Global Aspects of the N-Body Problem

牛顿的N体问题有两个困难是一般理论的典型动力学系统所没有的：使流动不完整的碰撞奇异性，以及使其所有周期轨道退化的对称性。调查员和他的同事排除了这两个困难。他们通过Levi-Civita正则化、Kuustaanheimo-Steifel正则化或麦基希blow-up来消除碰撞奇异性。 他们通过辛约化和不变量理论消除对称性。他们把得到的正则化约化流称为“全局正则化”。“这个流程是完整的， 对于编码N体问题的流，可能的最小自由度数。考虑到全局正则化，研究者和他的同事研究了N体问题中以前无法解决的问题，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或不存在，其3个符号是3种类型的syzygies（共线性）。 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。在宇宙大尺度动力学中占主导地位的力是引力的吸引力，它是天体大尺度旋转运动的原因。 N体方程是控制这种运动的方程。（“N”代表机构数量。） 当两个物体靠近时，力会变大，当两个物体碰撞时，力会变得无穷大。 这个无穷大被称为“碰撞奇点”。如果整个宇宙平移或旋转一定的量，运动保持不变，因此在通常的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来解决我们宇宙中发生的主要动力学行为之一的许多悬而未决的问题。

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.