On the eigenvalues of a central configuration

TL;DR

This paper proves that in a 5-body central configuration, the two nontrivial eigenvalues of the associated matrix are strictly greater than the three trivial ones, refining existing inequalities and enhancing understanding of configuration stability.

Contribution

It establishes a new inequality for eigenvalues in 5-body central configurations, improving previous bounds and clarifying related identities.

Findings

Nontrivial eigenvalues are greater than trivial ones in 5-body configurations.

Improved inequalities refine understanding of configuration shape constraints.

Clarified identities related to eigenvalue inequalities and stability analysis.

Abstract

The equations of the Newtonian $n$-body problem have a matrix form, where an $n\times n$ matrix depending on the masses and on the mutual distances appears as a factor. The $n$ eigenvalues of this matrix are real and nonnegative. In a motion of relative equilibrium, the configuration, called {\it central}, has constant mutual distances. The matrix is constant. We prove that in a relative equilibrium of 5 bodies the two nontrivial eigenvalues are strictly greater than the three trivial ones. This result improves published inequalities about the central configurations, which belong to two independent lines of research. One starts with Williams in 1938 and concerns constraints on the shape of the configuration. The other concerns the Hessian of the potential and its index, and applies to the linear stability of the self-similar motions and to the possible bifurcations. We also considerably clarify the very useful identities with which Williams discusses his inequalities.