The text provides an overview of a book on analytical mechanics, developed from a course taught over six years at the Moscow Physico-Technical Institute. It focuses on key foundational concepts for theoretical physics, including quantum mechanics and relativity, with detailed discussions on variational principles, integral invariants, canonical transformations, and systems with cyclic coordinates. The approach follows Poincaré and Cartan, treating integral invariants as central to the theory. Technical applications include constrained systems, electromechanical analogies, and the theory of stability and oscillations, with practical examples. The book is aimed at students and professionals with a background in theoretical mechanics and higher mathematics.
Translated from the Russian by George Yankovsky
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Preface 7
Chapter 1. THE DIFFERENTIAL EQUATIONS OF MOTION OF AN ARBITRARY SYSTEM OF PARTICLES 9
1. Free and Constrained Systems. Constraints and Their Classification
2. Possible and Virtual Displacements. Ideal Constraints
3. The General Equation of Dynamics. Lagrange’s Equations of the First Kind
4. The Principle of Virtual Displacements. D’Alembert’s Principle
5. Holonomic Systems. Independent Coordinates. Generalized Forces
6. Lagrange’s Equations of the Second Kind in Independent Coordinates
7. Investigating Lagrange’s Equations
8. Theorem on Variation of Total Energy. Potential, Gyroscopic and Dissipative Forces
9. Electromechanical Analogies
10. Appell’s Equations for Nonholonomic Systems. Pseudocoordinates
Chapter 2. THE EQUATIONS OF MOTION IN A POTENTIAL FIELD 66
11. Lagrange’s Equations for Potential Forces. The Generalized Potential. Nonnatural Systems
12. Canonical Equations of Hamilton
13. Routh’s Equations
14. Cyclic Coordinates
15. The Poisson Bracket
Chapter 3. VARIATIONAL PRINCIPLES AND INTEGRAL INVARIANTS 88
16. Hamilton’s Principle
17. Second Form of Hamilton’s Principle
18. The Basic Integral Invariant of Mechanics (Poincare-Cartan Integral Invariant)
19. A Hydrodynamical Interpretation of the Basic Integral Invariant. The Theorems of Thomson and Helmholtz on Circulation and Vortices
20. Generalized Conservative Systems. Whittaker’s Equations. Jacobi’s Equations. The Maupertuis-Lagrange Principle of Least Action
21. Inertial Motion. Relation to Geodesic Lines in the Arbitrary Motion of a Conservative System
22. The Universal Integral Invariant of Poincare. Lee Hwa-Chung’s Theorem
23. Invariance of Volume in the Phase Space. Liouville’s Theorem
Chapter 4. CANONICAL TRANSFORMATIONS AND THE HAMILTON-JACOBI EQUATION 128
24. Canonical Transformations
25. Free Canonical Transformations
26. The Hamilton-Jacobi Equation
27. Method of Separation of Variables. Examples
28. Applying Canonical Transformations to Perturbation Theory
29. The Structure of an Arbitrary Canonical Transformation
30. Testing the Canonical Character of a Transformation. The Lagrange Brackets
31. The Simplicial Nature of the Jacobian Matrix of a Canonical Transformation
32. Invariance of the Poisson Brackets in a Canonical Transformation
Chapter 5. STABILITY OF EQUILIBRIUM AND THE MOTIONS OF A SYSTEM 166
33. Lagrange’s Theorem on the Stability of an Equilibrium Position
34. Criteria of Instability of an Equilibrium Position. Theorems of Lyapunov and Chetayev
35. Asymptotic Stability of an Equilibrium Position. Dissipative Systems
36. Conditional Stability. General Statement of the Problem. Stability of Motion or of an Arbitrary Process. Lyapunov’s Theorem
37. Stability of Linear Systems
38. Stability in Linear Approximation
39. Criteria of Asymptotic Stability of Linear Systems
Chapter 6. SMALL OSCILLATIONS 202
40. Small Oscillations of a Conservative System
41. Normal Coordinates
42. The Effect of Periodic External Forces on the Oscillations of a Conservative System
43. Extremal Properties of Frequencies of a Conservative System. Rayleigh’s Theorem on Frequency Variation with Change in Inertia and Rigidity of the System. Superimposition of Constraints
44. Small Oscillations of Elastic Systems
45. Small Oscillations of a Scleronomic System under the Action of Forces Not Explicitly Dependent on Time
46. Rayleigh’s Dissipative Function. The Effect of Small Dissipative Forces on the Oscillations of a Conservative System
47. The Effect of a Time-Dependent External Force on Small Oscillations of a Scleronomic System. The Amplitude-Phase Characteristic
Chapter 7. SYSTEMS WITH CYCLIC COORDINATES 242
48. Reduced System. The Routh Potential. Hidden Motions. Hertz’s Conception of the Kinetic Origin of Potential Energy
49. Stability of Stationary Motions
References 259
Name Index 260
Subject Index 261
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