Families of periodic solutions of the 4- and 6-body problem using a gradient-free continuation method

TL;DR

This paper introduces a gradient-free stochastic method to find families of pseudo-periodic solutions in 4- and 6-body gravitational problems, demonstrating its effectiveness in constructing specific periodic orbits.

Contribution

A novel gradient-free stochastic approach is developed to compute families of periodic solutions in multi-body problems, avoiding traditional gradient-based methods.

Findings

Successfully constructed families of periodic solutions for 4- and 6-body problems.

Method relies solely on function evaluations, making it robust to complex landscapes.

Identified specific configurations with bodies moving on opposite sides or on equilateral triangles.

Abstract

In this paper, we describe a gradient-free method to solve a system of equations, and we use it to construct two families of pseudo-periodic planar solutions of the 4- and 6-body problem. The method is a stochastic black-box procedure that uses only function evaluations. For the 4-body problem, bodies 1 and 2 have mass 1 and move opposite to each other, and bodies 3 and 4 have mass $m_2$ and also move opposite to each other. For the 6-body problem, bodies 1, 2, and 3 have mass 1 and move on the vertices of an equilateral triangle centered at the origin, and bodies 4, 5, and 6 have mass $m_2$ and also move on the vertices of an equilateral triangle. In both cases, we compute families of periodic solutions by imposing return conditions up to rotation and relabeling.