Non-integrability of the $n$-body problem

TL;DR

This paper proves that the classical planar n-body problem is generally non-integrable under fixed energy and angular momentum levels, except when both are zero, using differential Galois theory.

Contribution

It establishes the non-integrability of the n-body problem for most energy and angular momentum levels, extending previous results with a novel application of differential Galois theory.

Findings

The n-body problem is non-integrable for all energy and angular momentum levels except zero.

Differential Galois theory is effectively used to prove non-integrability.

The result clarifies the conditions under which the n-body problem lacks additional integrals.

Abstract

We prove that the classical planar $n$-body problem when restricted to a common level of the energy and the angular momentum is not integrable except in the case when both values of these integrals are zero. In the proof of our theorem, we use methods of differential Galois theory.