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

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 $n\ge 3$, 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 $n\ge 3$ 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 $\mathfrak{e}\in [0,1)$.