Abstract
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 \({F = -\nabla_{M}U / ||\nabla_{M}U||_{M}}\) , 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.
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Ferrario, D.L. Fixed point indices of central configurations. J. Fixed Point Theory Appl. 17, 239–251 (2015). https://doi.org/10.1007/s11784-015-0246-z
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DOI: https://doi.org/10.1007/s11784-015-0246-z