CHAPTER 1. SURVEY OF THE ELEMENTARY PRINCIPLES 1

1-1 Mechanics of a particle 1
1-2 Mechanics of a system of particles 4
1-3 Constraints 10
1-4 D’Alembert’s principle and Lagrange’s equations 14

1-5 Velocity-dependent potentials and the dissipation function 18

1-6 Simple applications of the Lagrangian formulation 22

CHAPTER 2. VARIATIONAL PRINCIPLES AND LAGRANGE’S EQUATIONS 30

2-1 Hamilton’s principle 30
2-2 Some techniques of the calculus of variations 31
2-3 Derivation of Lagrange’s equations from Hamilton’s principle 36
2-4 Extension of Hamilton’s principle to nonconservative and non-holonomic systems 38
2-5 Advantages of a variational principle formulation 44
2-6 Conservation theorems and symmetry properties 47

CHAPTER 3. THE TWO-BODY CENTRAL FORCE PROBLEM 58

3-1 Reduction to the equivalent one-body problem 58
3-2 The equations of motion and first integrals 59

3-3 The equivalent one-dimensional problem, and classification of orbits 63
3-4 The virial theorem 69
3-5 The differential equation for the orbit and integrable power-law potentials 71
3-6 The Kepler problem: inverse square law of force 76
3-7 Scattering in a central force field 81

3-8 Transformation of the scattering problem to laboratory conditions 85

CHAPTER 4. THE KINEMATICS OF RIGID BODY MOTION 93

4-1 The independent coordinates of a rigid body 93
4-2 Orthogonal transformations 97
4-3 Formal properties of the transformation matrix 101
4-4 The Eulerian angles 107
4-5 The Cayley-Klein parameters 109
4-6 Euler’s theorem on the motion of a rigid body 118
4-7 Infinitesimal rotations 124
4-8 Rate of change of a vector 132
4-9 The Coriolis force 135

CHAPTER 5. THE RIGID BODY EQUATIONS OF MOTION 143

5-1 Angular momentum and kinetic energy of motion about a point 143
5-2 Tensors and dyadics 146
5-3 The inertia tensor and the moment of inertia 149
5-4 The eigenvalues of the inertia tensor and the principal axis transformation 151
5-5 Methods of solving rigid body problems and the Euler equations of motion 156
5-6 Force-free motion of a rigid body 159
5-7 The heavy symmetrical top with one point fixed 164
5-8 Precession of charged bodies in a magnetic field 176

CHAPTER 6. SPECIAL RELATIVITY IN CLASSICAL MECHANICS 185

6-1 The basic program of special relativity 185
6-2 The Lorentz transformation 187
6-3 Covariant four-dimensional formulations 194

6-4 The force and energy equations in relativistic mechanics 199
6-5 The Lagrangian formulation of relativistic mechanics 205
6-6 Covariant Lagrangian formulations 207

CHAPTER 7. THE HAMILTON EQUATIONS OF MOTION 215

7-1 Legendre transformations and the Hamilton equations of motion 215
7-2 Cyclic coordinates and Routh’s procedure 218
7-3 Conservation theorems and the physical significance of the Hamiltonian 220
7-4 Derivation from a variational principle 225
7-5 The principle of Least Action 228

CHAPTER 8. CANONICAL TRANSFORMATIONS 237

8-1 The equations of canonical transformation 237
8-2 Examples of canonical transformations 244
8-3 The integral invariants of Poincaré 247
8-4 Lagrange and Poisson brackets as canonical invariants 250
8-5 The equations of motion in Poisson bracket notation 255
8-6 Infinitesimal contact transformations, constants of the motion, and symmetry properties 258
8-7 The angular momentum Poisson bracket relations 263
8-8 Liouville’s theorem 266

CHAPTER 9. HAMILTON-JACOBI THEORY 273

9-1 The Hamilton-Jacobi equation for Hamilton’s principal function 273

9-2 The harmonic oscillator problem as an example of the Hamilton-Jacobi method 277

9-3 The Hamilton-Jacobi equation for Hamilton’s characteristic function 279
9-4 Separation of variables in the Hamilton-Jacobi equation 284
9-5 Action-angle variables 288
9-6 Further properties of action-angle variables 294
9-7 The Kepler problem in action-angle variables 299

9-8 Hamilton-Jacobi theory, geometrical optics, and wave mechanics 307

CHAPTER 10. SMALL OSCILLATIONS 318

10-1 Formulation of the problem 318

10-2 The eigenvalue equation and the principal axis transformation 321

10-3 Frequencies of free vibration, and normal coordinates 329
10-4 Free vibrations of a linear triatomic molecule 333
10-5 Forced vibrations and the effect of dissipative forces 338

