Central configurations of n point particles in \$\${E \approx \mathbb{R}^d}\$\$E≈<sup>Rd</sup> 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.

Ferrario, D. (2015). Fixed point indices of central configurations. JOURNAL OF FIXED POINT THEORY AND ITS APPLICATIONS, 17(1), 239-251 [10.1007/s11784-015-0246-z].

### Fixed point indices of central configurations

#### Abstract

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.
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Central configurations; n-body problem; relative equilibria; Geometry and Topology; Applied Mathematics; Modeling and Simulation
English
239
251
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Ferrario, D. (2015). Fixed point indices of central configurations. JOURNAL OF FIXED POINT THEORY AND ITS APPLICATIONS, 17(1), 239-251 [10.1007/s11784-015-0246-z].
Ferrario, D
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/10281/99631`