Gantmacher – Lectures in Analytical Mechanics

We now come to Lectures on Analytical Mechanics by F. Gantmacher.

The course of analytical mechanics is a foundation supporting such   divisions of theoretical physics as quantum mechanics, the special   and general theories of relativity, and so forth. For this reason, a   detailed presentation is given of variational principles and the   integral invariants of mechanics, canonical transformations, the   Hamilton-Jacobi equation, and systems with cyclic (ignorable)   coordinates (Chapters 2, 3, 4 and 7). Following the ideas of Poincare   and Cartan, the author takes the integral invariants of mechanics as   the basis of presentation. Here they do not represent an   embellishment of the theory but its actual workaday machinery. The   technical applications are associated with a consideration of   constrained systems, which are studied in detail inChapter 1. In a   special section of that chapter, which is devoted to   electromechanical analogies, the possibility is investigated of   extending the analytical methods of mechanics to electrical and   electromechanical systems. In Chapters 5 and 6 are given   applications of analytical mechanics to Lyapunov’s theory of   stability and the theory of oscillations. Elements of modern   frequency methods are given along with the classical problems in the   theory of linear oscillations. Problems in the dynamics of rigid   bodies are taken up in individual examples.

Whom this text is for and what are the prerequisites?

It is assumed the reader is acquainted with the general fundamentals
of theoretical mechanics and higher mathematics. The text is
designed for undergraduate and graduate students of
mechanico-mathematical, physical and physical-engineering
departments of universities, and also for research engineers and
other specialists who feel a need to extend and deepen their
knowledge in the field of mechanics.

This book was translated from the Russian by George Yankovsky and
was first published by Mir Publishers in 1975.

All credits to the original uploader.

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Table of Contents

Contents
Preface 7

Chapter 1
The Differential Equatlons Of Motion Of An Arbitrary System Of
Particles – 9

1. Free and Constrained Systems. Constraints and Their Classification 9
2. Possible and Virtual Displacements. Ideal Constraints 12
3. The General Equation of Dynamics. Lagrange’s Equations of the First Kind 20
4. The Principle of Virtual Displacements. D’Alembert’s Principle 25
5. Holonomic Systems. Independent Coordinates. Generalized Forces 33
6. Lagrange’s Equations of the Second Kind in Independent Coordinates 40
7. Investigating Lagrange’s Equations 45
8. Theorem on Variation of Total Energy. Potential, Gyroscopicand Dissipative Forces 48
9. Electromechanical Analogies 54
10. Appell’s Equations for Nonholonomic Systems. Pseudo coordi-nates 57

Chapter 2.
The Equations Of Motion In A Potential Field 66

11. Lagrange’s Equations for Potential Forces. The Generalized
Potential. Nonnatural Systems 66
12. Canonical Equations of Hamilton 71
13. Routh’s Equations 78
14. Cyclic Coordinates 80
15. The Poisson Bracket 83

Chapter3.
Variational Principles And Integral Invariants 88

16. Hamilton’s Principle 88
17. SecondFormofHamilton’sPrinciple 96
18. The Basic Integral Invariant of Mechanics (Poincare-Cartan
Integral Invariant) 98
19. A Hydrodynamical Interpretation of the Basic Integral
Invariant.The Theorems of Thomson and Helmholtzon Circulation and Vortices 105
20. Generalized Conservative Systems. Whittaker’s Equations. Jacobi’s
Equations. TheMaupertuis-Lagrange Principle of Least Action 111
21. Inertial Motion. Relation to Geodesic Lines in the Arbitrary
Motion of a Conservative System 117

22. The Universal Integral Invariant of Poincare. Lee Hwa-Chung’s Theorem 119

23. Invariance of Volume in the Phase Space. Liouville’s Theorem 125

Chapter4.
Canonical Transformations And The Hamilton-Jacobi Equation 128

24. Canonical Transformations 128
25. Free Canonical Transformations 132
26. The Hamilton-Jacobi Equation 135
27. Method of Separation of Variables. Examples 141
28. Applying Canonical Transformations to Perturbation Theory 150
29. The Structure of an Arbitrary Canonical Transformation 152
30. Testing the Canonical Character of a Transformation.The Lagrange Brackets 158
31. The Simplicial Nature of the Jacobian Matrix of a Canonical Transformation 161
32. Invariance of the Poisson Brackets in a Canonical Transformation 163

Chapter5.
Stability Of Equilibrium And The Motions Of A System 166

33. Lagrange’s Theorem on the Stability of an Equilibrium Position 166
34. Criteria of Instability of an Equilibrium Position. Theorems of
Lyapunov and Chetayev 173
35. Asymptotic Stability of an Equilibrium Position. Dissipative Systems 176
36. Conditional Stability. General Statement of the Problem. Stability of Motion of an Arbitrary Process. Lyapunov’sTheorem 182
37. Stability of Linear Systems 189
38. Stability in Linear Approximation 193
39. Criteria of Asymptotic Stability of Linear Systems 197

Chapter6.
Small Oscillations 202

40. Small Oscillations of a Conservative System 202
41. Normal Coordinates 211
42. The Effect of Periodic External Forces on the Oscillations of a
Conservative System 214
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. 216
44. Small Oscillations of Elastic Systems 222
45. Small Oscillations of a Scleronomic System under the Action of
Forces Not Explicitly Dependent on the Time 228
46. Rayleigh’s Dissipative Function, The Effect of Small Dissipative
Forces on the Oscillations of a Conservative System 231
47. The Effect of a Time-Dependent External Force on Small Oscillations of a Scleronomic System. The Amplitude-Phase Characteristic 235

Chapter7.
Systems With Cyclic Coordinates 242

48. Reduced System. The Routh Potential. Hidden Motions. Hertz
Conception of the Kinetic Origin of Potential Energy 242
49. Stability of Stationary Motions 252

References 259
Name Index 260
Subject Index 261

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