On the n-body problem in $\mathbb{R}^4$

TL;DR

This paper explores the dynamics of n particles under pairwise potentials in four-dimensional space, analyzing relative equilibria, invariant manifolds, and stability, with a focus on regular n-gons and the three-body case.

Contribution

It extends geometric mechanics methods to the four-dimensional n-body problem, characterizing invariant manifolds and stability of regular polygon configurations.

Findings

Regular n-gons form invariant manifolds in $\

Relative equilibria with equilateral configurations are generally unstable for n=3.

The dynamics can be reduced to a six degrees of freedom system for three particles.

Abstract

Using geometric mechanics methods, we examine aspects of the dynamics of n mass points in $\mathbb{R}^4$ with a general pairwise potential. We investigate the central force problem, set up the n-body problem and discuss certain properties of relative equilibria. We describe regular n-gons in $\mathbb{R}^4$ and when the masses are equal, we determine the invariant manifold of motions with regular n-gon configurations. In the case n=3 we reduce the dynamics to a six degrees of freedom system and we show that for generic potentials and momenta, relative equilibria with equilateral configuration are unstable.