Hamilton系统同宿轨的研究

Hamiltonian systems is an important field of research in nonlinear science. It comes from the geometrical optics, and has a natural connection with classical mechanics and celestial mechanics. Finding homoclinic orbits has been one of the most important problems in the study of Hamiltonian systems. This project will apply variational methods and critical point theory to study the existence and multiplicity of homoclinic orbits of Hamiltonian systems. The main contents include: (1)we will study the existence and multiplicity of homoclinic orbits of the nonperiodic first-order Hamiltonian systems with subquadratic potential or concave-convex nonlinearity; (2)we will study the existence and multiplicity of homoclinic orbits of the second-order Hamiltonian systems with sign-changing potential, and discuss the effect of different nonlinear assumptions, together with the change of parameter, on the existence of homoclinc orbits. We hope that the projet can enrich and develop the theories of Hamiltonian systems, and also provide theoretic references and technical supports for other scientific fields.

Hamilton系统是非线性科学的一个重要研究领域。它源于几何光学，同经典力学和天体力学有着自然的联系。寻找同宿轨是Hamilton系统研究问题中最重要的问题之一。本项目拟使用变分方法和临界点理论研究一阶、二阶Hamilton系统同宿轨的存在性和多重性，主要内容包括：(1)研究一阶非周期Hamilton系统分别在次二次情形和带凹凸非线性项情形下同宿轨的存在性、多重性以及无穷多条同宿轨的存在性；(2)研究具有变号位势的二阶Hamilton系统同宿轨的存在性和多重性，探讨不同非线性条件以及参数变化对同宿轨道存在个数的影响。期望本课题的研究能丰富和发展Hamilton系统理论，同时为其它相关科学领域提供理论依据和技术支持。