CAREER: Periodic Orbits of Hamiltonian Diffeomorphisms and Reeb Flows

哈密顿系统构成了一大类可以忽略能量耗散的力学系统。例如，天体力学中的行星运动、不可压缩理想流体的流动以及带电粒子在电磁场中的运动通常被视为哈密顿动力学系统。周期轨道的存在是这类系统动力学中最重要的问题之一，它与数学和物理学的许多其他分支有关。与循环运动相对应，这是平衡后最简单的动力学现象，对系统周期轨道的研究对于理解其整体行为至关重要。仅举几个例子，周期轨道的知识在天文学、粒子加速器和流体动力学中是至关重要的，或者可以用来理解大时间解的稳定性。哈密顿系统往往有许多周期轨道，但即使证明一个闭合轨道的存在也往往需要先进而强大的数学工具。本研究项目旨在为广泛的哈密顿系统建立无穷多个周期轨道的存在性，并分析这类系统之外的系统，这项工作将促进我们对保守系统动力学的理解，并导致适用于其他问题的新强大技术的发展。提案中考虑的许多系统（例如，磁流）在物理和工程领域都很有趣，其中一些项目有望在数学物理、几何力学和其他领域得到应用。该研究计划的重点是在各种情况下哈密顿动力系统存在无穷多个周期轨道的问题，以及PI目前正在开发的研究这一问题的新方法。该研究包括几个相互关联的项目，针对特定类别的哈密顿微分同态和Reeb流以及特定的哈密顿系统（如磁流）解决这个问题的各个方面。该项目还开辟了新的研究方向，如闭合流形上不可收缩周期轨道的研究。PI将采用辛拓扑方法解决这些问题，包括Floer和量子同调技术、接触和辛同调技术、j全纯曲线和谱不变量，并将继续开发针对周期轨道研究的新的Floer理论技术。这些项目也适用于其他感兴趣的问题，如辛和接触动力学和拓扑学、经典动力系统和黎曼几何。

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.