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ОбложкаGregory R.D. Classical mechanics: an undergraduate text. - Cambridge; New York: Cambridge University Press, 2006. - xii, 596 p.: ill. - Bibliogr.: p.589-590. - Ind.: p.591-596. - ISBN 978-0-521-53409-3
Шифр: (И/В2-G79) 02
 

Место хранения: 02 | Отделение ГПНТБ СО РАН | Новосибирск

Оглавление / Contents
 
 
   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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