三涡旋系统：Hopf纤维化、辛约化与近碰撞
The three-vortex system: Hopf fibration, symplectic reduction, and near-collisions
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中文总结 AI 辅助
本文研究平面三涡旋系统的碰撞与近碰撞动力学，通过约化坐标和辛约化方法分析，提出新的近碰撞结果并证明新的临界能量不等式，同时重现经典结论。
中文摘要 AI 辅助
我们研究平面中三个点涡旋系统中的碰撞与近碰撞动力学。该动力学已知是完全可积的，并可约化为二维辛叶上的哈密顿系统研究。这仍然允许解的有趣行为，其各个方面已在众多作者的大量工作中得到研究。我们的重点将放在碰撞及其通过扰动系统适当参数进行正则化的可能性上。我们将使用由 $\zeta=(z_2-z_1)/(z_3-z_1)$ 给出的约化坐标，其中 $z_1,z_2,z_3$ 是表示涡旋位置的复数。这个选择似乎非常自然，尽管我们在该主题的现有文献中未曾遇到。我们关于近碰撞行为的一些结果似乎是新的，并且我们还证明了在文献中未找到的临界能量值之间的不等式。我们为分析碰撞和近碰撞所做的计算也重现了先前由其他作者使用不同坐标获得的各种经典结果，因此论文的部分内容具有阐述性质。
英文摘要
We study the dynamics of collisions and near-collisions in the system of three point vortices in the plane. The dynamics is known to be completely integrable and can be reduced to the study of Hamiltonian systems on two-dimensional symplectic leaves. This still allows for interesting behavior of the solutions, various aspects of which have been investigated in a large body of work by many authors. Our focus will be on collisions and the possibility of their regularization through perturbations of suitable parameters of the system. We will use the reduced coordinate given by $ζ=(z_2-z_1)/(z_3-z_1)$, where $z_1,z_2,z_3$ are the complex numbers representing the positions of the vortices. This choice seems very natural, although we have not come across it in the existing literature on the topic. Some of our results concerning the near-collision behavior appear to be new, and we also prove inequalities between critical energy values that we have not found in the literature. The calculations we do to analyze the collisions and near-collisions also reproduce various classical results that were previously obtained by other authors using different coordinates, and parts of the paper therefore have an expository character.
