Fixed point indices of central configurations

Central configurations of n point particles in $${E \approx \mathbb{R}^d}$$E≈^Rd with respect to a potential function U are shown to be the same as the fixed points of the normalized gradient map $${F = -\nabla_{M}U / ||\nabla_{M}U||_{M}}$$F=-∇MU/||∇MU||M , which is an SO(d)-equivariant self-map defined on the inertia ellipsoid. We show that the SO(d)-orbits of fixed points of F are all fixed points of the map induced on the quotient by SO(d), and we give a formula relating their indices (as fixed points) with their Morse indices (as critical points). At the end, we give an example of a nonplanar relative equilibrium which is not a central configuration.