Equilateral Chains and Cyclic Central Configurations of the Planar Five-Body Problem

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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N>3$$\end{document}. In this paper, we focus on a particular but interesting class of configurations of the five-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 the point-vortex model), some constraints on their shapes, and we determine some exact solutions for particular N-body potentials.