Physics for Mathematicians — Index (Q–Z)

Q

quadratic form, associated with symmetric matrix, 568

quadratic functions, homogeneous, 44

• Euler's theorem on, 448

quadratures, 614, 634

quality of oscillation, 297

quantity

• of action, 497
• of matter, 8, 16
• of motion, 9

quantization, royal road to, 640

quantum mechanics, 48, 653

• relation to Hamiltonian mechanics

quantum theory, 501

quark, 24

quasi-linear first order PDE, 673

Quetelt, 499

R

radially symmetric central force, 74, 88

radius of earth, 16

radius of gyration, 213

radius of nucleus, 150

range of influence, 502, 506, 507

rank, of (skew)-inner product, 599

rate of nutation of top, compared to

• rate of spinning, 351

rate of precession of top, compared to

• rate of spinning, 351

rate of spinning, compared to rate of

• nutation, 351

rate of spinning, compared to rate of

• precession, 351

rate of spinning, compared to size of

• nutation, 351

Rauch-Wojciechowski, Sköldstam, and

• Glad, 432

ray, incident, 522

Rayleigh [Lord Rayleigh] (John

• William Strutt, Baron Rayleigh)
• 644, 645, 646

ray of light, 492

ray, refracted, 522

reach origin, 160

reciprocal spiral, 126

Reddingius, 407

reducedHamilton-Jacobi equation, 554

reducedmass, 116

reduction to one-dimensional problem, 128, 144

reflection

• angle of, 249, 487
• law of, 487, 492, 493

reflection and refraction, laws of, 500

refracted ray, 522

refraction

• conical, 526
• external, 526
• internal, 526

Refractiones, Analysis ad, 490

Refractiones, Synthesis ad, 490

refractive index, 497, 522

regular Lagrangian, 482, 517, 533

regular precession of top, 340

relative integral invariant of Poincaré

relative motion, 12

relativity, general, 39, 394

• Schwarzschild solution in, 640

relativity principle, Einstein's, 275

relativity principle, Galilean, 275

relativity, special, 12, 17, 34, 175, 200, 394, 487

Religion and Natural Science, 498

remainder, integral form for Taylor's

• formula, 659

repeating sections, for herpolhode, 338

repeating sections of orbits, 129

resolution of forces, 30

restitution, coefficient of, 94

rest mass, 43

restricted three-body problem, 396

revolution, ellipsoid of, 494

revolving elliptical orbit, 131,166

revolving orbits, 129

• Newton's theorem of, 130

Riccati, 270

Richer, 20

Riemannian manifolds, geodesics on

Riemannian metric, 476, 482

rigid body, 26, 81, 175ff., 491

• in equilibrium, 176
• bending of, 206
• collision of, 206
• in conservative force, 194
• in contact, 205ff.
• in equilibrium, 175
• kinetic energy of, 193
• motion, 332
• geometric description of, 336ff.
• non-planar, 177ff.
• possible motion of, 185
• rotation about an axis, 334
• work done by internal forces of, 195

rigid equilibrium, 176

rigid motion, 178

• rigid solution for, 185

rigor, gone with the wind, 653

ring, college, 426, 430

rising top, 354

rocket, 32

rocket launches, 117

rocket science, 32

Roemer, 491

rolling, 220, 222, 418

• and frictional forces, 221
• body cone on space cone, 341
• inertia ellipsoid, 337
• not, 417, 422
• velocity of contact point, 221, 222
• what is going on, 223

rolling condition, 222, 453

rolling disc, 230, 362

• along circle, 365
• along straight line, 364
• stability of 364
• down inclined plane, 235
• Lagrange's equations for, 452
• non-upright, 231

rolling friction, 411

rolling sphere, 240

rolling wheel, speed of, 226

Roll, Krotkov, and Dicke, 40

Roma caravans, 356

Ronchi, 487, 488, 489, 496

rope, 241

• wrapped around object, 245

rotating

• axes, within body, 424
• bucket, 393
• coordinate system, 332
• distinguishing from non-rotating
• earth, 376, 377
• elliptical path, 292
• liquid, 379
• as speedometer, 408
• observer's derivative, 333
• pendulum, 620
• principal vectors, Euler equations
• for, 362

