论文标题
非恒定尺寸项目的Clifford Tori上的正椭圆形旋转旋转器上的正常多边形
Positive elliptic-elliptic rotopulsators on Clifford tori of nonconstant size project onto regular polygons
论文作者
论文摘要
令$ q_ {1} $，...，$ q_ {n} $为弯曲$ n $ body问题的点质量的位置向量。考虑任何阳性椭圆形旋转旋转溶液美元 $i\in\{1,...,n\}$, 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 $和$ρ\ geq 0 $。我们证明，如果点质量的配置为非恒定尺寸，则为向量的配置 $（r \ cos {（θ+α_{i}）}，r \ sin {（θ+α_{i}）}）^{t} $是一个常规的多边形，向量的配置也是$（ρ\ cos {（ϕ+β_{i}）}，ρ\ sin {（ϕ+β_{i}）}）^{t} $，$ i \ in \ in \ in \ in \ in \ in \ {1，...，n \} $。
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 $(ρ\cos{(ϕ+β_{i})},ρ\sin{(ϕ+β_{i})})^{T}$, $i\in\{1,...,n\}$.
