Galoisian obstructions to integrability of Hamiltonian systems, II

Abstract

By applying the results of our previous paper [19], we obtain non-integrability results for the following four Hamiltonian systems: the Bianchi IX Cosmological Model, the Sitnikov system of celestial mechanics, the Spring-Pendulum system and a generalization of the homogeneous potentials considered by Yoshida. All these systems are considered over the complex domain (complex time and complex phase space) and the integrability is considered in the Liouville sense of existence of a maximal number of first integrals in involution. 1. Introduction. This paper is the continuation of our previous papers [19, 20] and is devoted to the applications of the results of [19] to some models: the Bianchi IX cosmological model, the Sitnikov system of celestial mechanics, the Spring-Pendulum system and a generalization of our results in [20] about the non-integrability of homogeneous potentials. From the differential Galois theory we will need only the algorithm of Kovacic and the theorem of Kimura about the solvability of the Riemann's hypergeometric differential equation. Then, using our results in [19], the methods proposed here are efficient, but they remain completely systematic and very elementary.