Central configurations, symmetries and fixed points

TL;DR

This paper introduces a topological fixed point approach to analyze planar central configurations in the n-body problem, proving key theorems, deriving properties, and providing numerical tables for equal masses.

Contribution

It defines a new map related to central configurations, proves a generalized Moulton-Smale theorem, and offers a topological framework for understanding configuration properties.

Findings

Proved the generalized Moulton-Smale theorem for collinear configurations.

Derived estimates on the number of central configurations for three bodies.

Provided numerical tables of central configurations for equal masses from 3 to 10 bodies.

Abstract

Planar central configurations can be seen as critical points of the reduced potential or solutions of a system of equations. By the homogeneity and invariance of the potential with respect to SO(2), it is possible to see that the SO(2)-orbits of central configurations are fixed points of a suitable map f. The purpose of the paper is to define this map and to derive some properties using topological fixed point theory. The generalized Moulton-Smale theorem for collinear configurations is proved, together with some estimates on the number of central configurations in the case of 3 bodies, using fixed point indexes. Well-known results such as the compactness of the set of central configuration can also be proved in an easy way in this topological framework. At the end of the paper some tables of (numerical) planar central configurations of n equal masses with Newtonian potential are given, for n=3,..., 10. They have been computed as the fixed points of a suitable self-map of R^{2(n-2)}.