Nearly Axisymmetric Systems

近轴对称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，Paul陷阱中的离子运动以及小行星卫星的复杂轨道。 在所有这些例子中的运动是很好地描述了一个自治的哈密顿与近似保守的正则动量，其中存在一个平均的有效势。 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。 这种效应对于长时间限制轨道具有重要影响，例如保罗阱中的离子对。 在这项研究中，我们将开发新的分析方法来描述这种过渡从二维到三维的行为，使用正则微扰理论，莫尔斯理论，和伯克霍夫规范形式。 临界点的全局分叉和Arnold扩散也将被研究。 这些结果预计将对正在积极研究的几个领域产生直接影响，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带电尘埃颗粒的限制，轴对称系统具有关于轴的圆对称性，绕地球地球同步轨道的通信卫星就是一个例子。 实际上，地球在经度方向上与球形有很小的偏差，这使得地球同步卫星在天空中徘徊，而不是直接在头顶上，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准二维行为的近轴对称系统。 本计画将研究这种近似轴对称系统的详细动力学。 由此产生的增加定量的理解将大大有助于设计，数值模拟和操作这样的系统。 其他（更复杂）的例子包括离子阱中的双离子运动，通常被构造成略椭圆形而不是圆形，以及最近在许多小行星周围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.