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

英文摘要：

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 $q_{i}^{T}=(r\cos{(θ+α_{i})},r\sin{(θ+α_{i})},ρ\cos{(ϕ+β_{i})},ρ\sin{(ϕ+β_{i})})$, $i\in\{1,...,n\}$, where $α_{1},...,α_{n},β_{1},...,β_{n}\in [0,2π)$ are constants, $ϕ$, $θ$, $r$ and $ρ$ are twice-differentiable, continuous, nonconstant functions, $r^{2}+ρ^{2}=1$, $r\geq 0$ and $ρ\geq 0$. We prove that the if the configuration of the point masses is of nonconstant size, the configuration of the vectors $(r\cos{(θ+α_{i})},r\sin{(θ+α_{i})})^{T}$ is a regular polygon, as is the configuration of the vectors $(ρ\cos{(ϕ+β_{i})},ρ\sin{(ϕ+β_{i})})^{T}$, $i\in\{1,...,n\}$.