Action minimizing solutions of the Newtonian n-body problem: from homology to symmetry

TL;DR

This paper discusses how action minimizing paths in the Newtonian n-body problem lead to collision-free solutions and enable the discovery of symmetric periodic solutions like the Eight and Hip-Hop orbits.

Contribution

It introduces a framework connecting homology and symmetry to find new periodic solutions in the n-body problem based on action minimization principles.

Findings

Existence of collision-free action minimizing solutions

Discovery of symmetric periodic solutions such as the Eight and Hip-Hop orbits

Generalization of solutions to n-body configurations

Abstract

An action minimizing path between two given configurations, spatial or planar, of the $n$-body problem is always a true -- collision-free -- solution. Based on a remarkable idea of Christian Marchal, this theorem implies the existence of new "simple" symmetric periodic solutions, among which the Eight for 3 bodies, the Hip-Hop for 4 bodies and their generalizations.