Fundamentals of Theoretical Physics: Volume 1 - Mechanics, Electrodynamics

425 Pages·1970·19.635 MB·English

by  I. V. Savelyev

I. V. Savelyev
FUNDAMENTALS
OF THEORETICAL
PHYSICS
Volume 1
Mechanics
Electrodynamics
Mir Publishers Moscow
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FUNDAMENTALS
OF THEORETICAL
PHYSICS
Volume1
Mechanics
Electrodynamics
Translated from the Russian
by
G. Leib
MIR PUBLISHERS MOSCOW
First published 1982
Revised from the 1975 Russian edition
TO THE READER
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Mir Publishers
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PREFACE
The bonk 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 sel out
in the three-volume course. Particularly. the experiments underlying
the advancing of physical ideas are not treated, and some of the re
sults obtained are not discussed.
In the part devoted to mechanics, unlike the established tradi
tions, Lagrange's equations are derived directly from Newton's equa
tions instead of from d 'Alembert' s principle. Among the books I have
acquainted myself with, such a derivation is given in A. S. Kompa
neits'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.
Sections 40 and 69 devoted to the energy-momentum tensor are
appreciably more complicated. They have been inelndccl in the book
becnuse they con tnin an excellent illustra timt of ho\v LngTnngian
formalism is generalized for non-mechanical systems. A reader to
whom these sections will seem too difficult may omit them without
any harm to his understanding the remaining sections of the book.
I have devoted much attention to the vnriational principl<>. with
the consistent use of the following procedure-first the required resnlt
is obtnined with tlte aid of mrthods which the render is alt·endr nc
quuintPtl with, and then the ~ame n'sult is obtained u~ing the varia
tional principle. The object here was to ensure the reader treating
the variational prlneiple as a quite reliable and powerful means of
research.
An appreciable difflculty appearing in studying theoretical physics
is the circumstance that quite often many malhematical topics have
eitl1er 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 tum to
6 PREFACE
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 informa
tion 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 details 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.
Igor Sauelyev
CONTENTS
Preface 5
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
8 CONTENTS
l 9. Oscillations of a System with Many Deg1·ees 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 lncrlia Tensor 88
24. Angular Momentum of u Higid Body 95
25. Free Axes of Rotation 99
26. Equation of Motion of u 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
Pari 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. Desq:iption 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
CONTENTS
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 Electromag-netic Field Energy 208
59. Momentum of Electromagnetic Field 211
Chapter X I. Equations ol" Eleetrodynarnit's in thP For11·-llinrcn"ional
Form 216
60. Four-Potential 2'16
61. Electromagnetic Field Tensor 219
fi2. Field Transformation Formulas 222
63. Field Invarinnt~ 225
u4. Maxwell's Equations in the Four-Dimen~ional 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
u7. Action for an Electromagnetic Field 234
G8. Derivation of Maxwell's Equations from the Principle of Least
Action 237
Gil. Energy-l\lornentnm Trn~ot· of an Electromagnrtic Firld 239
70. A Ch<trged Particle in an Electromagnetic Field 21,1,
Chapter XIII. Elrctromagnetic Waves 248
71. The Wave Equation 248
72. A Plane Electromagnetic Wave in a Homogeneous and Isotropic
Medium 250
7~. A Monochromatic Plane Wave 255
74. A Plane Monochromatic Wave in a Conducting 1\Iedium 260
75. Non-Monochromatic Waves 265
Chapter XIV. Hadiation of Electromagnetic Waves 269
7u. Hetnnled Potentials 269
77. Field of a Uniformly Moving Charge 272
78. Field of an Arbitrarily Moving Charge 276
79. Field Produced by a Sy~tem of Charges at Great Distances 283:
80. Dipole Hadiation 288
81. Magnetic Dipole and Quadrupole Hadiations 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
VI II. Determinants 338
IX. Quadratic Forms 347
10 CONTENTS
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