Symplectic Geometry and Dynamics

辛几何与动力学

• 批准号: 1104470
• 负责人: Helmut Hofer
• 金额: $ 21.01万
• 依托单位: Institute For Advanced Study
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-06-01 至 2015-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1104470&HistoricalAwards=false
• 关键词: Symplectic; Geometry; Dynamics

AbstractAward: DMS-1104470Principal Investigator: Helmut HoferThe first subproject will be concerned with the completion ofsymplectic field theory (SFT). This is currently the mostgeneral and most comprehensive theory of symplecticinvariants. There are three ingredients to this project, namelythe polyfold Fredholm theory, the scale-smooth analysis and theformulation of SFT as an algebraic presentation of the solutioncount of a polyfold Fredholm problem with operations. The secondsubproject, a by-product of the polyfold Fredholm theory,attempts to construct a homology theory based on equivalenceclasses of Fredholm problems. The resulting advantage would bethat transversality issues in studying nonlinear partialdifferential equations would actually not occur explicitly. Theywould be hidden in the (one-time) proof that the new homologytheory of a space would be naturally isomorphic to the rationalsingular homology. A third subproject is concerned with finiteenergy foliations, either in the context of area-preservingdisk-maps or the restricted planar three-body problem. Finiteenergy foliations were introduced by the PI and his collaboratorsand are an important tool in the study of the dynamics oflow-dimensional Hamiltonian systems. The restricted circularplanar three-body problem is not only interesting from a purelyacademic viewpoint, but is also used in the orbit design forscientific space missions. In certain regimes of the Jacobienergy, finite energy foliations allow to reduce the dynamics ofthe restricted three-body problem to that of the dynamics of anarea-preserving disk-map. Then again the same theory facilitatesthe understanding of the long-term behavior of iteratedarea-preserving disk maps. The project will study therelationship between the theory of finite energy foliations, asymplectic construction, and core dynamical systems notions likeentropy.It is a surprising fact that many physical systems evolving intime, allow a description as a Hamiltonian system. Systems ofthis kind describe the flow of an incompressible ideal fluid, themovement of a satellite under the gravitational forces ofcelestial bodies, or the movement of charged particles in amagnetic field. Hamiltonian systems are a very particular classof dynamical systems, which can be studied not only by themethods from dynamical systems theory, but also by a more exotickind of geometry called symplectic geometry. This is a geometrybased on the notion of area, in contrast to usual geometries,which have length and distance as their fundamentalnotions. Recent advances in this field already found somepractical uses. For example, this novel point of view has beenused in accelerator physics in algorithms controlling andinsuring stability of a beam of particles and called symplectictracking. It is the explicit purpose of this research tointegrate the approaches to Hamiltonian systems coming from thesetwo different perspectives. This should result in new methods,which should make it possible to attack problems, which so farseemed unreachable. For example it seems long term be feasible touse the new methods to develop algorithms, which would findfuel-efficient orbits for scientific space missions. Currenttechnology can verify the properties of proposed orbits extremelyfast. However, currently there is no good method for finding suchorbits. It rather compares to finding a needle in a haystack,using very large amounts of computing time.

AbstractAward：DMS-1104470主要研究者：Helmut Hofer第一个子项目将涉及辛场论（SFT）的完成。 这是目前最普遍和最全面的理论symplecticinvariants。这个项目有三个组成部分，即多重Fredholm理论，尺度光滑分析和SFT的制定作为一个代数表示的一个多重Fredholm问题的操作的解决方案计数。 第二个子项目是多重Fredholm理论的副产品，试图基于Fredholm问题的等价类构建同调理论。这样做的好处是，在研究非线性偏微分方程时，横截性问题实际上不会明确出现。它们将隐藏在（一次性）证明空间的新同调理论将自然同构于有理奇异同调中。 第三个子项目涉及有限能量叶理，无论是在面积保持磁盘映射还是限制性平面三体问题的背景下。能量叶理是由PI及其合作者提出的，是研究低维Hamilton系统动力学的重要工具。限制性圆平面三体问题不仅是一个纯学术性的问题，而且在空间科学任务的轨道设计中也有重要的应用。在雅可比能量的某些区域，有限能量叶理允许将限制性三体问题的动力学简化为面积保持圆盘映射的动力学。 同样的理论也有助于理解迭代面积保持圆盘映射的长期行为。该项目将研究有限能量叶状理论、辛结构和熵等核心动力系统概念之间的关系。令人惊讶的是，许多随时间演化的物理系统都允许描述为汉密尔顿系统。这类系统描述了不可压缩的理想流体的流动，卫星在天体引力作用下的运动，或者带电粒子在磁场中的运动。Hamilton系统是一类非常特殊的动力系统，它不仅可以用动力系统理论的方法来研究，而且还可以用一种更奇特的几何形式--辛几何来研究。这是一种基于面积概念的几何，与通常的几何不同，通常的几何以长度和距离为基本概念。这一领域的最新进展已经找到了一些实际用途。 例如，这种新颖的观点已经被用于加速器物理学中的控制和确保粒子束稳定性的算法中，称为辛波跟踪。 本研究的目的是将这两种不同的研究方法结合起来。这将产生新的方法，使人们有可能解决迄今为止似乎无法解决的问题。例如，从长远来看，使用新方法开发算法似乎是可行的，这将为科学太空任务找到燃料效率高的轨道。目前的技术可以非常快地验证所提出的轨道的性质。然而，目前还没有一个好的方法来寻找suchorbits。这相当于大海捞针，需要大量的计算时间。