Integrability conditions at order 2 for homogeneous potentials of degree -1

TL;DR

This paper establishes second-order integrability conditions for homogeneous potentials of degree -1, extending Morales-Ramis criteria, and explicitly computes the associated Galois group and Picard-Vessiot extension.

Contribution

It introduces a second-order meromorphic integrability condition for degree -1 homogeneous potentials, extending previous first-order Morales-Ramis conditions.

Findings

Second-order integrability condition proven

Galois group of second order variational equation is abelian under the criterion

Explicit computation of Galois group and Picard-Vessiot extension

Abstract

We prove a meromorphic integrability condition at order 2 near a homothetic orbit for a meromorphic homogeneous potential of degree -1, which extend the Morales Ramis conditions of order 1. Conversely, we prove that if this criterion is satisfied, then the Galois group of second order variational equation is abelian and we compute explicitly the Galois group and the Picard-Vessiot extension.