Symmetric solutions of the $n$-body problem: a numerical study of Floquet multipliers and Morse indices

TL;DR

This paper investigates symmetric periodic solutions in the n-body problem, analyzing their stability via Floquet multipliers and Morse indices using specialized algorithms, with numerical results in 2D and 3D configurations.

Contribution

It introduces algorithms for computing Floquet multipliers and Morse indices of symmetric n-body solutions, enhancing stability analysis methods.

Findings

Algorithms successfully compute stability indicators for symmetric solutions.

Numerical experiments demonstrate the methods in various dimensions and configurations.

Results provide insights into the stability of symmetric periodic orbits.

Abstract

In this paper, we consider periodic solutions of the $n$-body problem that satisfy symmetry constraints, expressed through invariance under finite group actions. We focus on their stability properties and present algorithms specifically designed for the computation of Floquet multipliers and Morse indices. Numerical results are provided to illustrate our methods in both two and three dimensional configuration spaces, and for different choices on the number of bodies.