CHAPTER 11. INTRODUCTION TO THE LAGRANGIAN AND HAMILTONIAN FORMULATIONS FOR CONTINUOUS SYSTEMS AND FIELDS 347

11-1 The transition from a discrete to a continuous system 347

11-2 The Lagrangian formulation for continuous systems 350

11-3 Sound vibrations in gases as an example of the Lagrangian formulation 355

11-4 The Hamiltonian formulation for continuous systems 359

11-5 Description of fields by variational principles 364

INDEX 385

Acceleration, centripetal, 25, 26, 67

linear, 2, 205, 212

of gravity, 25, 136

Acoustics, 46, 318, 338, 355ff

Action, 228, 282, 291

and reaction, 4

at a distance, 208, 210

variables, 288f, 291f

Action-angle variables, for general multiply periodic system, 296

for harmonic oscillator, 293f

history of applications of, 30Sf

in Kepler problem, 299ff

in presence of degeneracy, 297f, 304

Adiabatic gas law, 357

Adjoint matrix, 105

Alpha particle scattering, 81, 84, 88

Ames, J. S. and Murnaghan, F. D., 48

Analytical mechanics, 1

Angle of rotation of orthogonal matrix, 123f, 141-142

Angle variables, 292, 332

for Kepler problem, 305, 317

solution for, 292

Angular momentum, 2, 60, 144

as canonical momenta, 266, 272

connection with angular velocity, 145

conservation theorem for, 3, 6, 51f, 60f, 262, 300

electromagnetic, 6

generator of rotational motion, 263

and magnetic moment, 177

Poisson brackets involving, 263f, 272

as pseudovector, 131

resolution theorems for, 7,143

when independent of point of reference, 8, 176

with velocity-dependent potentials, 57

Angular velocity, 26, 133

precession of for force-free symmetrical rigid body, 162

in terms of Euler angles, 134, 141

vector, and direction of angular momentum, 160

Antecedent, 147

Antisymmetric matrix, 127, 141

as corresponding to three-dimensional

pseudovector, 129, 131

Apsidal distances, 66, 72

Arc length in configuration space, 232f, 310

Areal velocity, 60, 79

Astronomy, 61, 80, 163, 175, 30Sf, 31Sf

Atomic bomb, 204

Atwood’s machine, 23, 25

Axial vector, 130

Axis of rotation, and angular momentum direction, 161

Beats, 345

Becker, R., 176, 188, 192

Bergmann, P., 192, 201

Bernoulli, James, 15

Bernoulli, John, 36

Bôcher, M., 106

Body set of axes, 95

Bohr atom, 79, 80, 306f

Born, M., 296, 297

Born, M. and Jordan, P., 266

Boyle’s law, 70, 356

Brachistochrone, 36, 56

Brillouin, L., 310

Byerly, W. E., 48

Calculus of variations, 31ff, 36f, 38, 56

parameter technique in, 32, 226, 229, 351

Canonical equations of Hamilton, 217 (see also: Hamilton’s equations)

equivalence with Hamilton-Jacobi equation, 275—276

Canonical momentum

(see: Momentum, canonical)

Canonical transformations, Chapter 8, 239, 269

comparison of two procedures, 282f to constant coordinates and momenta, 260, 273ff

