On then-body problem in R4

Abstract

Using geometric mechanics methods, we examine aspects of the dynamics of n mass points in 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 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 6 d.f. system and we show that for generic potentials and momenta, relative equilibria with equilateral configuration are unstable. This article is part of the theme issue ‘Topological and geometrical aspects of mass and vortex dynamics’.