Fixed point indices of central configurations

Central configurations of $n$ point particles in $E\approx \mathbb{R}^d$ with respect to a potential function $U$ are shown to be the same as the fixed points of the normalized gradient map $F=-\nabla_M U / \lVert \nabla_M U \rVert_M$, which is an $SO(d)$-equivariant self-map defined on the intertia 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 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 non-planar relative equilibrium which is not a central configuration.