Physics for Mathematicians — Index (I–P)

I

Iceland spar, 494, 500, 526

ideal constraint, 210

ignorable coordinates, 445

image, double, 494

impact line, 423

impact parameter, 147

• smallest value of, 150

impulsive force, 12, 13, 57, 278, 279

incidence, angle of, 249, 487

incident ray, 522

inclination, 565

inclined plane, sliding down, 215

increased friction, 411

independence of integrals, 615

indeterminate, statically, 206, 252

index, refractive, 497

indicatrices, discontinuously changing

indicatrix, 495

inelastic, completely, 94, 96, 98, 102

inequality, Gronwall's, 165

inertia, 9, 11

• law of, 11
• moment of, 190, 191
• for continuous body,196, 203
• principal axis of, 189
• principal moment of, 189
• product of, 190

inertia ellipsoid, 189

• rolling, 337

inertial coordinates, free symmetric top

• in, 340

inertial system, 11

• existence of, 275

inertia tensor, 186ff., 189

• calculating, 189
• matrix of, 189
• of continuousbody, 195

infinite amplitude, 299

infinitesimal displacements, virtual, 178ff.

infinitesimal, virtual work, 181

influence, range of, 502, 506, 507

inhomogeneous equation, 298

inner product, 599

• matrix of 599
• rank of 599

Inquisition, 276

instrument, optical, 524

integrability, Liouville 614, 641

integrability theorem, Liouville, see Liouville integrability theorem

integral equation, 288, 318

integral form of remainder for Taylor's

• formula, 659

integral for solutions of Lagrange's

• equations, 468

integral invariant of Poincaré{Cartan

integral invariants, 572

• of Poincaré, 572
• relative, of Poincaré, 573

integral of motion, 605

integral(s)

• complete, 540, 541
• contour, 663
• first, 93
• Jellett's, 425
• independence of, 615

integration, contour, 636

interference experiment, Young's, 500

interior product, 512

intermolecular forces, 216

internal conical refraction, 526

internal forces, 29, 176

• for pendulum (tension), 210
• for rigid body,work done by, 195
• non-unique for rigid body,178, 184
• work done by, for rigid body,195

invariable plane, 337

invariance, Galilean, 273

invariance of Lagrange's equations, 463, 532

invariance of nature under orthogonal

• maps, 280

invariant definition of 1-form ω on

• cotangent bundle, 511

invariant definitions, 532, 533

invariant(s)

• adiabatic, 433
• integral, of Poincaré{Cartan, 574
• Jellett, 424
• adiabatic, 641, 644
• integral, 572
• of Poincaré, 572
• relative, of Poincaré, 573
• universal, 593, 598

invariant torus, 621, 623

• theorem, 621
• circular frequencies of, 621
• frequencies of, 621
• periods of, 621

inverse cube force, 126

• circular orbits not stable for, 126
• Newton's investigation of, 127{128

inverse fifth power, 70

inverse square force, 16, 58, 60, 67, 75, 123, 552, 553, 635, 637

• exactness of, 135
• for spherical bodies, 137

involute(s), 288, 327

• infinitely many, 327
• parallel curve, 327

involution, functions in, 614

Irish Academy, Transactions of the Royal

Islamic scholars, 487

isoperimetric problem, 657

isotropic \coordinate subspaces", 601

isotropic “coordinate subspaces”, 601

isotropic media, 494

isotropic subspace, 601

isotropy lemma, 615

• adiabatic invariance of, 651

J

J

Jacobi, 532, 540

• metric, 467
• 607, 610,613

Jacobi's

• Theorem, 547, 549, 585
• and mechanics, 549
• version of principle of least action

Jellett, 412

Jellett's

• constant, 424
• integral, 425
• rediscovery of, 430
• invariant, 424

Jost, 640, 641

Joule, 98

Journal of Irreproducible Results, 43

Jupiter, satellites of, 491

Just So Stories, 527

K

Karcher, 170

Kater pendulum, 258

kenyon.edu, 242

Kepler, 488

• problem, 162ff.

