关于平面四体问题中二元同时碰撞的$ C^{8/3} $正则化
On the $ C^{8/3} $-Regularisation of Simultaneous Binary Collisions in the Planar 4-Body Problem
摘要：
四体问题的动力学允许两个二元碰撞同时发生。众所周知，在共线四体问题中，这种同时二元碰撞（SBC）可以被块正则化，但是所得到的块映射仅是$C^{8/3}$可微的。本文证明了平面四体问题中SBC的$C^{8/3}$可微性。证明使用几个几何工具，即爆破，规范形式，动力学附近的正常双曲流形的平衡点，和杜拉克地图。
英文摘要：
The dynamics of the 4-body problem allows for two binary collisions to occur simultaneously. It is known that in the collinear 4-body problem this simultaneous binary collision (SBC) can be block-regularised, but that the resulting block map is only $C^{8/3}$ differentiable. In this paper, it is proved that the $C^{8/3}$ differentiability persists for the SBC in the planar 4-body problem. The proof uses several geometric tools, namely, blow-up, normal forms, dynamics near normally hyperbolic manifolds of equilibrium points, and Dulac maps.
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