哈密尔顿系统及多体问题的周期解

Hamiltonian systems originated in the formalization of Newtonian mechanics with the N-body problem as a principle motivating paradigm. Thestudy of Hamiltonian and N-body stystems promoted the development of new mathematical theories such as max/min principles and Morse theory. These theories continue to be relevant even outside their original motivations. A major focus of research is the study of periodic solutions for Hamiltonian systems in general, and N-body problems in particular. Under the supervision of Professor Shiqing Zhang, my master's and doctorate work was in new periodic solutions to Hamiltonian systems and N-body problems. I will use variational method to continue the development of study periodic solutions with an emphasis on the geometry of their configurations.

Hamiltonian系统与多体问题有紧密关系，Hamiltonian系统起源于Newton力学,多体问题是特殊的Hamiltonian系统，它们的发展也催生了很多新的数学分支，推动了极大极小原理，Morse理论等重要理论的发展。多体问题及Hamiltonian系统的周期解是重要的研究课题。申请人在攻读硕士、博士学位期间跟随张世清教授学习、研究了多体问题及Hamiltonian系统的新的周期解。本项目将利用变分方法进一步研究多体问题及Hamiltonian系统的新的周期解的存在性及其解的形状。

哈密尔顿系统周期解的存在性问题在上个世纪吸引了很多数学家的眼球，他们应用了极大极小理论，Morse理论，指标理论，临界点理论等方法来研究其周期解。最早，Seifert及Rabinowitz应用最小作用原理来研究不带奇异性的哈密尔顿周期解。基于他们的研究工作，本项目主要应用变分最小方法研究了哈密尔顿连接轨周期解的存在性。主要研究了在牛顿弱力势能的条件至少存在一条连接质心与无穷远点的轨道，也研究了在适当势能条件下二阶哈密尔顿系统同宿轨的存在性。