Kepler's

• equation, 163{164
• second law, 55
• Newton's geometric proof, 56ff.
• third law, 37, 67, 125, 162

Khein and Nelson, 661,663, 664

kinetic energy, 87, 88

• factor of in, 420
• 1=2
• of rigid body,193
• loss of, 97
• rotational part, 194, 204
• for continuous body,196, 204
• translational part, 194

kinetic friction, 410

• coefficient of, 411

Kipling, 527

Kirchhoff, 501, 563

Kirchhoff's formula, 506

Kittel, 388

Klein, 646

Klein, Painlevé-Klein problem, 412

Kleppner and Kolenkow, 298

König, 466

Kozlov, 573

Krim, 411

L

laboratory coordinates, 115

laboratory scattering angle, 115

lab top, 345

Lacroix, 499

Lagrange, 398, 466, 490, 499

Lagrange multipliers, 233, 451

• for conditional critical points, 471
• Euler's Rule for, 474
• finding constraint forces with, 238
• in holonomic case, 239
• for Euler equation, 271

Lagrange points, 398

• periodic orbits near, 400
• stability of, 399, 401

Lagrange's equations, 441, 443, 528

• constraint problems with, 445
• d'Alembert's Principle with, 441
• equal solutions for, 459
• equilibrium points for, 476
• for rolling disc, 452
• invariance of, 532
• same solutions for, 459
• solutions for, integral for, 468
• third law built into, 470

Lagrangian, 346, 443

• arbitrary, 445
• for tension on pendulum string, 457
• preserving, 467
• regular, 482, 517, 533
• time-dependent, 449
• conservation of energy for, 450

Lagrangian mechanics, 220, 226, 230, 437, 439

Laidler, 646

lamina, 202

Landau and Lifschitz, 319, 372, 539

Laplace, 167, 279, 384, 466, 499

Laplace transform, 319

latitude, 170

latus rectum, 61

law of areas, 85

law of gravity, 37

law of inertia, 11

law of the lever, 3, 30

laws of friction, 409

least action, principle of, 464, 490, 498, 522, 527, 592

• additional hypothesis for, 464
• Euler's role in, 466
• for Hamilton's principle 466
• Jacobi's version, 466

Lee, 593

Legendre transform, 513, 518, 597

• classical notation for, 515, 519
• fibre-wise, 517, 528
• generalized, 515, 516
• geometric definition of, 513, 535
• original form, 514
• involutive, 514

Lehrer, 599

Leibniz, 275, 466

Leibnizian notation, 75, 88, 121

• deviousness of, 122
• old-fashioned trick for, 75, 88, 100

length, wave, 313

lenses, 500

• image focusedfor, 488

Leutwyler, 430

level sets of V, 526

lever, 3, 30, 183

libration, 626

libration points, 398

• periodic orbits near, 400
• stability of, 399, 401

light, 487

• analogous to sound, 490
• mechanical analogies for, 487, 489
• resolving motion of, 487, 489
• speed of, comparison of in air and
• water, 391, 489, 49
• determined by Fiseau and
• Breguet, 489, 496
• first determined by Foucault, 489
• wave theory of, 500, 524

