Hamiltonian and Celestial Mechanics

哈密​​顿量和天体力学

• 批准号: 0200992
• 负责人: Richard Moeckel
• 金额: $ 11.22万
• 依托单位: University of Minnesota-Twin Cities
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-06-01 至 2006-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0200992&HistoricalAwards=false
• 关键词: Hamiltonian; Celestial; Mechanics

PI: Richard Moeckel, University of Minnesota - Twin CitiesDMS-0200992Project Abstract:Hamiltonian and Celestial MechanicsThis project is devoted to research in the general area of dynamical systems theory with emphasis on Hamiltonian and celestial mechanics. Several very different aspects of the subject will be studied. Part of the project deals with the special periodic solutions of the n-body problem arising from central configurations. It is a long-standing open problem to determine how many central configurations are possible, or even if the number is finite. Techniques from algebraic geometry and computational algebra will be used to attack this question. A second part of the project involves the construction of isolating blocks in the phase space of the three-body problem. It is known that it is possible to find simple, explicit isolating blocks at the collinear Lagrange points of the restricted three-body problem. These blocks can be used to study the complicated invariant set nearby. The project is to carry out such a construction near the collinear central configuration of the unrestricted three-body problem. Here the dimension of the phase space is higher and the geometry is much more complicated. A final part of the project is concerned with understanding the mechanism of Arnold diffusion for Hamiltonian systems. The approach taken here is based on the construction of an invariant Cantor set of annuli and subsequent analysis of the resulting dynamics.The gravitational n-body problem remains an active topic for mathematics research three centuries after Newton proposed it. Over the years it has been a stimulus for the development of new mathematics of wide applicability. It is the classic example of a nonlinear mechanical system and its solutions include orderly cyclical motions, multi-body collisions, and irregular, chaotic behavior. The simplest solutions are the rigidly rotating orbits arising from the central configurations. Central configurations are special arrangements of the masses such that the gravitational forces can be exactly balanced by centrifugal forces when the configuration rotates. Although these solutions are dynamically very simple, the problem of finding or even counting the central configurations turns out to be very difficult when there are four or more masses involved. Part of this project is about how to deal with very complicated algebraic problems such as this, perhaps using the help of computers. The central configurations are important landmarks which provide a starting point for further analysis. It turns out that there are many other interesting solutions near the simple, rigidly rotating ones. One way to trap and study these nearby orbits involves the construction of so-called isolating blocks. The geometry of these blocks provides qualitative information about the solutions inside and can also form the basis of numerical methods for approximating these solutions. Finally, there is the problem of understanding chaotic dynamics in mechanical systems and how such behavior can lead to large-scale instability. Instability in celestial mechanics can give rise to such phenomena as the slow drifting of the orbital parameters of planets or asteroids. It also occurs in a variety of other mechanical systems. This phenomenon of Arnold diffusion is only partially understood at present. The new approach which will be pursued here seems promising but much work remains to be done before it can be applied to complex systems like the n-body problem.

Pi：Richard Moeckel，明尼苏达大学双子城DMS-0200992项目摘要：哈密顿和天体力学这个项目致力于动力系统理论的一般领域的研究，重点是哈密顿和天体力学。我们将研究这门学科的几个非常不同的方面。该项目的一部分涉及由中心构型引起的n体问题的特殊周期解。确定有多少中央构型是可能的，或者即使这个数字是有限的，这是一个长期悬而未决的问题。代数几何和计算代数的技巧将被用来解决这个问题。该项目的第二部分涉及在三体问题的相空间中构造隔离块。众所周知，在限制性三体问题的共线拉格朗日点处可以找到简单、显式的隔离块。这些块可以用来研究附近的复杂不变集。该项目就是在无约束三体问题的共线中心构型附近进行这样的构造。这里相空间的维度较高，几何形状复杂得多。该项目的最后一部分是关于理解哈密顿系统的阿诺德扩散机制。这里采用的方法是基于环的不变康托集的构造和随后的动力学分析。自牛顿提出引力n体问题三个世纪以来，它仍然是数学研究的活跃主题。多年来，它一直是广泛适用的新数学发展的推动力。它是非线性力学系统的经典例子，它的解包括有序的循环运动、多体碰撞和不规则的混沌行为。最简单的解是中心构型产生的刚性旋转轨道。中心构型是质量的特殊排列，当构形旋转时，引力可以被离心力精确地平衡。虽然这些解在动力学上非常简单，但当涉及四个或更多质量时，寻找甚至计算中心构型的问题变得非常困难。这个项目的一部分是关于如何处理像这样的非常复杂的代数问题，也许是利用计算机的帮助。中心配置是重要的里程碑，为进一步分析提供了起点。事实证明，在简单的、严格旋转的解决方案附近，还有许多其他有趣的解决方案。捕捉和研究这些附近轨道的一种方法是建造所谓的隔离块。这些块的几何形状提供了关于内部解的定性信息，也可以形成近似这些解的数值方法的基础。最后，还有一个问题是理解机械系统中的混沌动力学，以及这种行为如何导致大规模不稳定。天体力学中的不稳定性会导致行星或小行星轨道参数缓慢漂移等现象。它也出现在各种其他机械系统中。目前，人们对阿诺德扩散现象的了解还很有限。将在这里推行的新方法似乎很有希望，但在将其应用于像n体问题这样的复杂系统之前，仍有许多工作要做。