Preface ..................................................... xi
I Newtonian mechanics of a single particle ..................... 1
1 The algebra and calculus of vectors .......................... 3
1.1 Vectors and vector quantities ........................... 3
1.2 Linear operations: a + b and λa ......................... 5
1.3 The scalar product a • b ............................... 10
1.4 The vector product a × b ............................... 13
1.5 Triple products ........................................ 15
1.6 Vector functions of a scalar variable .................. 16
1.7 Tangent and normal vectors to a curve .................. 18
Problems .................................................... 22
2 Velocity, acceleration and scalar angular velocity .......... 25
2.1 Straight line motion of a particle ..................... 25
2.2 General motion of a particle ........................... 28
2.3 Particle motion in polar co-ordinates .................. 32
2.4 Rigid body rotating about a fixed axis ................. 36
2.5 Rigid body in planar motion ............................ 38
2.6 Reference frames in relative motion .................... 40
Problems .................................................... 43
3 Newton's laws of motion and the law of gravitation .......... 50
3.1 Newton's laws of motion ................................ 50
3.2 Inertial frames and the law of inertia ................. 52
3.3 The law of mutual interaction; mass and force .......... 54
3.4 The law of multiple interactions ....................... 57
3.5 Centre of mass ......................................... 58
3.6 The law of gravitation ................................. 59
3.7 Gravitation by a distribution of mass .................. 60
3.8 The principle of equivalence and g ..................... 67
Problems .................................................... 71
4 Problems in particle dynamics ............................... 73
4.1 Rectilinear motion in a force field .................... 74
4.2 Constrained rectilinear motion ......................... 78
4.3 Motion through a resisting medium ...................... 82
4.4 Projectiles ............................................ 88
4.5 Circular motion ........................................ 92
Problems .................................................... 98
5 Linear oscillations ........................................ 105
5.1 Body on a spring ...................................... 105
5.2 Classical simple harmonic motion ...................... 107
5.3 Damped simple harmonic motion ......................... 109
5.4 Driven (forced) motion ................................ 112
5.5 A simple seismograph .................................. 120
5.6 Coupled oscillations and normal modes ................. 121
Problems ................................................... 126
6 Energy conservation ........................................ 131
6.1 The energy principle .................................. 131
6.2 Energy conservation in rectilinear motion ............. 133
6.3 General features of rectilinear motion ................ 136
6.4 Energy conservation in a conservative field ........... 140
6.5 Energy conservation in constrained motion ............. 145
Problems ................................................... 151
7 Orbits in a central field .................................. 155
7.1 The one-body problem-Newton's equations ............... 157
7.2 General nature of orbital motion ...................... 159
7.3 The patlfequation ..................................... 164
7.4 Nearly circular orbits ................................ 167
7.5 The attractive inverse square field ................... 170
7.6 Space travel - Hohmann transfer orbits ................ 177
7.7 The repulsive inverse square field .................... 179
7.8 Rutherford scattering ................................. 179
Appendix A The geometry of conies ......................... 184
Appendix В The Hohmann orbit is optimal ................... 186
Problems ................................................... 188
8 Non-linear oscillations and phase space .................... 194
8.1 Periodic non-linear oscillations ...................... 194
8.2 The phase plane ((x1,X2)-plane) ....................... 199
8.3 The phase plane in dynamics (x, ν)-plane) ............. 202
8.4 Poincare-Bendixson theorem: limit cycles .............. 205
8.5 Driven non-linear oscillations ........................ 211
Problems ................................................... 214
2 Multi-particle systems ..................................... 219
9 The energy principle ....................................... 221
9.1 Configurations and degrees of freedom ................. 221
9.2 The energy principle for a system ..................... 223
9.3 Energy conservation for a system ...................... 225
9.4 Kinetic energy of a rigid body ........................ 233
Problems ................................................... 241
10 The linear momentum principle .............................. 245
10.1 Linear momentum ....................................... 245
10.2 The linear momentum principle ......................... 246
10.3 Motion of the centre of mass .......................... 247
10.4 Conservation of linear momentum ....................... 250
10.5 Rocket motion ......................................... 251
10.6 Collision theory ...................................... 255
10.7 Collision processes in the zero-momentum frame ........ 259
10.8 The two-body problem .................................. 264
10.9 Two-body scattering ................................... 269
10.10 Integrable mechanical systems ........................ 273
