论文标题
$ n $ body问题的周期性轨道分叉，来自非几何解决方案家族
Bifurcation of periodic orbits for the $N$-body problem, from a non geometrical family of solutions
论文作者
论文摘要
给定两个正实数$ m $和$ m $和一个整数$ n> 2 $，众所周知，我们可以找到$（n+1）$的解决方案的家族，其中质量$ m $ $ m $ boty的身体处于原点和其他$ n $ bodies，而其他$ n $ bodies则与$ x $ $ $ $ $ $ $ $ $ $ eccente clused eellipse ecccentri cyccentri cycccentri contection the Origin和其他$ n $ bodies。可以预期，这个依赖$ e $的几何家庭具有一些分叉，可以产生解决方案，其中中心的尸体在$ z $轴上移动，而不是保持原状。 By doing an analytic continuation of a periodic numerical solution of the $4$-body problem --the one displayed on the video http://youtu.be/2Wpv6vpOxXk --we surprisingly discovered that the origin of this periodic solution is not part of the geometrical family of elliptical solutions parametrized by the eccentricity $e$.它来自不那么几何，但更容易描述家庭。在注意到这个新家庭之后，作者找到了这个新家庭中分叉点的确切公式，并使用它来显示任何一对群众$ m $，$ m $和任何整数$ n $的非平面周期性解决方案。作为一个特别的例子，我们找到了一个解决方案，其中三个巨大$ 3 $的尸体在一个质量$ 7 $的身体上移动，可以上下移动。
Given two positive real numbers $M$ and $m$ and an integer $n>2$, it is well known that we can find a family of solutions of the $(n+1)$-body problem where the body with mass $M$ stays put at the origin and the other $n$ bodies, all with the same mass $m$, move on the $x$-$y$ plane following ellipses with eccentri\-city $e$. It is expected that this geometrical family that depends on $e$, has some bifurcations that produce solutions where the body in the center moves on the $z$-axis instead of staying put in the origin. By doing an analytic continuation of a periodic numerical solution of the $4$-body problem --the one displayed on the video http://youtu.be/2Wpv6vpOxXk --we surprisingly discovered that the origin of this periodic solution is not part of the geometrical family of elliptical solutions parametrized by the eccentricity $e$. It comes from a not so geometrical but easier to describe family. Having noticed this new family, the authors find an exact formula for the bifurcation point in this new family and use it to show the existence of a non-planar periodic solution for any pair of masses $M$, $m$, and any integer $n$. As a particular example, we find a solution where three bodies with mass $3$ move around a body with mass $7$ that moves up and down.
