Fixed point indices of central configurations

2014/12/18 by Davide L. Ferrario, Ferrario, Davide L.
Mathematics · Physics and Astronomy · #37C25 secondary #55M20 primary #Algebraic Topology (math.AT) #FOS: Mathematics #FOS: Physical sciences #Geometric Analysis and Curvature Flows #Homotopy and Cohomology in Algebraic Topology #Mathematical Physics (math-ph) #Nonlinear Waves and Solitons

paper · pdf · doi:10.48550/arxiv.1412.5817

openalex publication_date 2014/12/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Central configurations of n point particles in E≈ 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=-∇_M U / ‖ ∇_M U ‖_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.

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