# Torus knot choreographies in the $n$-body problem

## TL;DR

This paper introduces a systematic, computer-assisted method to prove the existence of spatial choreographies in the gravitational n-body problem, focusing on torus knot solutions with specific symmetries.

## Contribution

It develops a constructive approach using delay differential equations and Fourier analysis to establish choreographies with prescribed topological and symmetry properties.

## Key findings

- Proved existence of spatial choreographies for n=4,5,7,9 bodies.
- Utilized computer-assisted proofs with analytic estimates.
- Reduced the problem to solving a delay differential equation in a Fourier coefficient space.

## Abstract

We develop a systematic approach for proving the existence of choreographic solutions in the gravitational $n$ body problem. Our main focus is on spatial torus knots: that is, periodic motions where the positions of all $n$ bodies follow a single closed which winds around a $2$-torus in $\mathbb{R}^3$. After changing to rotating coordinates and exploiting symmetries, the equation of a choreographic configuration is reduced to a delay differential equation (DDE) describing the position and velocity of a single body. We study periodic solutions of this DDE in a Banach space of rapidly decaying Fourier coefficients. Imposing appropriate constraint equations lets us isolate choreographies having prescribed symmetries and topological properties. Our argument is constructive and makes extensive use of the digital computer. We provide all the necessary analytic estimates as well as a working implementation for any number of bodies. We illustrate the utility of the approach by proving the existence of some spatial choreographies for $n=4,5,7$, and $9$ bodies.