N体形状空间中的全碰撞
Total Collisions in the N-Body Shape Space
摘要：
讨论了形空间上引力N体问题的全碰撞奇点。形空间是系统的关系构形空间，它是通过将普通构形空间相对于总平移、旋转和缩放的相似群展开而得到的。对于零能引力N体系统，可以显式地构造形状空间上的动力学，并可以详细地分析总碰撞点，即中心位形点和零形状动量点。事实证明，即使在形状空间中，尺度不是描述的一部分，运动方程也会在（且仅在）完全碰撞的点处发散。我们构造并研究了分层全碰撞流形，证明了在形空间上的全碰撞点处，奇点是必不可少的。因此，没有办法通过这些点来发展解决方案。这与广义相对论的大爆炸奇点非常相似，比安奇九世的宇宙学模型显示了大爆炸的本质奇点。对广义相对论模型的一个简单修改（增加一个刚性物质场）将系统变成一个形状动力学描述允许通过奇点确定性演化的系统。我们猜想，同样，为了正则化N体模型的总碰撞奇异性，需要对动力学进行一些修改。
英文摘要：
We discuss the total collision singularities of the gravitational N-body problem on shape space. Shape space is the relational configuration space of the system obtained by quotienting ordinary configuration space with respect to the similarity group of total translations, rotations, and scalings. For the zero-energy gravitating N-body system, the dynamics on shape space can be constructed explicitly and the points of total collision, which are the points of central configuration and zero shape momenta, can be analyzed in detail. It turns out that, even on shape space where scale is not part of the description, the equations of motion diverge at (and only at) the points of total collision. We construct and study the stratified total-collision manifold and show that, at the points of total collision on shape space, the singularity is essential. There is, thus, no way to evolve the solutions through these points. This mirrors closely the big bang singularity of general relativity, where the homogeneous-but-not-isotropic cosmological model of Bianchi IX shows an essential singularity at the big bang. A simple modification of the general-relativistic model (the addition of a stiff matter field) changes the system into one whose shape-dynamical description allows for a deterministic evolution through the singularity. We suspect that, similarly, some modification of the dynamics would be required in order to regularize the total collision singularity of the N-body model.
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