哈密顿系统L-Maslov型指标与哈密顿系统边值解若干问题的研究

Theory for Hamiltonian systems is an important tool to research classical mechanics, especially celestial mechanics. Periodic motion is the most simple way of celestial bodies movement, which corresponds to periodic solution of Hamiltonian systems. Many mathematicians concerned the topic of the existence and multiplicity and minimal of periodic solutions of Hamiltonian systems. With deepening of research, the open string is another important issue to Hamiltonian systems, as well as Hamiltonian systems with various boundary conditions, also become one important research direction of many mathematicians. In this program, we will further study the L-Maslov index iteration theory and its applications. We will be different several classes of nonlinear Hamiltonian systems with various boundary conditions as the research object, especially Superquadratic and Subquadratic Hamiltonian systems. We will mainly use the L-Maslov index theory and its iteration theory to study the multiplicity of solutions with Lagrangian boundary conditions, along with the minimal and multiplicity of brake orbits with given periodic boundary conditions.

哈密顿系统理论是研究经典力学，尤其是天体力学的重要工具，周期运动是天体运动的最简单方式，它对应着哈密顿系统的周期解。一般哈密顿系统周期解的存在性和多重性以及最小周期等理论，是很多数学家所关心的课题。随着研究的不断深入，开弦问题作为哈密顿系统的另外一种重要问题以及由此而产生出的哈密顿系统的各种边值问题，也成为了广大数学家的一个重要研究方向。本项目将深入研究哈密顿系统L-Maslov型指标的迭代理论及其应用。将几类非线性哈密顿系统的不同边值解作为研究对象，尤其是超二次及次二次哈密顿系统。主要运用L-Maslov型指标理论及其迭代理论来研究其拉格朗日边值解的多重性问题以及给定周期边值的闸轨道的最小周期和多重性等边值解问题。

本项目深入研究了哈密顿系统L-Maslov型指标理论及其迭代理论的应用，并运用该理论重点研究了非线性哈密顿系统的不同边值解的存在性、最小周期性和多重性等问题。本项目三年来取得了以下主要进展：运用L-Maslov型指标理论及其迭代理论，获得了非线性哈密顿系统的次调和闸解的存在性及几何相异性的结果；获得了非线性哈密顿系统的次调和解的存在性及几何相异性的结果；优化非线性哈密顿系统的给定周期边值闸轨道的最小周期估计的结果；并且尝试构造L-Maslov型的相对指标，以对于哈密顿系统的不同边值解的最小周期问题及多重性问题进行研究。