Existence of symmetric central configurations

TL;DR

This paper demonstrates a unified variational approach proving that for every symmetry type and symmetric mass distribution, a corresponding central configuration exists, extending to balanced configurations.

Contribution

It introduces a uniform variational method to establish the existence of symmetric and balanced central configurations for all symmetry types and mass arrangements.

Findings

Existence of symmetric central configurations for all symmetry types.

Extension of the method to balanced configurations.

Applicability to any symmetric mass distribution.

Abstract

Central configurations have been of great interest over many years, with the earliest examples due to Euler and Lagrange. There are numerous results in the literature demonstrating the existence of central configurations with specific symmetry properties, using slightly different techniques in each. The aim here is to describe a uniform approach by adapting to the symmetric case the well-known variational argument showing the existence of central configurations. The principal conclusion is that there is a central configuration for every possible symmetry type, and for any symmetric choice of masses. Finally the same argument is applied to the class of balanced configurations introduced by Albouy and Chenciner.