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

TL;DR

This paper proves that the regular n-gon elliptic relative equilibrium in the planar n-body problem is spectrally unstable for all n ≥ 3 and eccentricities in [0,1), extending previous results and introducing new estimation methods.

Contribution

It establishes spectral instability for all n ≥ 3 and all eccentricities in [0,1), and introduces the β-system and a testing method for hyperbolicity to analyze stability regions.

Findings

Spectral instability holds for all n ≥ 3 and eccentricities in [0,1).

The β-system relates the Lagrange solution to stability analysis.

Hyperbolic instability is confirmed for n=3,4,5 across all eccentricities.

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 $\beta$-system which related the Lagrange solution, and we developed an estimation method that, by testing the hyperbolicity of the $\beta$-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 $\mathfrak{e} \in [0,1)$.