A study of braids arising from simple choreographies of the planar Newtonian N-body problem

TL;DR

This paper investigates the braid types of periodic solutions in the planar Newtonian N-body problem, proving most are pseudo-Anosov and identifying those with extremal stretch factors, thus revealing the complexity of these choreographies.

Contribution

It proves that most braids from Yu's simple choreographies are pseudo-Anosov, except for circular motions, and identifies extremal stretch factor choreographies.

Findings

Most braids are pseudo-Anosov types.

Circular motions are the exception with non-pseudo-Anosov braids.

Identified choreographies with largest and smallest stretch factors.

Abstract

We study periodic solutions of the planar Newtonian $N$-body problem with equal masses. Each periodic solution traces out a braid with $N$ strands in 3-dimensional space. When the braid is of pseudo-Anosov type, it has an associated stretch factor greater than 1, which reflects the complexity of the corresponding periodic solution. For each $N \ge 3$, Guowei Yu established the existence of a family of simple choreographies to the planar Newtonian $N$-body problem. We prove that braids arising from Yu's periodic solutions are of pseudo-Anosov types, except in the special case where all particles move along a circle. We also identify the simple choreographies whose braid types have the largest and smallest stretch factors, respectively.