light ray, 492

limit, elastic, 292

linear first order PDE, 668

linearization, 324

• canonical form for, 324

linearize Lagrange equations near an

• equilibrium point, 477

linear triatomic molecule, 480

line of nodes, 342, 565

link of chain, 102{105

• yanked, 102, 105

Liouville, 536, 576, 614, 641

Liouville-Arnold, 641

Liouville-Arnold tori, 623

Liouville integrability, 614,641

Liouville integrability theorem, 616, 620, 628, 640

• classical version, 616
• time-independent, 619, 620

Liouville's volume theorem, 536, 538, in thermodynamics, 538

Liouville tori, 623

liquid, 378

• rotating, 379
• as speedometer, 408

Lissajous, 295

Lissajous figure, 294, 620

Lloyd, 526

load, normal, 412

Lobachevsky song, 599

logarithmic spiral, 126

longest axis, stable rotation about, 336

longitude of ascending node, 565

longitudinal waves, 501

Lorentz force law, 52, 199

low shot, billiard, 419

M

Mach, 5, 282, 393

• Einstein letter to, 394

Mach's principle, 393, 394

MacTutor History of Mathematics

Magic Formula, Cartan's, 576, 595

magnet, 9, 23

magnetic field, 52, 199

magnetism, 52

• modern conception of, 24

Malus, 498

Malus' theorem, 499, 523, 524

Marsden and Hughes, 257

Marsden and Ratiu, 576, 596, 658

Mars Trojans, 401

mass, 8, 16, 19, 37

• additivity of, 16, 35
• center of, 80, 190
• coordinates, 94
• distinction from weight, 20
• effective, 248
• having the same, 28
• measuring, 18
• of planets, 162
• operational definition of, 15
• point, 3, 10, 175
• reduced, 116
• rest, 43
• test, 52
• variable, 35, 100

matter, quantity of, 8, 16

Maupertuis, 464, 496

• biography of, 466

Maupertuis' principle of least action, 466, 522, 527, 592

maximis and minimis, 489

Maxwell, 167,501

McOwen, 503

measuring mass, 18, 20

Mωecanique Analytique, 439

mechanical analogies for light, 487, 489, 496

mechanical engineering, 110,206

mechanics

• analytical, 439
• specialized considerations for, 440
• applied, 110, 206, 417
• characteristic function for, 527
• continuum, 468
• derived from symmetry, 470
• engineering, 110, 417
• Hamiltonian, 487, 509, 532
• relation to quantum mechanics
• Lagrangian, 220, 226, 230, 437, 439
• Newtonian, 7
• quantum, 487, 653
• relation to Hamiltonian mechanics, 487
• wave, 641

mechanics and optics, 522, 555

Mehra and Rechnenberg, 646

metaphysical, 464

method of descent, 506

method of exhaustion, 4

Meyer, 410

Miller and Ross, 319

Mineur, 640, 641

mirages, 494

mirrors, 500

Misner, Thorne, and Wheeler, 40

misspent youth, 420

mode(s), normal, 303, 305, 311ff., 315, normal, by cleverness, 311ff., 314

• in sync or anti-sync, 479

modulusof elasticity, 253

Moffatt and Shimomura, 433

molecular adhesion, 410

molecule, linear triatomic, 480

moment(s), 3

• of force, 85
• of inertia, 190, 191
• for continuous body,196, 203
• of momentum, 86, 200
• principal, of inertia, 189

momentum, 9

• angular, 83
• conservation of, 470, 475
• of system, 83
• rotational, 84, 187
• change of, 12
• conjugate, 445, 634
• conservation of, 32, 80, 469
• as vector equation, 87
• first statement of, 273
• Newton's statement of, 273
• conserved, 22
• law, 80, 186
• moment of, 86
• total, 22, 29

Monge, 499

Monge cone, 679

monkey, 111

moon, 16, 25, 67, 169

• always facing earth, 170
• orbit of, 131
• tidal forces on, 170

Morin, 410

Moser, 60

motion

• absolute, 12
• friction in opposite direction of, 412
• integral of, 605
• quantity of, 9
• relative, 12
• resolving, for light, 487, 489
• rigid body, 332
• possible, 185
• geometric description of, 336ff.

motion of light, resolving, 487, 489

movement of gyroscope gimbal, 357

moving charged particle, 52

Muggles, 576

multiple-valued coordinates, 623

multiple-valued function

multiplier, Lagrange, 233, 451

• for Euler equation, 271

N

Nahin, 271

Napoleon, 494

Nature, 463, 489, 496, 497

Navarro and Perez, 646

negatively charged body, 559

Neher and Leighton, 11

Neptune Trojans, 401

Nestor, 400

neutral plane, 254

neutron, 24, 113

Newton, 7, 382, 489, 495, 50

• and \absolute space", 393

Newtonian Mechanics, 7

Newton's

• analysis of central forces, 55ff.
• cradle, 108, (also 491)
• first law, 10
• geometric proof concerning gravity
• 137, 138
• geometric proof of Kepler's second
• law, 56ff.
• Laws, 3, 7
• proof of parallelogram law, 278
• second law, 12
• statement of parallelogram law, 278
• theorem of revolving orbits, 130
• third law, 21
• for gravity, 39
• Newton's proof of, 276, 277
• strong form, 79, 84, 179, 199
• Three Laws, 10

node, ascending, 565

• longitude of, 565

node, descending, 565

nodes, line of, 342, 565

Noether's theorem, 467, 610

• Hamiltonian form of, 606

nondegenerate, 512,599

non-homogeneous media, 494

non-inertial coordinate system, 332

non-integrable distribution, 232

non-intuitive nature of third law, 23

non-isotropic media, 494

non-planar rigid body, 177ff.

