Fundamentals of Theoretical Physics Volume 1 Mechanics, Electrodynamics by I. V. SavelyevThe 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 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.Part One. Mechanics 11Chapter I. The Variational Principle in Mechanics 111. 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 33Chapter II. Conservation Laws 368. Energy Conservation 369. Momentum Conservation 37 10. Angular Momentum Conservation 39Chapter III. Selected Problems in Mechanics 4111. Motion of a Particle in a Central Force Field 41 12. Two-Body Problem 45 13. Elastic Collisions of Particles 49 14. Particle Scattering 5315. Motion in Non-Inertial Reference Frames 57Chapter IV. Small-Amplitude Oscillations 6416. Free Oscillations of a System Without Friction 64 17. Damped Oscillations 66 18. Forced Oscillations 7019. Oscillations of a System with Many Degrees of Freedom 72 20. Coupled Pendulums 77Chapter V. Mechanics of a Rigid Body 8221. 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 9926. 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 111Chapter VI. Canonical Equations 11530. Hamilton's Equations 115 31. Poisson Brackets 11932. The Hamilton-Jacobi Equation 121Chapter VII. The Special Theory of Relativity 12533. The Principle of Relativity 125 34. Interval 127 35. Lorentz Transformations 130 36. Four-Dimensional Velocity and Acceleration 13437. Relativistic Dynamics 13638. Momentum and Energy of a Particle 139 39. Action for a Relativistic Particle 143 40. Energy-Momentum Tensor 147Part Two. Electrodynamics 157 Chapter VIII. Electrostatics 15741. 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 175Chapter IX. Magnetostatics 17747. 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 194Chapter X. Time-Varying Electromagnetic Field 19953. Law of Electromagnetic Induction 19954. 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 211Chapter XI. Equations of Electrodynamics in the Four Dimensional Form 21660. 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 230Chapter XII. The Variational Principle in Electrodynamics 23266. 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 244Chapter XIII. Electromagnetic Waves 24871. 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 265Chapter XIV. Radiation of Electromagnetic Waves 26976. 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 28380. Dipole Radiation 28881. Magnetic Dipole and Quadrupole Radiations 291Appendices 297I. Lagrange's Equations for a Holonomic System with Ideal Non- Stationary Constraints 297II. Euler's Theorem for Homogeneous Functions 299 III. Some Information from the Calculus of Variations 300IV. Conics 309 V. Linear Differential Equations with Constant Coefficients 313VI. Vectors 316 VII. Matrices 330 VIII. Determinants 338 IX. Quadratic Forms 347X. Tensors 355 XI. Basic Concepts of Vector Analysis 370XII. Four-Dimensional Vectors and Tensors in Pseudo-Euclidean Space 393 XIII. The Dirac Delta Function 412 XIV. The Fourier Series and Integral 413Index 419