A new analysis of the three--body problem

TL;DR

This paper reviews recent mathematical results on the three-body problem, focusing on periodic oscillations and chaotic dynamics near equilibria, using the two-centre problem as a key analytical tool.

Contribution

It synthesizes recent proofs of periodic and chaotic motions in the three-body problem using topological and canonical methods.

Findings

Existence of periodic oscillations about equilibria.

Onset of topological horseshoe indicating chaos.

Use of the two-centre problem as a neighboring integrable system.

Abstract

In the recent papers~[18],~[5], respectively, the existence of motions where the perihelions afford periodic oscillations about certain equilibria and the onset of a topological horseshoe have been proved. Such results have been obtained using, as neighbouring integrable system, the so--called two--centre (or {\it Euler}) problem and a suitable canonical setting proposed in~[16],~[17]. Here we review such results.