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

TL;DR

This paper investigates the degeneracy of planar central configurations in the $N$-body problem, analyzing the Jacobian matrix to identify conditions for degeneracy and applying the method to classical configurations.

Contribution

It introduces four formulations to analyze degeneracy, systematically isolates trivial eigenvalues, and applies the approach to classical configurations, including the rhombus case.

Findings

Identifies specific mass values leading to degeneracy in classical configurations.

Establishes nondegeneracy of rhombus configurations for arbitrary masses.

Provides a systematic method to analyze degeneracy in the $N$-body problem.

Abstract

The degeneracy of central configurations in the planar $N$-body problem makes their enumeration problem hard and the related dynamics appealing. To truly understand the bifurcations of central configurations, we should work in the FULL configuration space which also facilitates the computer-aided methods. 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. Four distinct formulations of degeneracy are presented, each tailored to handle different forms of the system appeared in the literature. 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 also established.