A Treatise on Analytical Dynamics

Title Page......Page 1
Copyright......Page 2
Introduction......Page 5
Contents......Page 11
1.1 The free particle......Page 21
1.2 Rectilinear motion in a field......Page 23
1.3 Libration motion......Page 29
1.4 The given force cannot be a function of acceleration......Page 31
1.5 The constrained particle (i)......Page 32
1.6 The constrained particle (ii)......Page 34
1.7 The constrained particle (iii)......Page 35
1.8 Holonomic and non-holonomic systems......Page 36
1.9 Two constraints......Page 38
2.1 A simple example......Page 40
2.2 The dynamical system......Page 42
2.4 The forces of constraint......Page 44
2.5 The idea of a dynamical system......Page 45
3.1 The fundamental equation......Page 48
3.2 The conservation of momentum......Page 49
3.3 The catastatic system and the first form of the equation of energy......Page 50
3.4 Conservative forces and the second form of the equation of energy......Page 51
3.5 The third form of the equation of energy......Page 53
3.7 Hamilton's principle......Page 54
3.8 The varied path......Page 56
3.9 Continuous systems......Page 57
4.1 The second form of the fundamental equation......Page 60
4.2 The third form of the fundamental equation......Page 61
4.4 Applications of Gauss's principle......Page 62
4.5 The physical significance of Gauss's principle......Page 63
5.1 The idea of Lagrangian coordinates......Page 65
5.2 Some classical problems......Page 67
5.3 The spherical pendulum......Page 76
5.4 The problem of two bodies......Page 79
5.5 Kepler's equation......Page 82
5.6 Collision......Page 83
5.7 Lagrangian coordinates for a holonomic system......Page 84
5.8 Lagrangian coordinates for a non-holonomic system......Page 85
5.9 Rolling bodies......Page 87
5.11 The varied path in Hamilton's principle......Page 89
5.12 Summary......Page 91
6.1 The fourth form of the fundamental equation, Lagrangian coordinates......Page 93
6.2 Lagrange's equations......Page 95
6.3 Lagrange's equations deduced from Hamilton's Principle......Page 96
6.4 The form of the equations......Page 98
6.5 Conservative systems and other systems with a potential function......Page 99
6.6 The Lagrangian function......Page 101
6.7 Jacobi's integral......Page 102
6.8 The explicit form of Jacobi's integral......Page 103
6.9 An insidious fallacy......Page 106
6.10 The generalized momentum-components......Page 107
6.12 The invariance of Lagrange's equations......Page 108
7.2 Euler's theorem......Page 110
7.3 The matrix I and the vector T......Page 112
7.4 Generalization of Euler's theorem......Page 114
7.6 The rotation formula......Page 115
7.7 Half-turns and reflexions......Page 117
7.8 Quaternion form of the rotation formula......Page 118
7.9 Composition of rotations......Page 119
7.10 Angular velocity......Page 122
7.12 The orientation of a rigid body, the angles 92,, q?21 q?3......Page 123
7.14 Rotations about the fixed axes......Page 124
7.15 Angular velocity found from 1 and 1......Page 125
7.16 Components of angular velocity......Page 126
8.1 The differential equations......Page 128
8.2 Formulae for acceleration in general orthogonal coordinates......Page 129
8.4 Kinetic energy of a rigid body......Page 130
8.5 A problem of motion in two dimensions -......Page 131
8.6 The spinning top, the fundamental equations......Page 133
8.7 The spinning top, another method......Page 134
8.9 The spinning top, study of the motion......Page 135
8.10 Numerical example......Page 138
8.11 Rod in a rotating plane......Page 139
8.12 The rolling penny......Page 140
9.1 Oscillations about equilibrium......Page 143
9.2 Theory of the transformation to normal coordinates......Page 152
9.3 Application of the theory......Page 157
9.4 Imposition of a constraint......Page 160
9.5 Rayleigh's principle......Page 161
9.6 Stability of a steady motion......Page 163
9.7 Oscillations about steady motion......Page 167
9.8 Foucault's gyroscope......Page 170
9.9 The sleeping top......Page 172
