论文标题
调查牛顿N体问题的总碰撞
Investigating total collisions of the Newtonian N-body problem on shape space
论文作者
论文摘要
我们分析了牛顿重力系统在形状空间（系统的关系配置空间）的总碰撞点。尽管牛顿方程在绝对空间和时间方面提出的运动方程在当时牛顿电位的奇异性时是奇异的，但在不存在绝对规模的形状空间上，这并不是这种情况。我们调查是否采用了系统的关系描述，可以通过总碰撞的点来演化，这仅仅是角度及其共轭动量的自由度。不幸的是，事实并非如此。即使没有比例，在总碰撞点（仅在那里），运动方程也很单数。这是由形状动量的特殊行为进行的。尽管这种行为引起了奇异性，但同时，它对总碰撞提供了纯粹的形状动态描述。通过这一点，我们能够从形状空间上的非碰撞解决方案中辨别总碰撞解决方案，也就是说，没有参考（外部）规模。我们可以进一步使用形状 - 动力描述，以表明所有解决方案之间的总碰撞解决方案构成了一组零。
We analyze the points of total collision of the Newtonian gravitational system on shape space (the relational configuration space of the system). While the Newtonian equations of motion, formulated with respect to absolute space and time, are singular at the point of total collision due to the singularity of the Newton potential at that point, this need not be the case on shape space where absolute scale doesn't exist. We investigate whether, adopting a relational description of the system, the shape degrees of freedom, which are merely angles and their conjugate momenta, can be evolved through the points of total collision. Unfortunately, this is not the case. Even without scale, the equations of motion are singular at the points of total collision (and only there). This follows from the special behavior of the shape momenta. While this behavior induces the singularity, it at the same time provides a purely shape-dynamical description of total collisions. By help of this, we are able to discern total-collision solutions from non-collision solutions on shape space, that is, without reference to (external) scale. We can further use the shape-dynamical description to show that total-collision solutions form a set of measure zero among all solutions.
