Nonautonomous Hamiltonian Systems and Morales-Ramis Theory I. The Case $\ddot{x}=f(x,t)$

TL;DR

This paper uses Morales-Ramis theory to analyze the non-integrability of certain differential equations resembling Hamiltonian systems, including Painlevé II and Sitnikov, providing new insights into their complex dynamics.

Contribution

It applies Morales-Ramis theory to establish non-integrability of specific non-autonomous Hamiltonian-like systems, including new analyses of Painlevé II and Sitnikov equations.

Findings

Proves Painlevé II is non-integrable using three Hamiltonian formulations.

Demonstrates Sitnikov problem's non-integrability with numerical evidence.

Provides a framework for analyzing non-integrability in non-autonomous systems.

Abstract

In this paper we present an approach towards the comprehensive analysis of the non-integrability of differential equations in the form $\ddot x=f(x,t)$ 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 Painlev\'e II, Sitnikov and Hill-Schr\"odinger equation. We emphasize in Painlev\'e II, showing its non-integrability 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 almost-same title delivered by the author in SIAM Conference on Applications of Dynamical Systems 2007.