二阶哈密尔顿系统及多体问题的周期解的研究

Hamiltonian systems originated in the formalization of Newtonian mechanics with the N-body problem as a principle motivating paradigm. The study of Hamiltonian and N-nody systems promoted the development of new mathematical theories such as Algebra Topology and glocal variational methods. A major focus of research is the study of periodic solutions for second order Hamiltonian systems in general and N-body problems in particular. I will continue to study further direct applications of the Ekelend Variational Principle and minmax theorems with (CPS) condition to the existence of periodic solutions with an emphasis on the geometry of their configurations. We will also explore the use of non-smooth critical theorems to study Hamiltonian systems with a local Lipschitz functional.

哈密尔顿系统与多体问题有紧密关系，哈密尔顿系统起源于牛顿二体问题，而多体问题也是非线性的奇异哈密尔顿系统，对它们的研究催生出许多新的数学分支，如代数拓扑学，大范围变分方法等。本项目旨在前面已有工作的基础上应用直接变分方法，Ekeland变分原理、带（CPS）条件的极大极小定理进一步研究给定条件下的二阶哈密尔顿系统及多体问题的新的周期解的存在性及其解的形状等。本项目也将用非光滑的临界点理论来研究具有局部Lipschitz泛函的哈密尔顿系统。

多体问题及哈密尔顿系统周期解的研究一直是众多学者关心的问题。本项目基于Rabinowitz，Ambrosetti-Coti Zelati等专家的基础上，利用极大极小方法，研究了哈密尔顿系统其能量泛函在一些特定情况下，其新的周期解的存在性。并将其方法推广到多体问题势能上，证明新的周期解的存在性。