Integrable Hamiltonsche Systeme und algebraische Geometrie. (Integrable Hamiltonian systems and algebraic geometry). (German) Zbl 0601.58041

The paper gives a brief review of the latest achievements in the theory of integrable Hamiltonian systems with both finite and infinite degrees of freedom. The author emphasises the algebraic-geometric aspects of integrability. The mathematical pendulum behaviour is discussed as an introductory example and also some classical finite-dimensional integrable cases are considered. The Lax representation for the Toda lattice is presented producing a bridge to the theory of KdV equation; the formulas of the solutions obtained in terms of Riemann’s theta- functions by Novikov and others are given. The last section is devoted to the Kadomtsev-Petviashvili equation and its connections with the Schottky problem.
Reviewer: I.Ya.Dorfman


37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
14K25 Theta functions and abelian varieties
14H40 Jacobians, Prym varieties
35Q99 Partial differential equations of mathematical physics and other areas of application