Appendix A Modelling bodies by particles .................. 277
Problems ................................................... 279
11 The angular momentum principle ............................. 286
11.1 The moment of a force ................................. 286
11.2 Angular momentum ...................................... 289
11.3 Angular momentum of a rigid body ...................... 292
11.4 The angular momentum principle ........................ 294
11.5 Conservation of angular momentum ...................... 298
11.6 Planar rigid body motion .............................. 306
11.7 Rigid body statics in three dimensions ................ 313
Problems ................................................... 317
3 Analytical mechanics ....................................... 321
12 Lagrange's equations and conservation principles ........... 323
12.1 Constraints and constraint forces ..................... 323
12.2 Generalised coordinates ............................... 325
12.3 Configuration space (q-space) ......................... 330
12.4 D'Alembert's principle ................................ 333
12.5 Lagrange's equations .................................. 335
12.6 Systems with moving constraints ....................... 344
12.7 The Lagrangian ........................................ 348
12.8 The energy function h ................................. 351
12.9 Generalised momenta ................................... 354
12.10 Symmetry and conservation principles ................. 356
Problems ................................................... 361
13 The calculus of variations and Hamilton's principle ........ 366
13.1 Some typical minimisation problems .................... 367
13.2 The Euler-Lagrange equation ........................... 369
13.3 Variational principles ................................ 380
13.4 Hamilton's principle .................................. 383
Problems ................................................... 388
14 Hamilton's equations and phase space ....................... 393
14.1 Systems of first order ODEs ........................... 393
14.2 Legendre transforms ................................... 396
14.3 Hamilton's equations .................................. 400
14.4 Hamiltonian phase space ((q, p)-space) ................ 406
14.5 Liouville's theorem and recurrence .................... 408
Problems ................................................... 413
4 Further topics ............................................. 419
15 The general theory of small oscillations ................... 421
15.1 Stable equilibrium and small oscillations ............. 421
15.2 The approximate forms of T and V ...................... 425
15.3 The general theory of normal modes .................... 429
15.4 Existence theory for normal modes ..................... 433
15.5 Some typical normal mode problems ..................... 436
15.6 Orthogonality of normal modes ......................... 444
15.7 General small oscillations ............................ 447
15.8 Normal coordinates .................................... 448
Problems ................................................... 452
16 Vector angular velocity and rigid body kinematics .......... 457
16.1 Rotation about a fixed axis ........................... 457
16.2 General rigid body kinematics ......................... 460
Problems ................................................... 467
17 Rotating reference frames .................................. 469
17.1 Transformation formulae ............................... 469
17.2 Particle dynamics in a non-inertial frame ............. 476
17.3 Motion relative to the Earth .......................... 478
17.4 Multi-particle system in a non-inertial frame ......... 485
Problems ................................................... 489
18 Tensor algebra and the inertia tensor ...................... 492
18.1 Orthogonal transformations ............................ 493
18.2 Rotated and reflected coordinate systems .............. 495
18.3 Scalars, vectors and tensors .......................... 499
18.4 Tensor algebra ........................................ 505
18.5 The inertia tensor .................................... 508
18.6 Principal axes of a symmetric tensor .................. 514
18.7 Dynamical symmetry .................................... 516
Problems ................................................... 519
19 Problems in rigid body dynamics ............................ 522
19.1 Equations of rigid body dynamics ...................... 522
19.2 Motion of 'spheres' ................................... 524
19.3 The snooker ball ...................................... 525
19.4 Free motion of bodies with axial symmetry ............. 527
19.5 The spinning fop ...................................... 531
19.6 Lagrangian dynamics of the top ........................ 535
19.7 The gyrocompass ....................................... 541
19.8 Euler's equations ..................................... 544
19.9 Free motion of an unsymmetrical body .................. 549
19.10 The rolling wheel .................................... 556
Problems ................................................... 560
Appendix Centres of mass and moments of inertia ............... 564
A.1 Centre of mass ........................................ 564
A.2 Moment of inertia ..................................... 567
A.3 Parallel and perpendicular axes ....................... 571
Answers to the problems ....................................... 576
Bibliography ............................................... 589
Index ...................................................... 591
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