Nearly Axisymmetric Systems

近轴对称系统

• 批准号: 0209856
• 负责人: James Howard
• 金额: $ 9.4万
• 依托单位: University of Colorado at Boulder
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-09-01 至 2006-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0209856&HistoricalAwards=false
• 关键词: Nearly; Axisymmetric; Systems

Nearly axisymmetric systems occur in many physical problems, including the motion of charged dust grains in planetary magnetospheres, plasma fusion devices, ion motion in Paul traps, and the complex orbits of asteroidal moons. In all these examples the motion is well described by an autonomous Hamiltonian with an approximately conserved canonical momentum, for which an averaged effective potential exists. For small deviations from axisymmetry the motion remains essentially two-dimensional, simply averaging over nonaxisymmetric perturbations. This effect has major consequences for the long time confinement of orbits, such as pairs of ions in a Paul trap. In this research we will develop new analytic methods for describing this transition from two- to three-dimensional behavior using canonical perturbation theory, Morse theory, and Birkhoff normal forms. Global bifurcations of critical points and Arnold diffusion will also be investigated. The results are expected to have immediate impact on several fields under active investigation, such as high-performance computers utilizing quantum computation and the confinement of charged dust grains under the influence of solar radiation pressure.Axisymmetric systems have circular symmetry about an axis; a communication satellite in geosynchronous orbit about Earth is an example. In reality the Earth has small longitudinal deviations from sphericity, causing a geosynchronous satellite to wander in the sky rather than remain directly overhead, unless it is carefully placed at a stable longitude. Nevertheless, its orbit remains very nearly circular. This simple example illustrates a nearly axisymmetric system with quasi-two-dimensional behavior. This project investigates the detailed dynamics of such nearly axisymmetric systems. The resulting increased quantitative understanding will greatly aid in design, numerical simulation, and operation of such systems. Other (much more complex) examples include two-ion motion in ion traps, which are often constructed to be slightly elliptical rather than circular, and the orbits of small moonlets that have been recently discovered about many asteroids.

近轴对称系统存在于许多物理问题中，包括行星磁层中带电尘埃颗粒的运动，等离子体聚变装置，Paul陷阱中的离子运动，以及小行星卫星的复杂轨道。在所有这些例子中，运动都可以用一个具有近似守恒的正则动量的自治哈密顿量来很好地描述，对于这个正则动量，它存在一个平均有效势。对于较小的轴对称偏差，运动本质上保持二维，简单地平均非轴对称扰动。这一效应对轨道的长时间限制有重大影响，例如保罗陷阱中的离子对。在这项研究中，我们将利用正则微扰理论、Morse理论和Birkhoff范式开发新的分析方法来描述这种从二维行为到三维行为的转变。还将研究临界点的全局分叉和Arnold扩散。预计这些结果将对正在积极研究的几个领域产生立竿见影的影响，例如利用量子计算的高性能计算机和在太阳辐射压力影响下对带电尘埃颗粒的限制。轴对称系统围绕轴具有圆形对称性；地球同步轨道上的通信卫星就是一个例子。事实上，地球与球体的纵向偏差很小，这导致地球同步卫星在天空中徘徊，而不是直接停留在头顶，除非小心地将其放置在稳定的经度上。尽管如此，它的轨道仍然非常接近圆形。这个简单的例子说明了一个近轴对称系统具有准二维行为。这个项目研究了这类近轴对称系统的详细动力学。由此增加的定量理解将极大地有助于此类系统的设计、数值模拟和操作。其他(复杂得多)的例子包括离子陷阱中的两个离子运动，这些离子陷阱通常被构建为略微椭圆形而不是圆形，以及最近在许多小行星上发现的小卫星的轨道。