On the Degeneracy of the Central Configuration Formed by a Regular n-Gon with a Central Mass

TL;DR

This paper develops a representation-theoretic framework exploiting dihedral symmetry to explicitly determine degeneracy values in central configurations formed by a regular n-gon with a central mass.

Contribution

It introduces a systematic, symmetry-based method to analyze degeneracy in central configurations, revealing mode-specific degeneracy conditions and providing explicit formulas.

Findings

Degeneracy values are explicitly determined for each Fourier mode.

The number of degeneracy values increases with n, reflecting more symmetry modes.

Degeneracy is explained as a consequence of the underlying symmetry, not an isolated phenomenon.

Abstract

We investigate the degeneracy of the central configuration formed by a regular $n$-gon of equal masses together with an additional mass at the center. While degeneracy of such configurations has traditionally been studied through direct spectral computations, a systematic structural understanding of the origin and multiplicity of degeneracy values has remained incomplete. Exploiting the dihedral symmetry $D_n$, we develop a representation-theoretic framework that decomposes the Hessian of $\sqrt{IU}$ into invariant blocks associated with irreducible symmetry modes, reducing the degeneracy problem to a finite collection of low-dimensional determinants. In particular, this decomposition reveals a distinguished $3 \times 3$ block arising from the coupling between the central mass and the first Fourier mode. Within this framework, degeneracy is organized mode by mode: for each admissible Fourier mode $l \geq 2$, there exists at most one critical value of the central mass parameter at which degeneracy occurs, while the mode $l = 1$ exhibits a qualitatively different behavior. As a consequence, all degeneracy values can be determined explicitly, and their number increases with $n$, reflecting the growing number of independent symmetry modes. Our results provide a structural explanation for the multiplicity of degeneracy values and show that degeneracy is not an isolated phenomenon, but a consequence of the underlying symmetry. The approach also suggests a general framework for analyzing degeneracy in symmetric central configurations.