Integrability and Nonintegrability in Geometry and Mechanics

• 1. Some Equations of Classical Mechanics and Their Hamiltonian Properties.- §1. Classical Equations of Motion of a Three-Dimensional Rigid Body.- §2. Symplectic Manifolds.- §3. Hamiltonian Properties of the Equations of Motion of a Three-Dimensional Rigid Body.- §4. Some Information on Lie Groups and Lie Algebras Necessary for Hamiltonian Geometry.- 2. The Theory of Surgery on Completely Integrable Hamiltonian Systems of Differential Equations.- §1. Classification of Constant-Energy Surfaces of Integrable Systems. Estimation of the Amount of Stable Periodic Solutions on a Constant-Energy Surface. Obstacles in the Way of Smooth Integrability of Hamiltonian Systems.- §2. Multidimensional Integrable Systems. Classification of the Surgery on Liouville Tori in the Neighbourhood of Bifurcation Diagrams.- §3. The Properties of Decomposition of Constant-Energy Surfaces of Integrable Systems into the Sum of Simplest Manifolds.- 3. Some General Principles of Integration of Hamiltonian Systems of Differential Equations.- §1. Noncommutative Integration Method.- §2. The General Properties of Invariant Submanifolds of Hamiltonian Systems.- §3. Systems Completely Integrable in the Noncommutative Sense Are Often Completely Liouville-Integrable in the Conventional Sense.- §4. Liouville Integrability on Complex Symplectic Manifolds.- 4. Integration of Concrete Hamiltonian Systems in Geometry and Mechanics. Methods and Applications.- §1. Lie Algebras and Mechanics.- §2. Integrable Multidimensional Analogues of Mechanical Systems Whose Quadratic Hamiltonians are Contained in the Discovered Maximal Linear Commutative Algebras of Polynomials on Orbits of Lie Algebras.- §3. Euler Equations on the Lie Algebra so(4).- §4. Duplication of Integrable Analogues of the Euler Equationsby Means of Associative Algebra with Poincaré Duality.- §5. The Orbit Method in Hamiltonian Mechanics and Spin Dynamics of Superfluid Helium-3.- 5. Nonintegrability of Certain Classical Hamiltonian Systems.- §1. The Proof of Nonintegrability by the Poincaré Method.- §2. Topological Obstacles for Complete Integrability.- §3. Topological Obstacles for Analytic Integrability of Geodesic Flows on Non-Simply-Connected Manifolds.- §4. Integrability and Nonintegrability of Geodesic Flows on Two-Dimensional Surfaces, Spheres, and Tori.- 6. A New Topological Invariant of Hamiltonian Systems of Liouville-Integrable Differential Equations. An Invariant Portrait of Integrable Equations and Hamiltonians.- §1. Construction of the Topological Invariant.- §2. Calculation of Topological Invariants of Certain Classical Mechanical Systems.- §3. Morse-Type Theory for Hamiltonian Systems Integrated by Means of Non-Bott Integrals.- References.

'...we consider this book to be a valuable contribution to present mathematics. It is useful primarily for researchers in the field but it may just as ell be used by students of mathematics, mechanics, or physics and by anybody interested in the modern applications of geometry and topology.' Acta Applicandae Mathematicae 28 1992