Symmetry in n-body problem via group representations

TL;DR

This paper presents an algebraic approach using group representation theory to analyze local stability in symmetric n-body configurations, simplifying eigenvalue calculations for specific configurations like squares and triangles.

Contribution

It introduces a novel algebraic method leveraging group representations to study stability in symmetric n-body problems, demonstrated on specific configurations.

Findings

Eigenvalues of 8x8 Hessians explicitly computed

Stability properties of certain relative equilibria analyzed

Method applicable to various symmetric configurations

Abstract

We introduce an algebraic method to study local stability in the Newtonian $n$-body problem when certain symmetries are present. We use representation theory of groups to simplify the calculations of certain eigenvalue problems. The method should be applicable in many cases, we give two main examples here: the square central configurations with four equal masses, and the equilateral triangular configurations with three equal masses plus an additional mass of arbitrary size at the center. We explicitly found the eigenvalues of certain 8x8 Hessians in these examples, with only some simple calculations of traces. We also studied the local stability properties of corresponding relative equilibria in the four-body problems.