9.10 Forced oscillations......Page 176
10.1 The ignoration of coordinates......Page 179
10.2 Ignoration of a single coordinate......Page 180
10.3 Gyroscopic stability......Page 182
10.4 Explicit expression for R in the general case......Page 184
10.5 The spinning top......Page 185
10.6 Linear terms in L......Page 186
10.7 Motion relative to a moving base......Page 189
10.8 Motion of a particle near a given point on the Earth's surface......Page 192
10.9 Foucault's pendulum......Page 193
10.10 Projectiles......Page 194
10.11 Rayleigh's dissipation function......Page 199
10.12 Gyroscopic system with dissipation......Page 201
10.13 Hamilton's equations......Page 202
10.14 The equation of energy and the explicit form of H......Page 205
10.15 The principal solid......Page 207
11.1 Particles with variable mass, the Lagrangian function......Page 210
11.2 The kinetic energy......Page 211
11.4 The moving electron......Page 212
11.5 Electron in an electromagnetic field......Page 214
12.2 Quasi-coordinates......Page 217
12.3 The fifth form of the fundamental equation......Page 219
12.4 Determination of the acceleration......Page 220
12.5 The Gibbs-Appell equations......Page 221
13.1 Particle moving in a plane......Page 223
13.3 Two-dimensional problems......Page 224
13.4 Motion of a rigid body in space......Page 225
13.5 Sphere on turntable......Page 227
13.6 Sphere on a rotating inclined plane......Page 229
13.7 Sphere rolling on a fixed surface......Page 231
13.8 The spinning top......Page 233
13.9 The rolling penny......Page 234
13.10 Euler's equations......Page 235
13.11 The free body, the case of axial symmetry......Page 236
13.12 The free body, the general case......Page 238
13.13 The free body, orientation......Page 242
13.14 The theorems of Poinsot and Sylvester......Page 243
13.15 Ellipsoid rolling on a rough horizontal plane......Page 244
13.16 Stability of the spinning ellipsoid......Page 245
14.1 The theory of impulses......Page 248
14.2 Impulsive constraints......Page 250
14.3 Impulsive motion of a system, the fundamental equation......Page 251
14.4 The catastatic system......Page 252
14.5 The principle of Least Constraint for impulses......Page 253
14.6 The catastatic system, the superposition theorem......Page 254
14.7 The catastatic system, the six energy theorems......Page 255
14.8 Lagrangian coordinates and quasi-coordinates......Page 258
14.9 Lagrange's form of the impulse equations......Page 261
14.10 The energy theorems reconsidered......Page 262
14.11 Examples of impulsive motion......Page 263
14.12 Impulsive motion of a continuous system......Page 268
15.2 Immediate deductions......Page 273
15.3 The Routhian function......Page 275
15.4 d The theorem at (p,. bqr) = 6L......Page 276
15.5 The principal function......Page 278
15.6 Reflexions on the principal function......Page 279
15.7 Proof that aS/ato = Ho......Page 280
15.8 Properties of the principal function......Page 281
15.9 Examples of the direct calculation of the principal function......Page 285
16.1 Hamilton's partial differential equation......Page 288
16.2 The Hamilton-Jacobi theorem, first proof......Page 289
16.3 The equivalence theorem......Page 291
16.5 Reflexions on the Hamilton-Jacobi theorem......Page 295
16.6 Uniform field......Page 297
16.7 The harmonic oscillator......Page 299
16.8 Particle in a varying field At......Page 301
16.9 Central orbit......Page 302
16.10 The spherical pendulum......Page 303
16.11 The spinning top......Page 304
16.12 Rod in rotating plane......Page 305
16.13 Electron under a central attraction......Page 306
16.14 The Pfaffian form p, dq,. - H dt......Page 308
17.2 Two degrees of freedom, conditions for separability......Page 311
17.3 Study of the motion......Page 314
17.4 Classification of the orbits......Page 316
17.5 Stability......Page 318
17.6 Application of the theory......Page 319
17.7 Central attraction k/r"+i......Page 320
17.8 Central attraction klr5......Page 322
17.9 Newtonian attraction and uniform field......Page 326
