Rigid Body Dynamics: (52 De Gruyter Studies in Mathematical Physics)

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

Table 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.