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On the eigenvalues of a central configuration

Alain Albouy, Jiexin Sun

TL;DR

This paper proves that in any planar, noncollinear central configuration of five bodies, the two nontrivial eigenvalues of the Brehm-Wintner-Conley matrix exceed the common eigenvalue λ associated with the central configuration, with 0 being a simple eigenvalue. The authors provide a covector-based representation of triangle areas, introduce a shifted (amended) force framework, and derive Williams’ identities through multiple complementary approaches. They establish the positivity of key Williams’ parameter ν and, using known results on eigenvalues, deduce the required eigenvalue ordering, thereby resolving Williams’ conjecture for the 5-body case and clarifying the geometric-sign structure of central configurations. The work also connects these results to stability indexes and bifurcation considerations relevant to self-similar motions in celestial mechanics.

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.

On the eigenvalues of a central configuration

TL;DR

This paper proves that in any planar, noncollinear central configuration of five bodies, the two nontrivial eigenvalues of the Brehm-Wintner-Conley matrix exceed the common eigenvalue λ associated with the central configuration, with 0 being a simple eigenvalue. The authors provide a covector-based representation of triangle areas, introduce a shifted (amended) force framework, and derive Williams’ identities through multiple complementary approaches. They establish the positivity of key Williams’ parameter ν and, using known results on eigenvalues, deduce the required eigenvalue ordering, thereby resolving Williams’ conjecture for the 5-body case and clarifying the geometric-sign structure of central configurations. The work also connects these results to stability indexes and bifurcation considerations relevant to self-similar motions in celestial mechanics.

Abstract

The equations of the Newtonian -body problem have a matrix form, where an matrix depending on the masses and on the mutual distances appears as a factor. The 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.

Paper Structure

This paper contains 6 sections, 15 theorems, 60 equations, 6 figures.

Table of Contents

  1. Introduction
  2. The areas of the triangles in terms of covectors
  3. Shifted BWC matrix and minor determinants
  4. Another approach to identities (\ref{['EQW']})
  5. Some inequalities for a 5-body central configuration
  6. Proof of Williams' inequality

Key Result

Theorem 1.1

In a planar noncollinear central configuration of 5 bodies, 0 is a simple eigenvalue of ${\cal Z}$, the multiplier $\lambda>0$ is a double eigenvalue. The two other eigenvalues are strictly greater than $\lambda$.

Figures (6)

  • Figure 1: Icosidodecahedron drawn in Divina proportionePaci.
  • Figure 2: Four quadrants in the plane.
  • Figure 3: 5-body configuration which is not central whatever the masses.
  • Figure 4: Normalized 5-body configuration which is not central whatever the masses.
  • Figure 5: Central configuration whose convex hull is a triangle.
  • ...and 1 more figures

Theorems & Definitions (27)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • proof
  • Proposition 5.1
  • Theorem 5.2
  • Proposition 5.3
  • Proposition 5.4
  • proof
  • Proposition 5.5
  • ...and 17 more