Lectures in classical mechanics

Preface......Page 7
Acknowledgements......Page 9
Contents......Page 10
 1.1 Introduction......Page 14
 1.2 Operations with Vectors......Page 15
 1.3 Vector Operators......Page 17
 1.4 Change of Basis......Page 18
 1.5 Proposed Exercises......Page 21
 Further Reading......Page 28
 2.1 Introduction......Page 29
 2.2 Vector Operators......Page 33
 2.3 Cylindrical and Polar Coordinates......Page 34
 2.4 Spherical Coordinates......Page 36
 2.5 Proposed Exercises......Page 38
 Further Reading......Page 47
 3.1 Velocity and Acceleration......Page 48
 3.2 Frenet Equations......Page 51
 3.3 Proposed Exercises......Page 52
 Further Reading......Page 56
 4.1 Newton's Laws......Page 57
 4.2 Conservative and Central Forces......Page 59
  4.2.1 Gravitational Potential......Page 60
 4.3 Force, Energy, Work, and Energy Conservation......Page 62
  4.3.1 Conservative Forces......Page 63
  4.3.2 Conservative and Non-conservative Forces......Page 65
 4.4 Angular Momentum, Torque and Conservation......Page 66
 4.5 Galilean Relativity and Inertial Reference Frames......Page 67
 4.6 Proposed Exercises......Page 69
 Further Reading......Page 92
 5.1 Point-Like Particle Systems......Page 94
 5.2 Variable Mass Systems......Page 97
 5.3 Proposed Exercises......Page 98
 Further Reading......Page 116
 6.1 One-Dimensional Potentials......Page 117
 6.2 Central Potentials......Page 121
  6.2.1 The Two-Body Problem......Page 123
 6.3 Kepler's Potential......Page 125
 6.4 Proposed Exercises......Page 131
 Further Reading......Page 142
 7.1 Frontal Collisions......Page 143
  7.1.1 Elastic Collisions in the Center of Mass......Page 144
  7.1.2 Elastic Collisions in the Laboratory Frame......Page 145
  7.1.3 Relating Both Frames......Page 146
  7.1.4 Inelastic Collisions......Page 147
 7.2 Scattering by a Hard Sphere......Page 149
 7.3 Scattering by a Repulsive Potential......Page 151
 7.4 Proposed Exercises......Page 152
 Further Reading......Page 164
 8.1 Introduction......Page 165
 8.2 Potentials with Spherical Symmetry......Page 166
 8.3 Gravitational Field and Gauss's Law......Page 168
 8.4 Gauss's Law for Spherical Mass Distributions......Page 171
 8.5 Proposed Exercises......Page 172
 Further Reading......Page 191
 9.1 Velocity and Angular Velocity......Page 192
  9.1.1 The Heuristic Approach......Page 196
 9.2 Motion over the Earth's Surface......Page 197
 9.3 Free Fall......Page 200
 9.4 Foucault's Pendulum......Page 204
 9.5 Proposed Exercises......Page 206
 Further Reading......Page 214
 10.1 Discrete Case......Page 215
  10.1.1 Principal Axes......Page 220
  10.1.2 Huygens–Steiner Theorem......Page 221
 10.2 Continuum Generalization......Page 222
 10.3 Stable Solutions for the Torque-Free Motion......Page 223
  10.3.1 Earth's Precession......Page 226
 10.4 Free Symmetric Spinning Top......Page 227
 10.5 Heavy Symmetric Spinning Top......Page 230
 10.6 Proposed Exercises......Page 234
 Further Reading......Page 245
 11.1 Introduction......Page 246
 11.2 Lorentz–Poincaré Transformations......Page 247
 11.3 Velocity Addition Rules......Page 251
 11.4 Minkowski Space-Time and Four-Vectors......Page 252
  11.4.1 Covariant and Contravariant Transformations......Page 255
  11.4.2 Summary......Page 256
 11.5 Four-Velocity, Acceleration and Force......Page 258
  11.5.1 Massless Particles......Page 264
 11.6 Proposed Exercises......Page 265
 11.7 Final Comments......Page 278
 Further Reading......Page 279
 12.1 Conservation Laws and Kinematic Invariants......Page 280
 12.2 Decays......Page 281
 12.3 2 to 2 Frontal Collisions......Page 283
 12.4 Proposed Exercises......Page 289
 Further Reading......Page 295
 13.1 Lagrangian Formalism......Page 297
  13.1.1 Cyclic Coordinates and Constants of Motion......Page 301
  13.1.2 Lagrangian with Boundary Conditions......Page 302
 13.2 Hamiltonian Formalism......Page 303
 13.3 Stationary Action Principle......Page 304
 13.4 Noether's Theorem......Page 307
 13.5 Symmetries and Conservation......Page 308
  13.5.2 Spatial Translations......Page 309
  13.5.4 Galilean Transformations......Page 310
 13.6 Proposed Exercises......Page 311
 Further Reading......Page 321
Appendix A Tensor Formalism in Cartesian Coordinates......Page 322
A.1  Dual Space......Page 324
A.2  Covariant and Contravariant Laws of Transformation......Page 325
A.3  Rank-Two Covariant Tensor......Page 326
A.4  Metric Tensor in mathbbR3......Page 327
A.5  Metric Tensor in Special Relativity......Page 329
Appendix B Tensors in Curvilinear Coordinates......Page 332
B.1  Metric Tensor and Scalar Products......Page 337
B.2  Vector Operators......Page 338
B.3  Practical Exercise......Page 339
Appendix C Passive and Active Transformations......Page 342
Appendix D Vector Operators in Curvilinear Coordinates......Page 345
Appendix E Variable Mass Equation......Page 349
Appendix F Rotations, Euler Angles and Angular Velocity......Page 351
Appendix G Three-Body Particle Decays......Page 354
 Further Reading......Page 355
Index......Page 357