From the Lagrange polygon to the figure eight I: Numerical evidence extending a conjecture of Marchal

TL;DR

This paper uses numerical continuation methods to explore how regular n-gon solutions in the n-body problem connect to figure-eight choreographies, revealing a continuous transition through spatial choreographies with torus knot topology.

Contribution

It introduces a symmetrized delay differential equation approach that simplifies continuation from n-gon solutions to figure-eight choreographies for odd n, supporting a conjecture of their connectedness.

Findings

n-gon solutions can be continued to figure-eight solutions for odd n

The continuation passes through spatial choreographies with torus knot topology

Symmetrization reduces bifurcation complexity and simplifies the analysis

Abstract

The present work studies the continuation class of the regular $n$-gon solution of the $n$-body problem. For odd numbers of bodies between $n = 3$ and $n = 15$ we apply one parameter numerical continuation algorithms to the energy/frequency variable, and find that the figure eight choreography can be reached starting from the regular $n$-gon. The continuation leaves the plane of the $n$-gon, and passes through families of spatial choreographies with the topology of torus knots. Numerical continuation out of the $n$-gon solution is complicated by the fact that the kernel of the linearization there is high dimensional. Our work exploits a symmetrized version of the problem which admits dense sets of choreography solutions, and which can be written as a delay differential equation in terms of one of the bodies. This symmetrized setup simplifies the problem in several ways. On one hand, the direction of the kernel is determined automatically by the symmetry. On the other hand, the set of possible bifurcations is reduced and the $n$-gon continues to the eight after a single symmetry breaking bifurcation. Based on the calculations presented here we conjecture that the $n$-gon and the eight are in the same continuation class for all odd numbers of bodies.