to cyclic coordinates, 237f, 247, 273, 280ff

equations of, 240ff, 248f, 275

examples of, 244ff, 270

as governing motion in time, 259

for harmonic oscillator, 246

including time, 242f

infinitesimal, 258

invariants of, 247f, 250f, 254

inversion of equations of, 275

possessing the group-property, 272

Carathéodory, C., 239

Catenary, 36

Cayley-Klein parameters, 109f, 140, 179, 180

Center of force, 81

Center of gravity, 5

Center of mass, 5

angular momentum with respect to, 8

motion of, independent of internal forces, 5

Central force motion, Chapter 3, 221, 236, 285, 287f, 299f

degeneracy of, 302, 306

in Hamilton-Jacobi form, 287f, 299f

inverse square law force, 76ff, 299ff

Central forces, classifications of orbits for, 65

constants of integration for, 63

equation of orbit for, 71f, 287f

equivalent one-body problem for, 58

equivalent one-dimensional problem for, 64f, 90

first integrals for, 60f

formal solution for, 63, 287

integrable power law potentials for, 73f

orbits expressible in elliptic functions, 74f

scattering in, 81ff

in special relativity, 207, 303

in spherical polar coordinates, 60, 288, 299f

symmetry properties of, 59f

Centrifugal barrier, 65, 90

Centrifugal force, 25, 64, 136f

Change in a function, under infinitesimal contact transformation, 260f, 263

Characteristic equation, 120

Chasle’s theorem, 124, 143

Classical mechanics, 1, 63, 81, 82, 89, 185, 237, 266, 305

as species of geometrical optics, 312

Clausius, R. J. E., 70

Commutator, 255, 266

Complete degeneracy, for closed orbits, 297, 303, 306

Completely degenerate motion, 297

Compton scattering, 213

Conditionally periodic motion, 291, 295, 330

examples of, 29Sf

Condon, E. V. and Shortley, G. S., 266

Configuration space, 30

motion of system point in, 30, 228, 231, 233, 308f

Congruent transformation, 326 Conic sections, as orbits for inverse

square law of force, 78

Consequent, 147

Conservation theorems, and symmetry properties, 47f, 220, 261

angular momentum, 3, 6, 51f, 61, 220, 262f, 300

exceptions to, 6

combined momentum and energy, 203, 210

for continuous systems, 363f

generalized momentum, 49, 220, 262f

including electromagnetic forces, 49, 210

linear momentum, 2, 5, 50, 202, 372 exceptions to, 4

microscopic, 364

for a particle, 2f

for systems, 5f, 47f

total energy, 4,10, 53, 61f, 206f, 300 Conservative systems, 3

Constants of integration, in Hamilton-Jacobi equation, 275, 281

Constants of the motion, as ensemble densities, 268

as generators of infinitesimal contact transformations, 261

and Poisson brackets, 256, 261, 268, 271

and symmetry properties, 47f, 220. 261

Constraints, lOf, 211

differential equations of, 13, 40

difficulties produced by, 11, 211

examples of, 10—13, 26

holonomic, 11, 40, 43, 156

moving, 26, 54

nonholonomic, 11, 40, 43, 156

rolling as an example of, 12—13, 141, 156

nonintegrable, 13, 141

rheonomous, 11

scleronomous, 11

vanishing virtual work for, 15 Contact transformations, 239

(see also: Canonical transformations)

infinitesimal, 258f

Continuous systems, 47, 318, 343

covariant formulation for, 354

transition to from discrete systems, 343-5, 347ff, 359f

Conversion of mass into kinetic energy, 203f

Coriolis forces, 136f

and molecular spectra, 140, 141

Coulomb field, scattering of charged particles by, 83f

Covariance, 194f

of wave equation, 198f

Covariant field theories, 370

Covariant formulations, 194f, 207f, 354

Covariant Hamiltonian, 223

for electromagnetic forces, 224, 236

Covariant Lagrangian, arbitrariness in, 209, 236

for electromagnetic forces, 209f, 236

for free particle, 209

Covariant Lagrangian density, 354

Cyclic coordinates, 48, 164, 218

action variables for, 291f, 301

canonical transformation to, 237f, 247, 273, 280ff

and conservation of conjugate momentum, 49, 206, 220

separation of variables for, 285f

Cyclone, 138

D’Alembert, Jean, 15

D’Alembertian, 199

D’Alembert’s principle, 14ff, 16, 30, 205

Damping factor, 342

Damping of free vibrations, 341f

Degeneracy, 297f, 302f

in central force motion, 302, 306

complete, 297, 303, 306

examples of, 297, 302, 317

in Kepler problem, 303f, 306

rn-fold, 297

and nature of the orbit, 297, 303, 306

and separable coordinates, 297, 315, 317

Degenerate frequencies, in action-angle variables, 297f

in small oscillations, 329, 337

Degrees of freedom, 11

infinite number of, 347

for rigid bodies, 93f

Delaunay orbit elements, 305

Density in phase space, 267

Derivatives, in terms of Poisson brackets, 254f, 256, 364

Determinant, 106, 140, 248

of canonical transformation, 271f

equal to product of eigenvalues, 122

explicit formula for, 130

invariance under similarity transformation, 106

Jacobian, 248, 271

(see also: Jacobian determinant)

of orthogonal matrix, 106

secular, 119, 321f

Diagonalization, of kinetic and potential energies, 332, 344

of a matrix, 120, 322

simultaneous, of two matrices, 326

Differential principles, 30

Differential scattering cross section, 81

Dilatation of time scales, 192f, 197

Dirac 3-function, 368

Direction cosines, to specify orientation of rigid bodies, 95f

Directional gyro, 175

Dissipation function, Rayleigh’s, 19, 21, 340

in Lagrange’s equations, 22

Double pendulum, 12, 29, 345

Double-valued property of spinor rotation matrices, 117

Duality of particle and wave, 312, 314

Dyad, 147

Dyad multiplication, 148

Dyadics, 148f, 178

as tensor of the second rank, 148

Dyadic form of angular momentum Poisson brackets, 265

Dynamics, 1, 15

Earth, force-free precession of, 163, 181, 183

Eigenvalue equation, 119, 153, 321

Eigenvalue problem, 119, 140, 152f, 321f

as diagonalizing a matrix, 120, 323

simultaneous diagonalization of two matrices, 326

Eigenvalues, of hermitean matrix, 153, 323

of inertia tensor, 153

of real orthogonal matrix, 119

for small oscillations, 321f multiple roots, 327f

positive roots for stable equilibrium, 324

reality of, 323, 341

Eigenvectors, 119

for small oscillations, 323f

indeterminacy of, 321, 323, 324f, 327

orthogonality of, 326

Eikonal, 311f

equation, 312, 314

Einstein, Albert, 186

Einstein addition law for velocities, 193f, 211f

Einstein energy relation, 203

Einstein summation convention, 196

Elastic rod, discrete approximation to, 347

Lagrangian and equations of motion for, 349, 354f

Elastic solid, 347, 364

Electric circuits, 45f, 343

Electromagnetic field, 47, 344, 364, 366f

Lagrangian for, 47, 366, 372

Electromagnetic forces, 4, 6, 19f, 48, 50, 200f, 207, 209, 213f, 222, 368f

Lagrangian for, 21, 48, 207, 209

Electromagnetic momentum, 6, 49

Electromagnetic potentials, 20, 201, 366

Elliptic functions, 74, 159, 166, 182

Energy, as canonical momentum, 210, 243, 247, 278

Energy, conservation theorem for, 4, 10, 53, 61, 206, 300

equivalence with mass, 203

form of for conservative systems, 8f

and frequency, 313

for inverse square law orbits, 79

kinetic (see: Kinetic energy)

potential (see: Potential energy)

