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.ContentsRigid Body Equations of Motion and Their IntegrationThe Euler ¿ Poisson Equations and Their GeneralizationsThe Kirchhoff Equations and Related Problems of Rigid Body DynamicsLinear Integrals and ReductionGeneralizations of Integrability Cases. Explicit IntegrationPeriodic Solutions, Nonintegrability, and Transition to ChaosAppendix A : Derivation of the Kirchhoff, Poincaré ¿ Zhukovskii, and Four-Dimensional Top EquationsAppendix B: The Lie Algebra e(4) and Its OrbitsAppendix C: Quaternion Equations and L-A Pair for the Generalized Goryachev ¿ Chaplygin TopAppendix D: The Hess Case and Quantization of the Rotation NumberAppendix E: Ferromagnetic Dynamics in a Magnetic FieldAppendix F: The Landau ¿ Lifshitz Equation, Discrete Systems, and the Neumann ProblemAppendix G: Dynamics of Tops and Material Points on Spheres and EllipsoidsAppendix H: On the Motion of a Heavy Rigid Body in an Ideal Fluid with CirculationAppendix I: The Hamiltonian Dynamics of Self-gravitating Fluid and Gas Ellipsoids

Table of ContentsChapter 1. Rigid Body Equations of Motion and their Integration1.1. Poisson Brackets and Hamiltonian Formalism1.2. Poincar¿e and Poincar¿e-Chetaev Equations1.3. Various systems of variables in rigid body dynamics1. 4. Different Forms of Equations of Motion1.5. Equations of Motion of a Rigid Body in Euclidean Space1. 6. Examples and Similar Problems1. 7. Theorems on inerrability and methods of integrationChapter 2. The Euler-Poisson equations and their generalizations2.1. Euler-Poisson equations and integrable cases2.2. The Euler case2.3. The Lagrange case2.4. The Kovalevskaya case2.5. The Goryachev-Chaplygin case2.6. Partial solutions of the Euler-Poisson equations2.7. Equations of motion of a heavy gyrostat2.8. Systems of linked rigid bodies, a rotatorChapter 3. Kirchhoff Equations3.1. Poincar¿e-Zhukovskii Equations3.2. A Remarkable Limit Case of the Poincar¿e-Zhukovskii Equations3.3. Rigid body in an Arbitrary Potential FieldChapter 4. Linear Integrals and Reduction4.1. Linear Integrals in Rigid Body Dynamics4.2. Dynamical Symmetry and Lagrange Integral4.3. Generalizations of the Hess CaseChapter 5. Generalizations of Inerrability Cases5. 1. Various Generalizations of the Kovalevskaya and Goryachev-Chaplygin Cases5.2. Separation of Variables5.3. Isomorphism and Explicit Integration5.4. Doubly Asymptotic Motions for Integrable SystemsChapter 6. Periodic Solutions, Nonintegrability, and Transition to Chaos6. 1. Nonintegrability of Rigid Body Dynamics Equations6. 2. Periodic and Asymptotic Solutions in Euler-Poisson Equations and Related Problems6. 3. Absolute and Relative Choreographies in Rigid Body Dynamics6. 4. Chaotic Motions. Genealogy of Periodic Orbits6. 5. Chaos Evolution in the Restricted Problem of Heavy Rigid BodyRotation6. 6. Adiabatic Chaos in the Liouville Equations6. 7. Heavy Rigid Body Fall in Ideal Fluid. Probability Effects and Attracting SetsAppendixBibliography