Equivariant optimisation for the gravitational $n$-body problem: a computational factory of symmetric orbits

TL;DR

This paper introduces exttt{SymOrb.jl}, a Julia software that uses group theory and variational methods to find and classify symmetric periodic orbits in the $n$-body problem, aiding in the validation of the Poincaré conjecture.

Contribution

It presents a unified implementation of a novel approach combining symmetry and variational techniques for solving singular dynamical systems in celestial mechanics.

Findings

Successful computation of symmetric periodic orbits in the $n$-body problem.

Framework facilitates large-scale search and classification of orbits.

Provides theoretical and practical guidelines with examples.

Abstract

In this paper we present \texttt{SymOrb.jl}, a software which combines group representation theory and variational methods to provide numerical solutions of singular dynamical systems of paramount relevance in Celestial Mechanics and other interacting particles models. Among all, it prepares for large-scale search of symmetric periodic orbits for the classical $n$-body problem and their classification, paving the way towards a computational validation of Poincar\'e conjecture about the density of periodic orbits. Through the accessible language of Julia, \texttt{SymOrb.jl} offers a unified implementation of an earlier version. This paper provides theoretical and practical guidelines for the specific approach we adopt, complemented with examples.