Equilateral Chains and Cyclic Central Configurations of the Planar 5-body Problem

Abstract

Central configurations and relative equilibria are an important facet of the study of the $N$-body problem, but become very difficult to rigorously analyze for $N>3$. In this paper we focus on a particular but interesting class of configurations of the 5-body problem: the equilateral pentagonal configurations, which have a cycle of five equal edges. We prove a variety of results concerning central configurations with this property, including a computer-assisted proof of the finiteness of such configurations for any positive five masses with a range of rational-exponent homogeneous potentials (including the Newtonian case and point-vortex model), some constraints on their shapes, and we determine some exact solutions for particular N-body potentials.