17.10 Two fixed Newtonian centres......Page 329
17.11 The bounded orbits......Page 333
17.12 The equations of the orbits......Page 336
17.13 The unbounded orbits......Page 337
17.14 Systems that are separable in more than one way......Page 338
18.1 Liouville's system......Page 340
18.2 Stackel's theorem......Page 341
18.3 Discussion of the integrals......Page 344
18.4 Further comments on Stackel's theorem......Page 345
18.5 Quasi-periodic motions......Page 346
18.6 Angle variables......Page 349
18.7 The standard cube......Page 351
18.8 The constants I,.......Page 352
18.9 Relations connecting q's and v's......Page 354
18.10 Small oscillations......Page 355
18.11 The spherical pendulum......Page 357
18.12 The problem of two bodies......Page 359
18.13 Interpretation of the a's and fl's......Page 361
18.14 Expression of r as a function of t......Page 363
18.16 The constants I,.......Page 364
18.17 Perturbation......Page 367
18.18 Non-orthogonal and non-natural separable systems......Page 368
19.1 The differential equations......Page 369
19.2 Particle moving on a straight line......Page 373
19.4 Motion in the neighbourhood of a singular point, the linear approximation......Page 376
19.5 Stability of equilibrium, complete stability and instability......Page 383
19.6 Motion in the neighbourhood of a singular point, the general theory......Page 384
19.7 Motion near a node......Page 385
19.8 Motion near a saddle point......Page 388
19.9 Motion near a spiral point......Page 391
19.10 Motion near a vortex point......Page 392
19.11 Relation of the linear approximation to the general theory......Page 396
20.1 Index of a curve, and index of a singular point......Page 399
20.2 The positive limiting set......Page 401
20.3 Segment without contact......Page 403
20.4 Segment without contact through a point of A......Page 404
20.6 The Poincare-Bendixson theorem......Page 405
20.7 Application to a particular system......Page 407
20.8 Existence of the limit cycle......Page 409
20.9 Van der Pol's equation......Page 412
21.1 Integrals of the system of differential equations......Page 415
21.2 Transformation to new coordinates......Page 419
21.3 The operator Tt......Page 420
21.4 Solution in power series......Page 421
21.5 A formula for X(x) - X(a)......Page 424
21.6 Integral-invariants......Page 425
21.7 Integral-invariants of order m......Page 428
21.8 Properties of the multipliers......Page 430
21.9 Jacobi's last multiplier......Page 431
21.11 Stability of equilibrium......Page 433
21.12 Discrete stability......Page 435
21.13 Stability of transformations......Page 437
21.14 Application to the differential equations......Page 439
21.15 The Poincare-Liapounov theorem......Page 440
21.16 The critical case......Page 443
22.1 Hamilton's equations......Page 447
22.2 Poisson brackets......Page 448
22.3 Poisson's theorem......Page 449
22.4 Use of a known integral......Page 450
22.5 Poincare's linear integral-invariant......Page 453
22.6 Liouville's theorem......Page 454
22.7 Poincare's recurrence theorem......Page 455
22.8 Examples of invariant regions......Page 456
22.9 Ergodic theorems......Page 457
22.10 Concrete illustrations......Page 458
22.11 The set K8......Page 459
22.12 Proper segments......Page 460
22.13 Proof of the ergodic theorem, first stage......Page 461
22.14 Proof of the ergodic theorem, second stage......Page 462
22.15 Metric indecomposability......Page 463
22.17 A corollary to Liouville's theorem......Page 466
22.18 The last multiplier......Page 467
23.1 The variational equations......Page 473
23.2 Solution of the variational equations......Page 475
23.3 The case of constant coefficients......Page 478
23.4 The case of periodic coefficients......Page 481
23.5 Zero exponents......Page 483
23.6 Variation from the Hamiltonian equations......Page 485
23.7 Stability of trajectories (i)......Page 487
23.8 Stability of trajectories (ii)......Page 494
23.9 Stability of a periodic orbit......Page 495
23.10 Forced oscillations......Page 497
