Integrable Hamiltonian systems. Geometry, topology, classification. Translated from the 1999 Russian original.

*(English)*Zbl 1056.37075
Boca Raton, FL: Chapman & Hall/CRC; (ISBN 0-415-29805-9/hbk). xv, 730 p. (2004).

This book provides an introduction to the problem of classification of integrable Hamiltonian systems. It presents, in a systematic way, a great deal of material previously available only in journals. The authors intend the book for a broader audience then specialists, at least to include students of physics, mechanics, and mathematics. However, the book is long (700 pages in addition to the bibliography and index), fairly narrowly focused, and is likely to be challenging to the non-specialist.

The aim of the book is classification, and three kinds of equivalence relation are considered: conjugacy, orbital equivalence (both topological and smooth), and Liouville equivalence. The authors focus on orbital and Liouville equivalence and provide a complete solution for nondegenerate integrable Hamiltonian systems with two degrees of freedom. The basis of the authors’ classification is a new approach to the qualitative theory of integrable Hamiltonian systems proposed by A. T. Fomenko, and the theory of topological classification of such systems proposed by Zieschang, Matveev, Bolsinov and Brailov.

Fomenko assigns a graph as a topological invariant for each integrable Hamiltonian system. He calls this graph a “molecule”. With this invariant, it is possible to describe the structure of the foliation of the isoenergy surface into invariant Liouville tori, and thus to classify such systems up to Liouville equivalence. As an additional invariant, the so-called “market molecule” includes orbital invariants.

The first part of the book (nine chapters) presents introductory material and the foundations of the classification theory. The remaining seven chapters present applications of the classification theory in geometry, mechanics and physics. Two classes of integrable systems are studied in some detail. These are integrable cases in rigid body dynamics, and integrable geodesic flows of Riemannian metrics on surfaces.

The aim of the book is classification, and three kinds of equivalence relation are considered: conjugacy, orbital equivalence (both topological and smooth), and Liouville equivalence. The authors focus on orbital and Liouville equivalence and provide a complete solution for nondegenerate integrable Hamiltonian systems with two degrees of freedom. The basis of the authors’ classification is a new approach to the qualitative theory of integrable Hamiltonian systems proposed by A. T. Fomenko, and the theory of topological classification of such systems proposed by Zieschang, Matveev, Bolsinov and Brailov.

Fomenko assigns a graph as a topological invariant for each integrable Hamiltonian system. He calls this graph a “molecule”. With this invariant, it is possible to describe the structure of the foliation of the isoenergy surface into invariant Liouville tori, and thus to classify such systems up to Liouville equivalence. As an additional invariant, the so-called “market molecule” includes orbital invariants.

The first part of the book (nine chapters) presents introductory material and the foundations of the classification theory. The remaining seven chapters present applications of the classification theory in geometry, mechanics and physics. Two classes of integrable systems are studied in some detail. These are integrable cases in rigid body dynamics, and integrable geodesic flows of Riemannian metrics on surfaces.

##### MSC:

37J35 | Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests |

37-02 | Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory |

53D25 | Geodesic flows in symplectic geometry and contact geometry |

70G40 | Topological and differential topological methods for problems in mechanics |

70G45 | Differential geometric methods (tensors, connections, symplectic, Poisson, contact, Riemannian, nonholonomic, etc.) for problems in mechanics |

70H06 | Completely integrable systems and methods of integration for problems in Hamiltonian and Lagrangian mechanics |