rotation

• about an axis, 334
• stable, 335
• unstable, 335
• about axis of top, 349
• about longest axis, stable, 336
• in phase portrait, 626
• pure, in rolling, 221, 222
• rigid body about axis, 191,193

rotational

• angular momentum, 84, 187
• part of kinetic energy, 194, 204
• for continuous body,196, 204
• symmetry, 470, 475

rough, perfectly, 225, 249

Routh, 430

Routh integral, 430

Royal Irish Academy, 522

Royal Irish Academy, Transactions of, 527

royal road to quantization, 640

Royal Society, Philosophical Transactions

• of, 527, 540

rubber band, 295, 298

Rutherford, 126

• experiments of, 148
• scattering formula, 148

Rutherford scattering, 147

S

Sabra, 490

saddle point, 325

sagitta, 59

Sahl,, 488

“same” solutions for Lagrangians, 459

Santos-Benito and Gras-Mati, 258

scale, 14, 46

scattering angle, 114, 147

• laboratory, 115

scattering formula, Rutherford, 147

Schlebusch, 405

Schrödinger, 555

Schrödinger wave equation, 509, 555, Hamilton-Jacobi theory and, 555

Schulz-Dubois, 391

Schwarzschild, 640

Schwarzschild solution in general relativity, 640

seat-of-the-pants approximation for

• Tipped Top, 428

secondary wave, 491,563

second law, Kepler's, 55

second law, Newton's, 12

second order ODE's, 532

self-braking, 414

semimajor axis, 166

semi-period, 297

separation of variables, 75, 314, 540

separatrix, 626

shape of orbit, 75, 121

Shapiro, 392

sharp signals, 506

shattering experience, 104

shift

• lagging, 301
• leading, 301
• phase, 302

side english, 421

Siegel and Moser, 594

signals, nonsharp, 507 (also 502)

• sharp, 506

simple harmonic motion, 287, 289, electrical examples for, 294

simultaneous diagonalization of matrices, 478

sine law (Snell's law), 488, 522

singular solution, 543

size of nutation of top, compared to

• rate of spinning, 351

skew-adjoint transformations, 181

skew-inner product, 599

• matrix of 599
• rank of 599

skew-orthogonal, 601

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

sleeping top, 353

• stability of, 354

sliding, 215

• down inclined plane, 215
• down wedge, 217
• configuration space for, 217
• without friction, 216

sliding and spinning, 417

sliding friction, 410

slowness, vector of normal, 526

slow precession, 353

small angle, pendulum equation, 50

smallest value of impact parameter, 150

small oscillations, 476

• approximations to, 292
• about equilibrium point, 478
• Lagrangian mechanicstreatment,476
• of a system, 478
• of spherical pendulum, 291
• sum of, 478

Smith, 319

snakelike, 338

Snell, 488

sn, elliptic function, 372

Snell's law, 488

• Newton's argument for, 496

Snel, Willebrord van Royen, 488

soft body, 94

SOHO (Solar and Heliospheric

• Observatory), 404

solenoid, 52

solid body, bending of, 206

solutions for Lagrange's equations, integral for, 468

Sommerfeld, 35, 98, 102,287, 307, 636, 640, 646

sound, analogous to light, 490

southward deflection

• of falling body,386, 405
• of hanging body,380
• not possibleto detect, 381

space, absolute, 12, 275

space cone, 341

• body cone rolling on, 341

space, homogeneity of, 469

spar see Iceland spar

special relativity, 34, 175, 200, 394

specific heat, 98

spectacles (eye-glasses), 488

speed, finite propagation, 502

speed of light, 491

• in air and water, 391, 489, 496
• determined by Fiseau
• and Breguet, 496
• determined by Foucault, 496

speedometer, from rotating liquid, 408

sphere, rolling, 240

spherical bodies, attractive force of, 137

spherical coordinates, 290

spherical means, Poisson's method of

spherical pendulum, see pendulum, spherical

spin, 236

• opposite direction for gyroscope
• 257, 358

spinning and sliding, 417

spinning ball, 249

spinning football, 432

spiral

• Cotes', 127
• equiangular, 126
• hyperbolic, 126
• logarithmic, 126
• reciprocal, 126

Spivak, 101

spring, 9, 14, 15, 47

• connecting oscillators, 302
• constant, 47, 292
• equinox, 361
• stiffness, 316
• stretching, 292

sprocket wheel, chain and, 246

square, inverse force, 16, 58, 60, 67, 75

• exactness of, 135

stability, 213ff.

