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

Tingjie Zhou, Zhihong Xia

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 structural understanding of the origin and multiplicity of degeneracy values has remained unclear. 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. This reduces the degeneracy problem to a finite collection of low-dimensional determinants, including a distinguished $3 \times 3$ block arising from the coupling between the central mass and the first Fourier mode. Within this framework, we show that degeneracy is completely governed by symmetry modes: 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, the number of degeneracy values increases with $ n $, reflecting the growing number of independent symmetry modes. Our results provide a conceptual explanation for the multiplicity of degeneracy values and reveal that degeneracy is not an isolated phenomenon, but a structural consequence of the underlying group symmetry. The approach also suggests a general strategy for analyzing degeneracy in symmetric central configurations.

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 -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 structural understanding of the origin and multiplicity of degeneracy values has remained unclear. Exploiting the dihedral symmetry , we develop a representation-theoretic framework that decomposes the Hessian of into invariant blocks associated with irreducible symmetry modes. This reduces the degeneracy problem to a finite collection of low-dimensional determinants, including a distinguished block arising from the coupling between the central mass and the first Fourier mode. Within this framework, we show that degeneracy is completely governed by symmetry modes: for each admissible Fourier mode , there exists at most one critical value of the central mass parameter at which degeneracy occurs, while the mode exhibits a qualitatively different behavior. As a consequence, the number of degeneracy values increases with , reflecting the growing number of independent symmetry modes. Our results provide a conceptual explanation for the multiplicity of degeneracy values and reveal that degeneracy is not an isolated phenomenon, but a structural consequence of the underlying group symmetry. The approach also suggests a general strategy for analyzing degeneracy in symmetric central configurations.

Paper Structure

This paper contains 18 sections, 10 theorems, 85 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Symmetry and invariant decomposition
  3. The dihedral symmetry
  4. Equivariance of the Hessian and isotypic decomposition
  5. Fourier-type decomposition
  6. Refinement via reflection symmetry
  7. Block structure of the Hessian
  8. Degeneracy of the central $+$ regular $n$-gon configuration
  9. Notation
  10. Computation of the blocks
  11. Structure of the blocks
  12. Degeneracy
  13. Computation of the Hessian blocks
  14. Decomposition of the Hessian
  15. The matrices $\nabla(\sqrt{2I}) \nabla U^{T}$ and $\nabla U \nabla(\sqrt{2I})^{T}$
  16. ...and 3 more sections

Key Result

Theorem 1

In a symmetry-adapted basis, the Hessian $D^2f(z_0)$ is block-diagonal. The blocks correspond to symmetry modes and have the following structure: $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (17)

  • Remark 1
  • Theorem 1: Block structure
  • Remark 2: Coupling at $l=1$
  • Remark 3: Parity of $n$
  • Theorem 2: Main Theorem
  • Proposition 1
  • proof
  • Proposition 2: Structure of $A_l$ for $l\ge2$
  • Proposition 3: Structure of $A_1$
  • Proposition 4
  • ...and 7 more