N-body choreographies on a p-limacon curve

Abstract

We consider an $N$--body problem under a harmonic potential of the form $\frac{1}{2}\sum \kappa_{jl} |q_j-q_l|^2$. A $p$-lima\c{c}on curve is a planar curve parametrized by $t$ given by $a(\cos t,\sin t)+b(\cos pt, \sin pt)$, where $a,b\in \mathbb{R}$, $p \in \mathbb{Z}$, and $t \in [0,2\pi]$. We study $N$-body choreographic motions constrained to a $p$-lima\c{c}on curve and establish necessary and sufficient conditions for their existence. Specifically, we prove that choreographic motions exist if and only if $p/N, (p \pm 1)/N \notin \mathbb{Z}$. Under an additional symmetry assumption on the force coefficients, we further refine these conditions. We also analyze the occurrence of collisions, showing that for given $p$ and $N$, at most $2(N-1)$ choices of $a/b$ lead to collisions. Furthermore, we find additional conserved quantities.