Synopses & Reviews
Presenting a new approach to qualitative analysis of integrable geodesic flows based on the theory of topological classification of integrable Hamiltonian systems, this is the first book to apply this technique systematically to a wide class of integrable systems. The first part of the book provides an introduction to the qualitative theory of integrable Hamiltonian systems and their invariants (symplectic geometry, integrability, the topology of Liouville foliations, the orbital classification theory for integrable nondegenerate Hamiltonian systems with two degrees of freedom, obstructions to integrability, etc). In the second part, the class of integrable geodesic flows on two-dimensional surfaces is discussed both from the classical and contemporary point of view. The authors classify them up to different equivalence relations such as an isometry, the Liouville equivalence, the trajectory equivalence (smooth and continuous), and the geodesic equivalence. A new technique, which provides the possibility to classify integrable geodesic flows up to these kinds of equivalences, is presented together with applications. Together with systematic presentation of wide material on this subject, the book contains previously unpublished new results, and is enhanced with many original illustrations.
Table of Contents
Preface.
1. Basic Notions.
2. Topology of Foliations Generated by Morse Functions on Two-Dimensional Surfaces.
3. Rough Liouville Equivalence of Integrable Systems with Two Degrees of Freedom.
4. Liouville Equivalence of Integrable Systems with Two Degrees of Freedom.
5. Trajectory Classification of Integrable Systems with Two Degrees of Freedom.
6. Integrable Geodesic Flowson Two-Dimensional Surfaces.
7. Liouville Classification of Integrable Geodesic Flows on Two-Dimensional Surfaces.
8. Trajectory Classification of Integrable Geodesic Flows on Two-Dimensional Surfaces.
9. Maupertuis Principle and Geodesic Equivalence.
10. Euler Case in Rigid Body Dynamics and Jacob Problem About Geodesics on the Ellipsoid. Trajectory Isomorphism. References.