Group-Theoretic Methods in Mechanics and

Group analysis of differential equations has applications to various problems in nonlinear mechanics and physics. Group-Theoretic Methods in Mechanics and Applied Mathematics systematizes the group analysis of the main postulates of classical and relativistic mechanics. Exact solutions are given for the following equations: dynamics of rigid body, heat transfer, wave, hydrodynamics, Thomas-Fermi, and more. The author pays particular attention to the application of group analysis to developing asymptotic methods for problems with small parameters. This book is designed for a broad audience of scientists, engineers, and students in the fields of applied mathematics, mechanics and physics.

D. M. Klimov is the Director of the Institute for Problems in Mechanics of the Russian Academy of Sciences and the Head of the Chair of Mechanics of Controlled and Gyroscopic Systems at the Moscow Institute of Physics and Technology. Professor Klimov is an author of five books and more than 100 research papers V. Ph. Zhuravlev is a professor at the Institute for Problems in Mechanics and the Head of the Chair of Mechanics at the Moscow Institute of Physics and Technology. Professor Zhuravlev is an author of over 150 scientific publications, four monographs and a textbook

Foreword

vii

Authors

x

Basic Notions of Lie Group Theory

1

(46)

Notion of Group

1

(2)

Lie Group. Examples

3

(2)

Group Generator. Lie Algebra

5

(6)

One-Parameter Groups. Uniqueness Theorem

11

(2)

Liouville Equation. Invariants. Eigenfunctions

13

(4)

Linear Partial Differential Equations

17

(1)

Change of Variables. Canonical Coordinates of a Group

18

(2)

Hausdorff's Formula. Symmetry Groups

20

(9)

Principle of Superposition of Solutions and Separation of Motions in Nonlinear Mechanics

29

(2)

Prolongation of Groups. Differential and Integral Invariants

31

(5)

Equations Admitting a Given Group

36

(6)

Symmetries of Partial Differential Equations

42

(5)

Group Analysis of Foundations of Classical and Relativistic Mechanics

47

(18)

Axiomatization Problem of Mechanics

47

(1)

Postulates of Classical Mechanics

48

(2)

Projective Symmetries of Newton's First Law

50

(1)

Newton's Second Law. Galilean Symmetries

51

(3)

Postulates of Relativistic Mechanics

54

(1)

Group of Symmetries of Maxwell's Equations

54

(2)

Twice Prolonged Lorentz Group

56

(4)

Differential and Integral Invariants of the Lorentz Group

60

(2)

Relativistic Equations of Motion of a Particle

62

(1)

Noninertial Reference Frames

63

(2)

Application of Group Methods to Problems of Mechanics

65

(60)

Perturbation Theory for Configuration Manifolds of Resonant Systems

65

(8)

Statement of the Problem

65

(1)

The Case of Double Natural Frequency

66

(1)

The Manifold of Degenerate Forms. Local Evolution Basis

67

(2)

Algebra of Local Evolutions

69

(1)

Classification of Perturbations

70

(1)

The Problem of Stabilization of the Oscillation Shape

71

(2)

Poincare's Equation on Lie Algebras

73

(3)

Kinematics of a Rigid Body

76

(28)

Ways of Specifying the Orientation of a Rigid Body

76

(14)

Addition of Rotations

90

(6)

Topology of the Manifold of Rotations of a Rigid Body (Topology of the SO(3) group)

96

(1)

Angular Velocity of a Rigid Body

97

(7)

Problems of Mechanics Admitting Similarity Groups

104

(8)

Suslov Problem

104

(3)

The Problem of the Follower Trajectory

107

(2)

Rolling of a Homogeneous Ball Over a Rough Plane

109

(3)

Problems With Determinable Linear Groups of Symmetries

112

(13)

Falling of a Heavy Homogeneous Thread

112

(3)

Motion of a Point Particle Under the Action of a Follower Force

115

(5)

The Problem of an Optimal Shape of a Body in an Air Flow

120

(5)

Finite-Dimensional Hamiltonian Systems

125

(26)

Legendre Transformation

125

(1)

Hamiltonian Systems. Poisson Bracket

126

(2)

Nonautonomous Hamiltonian Systems

128

(1)

Integrals of Hamiltonian Groups. Noether's Theorem

129

(3)

Conservation Laws and Symmetries

132

(1)

Integral Invariants

133

(6)

Poincare--Cartan Invariants

134

(4)

Liouville's Theorem of the Phase Volume

138

(1)

Canonical Transformations

139

(6)

Hamilton-Jacobi Equation

145

(2)

Liouville's Theorem of Integrable Systems

147

(1)

The Angle-Action Variables

148

(3)

Asymptotic Methods of Applied Mathematics

151

(68)

Introduction

151

(2)

Normal Coordinates of Conservative Systems

153

(5)

Single-Frequency Method of Averaging Based on Hausdorff's Formula

158

(2)

Poincare Normal Form

160

(10)

The Averaging Principle

170

(19)

Averaging of Single-Frequency Systems

175

(8)

Multifrequency Systems. Resonance

183

(6)

Asymptotic Integration of Hamiltonian Systems

189

(15)

Birkhoff Normal Form

189

(11)

Averaging of Hamiltonian Systems in Terms of Lie Series

200

(1)

Artificial Hamiltonization

201

(3)

Method of Tangent Approximations

204

(4)

Classical Examples of Oscillation Theory

208

(11)

Van Der Pol's Equation

208

(1)

Mathieu's Equation

209

(3)

Forced Oscillations of Duffing's Oscillator

212

(3)

Forced Oscillations of Van Der Pol's Oscillator

215

(4)

Brief Historical Sketch

219

(6)

Index

225

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Group-Theoretic Methods in Mechanics and Applied Mathematics

Summary

Author Biography

Table of Contents

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