本项目研究天体力学中周期轨道的性质。周期轨道(或周期解)是天体力学中的重要的研究对象。由于天体力学系统的复杂性，寻找并研究周期轨道的有关性质就成了天体力学的重要课题之一。其中关于这些周期轨道的定性与定量性质，尤其是寻找某些周期轨道以及这些周期轨道稳定性方面还有许多未解决的问题,它们也是非线性分析、辛几何、动力系统等自然科学领域关心的重要问题。本项目致力于对各类天体力学系统，包括经典三体问题、限制性三体问题、更一般的N体问题中的周期轨道以及更一般的周期轨道进行研究，建立它们存在性条件、线性稳定性的相关条件，并结合对应的Maslov-型指标和其他一些相关的工具对它们的性质作出更好的刻画。此外，本项目还将利用自伴算子的谱理论、谱流理论、迹公式理论、Jcobi矩阵理论以及Sturm-Liouville方程理论的方法去研究这些周期轨道的更进一步的性质。

This program studies the periodic orbits of Celestial Mechanics. Periodic orbits (or periodic solutions) are important objects in Celestial Mechanics. Because of the complexity of celestial mechanical system, studies on the nature of the periodic orbits becomes one of the important topics in Celestial Mechanics. On the qualitative and quantitative nature of these periodic orbits, especially for the existence of some periodic orbits and the linear stabilities of the periodic orbits, there are many unresolved problems, which are also important topics in Nonlinear Analysis, Symplectic Geometry, Dynamical System, etc. In this program, we study the periodic solutions of the classical three-body problem, restricted three-body problem, N-body problem and more general periodic orbits of the celestial mechanical system. We establish their existing condition, the relevant conditions of linear stabilities, and use the method of Maslov-type index and other tools to make a better characterization. In addition, this program also use the spectral theory of self ad-joint operators, spectral flow theory, trace formula, Jacobi matrix theory and the theory of Sturm-Liouville equations to study more properties of the periodic orbits.