Symplectic topology of Hamiltonian systems with infinitely many periodic orbits

具有无限多个周期轨道的哈密顿系统的辛拓扑

• 批准号: 1207680
• 负责人: Basak Gurel
• 金额: $ 14.65万
• 依托单位: Vanderbilt University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-07-15 至 2014-01-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1207680&HistoricalAwards=false
• 关键词: Symplectic; topology; Hamiltonian; systems; infinitely

The primary goal of the proposed research is to investigate the phenomenon of existence of infinitely many periodic orbits for a variety of Hamiltonian dynamical systems and to understand the nature of the systems admitting finitely many periodic orbits. The proposal comprises several interconnected projects addressing these questions for certain classes of Hamiltonian diffeomorphisms and also for specific Hamiltonian systems such as magnetic flows. The PI will tackle these problems by employing methods from symplectic topology including Floer and quantum homological techniques, holomorphic curves, spectral invariants, Ljusternik-Schnirelman theory, as well as methods from differential geometry such as h-principles. The techniques utilized by the PI also have applications beyond the question of existence of infinitely many periodic orbits. In particular, the projects concerning the Poincaré recurrence, the Reeb flows and the coisotropic symplectic topology draw heavily on her recent works concerning periodic orbits. These projects have applications to measure-preserving and classical dynamical systems, and to some embedding problems in symplectic topology.Hamiltonian systems constitute a broad class of physical systems where dissipative forces can be disregarded. For example, the planetary motion in celestial mechanics, the flow of an incompressible ideal fluid, and the motion of a charged particle in a magnetic field are usually treated as Hamiltonian systems. One general, but not universal, feature of such systems is that they tend to have numerous periodic orbits. Corresponding to the 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, fluid dynamics (e.g., statistics of turbulent flow) or can be used to understand stability of solutions for large times. In all but the simplest cases, establishing existence of periodic orbits often requires advanced and powerful mathematical tools. For a broad class of Hamiltonian systems, the number of periodic orbits is known to be infinite and this is thought to be the case for many, but not all, Hamiltonian systems. The proposal focuses on the problem of existence of infinitely many periodic orbits for Hamiltonian dynamical systems in a variety of settings and on applications of the techniques used by the PI to attack this problem to some other related questions. The projects in the last part of the proposal concern a certain class of spaces which arise, for instance, in the study of Hamiltonian systems with symmetries. The proposed work is related to and has potential applications in mathematical physics, and geometric and quantum mechanics.

本研究的主要目的是研究各种Hamilton动力系统存在无穷多个周期轨道的现象，并了解允许无穷多个周期轨道的系统的性质。该建议包括几个相互关联的项目，解决这些问题的某些类的哈密顿同构，也为特定的哈密顿系统，如磁流。PI将通过采用辛拓扑的方法来解决这些问题，包括Floer和量子同调技术，全纯曲线，谱不变量，Ljusternik-Schnirelman理论，以及微分几何的方法，如h-原理。PI所使用的技术也有超越无限多个周期轨道存在的问题的应用。特别是，项目有关的庞加莱复发，Reeb流和各向同性辛拓扑严重借鉴她最近的作品有关周期轨道。这些项目有应用保测和经典动力系统，并在辛拓扑中的一些嵌入问题。哈密顿系统构成了一个广泛的一类物理系统，其中耗散力可以忽略不计。例如，天体力学中的行星运动，不可压缩理想流体的流动，带电粒子在磁场中的运动通常被视为哈密顿系统。此类系统的一个一般但非普遍的特征是它们往往具有许多周期轨道。对应于循环运动，这是平衡后最简单的动力学现象，并且研究系统的周期轨道对于理解其全局行为至关重要。仅举几个应用，周期轨道的知识在天文学、流体动力学（例如，湍流的统计），或者可以用来理解解的稳定性。除了最简单的情况之外，在所有情况下，确定周期轨道的存在通常需要先进且强大的数学工具。对于一类广泛的哈密顿系统，已知周期轨道的数量是无限的，这被认为是许多但不是所有哈密顿系统的情况。 该建议的重点是问题的存在无穷多个周期轨道的哈密顿动力系统在各种设置和应用程序的技术所使用的PI攻击这个问题的一些其他相关问题。该项目在最后一部分的建议涉及到某一类空间所产生的，例如，在研究哈密顿系统的对称性。所提出的工作与数学物理、几何和量子力学有关，并具有潜在的应用。