Lectures in Analytical Mechanics

Description

Title page Preface Chapter 1. THE DIFFERENTIAL EQUATIONS OF MOTION OF AN ARBITRARY SYSTEM OF PARTICLES 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 lN A POTENTIAL FIELD 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 16. Hamilton's Principle 17. Second Form of Hamilton's Principle 18. The Basic Integral Invariant of Mechanics (Poincaré-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. Whîttaker's Equations. Jacobi's Equations. The Maupertuis-Lagrange Principle of Least Action 21. Inertial Motion. Relation to Geodesie Lines in the Arbitrary Motion of a Conservative System 22. The Universal Integral Invariant of Poincaré. 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 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 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 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 the 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 48. Reduced System. The Routh Potential. Hidden Motions. Hertz' Conception of the Kinetic Origin of Potential Energy 49. Stahility of Stationary Motions References Name Index Subject Index