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.

拟议的研究的主要目的是研究多种汉密尔顿动态系统的无限周期性轨道存在的现象，并了解承认有限多个周期性轨道的系统的性质。该提案包括几个相互联系的项目，这些项目解决了某些类别的哈密顿二差异性以及针对特定的哈密顿系统（例如磁流）的问题。 PI将通过采用来自对称拓扑的方法来解决这些问题，包括漂浮物和量子同源技术，全体形态曲线，光谱不变式，ljusternik-Schnirelman理论，以及来自H-principles等微分几何的方法。 PI使用的技术还具有无限多个周期性轨道存在的问题。特别是，有关庞加莱复发的项目，Reeb流动和共同体Symplopic Symple Tostology在很大程度上借鉴了她最近关于周期性轨道的作品。这些项目具有衡量保护和经典动态系统的应用，以及对称拓扑中的一些嵌入问题。HamiltonianSystems构成了一类广泛的物理系统，可以忽略耗散力。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磁场中带电粒子的运动通常被视为哈密顿系统。这种系统的一般但不是普遍的特征是它们倾向于具有许多周期性轨道。对应于环状运动，这是平衡后最简单的动态现象，并且系统的周期性轨道投资对于理解其全球行为至关重要。为了提供一些应用，周期性轨道的知识对于天文学，流体动力学（例如，湍流的统计数据）至关重要，或者可用于了解大量解决方案的稳定性。除了最简单的情况外，确定周期性轨道的存在通常需要先进且强大的数学工具。对于大量的哈密顿系统，已知周期性轨道的数量是无限的，因此人们认为这是许多但不是全部的哈密顿系统的情况。该提案重点是在各种环境中无限期许多周期性轨道存在的问题，以及PI用于攻击此问题的技术的应用以及其他相关问题的应用。该提案的最后一部分项目涉及一定类别的空间，例如在研究具有对称性的汉密尔顿系统中。所提出的工作与数学物理学以及几何和量子力学有关，并具有潜在的应用。