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: 0906204
• 负责人: Basak Gurel
• 金额: $ 13.6万
• 依托单位: Vanderbilt University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-09-15 至 2013-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0906204&HistoricalAwards=false
• 关键词: Symplectic; Topology; Hamiltonian; Systems; Infinitely

AbstractAward: DMS-0906204Principal Investigator: Basak GurelThis award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).The main long-term objective of the proposed work is to examine thereason for the existence of infinitely many periodic orbits for avariety of Hamiltonian dynamical systems. The first part of theproposal is comprised of several interconnected projects addressingthis question for certain classes of Hamiltonian diffeomorphisms andalso for specific Hamiltonian systems such as twisted geodesicflows. The PI will tackle these problems by making use of methods fromsymplectic topology including Floer homological techniques, spectralinvariants, Hamiltonian Ljusternik-Schnirelman theory, the Seidelrepresentation as well as methods from differential geometry such ash-principles. Among more specific tools pertinent to thisinvestigation are local Floer homology, the mean index, the propertiesof action and index spectra and of action and index gaps, andgeometric and dynamical properties of the symplectically degeneratemaxima. The techniques utilized by the PI also have applicationsbeyond establishing the existence of infinitely many periodicorbits. In particular, the PI's approach to the projects in the secondpart of the proposal draws heavily on one of her recent works on theperiodic orbits problem. The PI outlines a method of proving that thediameter of the group of Hamiltonian diffeomorphisms is infinite formany symplectic manifolds, including some new instances. Another groupof problems considered in this proposal concerns the symplectictopology of coisotropic submanifolds and some of its applications.Hamiltonian systems constitute a broad class of physical systems wheredissipative forces can be disregarded. For example, the planetarymotion in celestial mechanics and the motion of a charged particle ina magnetic field are usually treated as Hamiltonian systems. Onegeneral, but not universal, feature of such systems is that they tendto have numerous periodic orbits. Corresponding to the cyclic motion,this is the simplest dynamical phenomenon after equilibrium and aninvestigation of periodic orbits of a system is crucial inunderstanding its global behavior. For a broad class of Hamiltoniansystems, the number of periodic orbits is known to be infinite and thisis thought to be the case for many (but not all) Hamiltoniansystems. Yet, establishing the existence of periodic orbits oftenrequires advanced and powerful mathematical tools. The proposalfocuses on the existence problem for infinitely many periodic orbits ofHamiltonian dynamical systems in a variety of settings and onapplications of the techniques used by the PI to attack this problemto some other related questions. The projects in the last part of theproposal concern a certain class of spaces which arise, for instance,in the study of Hamiltonian systems with symmetries. The proposed workis related to and has potential applications in mathematical physics,and geometric and quantum mechanics.

Abstractaward：DMS-0906204原理研究者：Basak Gurelthis奖是根据2009年的《美国回收与再投资法》（公法111-5）资助的。拟议工作的主要长期目标是研究无限期存在的许多周期性的Hamilton Expariatian Demansigation Demansigation demantimical dementical Systems。 Proposal的第一部分由几个相互联系的项目组成，解决了某些类别的汉密尔顿二型差异性的问题，以及针对特定的汉密尔顿系统（例如扭曲的地理素流量）的问题。 PI将通过利用来自彼此拓扑的方法来解决这些问题，包括浮动同源技术，Spectralinvariants，Hamiltonian ljusternik-Schnirelman理论，Seidel-ReSentation以及差异几何形状的方法。在与此进一步相关的更具体的工具中，有局部浮点同源性，平均索引，动作和索引光谱的属性以及动作和索引差距，以及符号脱发性脱脂性的几何和动态特性。 PI使用的技术还具有应用程序，以建立无限多个周期性的存在。特别是，PI在该提案第二部分中对项目的方法大大吸引了她最近关于旋转轨道问题的作品之一。 PI概述了一种证明哈密顿量差异的方法是无限的formanany symblectic歧管，其中包括一些新实例。该提案中考虑的另一组问题涉及索迪索西亚亚曼群及其某些应用的符号学。例如，天体力学中的行星变量和带电粒子INA磁场的运动通常被视为哈密顿系统。此类系统的单加（但不是通用）特征是它们倾向于具有多种周期性轨道。对应于循环运动，这是在系统的平衡和对周期轨道进行衡量之后最简单的动态现象，这对于其全局行为至关重要。对于大量的汉密尔顿系统系统，已知周期性轨道的数量是无限的，这对于许多（但不是全部）哈密顿系统的情况就是这种情况。然而，确定周期性轨道的存在通常会先进和强大的数学工具。在各种环境中，无限多个周期性轨道的生存问题的提案问题以及PI用于攻击此问题的技术的应用中，对其他相关问题攻击此问题的技术。 Proposal的最后一部分的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。所提出的与数学物理学以及几何和量子力学有关的工作和潜在应用。