Non-integrability of a Hamiltonian system and Legendre functions

TL;DR

This paper investigates the non-integrability of a specific two-dimensional Hamiltonian system with degree 6 homogeneous potential using advanced differential Galois theory and Legendre functions.

Contribution

It introduces a novel approach by reducing variation equations to Legendre equations, bypassing the need for Darboux points, and analyzes the Galois group structure for integrability.

Findings

The variation equations reduce to Legendre equations, simplifying analysis.

Conditions for non-commutative Galois groups are identified.

The approach clarifies the gray areas of classical integrability results.

Abstract

In this paper we are studying the meromorphic integrability of a two-dimensional Hamiltonian system with a homogeneous potential of degree 6. The approach used in this work is the theory of the Ziglin-Moralez-Ruiz-Ramis-Simo. Within the scope of this theory, the study of such systems is reduced to determining the differential Galois group of a linear differential equation, obtained as a projection onto the tangent bundle of the phase curve of its non-equilibrium solution - Variation Equations (VE). In the case of Hamiltonian systems with homogeneous potentials, the variation equations are hypergeometric. If a standard approach is used to study such a system, it is necessary to calculate a Darboux point, which is not always easy. In this paper we can skip this difficulty by reducing VE to a Legendre equation. We use the results for commutativity of the Galois group of the associated Legendre equation for study a Hamiltonian system with a homogeneous polynomial potential of degree 6. The approach is different and answers are sought as to what exactly is happening in the gray areas of the classical results. For the full study, the second variations are used and conditions for a non-zero logarithmic term in their solutions are found. This is exactly the case when VE is solvable, but the unit component of the Galois group is not commutative.