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

TL;DR

This paper investigates equilateral pentagonal central configurations in the planar 5-body problem, proving finiteness results, shape constraints, and providing exact solutions for specific potentials, advancing understanding of these complex configurations.

Contribution

It introduces new results on the finiteness, shape constraints, and exact solutions of equilateral pentagonal central configurations in the 5-body problem.

Findings

Finiteness of equilateral pentagonal configurations for positive masses.

Constraints on the shapes of these configurations.

Exact solutions for particular N-body potentials.

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.