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Lectures in Classical Mechanics for free

Lectures in Classical Mechanics for free
Lectures in Classical Mechanics

About this book
This exceptionally well-organized book uses solved problems and exercises to help readers understand the underlying concepts of classical mechanics; accordingly, many of the exercises included are of a conceptual rather than practical nature. A minimum of necessary background theory is presented, before readers are asked to solve the theoretical exercises. In this way, readers are effectively invited to discover concepts on their own. While more practical exercises are also included, they are always designed to introduce readers to something conceptually new.Special emphasis is placed on important but often-neglected concepts such as symmetries and invariance, especially when introducing vector analysis in Cartesian and curvilinear coordinates. More difficult concepts, including non-inertial reference frames, rigid body motion, variable mass systems, basic tensorial algebra, and calculus, are covered in detail. The equations of motion in non-inertial reference systems are derived in two independent ways, and alternative deductions of the equations of motion for variable mass problems are presented. Lagrangian and Hamiltonian formulations of mechanics are studied for non-relativistic cases, and further concepts such as inertial reference frames and the equivalence principle are introduced and elaborated on.

About the authors
Victor Ilisie, B.S., M.S., Ph.D., is a postdoctoral researcher at the Institute for Instrumentation in Molecular Imaging, Spanish National Research Council (CSIC) and associate professor at the University of Valencia (Spain). During his Ph.D., Dr. Ilisie’s research focused on the study of high-energy physics phenomena related to the Large Hadron Collider (Geneva) and Higgs physics. Since then, his research activities have contributed to various fields in particle physics and medical physics, and he has also authored a book on quantum field theory. His skills and experience have been highly useful in developing projects related to PET, SPECT, and image reconstruction. Most of his postdoctoral research has focused on the study of high-resolution and high-sensitivity PET that incorporates the Compton effect, in a project financed by the European Research Council under the European Union’s Horizon 2020 research and innovation program. He is also coordinating the development of a SPECT project in a collaboration between Bruker Corporation and the Institute for Instrumentation in Molecular Imaging, and is involved in a project on the design and development of a novel multi-pinhole high-sensitivity SPECT device.

Table of contents :
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

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