Linear Stability of an Elliptic Relative Equilibrium in the Spatial $n$-Body Problem via Index Theory

It is well known that a planar central configuration of the $n$-body problem gives rise to a solution where each particle moves in a Keplerian orbit with a common eccentricity $\mathfrak{e}\in[0,1)$. We call this solution an elliptic relative equilibrium (ERE for short). Since each particle of the ERE is always in the same plane, it is natural to regard it as a planar $n$-body problem. But in practical applications, it is more meaningful to consider the ERE as a spatial $n$-body problem (i. e., each particle belongs to $\mathbb{R}^3$). In this paper, as a spatial $n$-body problem, we first decompose the linear system of ERE into two parts, the planar and the spatial part. Following the Meyer – Schmidt coordinate [19], we give an expression for the spatial part and further obtain a rigorous analytical method to study the linear stability of the spatial part by the Maslov-type index theory. As an application, we obtain stability results for some classical ERE, including the elliptic Lagrangian solution, the Euler solution and the $1+n$-gon solution.