Energy equation in special relativity, 202

Ensemble, 266

Equation of continuity, for energy momentum density, 364

for gas flow, 377f

Equation of state, 70, 90

Equations of canonical transformation, 240ff, 25Sf

Equations of motion, for continuous systems, 352f

in Routh’s procedure, 219

for small oscillations, 320, 332, 338, 340, 342

for sound in gases, 359, 362, 365

in special relativity, 199f, 205, 212

in terms of angle variables, 292

in terms of Poisson brackets, 255f

Equilibrium, 15, 200, 268, 315f

labile or indifferent, 334

stable, 318

condition for, 318f, 324

statistical, 268

unstable, 318

Equivalence postulate, 186

Escape velocity, 28

Euler angles, 97, 107ff, 134, 140

definitions of in the literature, 108

for elliptic orbit in space, 305

matrix in terms of, 109, 141

Euler-Lagrange equations, 3Sf, 205

(see also: Lagrange’s equations)

Euler’s equations of motion, 157f, l61f

from Lagrange’s equations, 157

from torque equation, 158

Euler’s theorem, 118ff, 132

Evaluation of action integrals, by method of residues, 302, 315

Event, defined in special relativity, 198

Exchange transformation, 245, 271

Extremum problems, 31f, 38

Falling body, Coriolis deflection of, 139

Fermat’s principle, 231, 312

Fields, 47, 364ff, 370

Figure axis locus, for heavy top, 167f

First integrals, 47, 60

Flux density, in scattering problems, 81

Force, 1

defined as time derivative of momentum, 1, 201

effective, in rotating axes, 135

reversed effective, 15, 67

transformation properties in special relativity, 200f

Forced vibrations, amplitudes of, 339, 343

effect of frictional forces on, 342f

for nonsinusoidal forces, 346

in normal coordinates, 338

sinusoidal driving forces, 338f Forces, conservative, 3, 61, 156

electromagnetic, 4, 19f, 48, 50, 200f, 207, 209, 213f, 222f

external, 4

frictional, 21f, 70

generalized, 17

for conservative systems, 18

of constraint, 42

impulsive, 57

internal, 4

do no work in rigid bodies, 10

possible sources of, 200, 208, 224f

Formal solution of mechanical problems, 237f, 247, 260, 273ff, 277, 282f

Foucault gyrocompass, 175, 180, 183

Foucault pendulum, 139, 142

Four-dimensional space, 180, 188f

Four-gradient, 198

Four-vector, 195

as space-or timelike, 197

potential, 201

Four-velocity, 198

Fourier expansion, for periodic motion, 294f

Free vibrations, 329, 340

damping of by friction, 340f

Frequencies of periodic motion, from angle variables, 293

Frequencies, reduction of number in degeneracy, 298, 304

of small oscillation, 321, 324, 329f

effect of friction on, 341f

of waves associated with classical motion, 312f

zero, in small oscillations, 334f,346

zero, in degeneracy, 298

Friction, 21f, 70, 340f

Functional derivative, 353, 361

Fundamental Lagrange brackets, 251

Fundamental Poisson brackets, 254

Galilean transformation, 186

Gauge condition, 368

Generalized coordinates, 11

choices for, 238

for continuous systems, 349f

of electromagnetic field, 366

examples of, 12

motion when all cyclic, 237f, 247

relative to equilibrium, 319f

for rigid bodies, 107, 109

transformation equations for, 12

Generalized momentum, 48, 215

as independent variable, 215, 227, 245

Generating function, for infinitesimal rotation, 262

types of, 240

Geodesics, 35, 56, 233f

of a plane, 34f

of a sphere, 35, 234

Geometrical optics, 231, 307, 311f

compared with classical mechanics, 312f, 315

conditions for, 311f

Group-property, 272

Hamilton, Sir W. R., 314

Hamilton-Jacobi equation, 274ff

arbitrary constant momenta obtained from, 276, 281f

for central forces, 287f, 299f

compared with eikonal equation, 312, 314

equivalence with canonical equations, 275f

furnishes Hamilton’s characteristic function, 281

for harmonic oscillator, 277

mathematical nature of solution, 274

and orbit equations, 281

and quantum mechanics, 314

separation of variables in, 279, 284ff, 299f

and theory of partial differential equations, 276, 315

when H not explicit function of time, 279

Hamilton-Jacobi theory, Chapter 9, 269

comparison of Hamilton’s principal and characteristic function, 282f

Hamiltonian, 217ff

as a canonical momentum, 24 281, 286

change under infinitesimal contact transformation, 261

for charged particles, 222

as constant of the motion, 53, 206, 220, 363

for continuous systems, 360f

covariant, 223f, 236, 271

and cyclic coordinates, 218, 220

as fourth component of world momentum, 223

as a function of action variables, 291, 298

as generator of motion in time, 259

for infinite linear molecule, 359f

invariance of and constants of the motion, 261f

Poisson bracket with, 256, 364

Hamiltonian, relativistic, 222

time derivatives of, 217, 220,224, 256, 363

as total energy, 54, 207, 220f, 222

transformed under canonical transformation, 239