24.1 Contact transformations......Page 504
24.2 Explicit formulae for a contact transformation......Page 506
24.3 Other formulae......Page 508
24.4 Extended point transformations and other homogeneous contact transformations......Page 510
24.6 The extension of Liouville's theorem......Page 512
24.8 Relations between the two sets of derivatives......Page 513
24.10 Relations between Lagrange brackets and Poisson brackets......Page 515
24.12 Invariance of a Poisson bracket......Page 516
24.13 Another form of the conditions for a contact transformation......Page 517
24.14 Functions in involution......Page 518
24.15 Some concrete examples......Page 519
25.1 The equations of motion after a contact transformation......Page 522
25.2 The variation of the elements......Page 524
25.3 The variation of the elliptic elements......Page 528
25.4 Other proofs of Jacobi's theorem......Page 531
25.5 The constancy of Lagrange brackets......Page 535
25.6 Infinitesimal contact transformations......Page 536
25.7 Integrals in involution......Page 537
25.8 Lie's theorem on involution systems......Page 540
25.9 Integrals linear in the momenta......Page 541
25.10 The case of a Hamiltonian function which is a homogeneous quadratic form......Page 543
26.1 Hamilton's principle......Page 548
26.2 Livens's theorem......Page 550
26.3 Minima and saddle points......Page 551
26.4 Non-contemporaneous variations, Holder's principle......Page 553
26.5 Voss's principle......Page 554
26.6 The generalization of Hamilton's principle......Page 555
26.7 Change of the independent variable......Page 556
26.8 Normal form for a system with two degrees of freedom......Page 557
26.9 Liouville's system......Page 558
26.10 Conformal transformation......Page 560
27.1 The variation of the Action......Page 563
27.2 The principle of Least Action......Page 564
27.3 Jacobi's form of the principle of Least Action......Page 566
27.4 Whittaker's theorem......Page 568
27.5 The ignoration of coordinates......Page 570
27.6 The characteristic function......Page 571
27.7 The configuration space......Page 572
27.8 System with two degrees of freedom......Page 573
27.9 Kelvin's theorem......Page 574
27.10 Uniform field......Page 576
27.11 Tait's problem, direct solution......Page 578
27.12 Tait's problem, envelope theory......Page 579
28.1 The problem of three bodies......Page 582
28.2 The restricted problem, the equations of motion......Page 583
28.3 Positions of equilibrium......Page 584
28.4 Equilibrium points on AB......Page 585
28.5 Equilibrium points not on AB......Page 587
28.6 The surface z = U......Page 588
28.7 Motion near a point of equilibrium......Page 589
28.8 Lunar theory......Page 591
29.1 The classical integrals......Page 593
29.2 The case of vanishing angular momentum......Page 595
29.3 Lagrange's three particles......Page 596
29.4 Fixed-shape solutions......Page 598
29.5 Motion in a plane......Page 600
29.7 Motion near the equilibrium solution......Page 602
29.8 Reduction to the sixth order......Page 605
29.9 Stability of Lagrange's three particles......Page 607
29.10 Reduced form of the equations of motion......Page 608
29.11 Lagrange's three particles reconsidered......Page 610
29.12 Reduction to the eighth order......Page 612
29.13 Impossibility of a triple collision......Page 616
29.14 Motion in a plane, another method of reduction to the sixth order......Page 619
29.15 Equilibrium solutions......Page 621
30.2 Periodic motion near a singular point......Page 624
30.3 Reality conditions......Page 627
30.4 Hamiltonian equations......Page 628
30.5 Convergence......Page 631
30.6 Lagrange's three particles......Page 633
30.7 Systems involving a parameter......Page 635
30.8 Application to the restricted problem of three bodies......Page 638
30.10 Poincare's ring theorem......Page 641
30.11 Periodic orbits and the ring theorem......Page 642
30.12 Proof of Poincare's ring theorem......Page 646
Notes......Page 650
Bibliography......Page 655
Index......Page 659