Completely integrable systems: Jacobi's heritage

During the last few decades, algebraic geometry has become a tool for solving differential equations and spectral questions of mechanics and mathematical physics. This paper deals with the study of the integrable systems from the point of view of algebraic geometry, inverse spectral problems and mechanics from the point of view of Lie groups. Section 1 is preliminary giving a little background. In Section 2, we study a Lie algebra theoretical method leading to completely integrable systems, based on the Kostant-Kirillov coadjoint action. Section 3 is devoted to illustrate how to decide about the algebraic complete integrability (a.c.i.) of Hamiltonian systems. Algebraic integrability means that the system is completely integrable in the sense of the phase space being foliated by tori, which in addition are real parts of a complex algebraic tori (abelian varieties). Adler-van Moerbeke's method is a very useful tool not only to discover among families of Hamiltonian systems those which are a.c.i., but also to characterize and describe the algebraic nature of the invariant tori (periods, etc.) for the a.c.i. systems. Some integrable systems, such as Kortewege—de Vries equation, Toda lattice, Euler rigid body motion, Kowalewski's top, Manakov's geodesic flow on S O (4), etc. are treated.