Positive elliptic-elliptic rotopulsators on Clifford tori of nonconstant size project onto regular polygons

Abstract

Let $q_1$,...,$q_n$ be the position vectors of the point masses of the curved $n$-body problem. Consider any positive elliptic-elliptic rotopulsator solution $q_i^T=(r\cos (\theta +{\alpha }_i),r\sin (\theta +{\alpha }_i),\rho \cos (\varphi +{\beta }_i),\rho \sin (\varphi +{\beta }_i))$, $i\in \{1,...,n\}$, where ${\alpha }_1,...,{\alpha }_n,{\beta }_1,...,{\beta }_n\in [0,2\pi )$ are constants, $\varphi$, $\theta$, $r$ and $\rho$ are twice-differentiable, continuous, nonconstant functions, $r^2+{\rho }^2=1$, $r\ge 0$ and $\rho \ge 0$. We prove that the if the configuration of the point masses is of nonconstant size, the configuration of the vectors $(r\cos (\theta +{\alpha }_i),r\sin (\theta +{\alpha }_i){)}^T$ is a regular polygon, as is the configuration of the vectors $(\rho \cos (\varphi +{\beta }_i),\rho \sin (\varphi +{\beta }_i){)}^T$, $i\in \{1,...,n\}$.