when not the total energy, 54f, 221

Hamiltonian density, 360

as an energy density, 362

for sound field in gases, 362, 366

Hamiltonian formulation, for electromagnetic field, 368

nature of, 215

Hamilton’s characteristic function, 280

as indefinite integral of action, 282

periodic function obtained from, 296

separated form for, 284

Hamilton’s equations, for continuous systems, 362, 371

covariant, 224

derivation of, 217, 225f, 360

procedure for obtaining, 217f, 221

with auxiliary conditions, 271

Hamilton’s principal and characteristic function, comparison of, 282f

Hamilton’s principal function, 274

as indefinite integral of L, 276f, 279

and particle trajectories, 309f

propagation in configuration space, 308f, 316

Hamilton’s principle, 30ff, 39, 205, 225f, 276

for continuous systems, 350f

in covariant form, 208

for both electromagnetic field and charged particles, 369

and field theories, 365f

for nonconservative and nonholonomic systems, 38

and theory of partial differential equations, 235, 270, 276, 315

Harmonic oscillator, 46, 68, 246f, 277f, 288, 293f, 296f, 316f, 333

Hermitean property, 112, 153, 322f

invariance under similarity transformation, 112, 141

Herpoihode, 161

Hertz’ principle of least curvature, 234

Homographic transformations, and Cayley-Klein parameters, 141, 179

Hooke’s law, 46, 348

Hydrogenic atoms, 80, 306f

Identity transformation, 103, 244, 271, 286

Ignorable coordinate, 48, 218

Impact parameter, 82

Improper rotation, 122, 129, 131

Impulse, 57

Inclined axes, 327

Index of refraction, 310f

Inertia ellipsoid, 155f, 159f, 332

motion on invariable plane, 160f

Inertia tensor, 146f, 149, 178, 332

eigenvalue problem for, 152f

hermitean property of, 151

orthogonality of eigenvectors, 154

shift of point of reference for, 180

Inertial system, 135, 185f

Infinitesimal contact transformation, 258f

equations of, 259

functional change under, 260f

Infinitesimal rotation, 124f, 258, 262

generated by angular momentum, 262f

in a plane, 126

represented by vector, 128, 131—2

Initial conditions for heavy symmetrical top, 168

Instantaneous axis of rotation, 133

Integral invariants of Poincaré, 247f

Integral principles, 30

Intensity, in scattering problems, 81

Interaction, between charged particles and field, 370

Invariable plane, 161

Invariance of physical laws under Lorentz transformation, 186f, 196, 198, 207f

Ivariance of physical laws under rotation, 194f

Invariants of canonical transformations, 247f, 250f, 254

Inverse cube law of force, 74, 91, 303f

Inverse matrix, 102

Inverse square law of force, 61, 67, 71, 74, 76f, 83, 299f

Inversion, 122, 129, 141

Isomorphic sets of matrices, 113, 117

Jacobi, C. G. J., 277

Jacobian determinant, invariance under canonical transformation, 248f, 250

(see also: determinant, Jacobian)

Jacobi’s form of least action principle, 233

Jacobi’s identity, 256f

as generating constants of the motion, 258, 265, 272

Jet propulsion, 5

Kepler problem, 76f, 299f, 315

degeneracy of orbits in, 303f, 306f

energy of elliptical orbits, 304

relativistic, 317

Kepler’s laws of planetary motion, 61, 80, 304

Kinematics, 93

Kinetic energy, 3

conservation of in special relativity, 203

density of in gases, 355

form of for small oscillations, 320, 331

homogeneous quadratic form for, 23, 54, 232

resolution theorem for systems, 8, 58, 143

in special relativity, 202f

in terms of inertia dyadic, 149

Kinetic theory of gases, 70, 266f

Klein, Felix, 109

Klein, Felix and Sommerfeld, A., 161, 166, 171, 175

Laboratory coordinates, transformation to, 85f

Lagrange brackets, 250f

connection with Poisson brackets, 252f

fundamental, 251, 271

invariance of, 251

Lagrange multipliers, 13, 41, 156, 181

Lagrange’s equations, 14f, 18, 36f

for continuous systems, 352f, 354

in relativistic form, 354

covariant, 208

for electromagnetic field, 367

including frictional forces, 22

Lagrangian, 18

for continuous systems, 350

covariant, 207f, 236, 354

for electrical circuits, 45

for both electromagnetic field and charged particle, 369

for relativistic motion, 206f

for small oscillations, 320, 332

for systems of charged particles, 21, 207, 209, 369

Lagrangian density, 350

for electromagnetic field, 866, 372

for the Schr#dinger equation, 372

of sound vibrations in gases, 358, 365

Lagrangian formulation, for continuous systems and fields, 350f

for nonmechanical systems, 45, 365f

in special relativity, 205f

Larmor frequency, 177

Larmor’s theorem, 177f, 183, 306

Least action principle, 228f

(see also: Principle of least action)

Legendre transformation, 21Sf, 235, 241f, 270, 296

Levi-Civita density, 129

Lewis, G. N. and Tolman, R. C., 201

Libration, 288, 294

Light

(see: Geometrical optics or Wave equation)

