Spectral instability of the regular n-gon elliptic relative equilibrium in the planar n-body problem

TL;DR

This work resolves the question of spectral stability for the regular $n$-gon elliptic relative equilibrium (ERE) in the planar $n$-body problem by proving spectral instability for all $n\ge 3$ and eccentricities $\mathfrak{e}\in[0,1)$. It achieves this through a Meyer–Schmidt based reduction that decomposes the linearized system into translation, dilation/rotation, and an essential core, and by introducing the $\beta$-system to relate to the Lagrange solution. A finite-point hyperbolicity testing method is developed, enabling extensive hyperbolic regions to be certified; in particular, the regular $3$-, $4$-, and $5$-gon EREs are shown to be hyperbolic for all $\mathfrak{e}$. The results extend Moeckel’s instability at $\mathfrak{e}=0$ to all eccentricities and provide a quantitative path to hyperbolicity in low-$n$ cases via Sturm–Liouville positivity. The approach blends spectral analysis of the reduced blocks with a propagation principle for positivity, yielding both instability and broad hyperbolicity regions.

Abstract

The regular $n$-gon elliptic relative equilibrium (ERE) is a Kepler homographic solution generated by the regular $n$-gon central configuration, and its linear stability depends on the eccentricity $\mathfrak{e}\in[0,1)$. While Moeckel \cite{Moe1} established the spectral instability for this solution at $\mathfrak{e}=0$ for all $n\geq3$, it remained unknown whether instability persists for $\mathfrak{e} \in (0,1)$. This paper resolves this problem: we prove that the regular $n$-gon ERE is spectral instability for all $n\geq 3$ and $\mathfrak{e} \in [0,1)$. Furthermore, we introduce the $β$-system which related the Lagrange solution, and we developed an estimation method that, by testing the hyperbolicity of the $β$-system at a finite number of points alone, allows us to obtain extensive hyperbolic regions. As a corollary, for $n=3,4,5$, we uniformly demonstrate that the instability is hyperbolic (and hence stronger) for all $\mathfrak{e} \in [0,1)$.

Figures (3)

• Figure 1: Hyperbolic region $\mathfrak{U}$ (gray)
• Figure :
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Theorems & Definitions (30)

• Theorem 1.1
• Remark 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Corollary 1.6
• Remark 1.7
• Remark 1.8
• Lemma 2.1
• proof