CAREER: Periodic Orbits of Hamiltonian Diffeomorphisms and Reeb Flows

职业：哈密顿微分同胚和 Reeb 流的周期轨道

• 批准号: 1454342
• 负责人: Basak Gurel
• 金额: $ 40万
• 依托单位: The University of Central Florida Board of Trustees
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-09-15 至 2023-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1454342&HistoricalAwards=false
• 关键词: CAREER; Periodic; Orbits; Hamiltonian; Diffeomorphisms

Hamiltonian systems constitute a broad class of mechanical systems where energy dissipation can be neglected. For example, the planetary motion in celestial mechanics, the flow of an incompressible ideal fluid, and the motion of a charged particle in an electro-magnetic field are usually treated as Hamiltonian dynamical systems. One of the most important questions concerning the dynamics of such systems and connected to many other branches of mathematics and physics is the existence of periodic orbits. Corresponding to cyclic motion, this is the simplest dynamical phenomenon after equilibrium, and an investigation of periodic orbits of a system is crucial in understanding its global behavior. To give but a few applications, the knowledge of periodic orbits is crucial in astronomy, particle accelerators, and fluid dynamics or can be used to understand stability of solutions for large times. Hamiltonian systems tend to have numerous periodic orbits, but proving the existence of even one closed orbit often requires advanced and powerful mathematical tools. This research project aims at establishing the existence of infinitely many periodic orbits for a broad class of Hamiltonian systems and analyzing the systems that fall outside this class, and the work will advance our understanding of the dynamics of conservative systems and result in the development of new powerful techniques applicable to other questions. Many of the systems considered in the proposal (e.g., magnetic flows) are of interest in physics and engineering, and some of the projects are expected to have applications in mathematical physics, geometric mechanics, and other areas.The research program focuses on the question of the existence of infinitely many periodic orbits for Hamiltonian dynamical systems in a variety of settings and on new methods to investigate this question that are currently being developed by the PI. The research comprises several interconnected projects addressing various aspects of this question for certain classes of Hamiltonian diffeomorphisms and Reeb flows and also for specific Hamiltonian systems such as magnetic flows. The project also opens up new research directions such as the study of non-contractible periodic orbits on closed manifolds. The PI will tackle these problems employing symplectic topological methods including Floer and quantum homological techniques, contact and symplectic homology, J-holomorphic curves, and spectral invariants, and will continue to develop new Floer theoretic techniques tailored for the study of periodic orbits. The projects have applications to other questions of interest in symplectic and contact dynamics and topology, classical dynamical systems, and Riemannian geometry.

哈密​​顿系统构成了可以忽略能量耗散的广泛机械系统。例如，天体力学中的行星运动，不可压缩的理想流体的流动以及电磁场中带电粒子的运动通常被视为哈密顿动力学系统。有关此类系统动力学并与数学和物理学的许多其他分支相关的最重要问题之一是存在周期性轨道。对应于循环运动，这是平衡后最简单的动态现象，并且对系统的周期性轨道进行研究对于理解其全局行为至关重要。为了提供一些应用，周期性轨道的知识在天文学，粒子加速器和流体动力学中至关重要，或者可用于在很大程度上理解解决方案的稳定性。哈密​​顿系统往往具有多种周期性轨道，但是证明即使是一个封闭的轨道的存在通常都需要先进且强大的数学工具。该研究项目旨在建立一个无限多个周期性轨道的存在，用于广泛的哈密顿系统，并分析属于此类班级的系统，这项工作将提高我们对保守系统动态的理解，并导致开发适用于其他问题的新的强大技术。该提案中考虑的许多系统（例如磁流）在物理和工程上都感兴趣，预计某些项目将在数学物理学，几何力学和其他领域中应用。研究计划的重点是无限期地存在于各种设置和新问题上的汉密尔顿动态系统的无限期周期性轨道存在的问题。该研究包括几个相互联系的项目，这些项目针对某些类别的哈密顿二型差异和REEB流，以及对于特定的哈密顿系统（例如磁流量），解决了该问题的各个方面。该项目还打开了新的研究方向，例如对封闭流形的非收集周期轨道进行研究。 PI将采用符合性拓扑方法（包括浮动和量子同源技术，接触和符号同源性，J旋晶曲线以及光谱不变性）来解决这些问题，并将继续开发针对周期性轨道研究的新型理论技术。这些项目在符号和接触动力学和拓扑，经典动力学系统和Riemannian几何形状中的其他感兴趣问题上都有应用。