Periodic Orbits of Hamiltonian Systems, Cobordisms and Geometric Quantization, and Poisson Geometry

DMS-0072202 Viktor L.Ginzburg本提案侧重于几个长期项目，并继续首席研究员由NSF拨款资助的先前工作。第一个问题是哈密顿Seifert猜想，或者更具体地说，在正则能级序列上没有周期轨道的哈密顿动力系统的存在性问题。哈密尔顿塞弗特猜想与提案中考虑的下一组问题密切相关。这些问题存在于辛拓扑领域，涉及描述电荷在磁场中运动的哈密顿系统的周期轨道的存在性。提案的第二部分包括泊松几何中的一系列问题。在这些问题中，例如，等变Poisson矩映射的存在性问题和对应于辛叶的叶的Poisson迹的构造。应用等变余弦论来研究紧群的哈密顿作用的一般程序是该提议的结论部分的主题。哈密顿动力系统描述了许多类可以忽略能量耗散的物理过程。例如，天体力学中的行星运动和一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。周期运动是继平衡之后最简单、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。第一个问题是构造无周期轨道的哈密顿系统。这对于哈密顿动力系统理论来说是一个相当重要的问题，因为这类系统的例子将进一步促进我们对哈密顿动力学的理解。第二个问题涉及描述电荷在磁场中运动的哈密顿系统的周期轨道的存在性。这类哈密顿系统在物理和力学中的应用是自然而然的。然而，最近在辛几何中发展起来的非常强大的通用方法中，很少有方法适用于这类系统。对这些系统的研究应该扩展现有方法的范围，并导致新方法的发展。提案中考虑的其他问题涉及研究经典力学系统的几何性质与某些量子力学现象之间的联系。

DMS-0072202Viktor L. GinzburgThe present proposal focuses on several long-term projectsand continues principal investigator's previous work funded by an NSF grant. The first question addressed in the proposalis the Hamiltonian Seifert conjecture or, more specifically, the existence problem for Hamiltonian dynamical systems without periodic orbits on a sequence of regular energy levels. The Hamiltonian Seifert conjecture is closely related to the next group of questions considered in the proposal. These questions lie in the area of symplectic topology and concern the existence of periodic orbits for Hamiltonian systems describing the motion of a charge in a magnetic field. The second part of the proposal includes a series of problems in Poisson geometry. Among these problems are, for example, the existence questions for equivariant Poisson moment maps and the construction of Poisson traces corresponding to the leaves of the symplectic foliation. A general program relying on applications of equivariant cobordisms to the study of Hamiltonian actions of compact groups is the subject of the concluding part of the proposal.Hamiltonian dynamical systems describe many classes of physical processes in which dissipation of energy 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 in the theory of dynamical systems is the study of periodic orbits (i.e. cyclic motions). Periodic motion is the simplest and most common type of motion after equilibrium.It is believed that a vast majority of Hamiltonian systems have periodic orbits. The first problem addressed in theproposal is the construction of Hamiltonian systems without periodic orbits. This is a question of considerable importance for the theory of Hamiltonian dynamical systems because examples of such systems would further advance our understanding of Hamiltonian dynamics. The next problem concerns the existence of periodic orbits for Hamiltonian systems describing the motion of a charge in a magnetic field. This class of Hamiltonian systems naturally arises in applications in physics and mechanics. However, few of the extremely powerful general methods that have been recently developed in symplectic geometry are applicable to this class of systems. The investigation of these systems should extend the limits of existing methods and result in the development of novel ones. Other problems considered in the proposal concern the study of connections betweengeometrical properties of classical-mechanical systems and certain quantum-mechanical phenomena.