A Reduced Form for Linear Differential Systems and its Application to Integrability of Hamiltonian Systems

TL;DR

This paper introduces a method to compute reduced forms of symplectic differential systems, aiding in the analysis of Hamiltonian system integrability through an effective application of the Morales-Ramis theorem.

Contribution

It develops a technique to find reduced forms of symplectic systems, enhancing the analysis of Hamiltonian integrability by simplifying variational equations.

Findings

Computed reduced forms for symplectic differential systems.

Applied reduced forms to analyze Hamiltonian system integrability.

Provided an effective version of the Morales-Ramis theorem.

Abstract

Let $[A]: Y'=AY$ with $A\in \mathrm{M}_n (k)$ be a differential linear system. We say that a matrix $R\in {\cal M}_{n}(\bar{k})$ is a {\em reduced form} of $[A]$ if $R\in \mathfrak{g}(\bar{k})$ and there exists $P\in GL_n (\bar{k})$ such that $R=P^{-1}(AP-P')\in \mathfrak{g}(\bar{k})$. Such a form is often the sparsest possible attainable through gauge transformations without introducing new transcendants. In this article, we discuss how to compute reduced forms of some symplectic differential systems, arising as variational equations of hamiltonian systems. We use this to give an effective form of the Morales-Ramis theorem on (non)-integrability of Hamiltonian systems.