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.

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）上的全部碰撞点。虽然相对于绝对空间和时间制定的牛顿运动方程在总碰撞点由于牛顿电势的奇异性而在奇异点处是奇异的，但在形状空间中绝对比例不存在这种情况就不必如此。不存在。我们研究，通过采用系统的关系描述，形状自由度（仅是角度及其共轭力矩）是否可以通过总碰撞点来演化。不幸的是，这种情况并非如此。即使没有比例尺，运动方程在总碰撞点（也仅在那里）都是奇异的。这是由于形状动量的特殊行为引起的。尽管此行为引起奇异性，但同时它提供了总碰撞的纯粹形状动力学描述。借助于此，我们能够区分形状空间上的非碰撞解决方案与总碰撞解决方案，即无需参考（外部）比例尺。我们可以进一步使用形状动力学描述来表明，总碰撞解决方案在所有解决方案中形成了一组零度量。