论文标题
2n体问题中的周期性振荡
Periodic oscillations in a 2N-body problem
论文作者
论文摘要
$ 2N $ - 体问题的嘻哈解决方案是在每个时间都满足的解决方案，具有相同质量$ m $的$ 2N $的身体处于两个常规$ n $ gons的顶点，每个$ n $ gons中的每一个都是固定平面$π_0$π_0$π_0$π_0$。在本文中，我们首先证明每$ n $和每$ m $都有一个定期嘻哈解决方案。对于这些家庭中的每个解决方案，我们称之为$ d（t）$的平面$π_0$的距离是一个奇怪的功能，甚至对于某些$ t> 0的$ t = t $也是如此。通过更仔细地探索我们的初始定期解决方案集，我们从数值上表明，存在我们存在定理中的某些分支具有分叉性，可以通过属性生成解决方案分支的分支，即定向距离函数$ d（t）$也不相对于任何$ t> 0 $，我们称这些解决方案单一的Symmetry Sysmetry Solutions。我们证明没有单个对称解是编舞。我们还展示了编舞的明确双对称解决方案。
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.
