PICARD-VESSIOT THEORY AND ZIGLIN THEOREM

Given a Hamiltonian system, Ziglin's theorem is one of the criteria us ed to study the integrability of the system. For this we make use of t he variational equations along a known solution. In this paper we obta in some consequences of the application of Picard-Vessiot differential Galois theory to Ziglin's theorem. First we present the general resul ts for the case of a finite arbitrary number of degrees of freedom. Th en we pass to the case of two degrees of freedom where the results can be given with more detail. Under the hypothesis of Ziglin's theorem a nd some additional technical assumptions the main results (Theorems 1 and 2) relate the integrability of the Hamiltonian with the properties of the differential Galois group of the Picard Vessiot extension asso ciated to the normal reduced variational equations. For two degrees of freedom it is possible to study also the resonant case. Theorems 4 an d 5 give the full classification of the Galois groups and the related extensions in the integrable case. A couple of applications are made t o recover Ito's theorem and to study the Lame equation. (C) 1994 Acade mic Press, Inc.