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 $q_{i}^{T}=(r\cos{(\theta+\alpha_{i})},r\sin{(\theta+\alpha_{i})},\rho\cos{(\phi+\beta_{i})},\rho\sin{(\phi+\beta_{i})})$, $i\in\{1,...,n\}$, where $\alpha_{1},...,\alpha_{n},\beta_{1},...,\beta_{n}\in [0,2\pi)$ are constants, $\phi$, $\theta$, $r$ and $\rho$ are twice-differentiable, continuous, nonconstant functions, $r^{2}+\rho^{2}=1$, $r\geq 0$ and $\rho\geq 0$. 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 $(\rho\cos{(\phi+\beta_{i})},\rho\sin{(\phi+\beta_{i})})^{T}$, $i\in\{1,...,n\}$.