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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