Hamiltonian and Celestial Mechanics

项目摘要：哈密顿量和天体力学本项目致力于动力系统理论一般领域的研究，重点是哈密顿量和天体力学。这个主题的几个非常不同的方面将被研究。项目的一部分处理由中心构型引起的n体问题的特殊周期解。确定有多少个可能的中心构型是一个长期存在的开放性问题，甚至是在数量有限的情况下。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建隔离块。已知有可能在受限三体问题的共线拉格朗日点处找到简单、显式的隔离块。这些块可以用来研究附近的复杂不变量集。本课题就是在近共线中心构型的无限制三体问题上进行这样的构造。这里相空间的维数更高，几何结构也更复杂。项目的最后一部分是了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空的不变康托集的构造和随后对结果动力学的分析。在牛顿提出引力n体问题三个世纪后，它仍然是数学研究的一个活跃话题。多年来，它一直激励着广泛适用的新数学的发展。它是非线性机械系统的经典例子，它的解包括有序的周期运动、多体碰撞和不规则的混沌行为。最简单的解是由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，当构型旋转时，引力可以被离心力精确地平衡。尽管这些解在动力学上非常简单，但当涉及到四个或更多质量时，找到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，可能会用到计算机的帮助。中心结构是重要的标志，为进一步的分析提供了起点。事实证明，在简单的、刚性旋转的解附近，还有许多其他有趣的解。捕获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的不稳定。天体力学的不稳定性会引起行星或小行星轨道参数缓慢漂移等现象。它也发生在其他各种机械系统中。目前对阿诺德扩散现象的认识还不完全清楚。这里将采用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，还有许多工作要做。

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.