Periodic orbits of Hamiltonian systems and symplectic topology of coisotropic submanifolds

哈密​​顿系统的周期轨道和各向同性子流形的辛拓扑

• 批准号: 1007149
• 负责人: Viktor Ginzburg
• 金额: $ 20.36万
• 依托单位: University of California-Santa Cruz
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-07-01 至 2014-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1007149&HistoricalAwards=false
• 关键词: Periodic; orbits; Hamiltonian; systems; symplectic

AbstractAward: DMS-1007149Principal Investigator: Viktor GinzburgThe present proposal focuses on several projects closely related to the PI's previous work. The first group of problems addressed concerns generalizations of the Conley conjecture. Thisconjecture asserts the existence of infinitely many periodic points of a Hamiltonian diffeomorphism of a symplectically aspherical, closed manifold. The Conley conjecture has been established by Hingston (for tori), the PI and eventually generalized to all symplectic manifolds with zero Chern class. However, many aspects of the problem require further investigation. The PI outlines an approach to the proof of the Conley conjecture for manifolds with large minimal Chern class and to some classes of symplectomorphisms and Hamiltonian diffeomorphisms with "too many" fixed points. A refinement of the almost existence theorem for periodic orbits of Hamiltonian systems and an investigation of periodic orbits of twisted geodesic flows, closely related to the Conley conjecture, are also considered in the proposal. The second part of the proposal focuses on symplectic topological properties of coisotropic submanifolds. These properties generalize the Lagrangian intersection property and the Liouville and Maslov class rigidity to a certain class of coisotropic submanifolds, the so-called stable submanifolds. Moreover, a general picture has emerged, enabling one to treat such facts as non-existence of exact Lagrangian embeddings and the existence of closed characteristics on a contact type hypersurface as particular cases of one phenomenon. The main goal of the program started by the PI and continued in the proposal is to further analyze this picture and to extend coisotropic rigidity results to a broader class of submanifolds.Hamiltonian dynamical systems describe many classes of physical processes in which dissipative forces can be neglected. For example, planetary motion in celestial mechanics and some electro- or magneto-dynamical processes can be, and usually are, treated as Hamiltonian dynamical systems. One of the classical subjects lying at the very core of modern theory of Hamiltonian dynamical systems and symplectic geometry is the study of periodic orbits (i.e., cyclic motions). Periodic orbits are ubiquitous: a vast majority of Hamiltonian systems have periodic orbits and the number of distinct periodic orbits is infinite for a broad class of systems. The analysis of this phenomenon, building on the PI's recent work, is among the main objectives of the proposed research. For instance, the PI proposes to show that Hamiltonian systems of a certain type have infinitely many periodic orbits. The class of dynamical systems in question includes those describing the motion of a charge in a magnetic field and the proposed research has potential applications to physics and mathematical aspects of mechanics.

Abstractaward：DMS-1007149原理研究人员：Viktor Ginzburgth当前的建议重点介绍了与PI先前工作密切相关的几个项目。 第一组问题解决了康利猜想的概括。该注射剂断言，在符合性非球面，封闭的歧管的哈密顿二差异的许多周期性点上存在。 Conley猜想是由Hingston（对于Tori）建立的PI，并最终概括为Chern类别为零的所有符号歧管。但是，问题的许多方面需要进一步研究。 PI概述了一种方法证明Conley猜想的方法，这些conley猜想具有很大的Chern类，以及某些类别的符号术和Hamiltonian diffeferismiss的方法，其固定点太多了。在该提案中也考虑了几乎存在的哈密顿系统周期性轨道和对扭曲的大地测量流的周期性轨道的调查，与Conley猜想密切相关。该提案的第二部分重点介绍了共髓质亚策略的符号拓扑特性。这些属性概括了拉格朗日交叉路口的属性以及liouville和Maslov类的刚性，以某种类别的坐骨submanifolds，即所谓的稳定的submanifolds。 此外，已经出现了一般图片，使人们能够将诸如精确拉格朗日嵌入不存在的事实以及接触类型超表面上的封闭特征的存在为一种现象的特定情况。该计划的主要目标是由PI启动的，并在该提案中继续进行，是进一步分析了这一情况，并将共同体刚性结果扩展到更广泛的Submanifolds.Hamiltonian Dynalical Systems。描述了许多可以忽略耗散力的物理过程。例如，天体力学中的行星运动以及某些电磁或动力学过程可以并且通常被视为哈密顿动力学系统。位于哈密顿动力学系统现代理论和象征性几何学现代理论的核心的经典主题之一是对周期轨道的研究（即环状运动）。周期性的轨道无处不在：绝大多数哈密顿系统具有周期性轨道，并且广泛的系统的不同周期性轨道的数量是无限的。基于PI最近的工作，对这种现象的分析是拟议研究的主要目标之一。例如，PI提出表明某种类型的汉密尔顿系统具有无限的周期性轨道。所讨论的动态系统类别包括描述磁场中电荷运动的那些类别，并且所提出的研究可能在机械的物理和数学方面有潜在的应用。