Periodic Orbits of Hamiltonian Systems, the Almost Existence Theorem, and Poisson Topology

哈密​​顿系统的周期轨道、几乎存在定理和泊松拓扑

• 批准号: 0307484
• 负责人: Viktor Ginzburg
• 金额: $ 16.27万
• 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: 2003
• 资助国家: 美国
• 起止时间: 2003-07-01 至 2007-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0307484&HistoricalAwards=false
• 关键词: Periodic; Orbits; Hamiltonian; Systems; Almost

AbstractAward: DMS-0307484Principal Investigator: Viktor GinzburgThe present proposal focuses on two projects closely related tothe principal investigator's previous work funded by NSFgrants. These projects are the problem of existence of periodicorbits for Hamiltonian dynamical systems and the study oftopological properties of certain Poisson manifolds. The firstproblem Viktor Ginzburg addresses in this proposal is theinvestigation of the size of the set of regular energy values onwhich a Hamiltonian system does not have periodic orbits. By thealmost existence theorem, this set must be of zero measure andfrom counterexamples to the Hamiltonian Seifert conjecture it isknown that this set may be non-empty. Thus the question is tobridge the gap between these two results. Another series ofproblems discussed in the proposal concerns the existence ofperiodic orbits for Hamiltonian systems of a special nature,including those describing the motion of a charge in a (strong)magnetic field or, more generally, the existence of periodicorbits near Morse-Bott non-degenerate symplectic extrema. Theseproblems are closely related to the investigation of the(relative) Hofer-Zehnder capacity function and the Floer homologyof certain Hamiltonians. The objective of the proposed researchin the area of Poisson topology is to study connections betweenthe geometry of Poisson structures and topology of underlyingmanifolds.Hamiltonian dynamical systems describe many classes of physicalprocesses in which dissipative forces can be neglected. Forexample, planetary motion in celestial mechanics and someelectro- or magneto-dynamical processes can be, and usually are,treated as Hamiltonian dynamical systems. One of the classicalsubjects in the theory of dynamical systems is the study ofperiodic orbits (i.e. cyclic motions). Periodic motion is thesimplest and most common type of motion after equilibrium. It isbelieved that a vast majority of Hamiltonian systems haveperiodic orbits and systems without such orbits have only beenrecently discovered. Yet, in all but simplest problems, findingperiodic orbits requires the use of advanced and powerfulmathematical methods. The investigation of periodic orbits liesat the very core of the modern theory of Hamiltonian dynamicalsystems. One of the main themes of the proposal is determininghow large the collection of periodic/aperiodic energy values canbe and showing that systems of a particular type carry periodicorbits of all energies. This class of systems includes thosedescribing the motion of a charge in a magnetic field and theproposed research has potential applications to physics andmathematical aspects of mechanics. The last part of the proposalconcerns the investigation of connections between geometrical andtopological properties of a certain class of spaces arising inthe study of systems with symmetries and in quantum mechanics.

摘要奖：DMS-0307484首席调查员：Viktor Ginzburg目前的建议集中在两个项目上，这两个项目与首席调查员以前的工作密切相关，由国家科学基金会拨款资助。这些项目是哈密顿动力系统周期轨道的存在问题和某些Poisson流形的拓扑性质的研究。Viktor Ginzburg在这个提议中解决的第一个问题是研究哈密顿系统没有周期轨道的规则能量值集合的大小。根据几乎存在定理，这个集合一定是零度量的，从哈密尔顿Seifert猜想的反例可以知道，这个集合可能是非空的。因此，问题是弥合这两个结果之间的差距。文中讨论的另一系列问题涉及特殊性质的哈密顿系统的周期轨道的存在，包括描述电荷在(强)磁场中的运动的问题，或者更一般地，在Morse-Bott非简并辛极值附近存在周期轨道的问题。这些问题与对(相对)Hofer-Zehnder容量函数和某些哈密顿量的Floer同调的研究密切相关。Poisson拓扑学研究的目的是研究Poisson结构的几何和下面流形的拓扑之间的联系。哈密顿动力系统描述了许多类可以忽略耗散力的物理过程。例如，天体力学中的行星运动和某些电磁或磁动力学过程可以而且通常被视为哈密顿动力系统。动力系统理论中的经典课题之一是周期轨道(即循环运动)的研究。周期运动是平衡后最简单、最常见的运动类型。人们认为，绝大多数哈密顿系统都有周期轨道，而没有周期轨道的系统是最近才发现的。然而，在除了最简单的问题之外的所有问题中，寻找周期轨道需要使用先进而强大的数学方法。周期轨道的研究是现代哈密顿动力系统理论的核心。该提案的主要主题之一是确定周期/非周期能量值的集合可以有多大，并表明某一特定类型的系统具有所有能量的周期轨道。这类系统包括描述电荷在磁场中运动的系统，所提出的研究在物理学和力学的数学方面具有潜在的应用。该建议的最后部分涉及到在具有对称性的系统的研究和量子力学中出现的一类空间的几何和拓扑性质之间的联系的研究。