论文标题
$ s^2 $上的碰撞轨迹和两体问题的正规化
Collision trajectories and regularisation of two-body problem on $S^2$
论文作者
论文摘要
在本文中，我们研究了放置在二维球体表面上的两个相同物体的碰撞轨道，并通过吸引$ v（q）= - \ cot（q）$的吸引潜力相互作用，其中$ q $是由两个身体的位置向量形成的角度。我们描述了与导致碰撞的初始数据相对应的符合性降低的系统中的$ω$限制集。此外，我们提供了动力学的几何描述。最后，我们将系统正规化并研究其在近碰撞轨道上的行为。这涉及对完全退化平衡的研究以及使用高维非雄源性爆炸的使用。
In this paper, we investigate collision orbits of two identical bodies placed on the surface of a two-dimensional sphere and interacting via an attracting potential of the form $V(q)=-\cot(q)$, where $q$ is the angle formed by the position vectors of the two bodies. We describe the $ω$-limit set of the variables in the symplectically reduced system corresponding to initial data that lead to collisions. Furthermore we provide a geometric description of the dynamics. Lastly, we regularise the system and investigate its behaviour on near collision orbits. This involves the study of completely degenerate equilibria and the use of high-dimensional non-homogenous blow-ups.
