Symmetries and periodic orbits for the $n$-body problem: about the computational approach

TL;DR

This paper discusses a computational approach to identifying and analyzing symmetric periodic orbits in the $n$-body problem, focusing on methods to prove their existence and describe their properties using specialized software.

Contribution

It introduces a software package that integrates symbolic algebra, numerical methods, and visualization to study symmetric orbits in the $n$-body problem.

Findings

Development of a computational framework for symmetric orbits

Application of the software to classify special symmetric critical points

Insights into the properties of symmetric periodic solutions

Abstract

The main problem is to understand and to find periodic symmetric orbits in the $n$-body problem, in the sense of finding methods to prove or compute their existence, and more importantly to describe their qualitative and quantitative properties. In order to do so, and in order to classify such orbits and their symmetries, computers have been extensively used in many ways since decades. We will focus on some very special symmetric orbits, which occur as symmetric critical points (local minimizers) of the gravitational Lagrangean action functional. The exploration of the loop space of the $n$-point configuration space, raised some computational and mathematical questions that couldd be interesting. The aim of the article is to explain how such questions and issues were % considered in the development of a software package that combined symbolic algebra, numerical and scientific libraries, human interaction and visualization.