This book considers the larger class of systems which are not (at least a priori) Hamiltonian but possess tensor invariants, in particular, an invariant measure. Several integrability theorems related to the existence of tensor invariants are formulated, and the authors illustrate the geometrical background of some classical and new hierarchies of integrable systems and give their explicit solution in terms of theta-functions. Most of the results discussed have not been published before, making this book immensely useful both to specialists in analytical dynamics who are interested in integrable problems and those in algebraic geometry who are looking for applications.
Ocenterlinea MEMOIRS ON INTEGRABLE SYSTEMSu Omedskip Ocenterlinea TABLE OF CONTENTSu Obigskip Chapter I. CLASSICAL MECHANICS AND LIE GROUPS. 1.1. Momentum theorem 1.2. Multi-dimensional dynamics 1.3. The Euler--PoincarOe equations Obigskip Chapter II. SYSTEMS WITH AN INVARIANT MEASURE. 2.1. Integral invariants 2.2. Integrability 2.3. The Kowalewski--Lyapunov method 2.4. Examples of systems with an invariant measure 2.5 Systems with an invariant measure on Lie groups Obigskip Chapter III. INTEGRABLE SYSTEMS, LAX PAIRS AND CONFOCAL QUADRICS 3.1. Geometry 3.2. Lax pairs and hierarchies of integrable Hamiltonian systems 3.3. Hierarchy of the Frahm--Manakov and Clebsch--Perelomov systems 3.4. Hierarchy of the Steklov--Lyapunov--Rubanovsky systems 3.5. The FMCP hierarchy and common tangent linear spaces of confocal quadrics 3.6. Complete KO"otters solution for the Clebsch case 3.7. Integrable nonholonomic systems on $so(n)$ 3.8. The Steklov--Lyapunov systems and pencils of lines Obigskip Chapter IV. EXPLICIT SOLUTIONS 4.1. Abelian tori and Jacobians 4.2. Theta-functional solutions 4.3. Solutions in generalized theta-functions. Obigskip References Index
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