• for solutions of differential equations
• 324, 326

stable equilibrium, 215,476

stable rotation about an axis, 335, 336

standing wave, 317

statically indeterminate, 206, 252

static friction, 410, 411

“statics, dynamics reduces to”, 184

stationary orbit, 396

steady state, 301

Stefan-Boltzmann law, 646

Steiner's theorem, 191

stiffness of spring, 316

straight down, 381

strain, 253

stress, 253

stretching, spring, 292

stretching, wire, 292

strike, preeömptive, 8, 567

string

• continuous, 314
• vibrating, 310ff., 557
• continuousand discrete cases
• 310ff.

strip condition, 683

strip manifold condition, 691

strong form of Newton's third law, 25, 79, 84, 179, 199, 470

• Rn

subgroup, discrete, 622, 641

subspace, isotropic, 601

Superball, bouncing, 249

SuperBall (Wham-O, 119

superconductivity, 391

surfaces, envelope of family of, 323

surfaces, polished, 411

suspension bridge, 269

suspension, Cardan, 356

swivel top, 345

symmetrical top, heavy, 345ff.

symmetry, 25, 200, 469

• rotational, 470, 475
• used to derive laws of mechanics

symplectic

• area-preserving in 1 dimension, 630
• as canonical, 603
• basis, 601
• coordinate system, 604
• diffeomorphism, 608
• manifold, 599, 604
• map, 608
• transformation, 601
• vector space, 599

sync, 479

sync, oscillations in, 303

Synge and Griffith, 257, 363, 372

Synthesis ad Refractiones, 490

T

Tantalus, 111

taut, 242

tautochrone problem, 318

tautochronous, 288

Taylor's theorem, 314

• integral form of remainder, 659

teleological principles, 463, 490, 498

telescopes, 488

temperature change of Carnot energy

• loss, 98

tension, 243, 267, 308, 310

• on pendulum string, using
• Lagrangians, 457
• on spherical pendulum string, using
• Lagrangians, 457
• on string for doublependulum, 480

Terrall, 466

test mass, 52

thermodynamics, 98, 538, 644

thin plank bending, 256

third law

• built into Lagrange's equations, 470
• for gravity, 39
• for rockets, 32
• Kepler's, 37, 67, 125, 162
• Newton's, 21
• proof of, 276, 277
• non-intuitive nature, 23
• reason for in gravitation, 25
• strong form of, 25, 79, 84, 179, 199

Thompson, 53

three-bodyproblem, 396, 532, 644

• restricted, 396, 559

tidal forces, 168

• on moon, 170

tides, 25, 168

time, as another variable, 227

time-dependent

• constraints, 227, 449
• Hamiltonians, 590
• Lagrangians, 449
• conservation of energy for, 450

time-independent Hamiltonians, 590

time of descent, 289, 385

Tippe Top, 426

Tobin, 391, 392, 496

top, heavy, 339, 445

• conditionally periodic motion of, 621
• cuspidal case, 349ff., 356
• fast, 349ff.
• symmetrical, 345ff.
• phase portrait for, 628
• lab, 345
• nutation, 349
• rate of compared to rate of spinning, 351
• size of compared to rate of spinning, 351
• perpendicularity of curve traced by
• axis in cuspidal case, 349
• polar cuspidal, 355
• precessing, 352
• stability of, 353
• without nutation, 352
• precession, 349
• rate of precession, compared to rate
• of spinning, 351
• regular precession of, 340
• rising, 354
• rotation about axis, 349
• sleeping, 353
• stability of, 354
• start precessing on its own, 352
• swivel, 345
• symmetric, 339
• free, 339
• in bodycoordinates, 339
• in inertial coordinates, 340

