# Symmetries and choreographies in families bifurcating from the polygonal relative equilibrium of the $n$-body problem

## TL;DR

This paper investigates periodic orbits called Lyapunov families in the $n$-body problem, showing how they relate to choreographies through bifurcation analysis and numerical continuation, for various numbers of bodies.

## Contribution

It introduces a numerical method to identify choreographies arising from Lyapunov families in the $n$-body problem, including cases with a central non-participating body.

## Key findings

- Dense set of Lyapunov orbits correspond to choreographies.
- Numerical examples for $n=4,6,7,8,9$ bodies.
- Choreographies persist even with a central non-participating body.

## Abstract

We use numerical continuation and bifurcation techniques in a boundary value setting to follow Lyapunov families of periodic orbits. These arise from the polygonal system of $n$ bodies in a rotating frame of reference. When the frequency of a Lyapunov orbit and the frequency of the rotating frame have a rational relationship then the orbit is also periodic in the inertial frame. We prove that a dense set of Lyapunov orbits, with frequencies satisfying a diophantine equation, correspond to choreographies. We present a sample of the many choreographies that we have determined numerically along the Lyapunov families and along bifurcating families, namely for the cases $n=4,~6,~7,~8$, and $9$. We also present numerical results for the case where there is a central body that affects the choreography, but that does not participate in it. Animations of the families and the choreographies can be seen at the link: http://mym.iimas.unam.mx/renato/choreographies/index.html