Non-Autonomous Hamiltonian Systems and Morales-Ramis Theory

In this paper we present an approach towards the comprehensive analysis of the non-integrability of differential equations in the form ẍ = f(x, t) which is analogous to Hamiltonian systems with 1 + 1/2 degree of freedom. In particular, we analyze the non-integrability of some important families of differential equations such as Painlevé II, Sitnikov and HillSchrödinger equation. We emphasize in Painlevé II, showing its nonintegrability through three different Hamiltonian systems, and also in Sitnikov in which two different version including numerical results are shown. The main tool to study the non-integrability of these kind of Hamiltonian systems is Morales-Ramis theory. This paper is a very slight improvement of the talk with the same title delivered by the author in SIAM Conference on Applications of Dynamical Systems 2007.