Integrability and non integrability of some n body problems

TL;DR

This paper proves the non-integrability of colinear 3 and 4 body problems with positive masses, introduces integrability criteria for 3D potentials, and identifies some integrable cases within the n-body problem.

Contribution

It establishes non-integrability results for specific n-body configurations and develops new criteria for integrability of homogeneous potentials.

Findings

Non-integrability of colinear 3 and 4 body problems with positive masses.

Integrability criteria for 3D homogeneous potentials of degree -1.

Identification of some integrable cases within the n-body problem.

Abstract

We prove the non integrability of the colinear $3$ and $4$ body problem, for any masses positive masses. To deal with resistant cases, we present strong integrability criterions for $3$ dimensional homogeneous potentials of degree $-1$, and prove that such cases cannot appear in the $4$ body problem. Following the same strategy, we present a simple proof of non integrability for the planar $n$ body problem. Eventually, we present some integrable cases of the $n$ body problem restricted to some invariant vector spaces.