Central configurations, Morse and fixed point indices

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 func-tionŪ 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. MSC Subject Class: 70F10, 55M20

Pfaffians and the inverse problem for collinear central configurations

“…A central configuration is a configuration that yields a relative equilibrium solution of the Newton equations of the n-body problem with potential function U , and can be shown (cf. [13], [12], [1], [6], [8], [7]) that it is a solution of the following n equations (1.1) λm j q j = −α k =j m j m k q j − q k q j − q k α+2 .…”