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

AbstractAward：DMS-0307484首席研究员：Viktor Ginzburg目前的建议集中在两个项目密切相关的首席研究员的以前的工作由NSF资助。这两个项目是Hamilton动力系统的非线性轨道的存在性问题和某些Poisson流形的拓扑性质的研究。第一个问题Viktor Ginzburg地址在这个建议是调查的大小的一套正规的能量值上的哈密尔顿系统没有周期轨道。根据几乎存在定理，这个集合必须是零测度的，并且从Hamiltonian Seifert猜想的反例中可知，这个集合可能是非空的。因此，问题是要弥合这两个结果之间的差距。 另一系列ofproblems中讨论的建议关注的存在ofperiodic轨道的哈密顿系统的一个特殊性质，包括那些描述运动的电荷在一个（强）磁场，或更一般地说，存在的轨道附近Morse-Bott非退化辛极值。这些问题与某些Hamilton算子的（相对）Hofer-Zehnder容量函数和Floer同调的研究密切相关。Poisson拓扑领域的研究目标是研究Poisson结构的几何和基础流形的拓扑之间的联系，Hamilton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动力学或磁动力学过程可以而且通常被视为哈密顿动力学系统。动力系统理论的经典课题之一是研究周期轨道（即循环运动）。周期运动是平衡后最简单也是最常见的运动形式。 人们相信，绝大多数的哈密顿系统具有周期轨道和系统没有这样的轨道只是最近才被发现。然而，除了最简单的问题，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。该提案的主题之一是确定周期性/非周期性能量值的集合可以有多大，并表明特定类型的系统携带所有能量的轨道。这类系统包括那些描述磁场中电荷运动的系统，并且所提出的研究在力学的物理和数学方面具有潜在的应用。 最后一部分的proposalconcerns的调查之间的联系几何和拓扑性质的某一类空间所产生的研究系统的对称性和量子力学。

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.