nonsharp signals, 507, (also 502)

non-upright rolling disc, 231

normal component of force, in friction

normal load, 412

normal mode(s), 303, 305, 311ff., 315, 478, by cleverness, 311ff., 314

• in sync or anti-sync, 479

normal slowness, vector of, 526

normal, unit, 139

• inward or outward pointing, 139

notation, condensed, 529, 531

nucleus, helium, 148

nucleus, radius of, 150

number, angular wave, 313

number, wave, 313

nutation of gyroscope, 357

nutation of top, 349

• rate of compared to rate of spinning
• size of compared to rate of spinning
• precession without, 352

O

objects falling, 276

ODE's, second order, 532

Odysseus, 400

Olbers, 385

Olbers' paradox, 385

old fogeys, 412

one-bodyproblem, 120

one-dimensional problem, reduction to, 128, 144

Onnes, 391, 392, 664

operational definition, 8

• of mass, 15

opposite electric charges, 24

optical instrument, 524

optical length, 523

optics

• and mechanics, 522, 555
• characteristic function in, 524, 527
• geometrical, 500, 524, 557
• in antiquity, 487
• wave theory of, 557

Optics of Euclid, 487

Optiks of Newton, 495

• Queries at end of, 496

orbit

• of moon, 131
• centered at focus, conic section, 124
• circular, 67
• conic section, centered at focus, 124
• elliptical, 16
• centered at focus, 124
• centered at origin, 123
• general nature of, 128
• hyperbolic, 126
• centered at focus, 124
• parabolic, 73
• centered at focus, 124
• period of, 125
• depending on energy, 125
• depending on length of semimajor
• axis, 125
• repeating sections of, 129
• revolving, 129
• shape of, 75, 121
• stationary, 396
• with parameterization, 121

orbital plane, 565

origin, reach, 160

orthogonalization, Gramm-Schmidt

orthogonal map, invariance of nature

• under, 280

orthogonal transformation, derivative

• of 181

oscillation(s), 287, 296

• damped, 295, 390
• forced, 298
• damped, 300
• underdamped case, 301
• in anti-sync, 303
• in sync, 303
• quality of, 297
• small, 476
• about equilibrium point, 478
• Lagrangian mechanics treatment
• of, 476
• of a system, 478
• of spherical pendulum, 291
• sum of, 478

oscillator, harmonic, see harmonic

• oscillator

oscillators, connected by spring, 302

oscillators, coupled, 302, 479

oscilloscope, 293

osculating circle, 62

Osgood, 265

Ostrogradsky's theorem, 138

overdamped, 296

Österberg, 426

P

Painlevé-Klein, 412

Palais and Palais, 164, 326

Panlevé paradoxes, 412, 414

Panthéon, 387

parabola, projectile's path, 279

parabolic orbit, 73

• centered at focus, 124

parallel axis theorem, 191, 196

parallel curves, for involutes, 327

parallelogram construction, 28

parallelogram law, 278

• essential hypothesis for all proofs
• 280ff.
• Newton's statement and proof, 278