Lindsay, R. B., 70

Line of nodes, 108, 164, 305

Linear molecule, infinite, 347f

Hamiltonian for, 359f

Lagrangian and equations of motion for, 348

Linear momentum, 1f

conservation theorems for, 2, 4, 50f, 202f, 372

of fields, 372

in special relativity, 202

Linear transformations, 97

Linear triatomic molecule, 333f, 345

degenerate modes of, 337f

frequencies of, 334

Liouville’s theorem, 266f, 269

Lissajous figures, 69, 291, 296, 337

Longitudinal mass, 205

Logarithmic potential, 73

Lorentz-Fitzgerald contraction, 192

Lorentz force, 19—20

Lorentz transformation, 186f

derivation of, 187f

equations of, 191

as including spatial rotation, 188

inverse, 191

matrix for, 191

pure, 188

reality conditions on matrix elements of, 189

successive, 193

Macmillan, W. D., 166

Magnetic field, 19, 176, 213, 306

Magnetic moment, 176

and angular momentum, 177

Magnetic quantum number, 307

Magnetic rigidity, 213

Major axis of elliptic orbit, 79, 304

Mass, 1

Matrix, inverse, 102

one-column, 102

unit, 103

Matrix addition, 102

Matrix elements, 98

Matrix multiplication, 101, 141

Matrix of transformation, 98

Maupertuis, Pierre de, 231 Maxwell’s equations, 19, 47, 366 from a Lagrangian, 367

Mechanics

(see under qualifying adjective: analytical, classical, quantum, statistical)

Mercury, precession of the perihelion of, 91, 214

Mesons, 213, 370

Metric tensor, 233, 310, 326

Microcanonical ensemble, 268

Million electron volt, 27

Minimum surface of revolution, 35

Minkowski force, 200f, 210

Minkowski space, 188

Minor axis of elliptic orbit, 79

Moderator, 89

Modified Hamilton’s principle, 225f, 239, 276, 371

Molecular spectra, 140, 318, 345

Molecular vibration, 333f, 345

Moment of force, 2

Moment of inertia, 149

about parallel axes, 150f

coefficients, 145

equivalent definitions for, 150

principal, 154

Momentum, as independent variable in Hamiltonian formulation, 227, 237f, 245

and wave length, 313

Momentum, angular

(see: Angular momentum)

canonical, 48, 206, 261, 360

conjugate, 48

for electromagnetic forces, 48, 207, 209, 223

four-vector, 203

linear

(see: Linear momentum)

Momentum density, 360

for electromagnetic field, 368

Multiple Fourier expansion, 295,318,330

Multiple toots, obtaining orthogonal eigenvectors for, 154, 328f

of secular equation, 123, 154, 327

Multiply periodic motion, 295f, 314, 315, 318, 330

Neutron-proton scattering, 88

Neutron pile, 89

Newton’s Second Law of Motion; 1, 135, 185

in special relativity, 200f

Newton’s Third Law of Motion, 4

electromagnetic forces as exception to, 4

Nonholonomic constraints, 11, 40

Nonion form of dyads, 148

Normal coordinates, 331f

in forced vibrations, 338f

kinetic and potential energy in terms of, 331

Lagrangian and equations of motion in, 332

Normal modes of vibration, 332f

Normal modes, corresponding to zero

frequencies, 334f

effect of frictional forces on, 340f

of linear triatomic molecule, 335f

Nutation, 168f, 175

astronomical, 175

Olson, H. F., 46

Operational calculus, 346

Optics, and classical mechanics, 92, 231, 307f

(see also: Geometrical optics)

Optical path length, 311

Orbit equations, from least action principle, 234

from Hamilton-Jacobi equation, 281, 287f, 317

Orbits, bounded, 66

circular, conditions for, 67, 78

classification of, 6Sf, 78f

closed, 66, 297

differential equations for, 71f

for inverse square law of force, 67, 76f, 304f, 317

number of constants of integrations in, 77f

in phase space, types of periodic, 288f

in scattering by central force, 81

Order of finite rotations, 124f

Orthogonal matrix, 98

as diagonalizing inertia tensor, 153

of rigid body orientation, 98, 109, 114, 118

as rotation operator, 118

Orthogonal transformations, 97f, 188, 195, 238, 244f

Orthogonality conditions, 96, 98, 104f, 325

in curvilinear space, 325f

Orthogonality of eigenvectors, of inertia tensor, 153f

for small oscillations, 323, 324f

Orthogonalization of eigenvectors for multiple roots, 328

Orthonormal eigenvectors, 328

Oscillations, about stable equilibrium, 318ff

about stable motion, 318, 344, 346

small

(see: Small oscillations)

(see also: Vibrations)

