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首席研究员：Basak Gurel该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。拟议工作的主要长期目标是研究各种Hamilton动力系统存在无穷多个周期轨道的原因。该建议的第一部分是由几个相互关联的项目addressingthis问题的某些类的哈密尔顿代数同态，也为特定的哈密尔顿系统，如扭曲geodesicflows。PI将通过使用辛拓扑的方法来解决这些问题，包括Floer同调技术，谱不变量，Hamiltonian Ljusternik-Schnirelman理论，Seidel表示以及微分几何的方法，如灰原理。与本研究相关的更具体的工具包括局部Floer同调、平均指数、作用谱和指数谱的性质、作用间隙和指数间隙的性质以及辛简并极大值的几何和动力学性质。PI所使用的技术也有应用程序超越建立无限多个轨道的存在。特别是，PI对提案第二部分中的项目的方法大量借鉴了她最近关于周期轨道问题的一项工作。PI给出了一种证明任何辛流形上Hamilton同态群的直径为无穷大的方法，包括一些新的例子。在这个建议中考虑的另一组问题是关于余各向同性子流形的辛拓扑及其应用. Hamilton系统构成了一个广泛的物理系统，其中耗散力可以忽略不计。例如，天体力学中的行星运动和带电粒子在磁场中的运动通常被视为哈密顿系统。这类系统的一个普遍但不普遍的特征是它们倾向于有许多周期轨道。与周期运动相对应，这是平衡后最简单的动力学现象，研究系统的周期轨道对于理解系统的全局行为至关重要。对于一类广泛的哈密顿系统，周期轨道的数量是无穷的，这被认为是许多（但不是所有）哈密顿系统的情况。然而，建立周期轨道的存在往往需要先进和强大的数学工具。该proposalfocus的存在性问题的无穷多个周期轨道的Hamilton动力系统在各种设置和应用的技术所使用的PI攻击这个问题的其他一些相关的问题。该项目在最后一部分的建议关注的某一类空间出现，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。