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Degeneracy of Planar Central Configurations in the $N$-Body Problem

Shanzhong Sun, Zhifu Xie, Peng You

TL;DR

This paper addresses the degeneracy of planar central configurations in the $N$-body problem by directly removing symmetry-induced zero eigenvalues from the Jacobian $Jac(F)$ of the central-configuration equations. It develops three equivalent forms to isolate the nontrivial degeneracy: Form I (two zeros from rotation and scaling), Form II (three zeros from translation and rotation), and Form III (four zeros from all symmetries), each yielding a reduced matrix $J_2$ whose determinant tests nondegeneracy. Applying these forms, the authors prove nondegeneracy for Lagrange’s equilateral-triangle configurations for arbitrary masses and establish nondegeneracy for rhombus 4-body configurations with masses $[m_1,1,m_1,1]$ using interval arithmetic to certify $ abla$-positive determinants across admissible parameter ranges. The combination of symmetry-aware reduction and interval analysis provides a rigorous tool for probing degeneracy and potential bifurcations in the full configuration space, with implications for understanding central-configurations bifurcations beyond classical subspaces.

Abstract

The degeneracy of central configurations in the planar $N$-body problem makes their enumeration problem hard and the related dynamics appealing. The degeneracy is always intertwined with the symmetry of the system of central configurations which makes the problem subtle. By analyzing the Jacobian matrix of the system, we systematically explore the direct method to single out trivial zero eigenvalues associated with translational, rotational and scaling symmetries, thereby isolating the non-trivial part of the Jacobian to study the degeneracy. Three distinct formulations of degeneracy are presented, each tailored to handle different formulation of the system. The method is applied to such well-known examples as Lagrange's equilateral triangle solutions for arbitrary masses, the square configuration for four equal masses and the equilateral triangle with a central mass revealing specific mass values for which degeneracy occurs. Combining with the interval algorithm, the nondegeneracy of rhombus central configurations for arbitrary mass is established.

Degeneracy of Planar Central Configurations in the $N$-Body Problem

TL;DR

This paper addresses the degeneracy of planar central configurations in the -body problem by directly removing symmetry-induced zero eigenvalues from the Jacobian of the central-configuration equations. It develops three equivalent forms to isolate the nontrivial degeneracy: Form I (two zeros from rotation and scaling), Form II (three zeros from translation and rotation), and Form III (four zeros from all symmetries), each yielding a reduced matrix whose determinant tests nondegeneracy. Applying these forms, the authors prove nondegeneracy for Lagrange’s equilateral-triangle configurations for arbitrary masses and establish nondegeneracy for rhombus 4-body configurations with masses using interval arithmetic to certify -positive determinants across admissible parameter ranges. The combination of symmetry-aware reduction and interval analysis provides a rigorous tool for probing degeneracy and potential bifurcations in the full configuration space, with implications for understanding central-configurations bifurcations beyond classical subspaces.

Abstract

The degeneracy of central configurations in the planar -body problem makes their enumeration problem hard and the related dynamics appealing. The degeneracy is always intertwined with the symmetry of the system of central configurations which makes the problem subtle. By analyzing the Jacobian matrix of the system, we systematically explore the direct method to single out trivial zero eigenvalues associated with translational, rotational and scaling symmetries, thereby isolating the non-trivial part of the Jacobian to study the degeneracy. Three distinct formulations of degeneracy are presented, each tailored to handle different formulation of the system. The method is applied to such well-known examples as Lagrange's equilateral triangle solutions for arbitrary masses, the square configuration for four equal masses and the equilateral triangle with a central mass revealing specific mass values for which degeneracy occurs. Combining with the interval algorithm, the nondegeneracy of rhombus central configurations for arbitrary mass is established.

Paper Structure

This paper contains 10 sections, 5 theorems, 86 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Central configuration and its symmetries
  3. Degeneracy
  4. Definition of degeneracy in general
  5. Form I: Two Trivial Zero Eigenvalues
  6. Form II: Three Trivial Zero Eigenvalues
  7. Form III: Four Trivial Zero Eigenvalues
  8. Non-degeneracy of Lagrange Central Configurations
  9. Non-degeneracy of Rhombus Central Configurations of 4-body Problem
  10. Conclusion

Key Result

Proposition 3.3

Let $q_0$ be a central configuration of $m_0$ defined by equation CF2. Let where $[B_1,B_2]^T=[A'(0)q_0, q_0]$ is constructed by the two zero eigenvectors corresponding to rotation and scaling, such that $P$ is invertible. Then $P^{-1}Jac(F)|_{(q_0,m_0)} P$ has the form$\left[\right]$ with $J_2$ a $(2N-2)\times (2N-2)$ matrix.

Figures (2)

  • Figure 1: Three rhombus central configurations for three different masses $m_1$.
  • Figure 2: The graph shows that the determinant of $J_2$ is positive for $a\in (\frac{\sqrt{3}}{3}, \sqrt{3})$, which means that the rhombus central configurations are nondegenerate.

Theorems & Definitions (18)

  • Definition 3.1: Degeneracy
  • Remark 3.2
  • Proposition 3.3
  • Definition 3.4: Nondegeneracy of a Central Configuration (Form I)
  • Example 3.5: Square Central Configuration
  • Example 3.6: Equilateral triangle plus one at center
  • Proposition 3.7
  • Remark 3.8
  • Definition 3.9: Nondegeneracy of a Central Configuration (Form II)
  • Example 3.10: Square Central Configuration
  • ...and 8 more