Braids, metallic ratios and periodic solutions of the $2n$-body problem

TL;DR

This paper classifies certain periodic solutions of the planar 2n-body problem using braid theory, proving they are pseudo-Anosov with stretch factors expressed as metallic ratios, and introduces new numerical solutions.

Contribution

It proves that a family of solutions from previous work are pseudo-Anosov braids with metallic ratio stretch factors, and provides new numerical periodic solutions.

Findings

Solutions are of pseudo-Anosov type with metallic ratio stretch factors.

Established a link between braid types and periodic solutions in the 2n-body problem.

Presented new numerical periodic solutions for the planar 2n-body problem.

Abstract

Periodic solutions of the planar $N$-body problem determine braids through the trajectory of $N$ bodies. Braid types can be used to classify periodic solutions. According to the Nielsen-Thurston classification of surface automorphisms, braids fall into three types: periodic, reducible and pseudo-Anosov. To a braid of pseudo-Anosov type, there is an associated stretch factor greater than 1, and this is a conjugacy invariant of braids. In 2006, the third author discovered a family of multiple choreographic solutions of the planar $2n$-body problem. We prove that braids obtained from the solutions in the family are of pseudo-Anosov type, and their stretch factors are expressed in metallic ratios. New numerical periodic solutions of the planar $2n$-body problem are also provided.