Computing necessary integrability conditions for planar parametrized homogeneous potentials

TL;DR

This paper presents an algorithm to compute necessary integrability conditions for planar homogeneous potentials, enabling analysis of complex systems like polynomial potentials up to degree 9 and the collinear three body problem.

Contribution

It introduces a novel algorithm that derives polynomial necessary conditions for integrability of parametrized potentials, including applications to previously intractable problems.

Findings

Algorithm successfully computes integrability conditions for complex potentials.

Proves non-integrability of polynomial potentials up to degree 9.

Provides the first complete proof of non-integrability for the collinear three body problem.

Abstract

Let $V\in\mathbb{Q}(i)(\a_1,\dots,\a_n)(\q_1,\q_2)$ be a rationally parametrized planar homogeneous potential of homogeneity degree $k\neq -2, 0, 2$. We design an algorithm that computes polynomial \emph{necessary} conditions on the parameters $(\a_1,\dots,\a_n)$ such that the dynamical system associated to the potential $V$ is integrable. These conditions originate from those of the Morales-Ramis-Sim\'o integrability criterion near all Darboux points. The implementation of the algorithm allows to treat applications that were out of reach before, for instance concerning the non-integrability of polynomial potentials up to degree $9$. Another striking application is the first complete proof of the non-integrability of the \emph{collinear three body problem}.