Hamiltonian and Celestial Mechanics

主要研究者：理查德Moeckel，明尼苏达大学双城DMS-0200992项目摘要：哈密尔顿和天体力学这个项目致力于研究一般领域的动力系统理论，重点是哈密尔顿和天体力学。 将研究这个问题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。 这是一个长期的开放性问题，以确定有多少中心配置是可能的，甚至如果数量是有限的。 从代数几何和计算代数的技术将被用来攻击这个问题。 该项目的第二部分涉及在三体问题的相空间中建造隔离块。 众所周知，在限制性三体问题的共线拉格朗日点上，可以找到简单的、明确的隔离块。 这些块可以用来研究附近复杂的不变集。 该项目是进行这样的建设附近的共线中心配置的无限制三体问题。 这里相空间的维数更高，几何形状也复杂得多。 该项目的最后一部分是关于了解哈密顿系统的阿诺德扩散机制。这里所采用的方法是基于一个不变的康托环集的建设和随后的动力学分析。引力n-body问题仍然是一个活跃的主题，数学研究的三个世纪后，牛顿提出了它。多年来，它一直是一个刺激的发展新的数学的广泛适用性。 它是非线性力学系统的经典例子，其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布置，使得当构型旋转时，重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，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非常复杂的代数问题，也许可以使用计算机的帮助。 中心构型是重要的标志，为进一步分析提供了起点。 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，也可以形成用于近似这些解的数值方法的基础。最后，还有一个问题是如何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性。 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。 它也发生在各种其他机械系统中。 阿诺德扩散的这种现象目前只被部分理解。 新的方法，将在这里追求似乎有前途，但仍有许多工作要做，才能将其应用于复杂的系统，如n体问题。

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.