top, see, in particular, free symmetric

• top

Top, Tippe, 426

top, toy, 345

top, with spherical bottom, 431

tori

• Arnold, 623
• invariant, 621, 623
• Liouville, 623
• Liouville-Arnold, 623
• periodic vs. dense trajectories on
• 627, 636

torque, 84, 85

torque coming from friction, 427

torque for central force, 85

• 0

torsion balance, 39

torus, invariant, see invariant torus

total momentum, 22, 29

toy top, 345

Traitωede lumiere, 490

Transactions of the Royal Irish Academy

transfer of energy, 305, 309

• not complete, 306

transform, Legendre, see Legendre

• transform

transient, 301

translational force, 377

translational part of kinetic energy, 194

transversal waves, 501

Treatiseon Light, 490

Trefethen, 392

triatomic molecule, linear, 480

tribology, 409

trick

• old-fashioned, for Leibnizian notation, 75, 88, 100
• physics, 378

Trojan asteroids, 396

Trojans, Mars, 401

Trojans, Neptune, 401

Trojan war, 400

true anomaly, 163, 643

true precession, 353

Truesdell, 200

tube, 574

tubs, bath-, 392

tuning fork, 295

• frequency of, 295
• standard, 298

two-bodyproblem, 120, 136, 162

two fixed masses, 559

type 1 generating function, 578

type 2 generating function, 580 617, 633, 652

type 3 generating function, 588

type 4 generating function, 588

U

underdamped, 296

UNESCO Courier, 356

uniform gravity, 13, 14, 204

unit normal, 139

• inward or outward pointing, 139

universal invariants, 593, 598

University of Dublin, 522

unstable equilibrium, 477

unstable rotation about an axis, 335

upright rolling disc, 230

• compared to rolling sphere, 230
• down inclined plane, 235

upwards, shooting cannonball, 386

V

van der Waerden, 6

variable mass, 35, 100

variables, separation of, 75, 314, 540

variational principle, 487, 461

variation (of a function), 461

variation of parameters, 298

variation vector field, 467

vector of normal slowness, 526

vector(s), 9

• cross-productof, 55

vector space structure, 28

velocity

• vector, 9
• angular, 83
• escape, 117

velocity of contact point, in rolling, 221, 222

vernal equinox, 565

vibrating string, 310ff., 557

• continuous and discrete cases, 310ff.

virtual infinitesimal displacements, 178ff.

virtual work, principle of, 181

viscous friction, 411

vision, 487, 488

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

Vorlesungen uéber Dynamik, 466, 549

W

Wallis, 22, 273

water waves, 507

water-wheels, 381

wave, 317, 555, 490

• electromagnetic, 501
• equation
• 1-dimensional, 316
• Schrödinger, 509, 555
• Hamilton-Jacobi theory and
• front, 524, 526, 562
• length, 313
• longitudinal, 501
• mechanics, 641
• number, 313
• angular, 313
• secondary, 491
• standing, 317
• theory of light, 500, 524, 557
• transversal, 501
• water, 507

weak coupling, 304

wedged, 414

wedge, sliding down, 217

• configuration space for, 218

weight, 19, 37

• decrease approaching equator, 380
• depending on distance from center
• of earth, 20
• distinction from mass, 20

Weinstein, 599, 604

Wells and Siklos, 648

• R R

Wham-O SuperBall, 119

wheel(s)

• paradoxes of, 220
• Galileo, 220, 224
• rolling, speed of, 226
• sprocket, chain and, 246
• water-, 381

whip, 100, 106

Whittaker, 572, 590, 640

Wikipedia, 119

Wintner, 283

WMAP (Wilkinson Microwave

• Anisotropy Probe), 404

wobble,Chandler, 340

Wolf, 400

Wong and Yasui, 107

work, 91, 181

• done by internal forces of rigid body
• modern meaning of, 420
• virtual infinitesimal, 181
• virtual, principle of, 181

wrapped around object, 245, 267

Wren, 22, 67, 273

wrench, 201

Y

yanked link of chain, 102, 105

Yogi Berra, 461

Young, 500, 524

Young's inequality, 520

Young's interference experiment, 500

youth, misspent, 420