Simple choreographies of the planar Newtonian $N$-body Problem

TL;DR

This paper proves that for the planar Newtonian equal-mass N-body problem with N ≥ 3, there are exponentially many distinct simple choreographies, confirming a conjecture about the abundance of such solutions.

Contribution

It establishes a lower bound on the number of simple choreographies in the N-body problem, confirming a conjecture and expanding understanding of periodic solutions.

Findings

At least 2^{N-3} + 2^{[(N-3)/2]} simple choreographies exist for N ≥ 3.

The result confirms a conjecture by Chenciner et al. about the multiplicity of choreographies.

Provides a combinatorial lower bound on the number of simple choreographies.

Abstract

In the $N$-body problem, a simple choreography is a periodic solution, where all masses chase each other on a single loop. In this paper we prove that for the planar Newtonian $N$-body problem with equal masses, $N \ge 3$, there are at least $2^{N-3} + 2^{[(N-3)/2]}$ different main simple choreographies. This confirms a conjecture given by Chenciner and etc. in \cite{CGMS02}.