Integrable Hamiltonian systems have been of growing interest over the past 30 years and represent one of the most intriguing and mysterious classes of dynamical systems. This book explores the topology of integrable systems and the general theory underlying their qualitative properties, singularites, and topological invariants.
The authors, both of whom have contributed significantly to the field, develop the classification theory for integrable systems with two degrees of freedom. This theory allows one to distinguish such systems up to two natural equivalence relations: the equivalence of the associated foliation into Liouville tori and the usual orbital equaivalence. The authors show that in both cases, one can find complete sets of invariants that give the solution of the classification problem.
The first part of the book systematically presents the general construction of these invariants, including many examples and applications. In the second part, the authors apply the general methods of the classification theory to the classical integrable problems in rigid body dynamics and describe their topological portraits, bifurcations of Liouville tori, and local and global topological invariants. They show how the classification theory helps find hidden isomorphisms between integrable systems and present as an example their proof that two famous systems--the Euler case in rigid body dynamics and the Jacobi problem of geodesics on the ellipsoid--are orbitally equivalent.
Integrable Hamiltonian Systems: Geometry, Topology, Classification offers a unique opportunity to explore important, previously unpublished results and acquire generally applicable techniques and tools that enable you to work with a broad class of integrable systems.
Basic Notions. The Topology of Foliations on Two-Dimensional Surfaces Generated by Morse Functions. Rough Liouville Equivalence of Integrable Systems with Two Degrees of Freedom. Liouville Equivbalence of Integrable Systems with Two Degrees of Freedom. Orbital Classification of Integrable Systems with Two Degrees of Freedom. Classification of Hamiltonian Flows on Two-Dimensional Surfaces up to Topological Conjugacy. Smooth Conjugacy of Hamiltonian Flows on Two-Dimensional Surfaces. Orbital Classification of Integrable Hamiltonian Systems with Two Degrees of Freedom. The Second Step. Liouville Classification of Integrable Systems with Two Degrees of Freedom in Four-Dimensional Neighborhoods of Singular Points. Methods of Calculation of Topological Invariants of Integrable Hamiltonian Systems. Integrable Geodesic Flows on Two-Dimensional Surfaces. Liouville Classification of Integrable Geodesic Flows on Two-Dimensional Surfaces. Orbital Classification of Integrable Geodesic Flows on Two-Dimensional Surfaces. The Topology of Liouville Foliations in Classical Integrable Cases in Rigid Body Dynamics. Maupertuis Principle and Geodesic Equivalence.