论文标题
从拉格朗日多边形到图8 i：数值证据，扩展了猜想
From the Lagrange polygon to the figure eight I: Numerical evidence extending a conjecture of Marchal
论文作者
论文摘要
目前的工作研究了$ n $ body问题的常规$ n $ gon解决方案的延续课。对于$ n = 3 $和$ n = 15 $之间的奇数，我们将一个参数数值延续算法应用于能量/频率变量，并从常规的$ n $ -gon开始找到图八个编排。延续使$ n $ gon的飞机通过圆环结的拓扑结束了空间编舞的家庭。 $ n $ gon解决方案的数值延续使线性化的内核具有很高的尺寸，这是复杂的。我们的工作利用了该问题的对称版本，该版本接受了密集的编舞解决方案集，并且可以将其写入其中一个身体的延迟微分方程。这种对称设置以多种方式简化了问题。一方面，内核的方向由对称性自动确定。另一方面，一组可能的分叉减少了，在一次对称性破坏分叉之后，$ n $ - 贡族继续持续到八个。根据此处提出的计算，我们猜测$ n $ -gon和八个在所有奇数尸体的同一延续类中。
The present work studies the continuation class of the regular $n$-gon solution of the $n$-body problem. For odd numbers of bodies between $n = 3$ and $n = 15$ we apply one parameter numerical continuation algorithms to the energy/frequency variable, and find that the figure eight choreography can be reached starting from the regular $n$-gon. The continuation leaves the plane of the $n$-gon, and passes through families of spatial choreographies with the topology of torus knots. Numerical continuation out of the $n$-gon solution is complicated by the fact that the kernel of the linearization there is high dimensional. Our work exploits a symmetrized version of the problem which admits dense sets of choreography solutions, and which can be written as a delay differential equation in terms of one of the bodies. This symmetrized setup simplifies the problem in several ways. On one hand, the direction of the kernel is determined automatically by the symmetry. On the other hand, the set of possible bifurcations is reduced and the $n$-gon continues to the eight after a single symmetry breaking bifurcation. Based on the calculations presented here we conjecture that the $n$-gon and the eight are in the same continuation class for all odd numbers of bodies.
