Fundametals Of Theoretical Physics Vol 1 | I. V. Savelyev

Fundamentals of Theoretical Physics Volume 1 Mechanics, Electrodynamics by I. V. Savelyev

The book was translated from the Russian by G. Leib. The book was first published in 1982, revised from the 1975 Russian edition by Mir Publishers.

The book being offered to the reader is a logical continuation of the author’s three-volume general course of physics. Everything possible has been done to avoid repenting what has been set out in the three-volume course. Particularly. the experiments underlying the advancing of physical ideas are not treated, and some of the results obtained are not discussed.
In the part devoted to mechanics, unlike the established traditions, Lagrange’s equations are derived directly from Newton’s equations instead of from d’Alembert’s principle. Among the books I have acquainted myself with, such a derivation is given in A. S. Kompaneyts’s book Theoretical Physics (in Russian) for the particular case of a conservative system. In the present book, I have extended this method of exposition to systems in which not only conservative, but also non-conservative forces act.

The treatment of electrodynamics is restricted to a consideration of media with a permittivity c and a permeability ~t not depending on the fields E and B.

An appreciable difficulty appearing in studying theoretical physics is the circumstance that quite often many mathematical topics have earlier never been studied by the reader or have been forgotten by him fundamentally. To eliminate this difficulty, I have provided the book with detailed mathematical appendices. The latter are sufficiently complete to relieve the reader of having to turn to mathematical aids and find the required information in them. This information is often set out in these aids too complicated for the readers which the present book is intended for. Hence, the information on mathematical analysis contained in a college course of higher mathematics is sufficient for mastering this book.

The book has been conceived as a training aid for students of non- theoretical specialities of higher educational institutions. I had in mind readers who would like to grasp the main ideas and methods of theoretical physics without delving into the download that are of interest only for a specialist. This book will be helpful for physics instructors at higher schools, and also for everyone interested in the subject but having no time to become acquainted with it (or re- store it in his memory) according to fundamental manuals.

Part One. Mechanics 11

Chapter I. The Variational Principle in Mechanics 11

1. Introduction 11
2. Constraints 13
3. Equations of Motion in Cartesian Coordinates 16
4. Lagrange’s Equations in Generalized Coordinates 19
5. The Lagrangian and Energy 24 6. Examples of Compiling Lagrange’s Equations 28
7. Principle of Least Action 33

Chapter II. Conservation Laws 36

8. Energy Conservation 36
9. Momentum Conservation 37
10. Angular Momentum Conservation 39

Chapter III. Selected Problems in Mechanics 41

11. Motion of a Particle in a Central Force Field 41
12. Two-Body Problem 45
13. Elastic Collisions of Particles 49
14. Particle Scattering 53
15. Motion in Non-Inertial Reference Frames 57

Chapter IV. Small-Amplitude Oscillations 64

16. Free Oscillations of a System Without Friction 64
17. Damped Oscillations 66
18. Forced Oscillations 70
19. Oscillations of a System with Many Degrees of Freedom 72
20. Coupled Pendulums 77

Chapter V. Mechanics of a Rigid Body 82

21. Kinematics of a Rigid Body 82
22. The Euler Angles 85
23. The lnertia Tensor 88
24. Angular Momentum of a Rigid Body 95
25. Free Axes of Rotation 99
26. Equation of Motion of a Rigid Body 101
27. Euler’s Equations 105
28. Free Symmetric Top 107
29. Symmetric Top in a Homogeneous Gravitational Field 111

Chapter VI. Canonical Equations 115

30. Hamilton’s Equations 115 31. Poisson Brackets 119
32. The Hamilton-Jacobi Equation 121

Chapter VII. The Special Theory of Relativity 125

33. The Principle of Relativity 125
34. Interval 127
35. Lorentz Transformations 130
36. Four-Dimensional Velocity and Acceleration 134
37. Relativistic Dynamics 136
38. Momentum and Energy of a Particle 139
39. Action for a Relativistic Particle 143
40. Energy-Momentum Tensor 147

Part Two. Electrodynamics 157

Chapter VIII. Electrostatics 157

41. Electrostatic Field in a Vacuum 157
42. Poisson’s Equation 159
43. Expansion of a Field in Multipoles 161
44. Field in Dielectrics 166
45. Description of the Field in Dielectrics 170
46. Field in Anisotropic Dielectrics 175

Chapter IX. Magnetostatics 177

47. Stationary Magnetic Field in a Vacuum 177
48. Poisson’s Equation for the Vector Potential 179
49. Field of Solenoid 182
50. The Biot-Savart Law 186
51. Magnetic Moment 188
52. Field in Magnetics 194

Chapter X. Time-Varying Electromagnetic Field 199

53. Law of Electromagnetic Induction 199
54. Displacement Current 200
55. Maxwell’s Equations 201
56. Potentials of Electromagnetic Field 203
57. D’Alembert’s Equation 207
58. Density and Flux of Electromagnetic Field Energy 208
59. Momentum of Electromagnetic Field 211

Chapter XI. Equations of Electrodynamics in the Four Dimensional Form 216

60. Four-Potential 216
61. Electromagnetic Field Tensor 219
62. Field Transformation Formulas 222
63. Field Invariant 225
64. Maxwell’s Equations in the Four-Dimensional Form 228
65. Equation of Motion of a Particle in a Field 230

Chapter XII. The Variational Principle in Electrodynamics 232

66. Action for a Charged Particle in an Electromagnetic Field 232
67. Action for an Electromagnetic Field 234
68. Derivation of Maxwell’s Equations from the Principle of Least Action 237
69. Energy-Momentum Tensor of an Electromagnetic Field 239
70. A Charged Particle in an Electromagnetic Field 244

Chapter XIII. Electromagnetic Waves 248

71. The Wave Equation 248
72. A Plane Electromagnetic Wave in a Homogeneous and Isotropic Medium 250
73. A Monochromatic Plane Wave 255
74. A Plane Monochromatic Wave in a Conducting Medium 260
75. Non-Monochromatic Waves 265

Chapter XIV. Radiation of Electromagnetic Waves 269

76. Retarded Potentials 269
77. Field of a Uniformly Moving Charge 272
78. Field of an Arbitrarily Moving Charge 276
79. Field Produced by a System of Charges at Great Distances 283
80. Dipole Radiation 288
81. Magnetic Dipole and Quadrupole Radiations 291

Appendices 297

I. Lagrange’s Equations for a Holonomic System with Ideal Non- Stationary Constraints 297
II. Euler’s Theorem for Homogeneous Functions 299
III. Some Information from the Calculus of Variations 300
IV. Conics 309
V. Linear Differential Equations with Constant Coefficients 313
VI. Vectors 316
VII. Matrices 330
VIII. Determinants 338
IX. Quadratic Forms 347
X. Tensors 355
XI. Basic Concepts of Vector Analysis 370
XII. Four-Dimensional Vectors and Tensors in Pseudo-Euclidean Space 393
XIII. The Dirac Delta Function 412
XIV. The Fourier Series and Integral 413

Index 419