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This book provides an up-to-date overview of results in rigid body dynamics, including material concerned with the analysis of nonintegrability and chaotic behavior in various related problems. The wealth of topics covered makes it a practical reference for researchers and graduate students in mathematics, physics and mechanics.
Contents
Rigid Body Equations of Motion and Their Integration
The Euler - Poisson Equations and Their Generalizations
The Kirchhoff Equations and Related Problems of Rigid Body Dynamics
Linear Integrals and Reduction
Generalizations of Integrability Cases. Explicit Integration
Periodic Solutions, Nonintegrability, and Transition to Chaos
Appendix A : Derivation of the Kirchhoff, Poincaré - Zhukovskii, and Four-Dimensional Top Equations
Appendix B: The Lie Algebra e(4) and Its Orbits
Appendix C: Quaternion Equations and L-A Pair for the Generalized Goryachev - Chaplygin Top
Appendix D: The Hess Case and Quantization of the Rotation Number
Appendix E: Ferromagnetic Dynamics in a Magnetic Field
Appendix F: The Landau - Lifshitz Equation, Discrete Systems, and the Neumann Problem
Appendix G: Dynamics of Tops and Material Points on Spheres and Ellipsoids
Appendix H: On the Motion of a Heavy Rigid Body in an Ideal Fluid with Circulation
Appendix I: The Hamiltonian Dynamics of Self-gravitating Fluid and Gas Ellipsoids
List of contents
Table of Contents
Chapter 1. Rigid Body Equations of Motion and their Integration
1.1. Poisson Brackets and Hamiltonian Formalism
1.2. Poincar´e and Poincar´e-Chetaev Equations
1.3. Various systems of variables in rigid body dynamics
1. 4. Different Forms of Equations of Motion
1.5. Equations of Motion of a Rigid Body in Euclidean Space
1. 6. Examples and Similar Problems
1. 7. Theorems on inerrability and methods of integration
Chapter 2. The Euler-Poisson equations and their generalizations
2.1. Euler-Poisson equations and integrable cases
2.2. The Euler case
2.3. The Lagrange case
2.4. The Kovalevskaya case
2.5. The Goryachev-Chaplygin case
2.6. Partial solutions of the Euler-Poisson equations
2.7. Equations of motion of a heavy gyrostat
2.8. Systems of linked rigid bodies, a rotator
Chapter 3. Kirchhoff Equations
3.1. Poincar´e-Zhukovskii Equations
3.2. A Remarkable Limit Case of the Poincar´e-Zhukovskii Equations
3.3. Rigid body in an Arbitrary Potential Field
Chapter 4. Linear Integrals and Reduction
4.1. Linear Integrals in Rigid Body Dynamics
4.2. Dynamical Symmetry and Lagrange Integral
4.3. Generalizations of the Hess Case
Chapter 5. Generalizations of Inerrability Cases
5. 1. Various Generalizations of the Kovalevskaya and Goryachev-
Chaplygin Cases
5.2. Separation of Variables
5.3. Isomorphism and Explicit Integration
5.4. Doubly Asymptotic Motions for Integrable Systems
Chapter 6. Periodic Solutions, Nonintegrability, and Transition to Chaos
6. 1. Nonintegrability of Rigid Body Dynamics Equations
6. 2. Periodic and Asymptotic Solutions in Euler-Poisson Equations and Related Problems
6. 3. Absolute and Relative Choreographies in Rigid Body Dynamics
6. 4. Chaotic Motions. Genealogy of Periodic Orbits
6. 5. Chaos Evolution in the Restricted Problem of Heavy Rigid Body
Rotation
6. 6. Adiabatic Chaos in the Liouville Equations
6. 7. Heavy Rigid Body Fall in Ideal Fluid. Probability Effects and Attracting Sets
Appendix
Bibliography
About the author
Alexey V. Borisov and Ivan S. Mamaev, Udmurt State University, Russia.
