Nearly Axisymmetric Systems

几乎轴对称的系统出现在许多物理问题中，包括行星磁层中带电尘埃颗粒的运动，等离子体聚变装置，保罗陷阱中的离子运动，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正则动量的自治哈密顿量来很好地描述，并且存在一个平均有效势。对于轴对称的小偏差，运动基本上保持二维，简单地对非轴对称扰动进行平均。这种效应对轨道的长时间限制有重要影响，比如保罗阱中的离子对。在这项研究中，我们将开发新的解析方法来描述这种从二维到三维行为的转变，使用正则摄动理论，莫尔斯理论和伯克霍夫范式。临界点的全局分岔和阿诺德扩散也将被研究。预计这一结果将对利用量子计算的高性能计算机和太阳辐射压力影响下带电尘埃颗粒的限制等几个正在积极研究的领域产生直接影响。轴对称系统围绕一个轴具有圆对称；地球同步轨道上的通信卫星就是一个例子。实际上，地球在纵向上与球形有很小的偏差，这导致地球同步卫星在天空中徘徊，而不是保持在头顶，除非它被小心地放置在一个稳定的经度上。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具有准二维特性的近轴对称系统。本项目研究了这种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帮助这类系统的设计、数值模拟和操作。其他（复杂得多的）例子包括离子阱中的双离子运动，它们通常被构造成略呈椭圆形而不是圆形，以及最近在许多小行星上发现的小卫星的轨道。

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.