parameter, impact, 147

parameters, variation of, 298

• 1851

Paris exhibition of, 387

• Panthéon, 387

Pars, 559, 594, 640

particle, 10

particles of ether, 491,563

Patroclus, 400

Pauli, 427

PDE

• first order, 540, 542, 668, 677
• linear, 668
• first order quasi-linear, 673

PDE Primer, 542, 667

pendulum, 20, 22, 287, 625

• equation, 47, 209
• compound, 211
• configuration space for, 209
• coordinate system on, 209
• cycloidal, 288
• double, 229, 307, 480
• tension on string for, 480
• for gravitational acceleration, 68
• Foucault's, 387, 661, 664
• intuitive, geometric, explanation
• for, 388
• path of, 389
• cusps in, 390
• loops in, 390
• popular explanation for, 387
• small oscillations of, 389
• interchanging potential and kinetic
• energy, 93
• Kater, 258
• period of, 50
• phase portrait of, 625, 626
• physical, 211
• rotating, 620
• spherical, 112, 290, 390, 479, 620
• phase portrait of, 627-628
• small oscillations of, 291
• swinging in plane, 112
• using Lagrangians, 457
• string, tension on, 210
• using Lagrangians, 457
• using constraints, 208

Perez, 646

perfectly elastic, 206, 249, 491

perfectly rough, 225, 249

periapsis, 128

• smallest, 174

pericenter, 128

perigee, 128

perihelion, 128, 565

• argument of 565

period, 287

periodic, conditionally, 620

periodic orbits, near Lagrange points, 400, 401

Period Lemma, 133, 151

period of orbit, 125

• depending on energy, 125
• depending on length of semimajor
• axis, 125

period of pendulum, 50

perpendicular axis theorem, 202

perpendicular, curve traced by axis of

• top, 349

Persson, 381, 392

perturbation, 132, 646

phase, 287

• difference, 294
• portrait, 624
• shift, 302

philosophical questions, 273

Philosophical Transactions of the Royal

• Society, 527, 540

physical pendulum, 211

physicist's way of looking at things, 378

physics trick, 378

piano, 316

Picard, 68

pin-hole camera, 487

Pisces, 361

Plague Years, 67

Planck, 498

Planck's constant, 498, 555, 557

plane

• invariable, 337
• neutral, 254
• of the ecliptic, 565

planet, 16

• elliptical orbits of, 37
• hidden, 404
• mass of, 162

Planet-X, 404

plank, supported, 254

plank, thin, bending, 256

platonic solid, 204

Pliskin, 428

plumb bob, 381

• deflection due to bulging of earth at
• equator, 381
• real versus theoretical

Poincaré{Cartan, integral invariant of

Poincaré, integral invariants of, 572

Poinsot, 336, 391

point mass, 3, 10, 175

point transformation, 596

Poisson, 279, 387, 466

• bracket, 605, 608
• invariant definition of, 606
• well-defined, 606
• fundamental, 607, 609

Poisson's method of spherical means

Poisson's theorem, 608, 610

polar coordinates, area in, 56, 630, 631

polar coordinates, length in, 160

polar cuspidal top, 355

polarization, 495, 500

• identity, 600

polhode, 337, 366

• cone, 366

polished surfaces, 411

portrait, phase, 624

positively charged body, 559

potential energy, 87, 89, 90

• effective, 144
• in Principia, 142
• only determined up to a constant

potential function, 270

• and equilibrium, 214

potential, Higgs, 166

Pourciau, 57, 60, 73, 279, 283

• (nth)

power force, 122ff., 135

power force laws, duality, 154

precessing tops, 352

• stability of, 353

precession, Euler, 339, 381

precession of equinoxes, 340, 361

• by d'Alembert, 361
• exercise in Goldstein [1], 361
• Newton's calculations, 361

precession of gyroscope, 356, 357

precession of top, 349

• \fast", 353
• rate of compared to rate of spinning
• \slow", 353
• starting on its own, 352
• without nutation, 352

precession, regular of top, 340

precession, true, 353

preeömptive strike, 8, 567

preserving Lagrangian, 467

Priamus, 400

Princeton, 108

principal, axis of inertia, 189

principal function, 527

• partial derivatives of, 530

principal moment, of inertia, 189

principal vectors, rotating, Euler equations for, 362

Principia, 7, 141, 490

• conservation of energy in, 141
• potential energy in, 142

product, interior, 512

productneighborhood lemma, 628

productof inertia, 190

propagation speed, finite, 502

proton, 24, 113

pulley, 241, 245

pure rotation, in rolling, 221, 222