N体问题中几类特殊周期轨道的存在性研究

In this project, we mainly concern the variational existence of several special kinds of periodic orbits in the N-body problem. We will study the application of variational methods in the N-body problem and introduce a new geometric method to eliminate possible collision singularities in the minimizer. More precisely,.1). The existence of two Henon-type orbits in the three-body problem. This is an open problem proposed by French mathematician Venturelli in 2003. One of the main challenges is to exclude possible collisions of a minimizer under order constraints. In order to study this problem, we will introduce a new geometric property of minimizers recently discovered by our group and apply the standard techniques in excluding collisions. We expect to make partial progress in the case of equal-masses..2). Variational existence of periodic orbits in the four-body and five-body problems. The main difficulty is to exclude possible collisions in the corresponding action minimizers. By analyzing the symmetry properties of the orbits, we plan to introduce new variational argument and extend the standard techniques in excluding collisions so that we can show the existence of several sets of orbits in the four-body and five-body problems..These studies will enable us to have an intensive understanding of the theory in eliminating collision singularities in action minimizers and related topics in the N-body problem.

本项目研究N体问题中几类特殊类型周期轨道的存在性。申请人将主要致力于变分法在N体问题中的应用以及引入新的几何方法来排除两点自由边值问题对应的泛函极小子中可能的碰撞奇点这两方面的研究。具体而言，.1）三体问题中两类Henon型周期轨道的存在性。该问题是2003年法国数学家Venturelli提出的公开问题。我们将引入新的几何方法来研究相应的泛函极小子，同时结合已有的排除碰撞奇点的方法，期望在该公开问题上取得进展；.2）寻找四体和五体的非平凡周期轨道并证明它们的存在性。证明周期轨道存在性的一个主要困难是排除两点自由边值问题的变分极小子中各种可能的碰撞奇点。通过对两个边值所在的空间引入合适的拓扑限制，我们将借助新的几何方法以及推广现有的排除碰撞方法，最终证明其中几类特定类型的周期轨道的存在性。.这些探索将帮助我们深入理解N体问题中极小子的碰撞奇点理论, 有助于更好的将变分法运用到N体问题中去。

本项目主要运用变分法、几何方法以及数值工具研究N体问题中的几类特殊的周期轨道。通过把Jacobi坐标和变分法相结合，我们成功在Venturelli2003年的一个公开问题上取得突破性进展。运用该技巧，我们也可以巧妙的证明三体问题中几类周期轨道的几何性质。另外，我们也运用变分法和数值工具研究了平面五体问题的周期轨道。