Year: 2014
Type: preprint
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 $F=-\nabla_M U / \lVert \nabla_M U \rVert_M$, 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.
Source: arXiv (Cornell University)
Author: Ferrario, Davide L.
Institution: University of Milano-Bicocca
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2953110787