Uniqueness of star central configurations in the $5$-body problem

TL;DR

This paper provides a rigorous analytical proof that the only symmetric five-body central configuration with equal masses arranged in a planar polygon is a regular pentagon, using algebraic techniques and domain analysis.

Contribution

It establishes the uniqueness of the regular pentagon configuration for equal-mass five-body central configurations through a purely analytical approach.

Findings

The regular pentagon is the only central configuration under the given symmetry.

The solution space was reduced to a two-variable system for analysis.

Explicit algebraic arguments ruled out other configurations within the domain.

Abstract

In this study, we present a rigorous analytical proof of the uniqueness of central configurations for the five-body problem, assuming that all five masses are equal and positioned at the vertices of a planar polygon. We consider configurations in which the bodies are equally spaced in angular position relative to the center of mass, and aim to determine whether a central configuration arises under these constraints. We prove that the only central configuration that satisfies these conditions occurs when the five bodies form a regular pentagon. Our approach is entirely analytical, relying on algebraic techniques rather than numerical approximations. By transforming the governing equations into a reduced system involving only two variables, we analyze the solution space over a significant and carefully bounded domain. This domain is divided into sixteen disjoint regions, within which we rule out additional solutions through explicit algebraic arguments. Our results confirm that the regular pentagonal configuration is the only central configuration in this symmetric five-body scenario.