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

Leasly A. Campa-Raymundo, Luis Franco-Pérez

TL;DR

This work proves, using a fully analytic approach, that the only star central configuration (SCC) for the planar five-body problem with equal masses positioned at equal angular separations is the regular pentagon. By reducing the CC conditions to a two-variable system in $(r_3,r_5)$ on a carefully bounded domain $\hat{S}$ and partitioning it into 16 regions, the authors rule out all nonregular configurations through explicit algebraic inequalities on auxiliary functions. The result complements computer-assisted classifications and establishes a clear uniqueness for $n\le 5$ SCCs in this symmetric setting, while indicating nonuniqueness for larger $n$. The methodology combines reduction to a low-dimensional, symmetry-constrained system with rigorous region-wise analysis, highlighting the deep structure of central configurations in the $5$-body problem.

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.

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

TL;DR

This work proves, using a fully analytic approach, that the only star central configuration (SCC) for the planar five-body problem with equal masses positioned at equal angular separations is the regular pentagon. By reducing the CC conditions to a two-variable system in on a carefully bounded domain and partitioning it into 16 regions, the authors rule out all nonregular configurations through explicit algebraic inequalities on auxiliary functions. The result complements computer-assisted classifications and establishes a clear uniqueness for SCCs in this symmetric setting, while indicating nonuniqueness for larger . The methodology combines reduction to a low-dimensional, symmetry-constrained system with rigorous region-wise analysis, highlighting the deep structure of central configurations in the -body problem.

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.

Paper Structure

This paper contains 8 sections, 24 theorems, 22 equations, 15 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Basic results
  4. $5$-body problem
  5. Uniqueness
  6. Properties of auxiliary functions of $J_1$
  7. Proofs of the regions $J_{n}$
  8. Conclusion

Key Result

Proposition 3.1

A configuration $\mathbf{q}\in S$ is a CC if and only if $|\mathbf{q}_1|^{-1} \bf{q}$ is a critical point of $U|_{\hat{S}}$.

Figures (15)

  • Figure 1: domain $\hat{S}$
  • Figure 2: regions $J_{1},...,J_{16}$.
  • Figure 5: proof parts of Proposition \ref{['aux5']}.
  • Figure 20:
  • Figure 21: functions $\lambda_{11_{\zeta}}(r_{3})$ and $\lambda_{52_{\zeta}}(r_{3})$ for $\eta=0.328794$
  • ...and 10 more figures

Theorems & Definitions (50)

  • Definition 2.1
  • Definition 2.2
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Theorem 5.1
  • proof
  • Theorem 5.2
  • Definition 5.3
  • ...and 40 more