Central configurations, Morse and fixed point indices

TL;DR

This paper links the fixed point index of non-degenerate central configurations in the n-body problem to the Morse index of the gravitational potential, using geometric analysis of orbit manifolds.

Contribution

It introduces a method to compute fixed point indices via Morse indices on configuration manifolds, connecting geometric and dynamical properties.

Findings

Fixed point index related to Morse index of potential

Analysis of orbit manifold geometry

Morse indices computed with respect to mass-metric form

Abstract

We compute the fixed point index of non-degenerate central configurations for the $n$-body problem in the euclidean space of dimension $d$, relating it to the Morse index of the gravitational potential function $\bar U$ induced on the manifold of all maximal $O(d)$-orbits. In order to do so, we analyze the geometry of maximal orbit type manifolds, and compute Morse indices with respect to the mass-metric bilinear form on configuration spaces.