2N体问题的周期振荡
Periodic oscillations in a 2N-body problem
摘要：
$2N$ -体问题的Hip-Hop解是在每个时刻都满足的解，即具有相同质量$m$的$2N$体位于两个规则$N$ -线的顶点，这些$N$-线中的每一个都位于与形成反棱柱的固定平面$Π_0$等距的平面上。本文首先证明了对每一个$N$和每一个$m$都存在一族周期hip-hop解。对于这些族中的每一个解，到平面$Π_0$的定向距离，我们称之为$d(t)$，是一个奇函数，对于某些$T>0.$，它相对于$t=T$也是偶数。通过更仔细地探索我们的初始周期解集，我们数值地证明了在我们的存在性定理中建立的一些分支具有分支，这些分支产生的解具有定向距离函数$d(t)$关于任何$T>0$都不是偶数的性质，我们称这些解为单对称解.我们证明，没有一个单一的对称性解决方案是一个编舞。我们还显示明确的双重对称的解决方案，编排。
英文摘要：
Hip-Hop solutions of the $2N$-body problem are solutions that satisfy at every instance of time, that the $2N$ bodies with the same mass $m$, are at the vertices of two regular $N$-gons, each one of these $N$-gons are at planes that are equidistant from a fixed plane $Π_0$ forming an antiprism. In this paper, we first prove that for every $N$ and every $m$ there exists a family of periodic hip-hop solutions. For every solution in these families the oriented distance to the plane $Π_0$, which we call $d(t)$, is an odd function that is also even with respect to $t=T$ for some $T>0.$ For this reason we call solutions in these families, double symmetric solutions. By exploring more carefully our initial set of periodic solutions, we numerically show that some of the branches stablished in our existence theorem have bifurcations that produce branches of solutions with the property that the oriented distance function $d(t)$ is not even with respect to any $T>0$, we call these solutions single symmetry solutions. We prove that no single symmetry solution is a choreography. We also display explicit double symmetric solutions that are choreographies.
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