平面三体问题的对称正则化、约化与爆破
Symmetric Regularization, Reduction and Blow-Up of the Planar Three-Body Problem
摘要：
我们进行了一系列的坐标变化的平面三体问题，依次消除平移和旋转对称性，正则化所有三个双碰撞奇点和爆破的三重碰撞。用三个相对位置向量对组态进行参数化，保持了质量之间的对称性，简化了二元碰撞的正则化。使用尺寸和形状坐标有利于通过旋转和三重碰撞的爆破来减少，同时强调形状球的作用。通过使用齐次坐标来描述位形空间为球面或射影空间的哈密顿系统，我们能够对这些熟悉的问题采取现代的、整体的方法。我们还展示了如何在几种方便的局部坐标系中得到约化和正则化的微分方程。
英文摘要：
We carry out a sequence of coordinate changes for the planar three-body problem which successively eliminate the translation and rotation symmetries, regularize all three double collision singularities and blow-up the triple collision. Parametrizing the configurations by the three relative position vectors maintains the symmetry among the masses and simplifies the regularization of binary collisions. Using size and shape coordinates facilitates the reduction by rotations and the blow-up of triple collision while emphasizing the role of the shape sphere. By using homogeneous coordinates to describe Hamiltonian systems whose configurations spaces are spheres or projective spaces, we are able to take a modern, global approach to these familiar problems. We also show how to obtain the reduced and regularized differential equations in several convenient local coordinates systems.
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