On the equilateral pentagonal central configurations

TL;DR

This paper classifies symmetric equilateral pentagon configurations as central in the 5-body problem, identifying two unique classes: one convex regular and one concave, using rational parameterizations to solve complex equations.

Contribution

It provides a complete classification of symmetric equilateral pentagonal central configurations in the 5-body problem, introducing a novel use of rational parameterizations for solving related equations.

Findings

Two unique classes of equilateral pentagonal central configurations identified

The regular convex pentagon forms one class of solutions

A concave pentagon also forms a distinct class of solutions

Abstract

An equilateral pentagon is a polygon in the plane with five sides of equal length. In this paper we classify the central configurations of the $5$-body problem having the five bodies at the vertices of an equilateral pentagon with an axis of symmetry. We prove that there are two unique classes of such equilateral pentagons providing central configurations, one concave equilateral pentagon and one convex equilateral pentagon, the regular one. A key point of our proof is the use of rational parameterizations to transform the corresponding equations, which involve square roots, into polynomial equations.