天体力学中两自由度问题的抛射-碰撞轨道
Ejection-collision orbits in two degrees of freedom problems in celestial mechanics
摘要：
在一般的两自由度哈密顿系统的情况下，假定经历势的某些性质，研究了系统的动力学接近奇异性和趋向奇异性，在$N$ -体问题的模型中，系统的动力学对应于完全碰撞.我们限制到潜力，表现出两个以上的奇点，可以被视为两种部分碰撞时，不是所有的机构都参与。正则化奇点后，总碰撞转化为二维不变流形。本文的目的是证明不同类型的弹射碰撞轨道的存在性，也就是说，轨道的开始和结束在总碰撞。这样的轨道被认为是两个平衡点之间的异宿联系，其主要特征是轨道在其途中发现的部分碰撞。它们的存在性的证明是基于二维不变流形的横截性和全碰撞流形上的动力学行为，两者都得到了充分的描述。
英文摘要：
In a general setting of a Hamiltonian system with two degrees of freedom and assuming some properties for the undergoing potential, we study the dynamics close and tending to a singularity of the system which in models of $N$-body problems corresponds to total collision. We restrict to potentials that exhibit two more singularities that can be regarded as two kind of partial collisions when not all the bodies are involved. Regularizing the singularities, the total collision transforms into a 2-dimensional invariant manifold. The goal of this paper is to prove the existence of different types of ejection-collision orbits, that is, orbits that start and end at total collision. Such orbits are regarded as heteroclinic connections between two equilibrium points and are mainly characterized by the partial collisions that the trajectories find on their way. The proof of their existence is based on the transversality of 2-dimensional invariant manifolds and on the behavior of the dynamics on the total collision manifold, both of them are thoroughly described.
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