Collision index and stability of elliptic relative equilibria in planar n-body problem

TL;DR

This paper introduces a collision index to analyze the linear stability of elliptic relative equilibria in the planar n-body problem, especially near collision scenarios, providing new criteria for hyperbolicity and stability.

Contribution

It develops a Maslov-type collision index for heteroclinic orbits in the n-body problem and applies it to establish hyperbolic stability conditions near collision cases.

Findings

ERE of minimal central configurations are generically hyperbolic near collision.

New hyperbolic criteria for elliptic relative equilibria.

Detailed analysis of Euler collinear orbits near collision.

Abstract

It is well known that a planar central configuration of the $n$-body problem gives rise to solutions where each particle moves on a specific Keplerian orbit while the totality of the particles move on a homographic motion. When the eccentricity $e$ of the Keplerian orbit belongs in $[0,1)$, following Meyer and Schmidt, we call such solutions elliptic relative equilibria (shortly, ERE). In order to study the linear stability of ERE in the near-collision case, namely when $1-e$ is small enough, we introduce the collision index for planar central configurations. The collision index is a Maslov-type index for heteroclinic orbits and orbits parametrised by half-lines that, according to the Definition given by authors in [16], we shall refer to as half-clinic orbits and whose Definition in this context, is essentially based on a blow up technique in the case $e=1$. We get the fundamental properties of collision index and approximation theorems. As applications, we give some new hyperbolic criteria and prove that, generically, the ERE of minimal central configurations are hyperbolic in the near-collision case, and we give detailed analysis of Euler collinear orbits in the near-collision case.