Collision Index and Stability of Elliptic Relative Equilibria in Planar ${n}$-body Problem

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 Hu and Portaluri (An index theory for unbounded motions of Hamiltonian systems, Hu and Portaluri (2015, preprint)), 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 a detailed analysis of Euler collinear orbits in the near-collision case.