Oscillatory motion, nature of, 288

Pair creation, 204

Particle, conservation theorems for, 2f

charge density for, 368f

mechanics of, 1f

Path in configuration space, equation for, 233f

Path length, element of, 233

Pauli spin matrices, 1 16f, 140, 142

Pendulum, simple, 289

Perihelion, 91, 214, 305, 306

Period of elliptic orbits, 79, 304

Periodic motion, 288f

Fourier expansion for, 294f

frequency of in terms from angle variables, 293

for one degree of freedom, 288f

oscillatory, 288

Perturbation theory, 305, 307, 315

Phase, 311, 313

Phase space, 247, 266f

invariance of volume in, 250, 268

Pierce, B.. 0., 77

Planck’s constant, 255, 306, 314

Plane waves, 310f

Planetary motion, 61, 80

Poincar#’s integral invariants, 247f, 268, 271

Poinsot’s construction, 159f, 179, 182

Point transformations, 238, 244, 297f

Poisson brackets, 252, 269, 363

algebraic properties of, 255

of angular momentum and a scalar, 265

as canonical invariants, 254f, 272

connection with Lagrange brackets, 252f

and constants of the motion, 256, 258, 272, 364

derivatives with respect to Q and P in terms of, 254f

equations of motion in terms of, 255f fundamental, 253f, 272

involving angular momentum, 263f, 269, 272

of two components of angular momentum, 265, 272

with H, time derivatives in terms of, 256f, 363f

Poisson’s theorem, 268, 265

Polhode, 161

Postfactor, 148

Postulate of equivalence, 186

Potential, 3

arbitrary zero of, 4

generalized, 19

velocity-dependent, 19

Potential energy, 3, 144

density of in gases, 355f, 358

at equilibrium, 318f, 324

form of for small oscillations, 320, 331

internal, 10

total, 10

Precession, astronomical, 163, 182f

"fast" and "slow," 173

of heavy symmetrical top, 168f

Larmor, 178, 306

pseudoregular, 171, 173, 179

Thomas, 212

in uniform magnetic field, 176f, 183f

Precession of axis of rotation, for force-free rigid body, 161f, 175, 181

Precession of the equinoxes, 163, 174f, 182f

of the North Pole, 163

of the perihelion, 91, 214, 306

Prefactor, 148

Principal axes, 154f, 327

Principal axis transformation, 154f

for small oscillations, 327, 332

Principal quantum number, 306

Principle of least action, 228f, 312

Jacobi’s form, 233

variations of, 231f

Products of inertia, 145

Projectiles, Coriolis forces on, 137, 142

Propagation of sound in gases, 355f, 371

Proper time, 196

Pseudoscalar, 131

Pseudovectors, 130f

Quadratic forms, simultaneous diagonalization of, 332, 344

Quantization of fields, 47, 333, 370

Quantum electrodynamics, 47, 333

Quantum mechanics, 63, 81, 82, 84f, 89, 97, 108, 110, 114, 119, 185, 223,237, 266, 269, 306f, 314

Quantum numbers, 306f

Quantum of action, 306

Quantum relations, for angular momenta, 266

Quimby, S. L., 365

Radial quantum number, 306

Radius of gyration, 156

Rayleigh, Lord, 343

Rays, 312

Recoil, 87f

Rectangular potential well, 92

Reduced mass, 59, 80

Reflection of orbits about turning points, 72

Reflection operator, 122

Relativistic Hamiltonian, 207, 222

Relativistic kinetic energy, 203

Relativistic mass, 204

Relativistic momentum, 201f, 204

Relativity (see: Special relativity)

Residues, method of, 302f, 315

Resonance phenomena, 339f, 343, 345

Resonant frequencies, 329 zero, 334f

Rest energy, 203

Rest mass, 203

Rigid body, 10, 211

angular momentum for, 144

kinetic energy for, 149

number of degrees of freedom for, 93f

orientation specified by Cayley-Klein parameters, 114

orientation specified by direction cosines, 95

orientation specified by Euler angles, 109, 141

symmetrical, force-free motion of, 161f

Rigid body modes of molecular vibration, 334, 345f

Rigid body motion, general procedures for solving, 156f

force-free, 159f, 178, 181, 182

in a plane, 156

resolution into rotation and translation, 143

Rocket propulsion, 5, 28, 213

Rolling, 12-13, 43, 141, 156, 161, 179, 181

Rotating coordinate axes, 55, 135f, 179, 236, 270

Rotating coordinates, transformation to, 132f, 212

Rotating earth, measurements on, 135f

Rotation, change of vector function under, 263f

generated by angular momentum, 263

invariance of physical laws under, 195

proper and improper, 122, 127

as type of periodic motion, 289, 295

Rotation of coordinate axes, two-dimensional, 99, 109, 262

Rotation operator, 100, 118

in spin or space, 116f

not a vector, 124

Rotational symmetry, 51f, 220, 262

Routh’s procedure for cyclic coordinates, 49, 219f

Routhian, 219

Rutherford scattering, 84f, 88, 91

Scalar, as a tensor of the zero rank, 146

Scattering, by Coulomb field, 83f

differences in classical and quantum mechanics, 81

elastic, 88, 201

inelastic, 90, 204

transfer of kinetic energy in, 88

Scattering angle, in center of mass coordinates, 81, 86

formal solution for, 92

in laboratory coordinates, 85

Scattering cross section, 81

in center of mass and laboratory coordinates, 88

Schering, E., 19

Schr#dinger equation, 47, 314, 372

Schwinger, J., 266

Second derivatives, in the Lagrangian, 56

Secular determinant, 119, 321

Secular equation, 120, 154, 321,. 327

Self-adjoint matrix, 112

Separation of variables, dependent on set of coordinates, 285

in Hamilton-Jacobi equation, 279, 284ff, 299f

in small oscillations, 330

when coordinates are cyclic, 285f

when possible, 285, 288, 315

Similarity transformation, 105, 322, 326

Simple harmonic motion, 68, 69, 333

Simply periodic motion, 288

Skew-symmetric matrix, 127

"Sleeping" top, 174

Small oscillations, for continuous systems, 318, 343f, Chapter 11

effects of constraints, 344, 345

elgenvalue problem for, 321f

and electrical network theory, 343

forced, 338f

free vibration solutions, 329f, 332

with frictional forces, 340f

of molecules, 333f

multiple roots in, 327f

normal coordinates for, 330f

normal modes of, 332f

rigid body normal modes, 334f, 345f

Sommerfeld, A., 239, 302, 306

Sound

(see: Acoustics)

