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

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.