Non integrability of the n body problem with non zero angular momentum

TL;DR

This paper establishes criteria for integrability of homogeneous rotationally invariant potentials and applies these to prove the non-integrability of the planar Newtonian n-body problem with equal masses when angular momentum is non-zero.

Contribution

It introduces new integrability and partial integrability criteria for homogeneous potentials and demonstrates their application to the non-integrability of the equal-mass n-body problem with angular momentum.

Findings

Proves meromorphic non-integrability of the n-body problem with equal masses in the plane.

Provides criteria for integrability of homogeneous rotationally invariant potentials.

Analyzes the n-body problem on a specific energy and angular momentum surface.

Abstract

We prove an integrability criterion and a partial integrability criterion for homogeneous potentials of degree -1 which are invariant by rotation. We then apply it to the proof of the meromorphic non-integrability of the n body problem with Newtonian interaction in the plane on a surface of equation $(H,C)=(H_0,C_0)$ with $(H_0,C_0) \neq (0,0)$ where C is the angular momentum and H the energy, in the case where the n masses are equal.