形空间上牛顿N体问题的全碰撞研究
Investigating total collisions of the Newtonian N-body problem on shape space
摘要：
本文分析了牛顿引力系统在形空间（系统的关系位形空间）上的全碰撞点。虽然相对于绝对空间和时间的牛顿运动方程在完全碰撞点处是奇异的，这是由于牛顿势在该点处的奇异性，但在绝对尺度不存在的形状空间中不一定是这种情况。我们调查是否，采用系统的关系描述，形状的自由度，这仅仅是角度和它们的共轭动量，可以通过总碰撞点的演变。不幸的是，情况并非如此。即使没有尺度，运动方程在完全碰撞的点处（也只有在那里）也是奇异的。这是由形状动量的特殊行为引起的。虽然这种行为会导致奇异性，但它同时提供了对总碰撞的纯粹形状动力学描述。借助于这一点，我们能够辨别全碰撞解决方案从非碰撞解决方案的形状空间，也就是说，不参考（外部）规模。我们可以进一步使用形状动力学描述，以表明全碰撞解决方案形成一个集的所有解决方案中的措施零。
英文摘要：
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.
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