Picard-Vessiot Theory and Ziglin′s Theorem

Given a Hamiltonian system, Ziglin′s theorem is one of the criteria used to study the integrability of the system. For this we make use of the variational equations along a known solution. In this paper we obtain some consequences of the application of Picard-Vessiot differential Galois theory to Ziglin′s theorem. First we present the general results for the case of a finite arbitrary number of degrees of freedom. Then 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 and 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 associated to the normal reduced variational equations. For two degrees of freedom it is possible to study also the resonant case. Theorems 4 and 5 give the full classification of the Galois groups and the related extensions in the integrable case. A couple of applications are made to recover Ito′s theorem and to study the Lamé equation.