Sound field, 355f, 364, 365f

Space like four-vector, 197

Spatial derivatives, in Lagrangian of continuous systems, 350

Special relativity, 97

included in classical mechanics, 1, 185

program of, 187

Spherical pendulum, 28, 69, 272

Spin, 9, 118, 177, 307

Spinor, 118

Spur of a matrix, 112

Statics, 15

Statistical mechanics, 237, 266f, 269

Stereographic projection, 302

Sturn-Liouville problem, 154

Stratton, J. A., 176

String, transverse vibrations of, 371, 372

Symmetrical top

(see: Top, heavy symmetrical)

Symmetry properties, of Hamiltonian, 261

of orbits in central forces, 72

of systems, and conservation theorems, 47f, 50, 53, 220f, 261

Systems, continuous (see: Continuous systems)

invariance properties of, 52f, 261f

Tensors, 146f, 195

Thermodynamics, 216, 235, 356f

Thomas precession, 212

Time, as canonical coordinate, 210, 243, 271, 281, 286

canonical transformations involving, 242f, 271

as fourth dimension in Minkowaki space, 188

transformation of in special relativity, 188, 192f

Time derivatives, as observed in space and body axes, 133f

Time like four-vector, 197

Time of transit an extremum, 231

Top, asymmetrical, 179, 180

charged, in magnetic field, 176f, 183f

fast, 169

heavy symmetrical, 164ff, 236, 316

average precession frequency, 171

condition for regular precession, 172f, 182

constants of the motion for, 165

extent of nutation of, 169f

fast and slow regular precession, 173

formal solution for, 166

frequency of nutation of, 170

initial motion of, 169, 171

initially vertical, 173f

magnitude of angular momentum, 182

nutation of, 168, 171

Top, "sleeping," 173f

Torque, 2

Torque equation, 3,176

Total scattering cross section, 84

infinite value in classical mechanics, 85

Trace of a matrix, 112, 124

invariance under similarity transformation, 112, 141

Trajectories, in configuration space, 309f

of light rays, 312

Transformation, between body and space set of coordinates, 95f, 132f

canonical from center of mass to laboratory coordinates, 87f

(see: Canonical transformations)

Transformation matrix, as operator on vector or coordinate axes, 100

Transients, 339, 346

Translational symmetry, 50, 334, 346

Transposed matrix, 104, 141

Transverse mass, 205

Turning points, 66. 77, 167

Uniformly moving systems, 186f

Unit dyadic, 148

Unit matrix, 103

Unit rotator, 117

Unitary matrix, 105, 110

Van Vieck, J. H., 301

Variation, 31, 33, 225

for constant time, 30, 225f

for continuous systems, 351

of end points, 38, 228f

including time, 228f

in terms of functional derivatives, 353f, 360f

Variational derivative, 353

Variational principles, advantages of, 44ff, 205, 225, 235, 370

for continuous systems, 350f

for fields, 364f, 370

Vector, as tensor of the first rank, 147

Vector functions, obeying angular momentum Poisson brackets, 264

Vector potential, 20, 366

Vector transformations, 97

Velocity, 1

of light, as maximum velocity, 193

experimental invariance, 186

of longitudinal elastic vibrations, 355

of sound in gases, 359

Velocity-dependent potentials, 19f

Vertical direction, definition of, 136

Vibrations, of a gas, 355f

of strings and membranes, 344, 345, 375

theory of, 46, Chapter 10

(see also: Oscillations)

Virial of Clausius, 70

Virial theorem, 69f, 78, 90, 91, 214

Virtual displacement, 14, 16, 33, 225

Virtual work, for moving constraints, 54

principle of, 15

Volume in phase space, 250, 267

Wave function, 118, 314

Wave equation, for light, 187, 198, 310f, 313

for sound in gases, 359, 365

Wave fronts, 308

Wave length, associated with classical mechanics, 312f

and momentum, 313

Wave mechanics, 307, 314

Wave motion, and classical mechanics, 307ff, 314

Wave number, 311

Wave velocity, and system velocity, 309

of surfaces of constant S, 308f

Waves, acoustic, 359

Weber’s electrodynamics, 19, 28, 236

Webster, A. G., 48, 161

Whittaker, E. T., 19, 31, 114,166

Whittaker, E. T. and Watson, G. N., 74

Wind circulation, 137-8

Winkelmann, M. and Grammel, R., 161

Work, 3

done by sound field in gases, 356f

World line, 196

World momentum, 203

World scalar, 195

World space, 188

World time, 196

World vector, 195

World velocity, 198

Young’s modulus. 349

Zeeman effect, 307

Zemansky, M. W., 357