Global Aspects of the N-Body Problem

牛顿n体问题遭受（或享受）一般理论的典型动力系统所没有的两个困难：使流动不完全的碰撞奇点和使其所有周期轨道简并的对称性。调查员和他的同事消除了这两个困难。它们通过Levi-Civita正则化、Kuustaanheimo-Steifel正则化或McGehee爆破来消除碰撞奇点。他们通过辛约化和不变量理论消除了对称性。他们将由此产生的正则化的减少流量称为“全局正则化”。该流是完整的，并且在编码n体问题的流中具有最小的自由度。考虑到全局正则化，研究者和他的同事研究了以前无法在n体问题中获得的问题，例如碰撞-碰撞解的动力学和变分性质，以及平面零角动量三体问题的符号动力学的存在与否，其中3个符号是3种类型的共线（共线）。想象一下，月亮绕着地球转，地球绕着太阳转，太阳绕着银河系转。在宇宙的大尺度动力学中占主导地位的力是引力，它负责天体的大尺度旋转运动。n体方程是控制这个运动的方程。（“N”代表尸体的数量。）当两个物体靠近时，力变得很大，以至于当两个物体碰撞时，力变得无穷大。这种无限性被称为“碰撞奇点”。如果整个宇宙被平移，或旋转一定的量，运动保持不变，所以在通常的n体方程中有额外的变量，即描述宇宙的“原点”和“轴”的变量。研究者和他的合作者消除了碰撞奇点，以及n体方程中的额外变量。这些消除产生了一个潜在的更有用的运动方程系统，一个没有无穷大的系统，它可以提供一个很好的平台，从这个平台上解决关于我们宇宙中发生的主要动力学行为之一的许多开放问题。

The Newtonian N-body problem suffers (or enjoys) two difficulties not shared by the typical dynamical systems of general theory: collision singularities which render the flow incomplete, and symmetries which make all its periodic orbits degenerate. The investigator and his colleague eliminate these two difficulties. They eliminate collision singularities by Levi-Civita regularization, Kuustaanheimo-Steifel regularization or McGehee blow-up. They eliminate the symmetries by symplectic reduction and invariant theory. They call the resulting regularized reduced flow the "global regularization." This flow is complete and has the minimum number of degrees of freedom possible among for a flow encoding the N-body problem. Given the global regularization, the investigator and his colleague study previously inaccessible problems within the N-body problem, such as the dynamical and variational nature of collision-collision solutions, and the existence or lack of existence of a symbolic dynamics for the planar zero-angular momentum three-body problem whose 3 symbols are the 3 types of syzygies (collinearities). Imagine the moon going around the earth, the earth going around the sun, and the sun spinning around the galaxy. The dominant force in the large scale dynamics of the universe is the attractive force of gravity and is responsible for the large scale rotational type motions of celestial bodies. The N-body equations are the equations governing this motion. ("N" stands for the number of bodies.) When two bodies get close the forces get large to the point that when the bodies collide the forces become infinity. This infinity is called a "collision singularity." If the whole universe is translated, or rotated some fixed amount, the motion remains identical and so there are extra variables in the usual N-body equations, namely the variables describing the "origin" and "axes" of the universe. The investigator and his collaborator eliminate the collision singularities, and the extra variables in the N-body equations. These eliminations yield a potentially more useful system of equations for the motion, a system free of infinities which may provide a good platform from which to solve some of the many open problems about one of the dominant dynamical behaviors going on in our universe.