Liouville integrability: an effective Morales-Ramis-Sim\'o theorem

TL;DR

This paper presents an effective method to analyze the integrability of complex Hamiltonian systems by reducing variational equations and applying the Morales-Ramis-Simó criterion, facilitating the testing of meromorphic Liouville integrability.

Contribution

It introduces a new procedure to efficiently reduce variational equations in Hamiltonian systems, enabling practical application of integrability criteria.

Findings

Provides an effective algorithm for variational equation reduction.

Enables practical testing of Liouville integrability.

Facilitates application of Morales-Ramis-Simó theorem.

Abstract

Consider a complex Hamiltonian system and an integral curve. In this paper, we give an effective and efficient procedure to put the variational equation of any order along the integral curve in reduced form provided that the previous one is in reduced form with an abelian Lie algebra. Thus, we obtain an effective way to check the Morales-Ramis-Sim\'o criterion for testing meromorphic Liouville integrability of Hamiltonian systems.