schwarzschild solution in, 640

generating function(s), 577, 583

• disembodied ghost of, 637, 638
• for Hamilton's equations, 585
• in time-independent case, 586ff.
• type 1, 578, 583
• type 2, 580, 584, 617, 633, 652, 653
• type 3 and type 4, 588, 589, 597
• mixture of types, 589
• other types, 588

geodesics, 39, 463, 524

• on Riemannian manifolds, 461
• standard from for equations, 482

geometric

• angle, 655
• definition of Legendre transform
• description of rigid bodymotion
• 336ff.
• proof of Kepler's second law, Newton's, 56ff.
• proof, Newton's concerning gravity
• 137, 138
• proofs for cycloid, 289

geometrically defined cross-product, 82

Geometrical optics, 500, 524, 557

Gergonne, 499

ghost, disembodied, 637, 639

Giaquinta and Hildebrant, 594

Gilbert and Sullivan, 420

gimbal, 355, 381

• direction of movement, 357

gliders, 217

Génot and Brogliato, 416

God (supreme being)

• existence of, 464, 498

Goldstein, 361, 567

grad, 90, 482

Gramm-Schmidt orthogonalization

gravitation force, normal component

• of, 225

gravity, 9, acceleration of, 14

• center of, 5, 81, 204
• force of, 14
• independent of velocity, 17
• law of, 37
• modern-day problem involving, 559
• third law for, 39
• uniform, 13, 14, 204

Gray and Nickel, 431

Green's theorem, 138

Grimaldi, 500, 501

Gronwall's inequality, 165

Guicciardini, 282

gyration, radius of, 213

gyrocompass, 358, 359

• period of, 360

gyroscope, 340, 355, 391

gyroscope, Anschütz-, 391

gyroscope, flflip over, 357, 358

gyroscope, Foucault's, 381

• gimbal, direction of movement, 357
• nutation of, 357
• opposite direction for spin, 357, 358
• precession of, 356

H

Hadamard, 506

Hair, 361

Hale, 659

Hall, 405

Halley, 20, 67

• th

Halphen (5 order equation of), 172

Hamel, 417

Hamilton, 279, 522, 528, 540

Hamiltonian, 533

• examples of, 539
• flows, 572
• form of Noether's theorem, 606
• local, 577
• mechanics, 487, 509, 532
• relation to quantum mechanics
• time-dependent, 590
• time-independent, 590
• vector field, 569, 608
• flflow of, 575, 576

Hamilton-Jacobi equation, 540, 546

• for Hamilton's characteristic function, 554
• in usual notation for first order
• PDE's, 546
• reduced, 554

Hamilton-Jacobi theory, 540

Hamilton's

• canonical equations, 540, 546, 547
• characteristic function, 554
• characteristic function, HamiltonJacobi equation for, 554
• equations, using generating functions
• to simplify, 585
• principle, 463, 498, 527
• use of \principle of least action"
• for, 466
• extended, 534, 583, 592
• paradoxical seeming fact about

Hand and Finch, 21, 361

hanging body

• deflection of, 379
• southward, 380
• not possible to detect, 381

hanging cable, 268{271

Hannay, 654, 655

• angle, 653, 655, 659, 664
• hoop, 656

hard boiled eggs, 426, 432

harmonic motion, 287, 289, 293

• electrical examples for, 294
• elliptical, 71

harmonic oscillator, 550, 623, 634, 637

harmonics, 316

Harriot, 488

heavy top, see top, heavy

height of cushions of billiard tables

Hektor, 400

helium nuclei, 148

Herivel, 67

Hero of Alexandria, 487

herpolhode, 338

• convexity of, 338, 366ff.
• repeating sections of, 338

Herrmann and Schmaölze, 110

Herrmann and Seitz, 110

hidden planet, 404

Higgs potential, 166

high shot, billiard, 418

hinges, 242

• and constraints, 227

historical questions, 273

Hogwarts, 576

holonomic constraints, 230ff.

homogeneity of space, 469

homogeneous, 116

homogeneous canonical transformation, 596

homogeneous media, 494

homogeneous quadratic functions, 447

• Euler's theorem on, 448

homo sapiens, 220

Hooke, 67, 382, 383

Hooke's law, 47, 252, 292

• for plank or rod, 253

“hook” notation, 512

Hugenholtz, 430

hurricanes, 392

Huygens, 22, 26, 112, 273, 275, 288, 490, 494

Huygens'

• construction, 492, 495, 501, 502
• 507, 524, 562
• principle, 502, 507, 562
• secondary waves, 563

hydrodynamics, 573

hyperbola

• conjugate, 159

hyperbolic equation, 316

• alternate normal form for, 316
• normal form for, 316

hyperbolic, orbit, 126

• centered at focus, 124

hyperbolic spiral, 126

hypergeometric equation, 455