A note on algebraic potentials and Morales-Ramis theory

TL;DR

This paper explores algebraic potentials in celestial mechanics, demonstrating that Morales-Ramis theorems can be applied beyond meromorphic cases to establish non-integrability results.

Contribution

It extends Morales-Ramis theory to algebraic potentials, enabling new non-integrability proofs in celestial mechanics involving algebraic functions.

Findings

Morales-Ramis theorems apply to algebraic potentials

Enhanced non-integrability proofs in celestial mechanics

Broader applicability of integrability analysis methods

Abstract

We present various properties of algebraic potentials, and then prove that some Morales-Ramis theorems readily apply for such potentials even if they are not in general meromorphic potentials. This allows in particular to precise some non-integrability proofs in celestial mechanics, where the mutual distances between the bodies appear in the potentials, and thus making this analysis unavoidable.