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

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.