Automated Perturbation Theory for Hamiltonian Systems

动力学中的许多非线性问题都属于一般摄动理论的范畴。重要的例子包括卫星的姿态和轨道稳定性，用光谱学识别分子，以及加速器中光束的稳定性。这些多维系统是不可积的，但可以应用摄动方法将它们转换为具有更少自由度的近似。大多数情况下，这种近似是通过没有复杂数学知识的基本程序来实现的，其结果是计算变得非常庞大。这个项目的目标是建立一个求解动力系统的工具箱，特别是针对哈密顿系统的摄动方法。工具箱将足够灵活，可以有效地处理不同类型的系统，这将允许研究人员探索新的问题，实验新的转换，并验证已发表的结果。为了证明有用性、灵活性和效率，将为低空月球卫星发展一种分析理论。月球勘测任务的规划者，例如1994年的克莱门汀号任务，寻找适合侦察的轨道。他们需要预测卫星的长期行为，以便选择一个低偏心（为了保持传感器高度不变）和固定中心（为了节省机动燃料）的轨道。目前，轨道选择在很大程度上是一种猜测。规划者使用粗略的近似，选择可能的轨道，然后用数值积分来检验。如果没有长期动态的全球视角，任务规划者只能希望他们的抽样是明智的。从分析模型生成长期相位空间的“地图”将消除猜测。这种地图已被证明对地球卫星的任务规划很有用。

Many non-linear problems in dynamics fall under the general theory of perturbations. Important examples include the attitude and orbital stability of satellites, the identification of molecules by spectroscopy, and the stability of beams in accelerators. These multidimensional systems are not integrable, but perturbation methods can be applied to transform them to approximations with fewer degrees of freedom. Most of the time, such approximations are performed by elementary procedures with no mathematical sophistication, with the result that the calculations become enormous. The goal of this project is to build a toolbox for solving dynamical systems, specifically targeted to perturbation methods for Hamiltonian systems. The toolbox will be flexible enough to treat efficiently different classes of systems will allow researchers to explore new problems, experiment with new transformations, and verify published results. As a demonstration of usefulness, flexibility and efficiency, an analytical theory will be developed for lunar satellites at low altitude. Planners for lunar survey missions, such as the Clementine mission in 1994, seek orbits appropriate for reconnaissance. They need to predict the long-term behavior of a satellite in order to select an orbit with low eccentricity (for constant sensor altitude) and fixed pericenter (to economize fuel in maneuvers). At present, orbit selection is largely a matter of guesswork. Using crude approximations, planners choose possible orbits which are then checked by numerical integration. Without a global view of the long-term dynamics, mission planners can only hope they sampled wisely. Generating ``maps'' of long-term phase space from an analytical model will remove the guesswork. Such maps have already proved useful in mission planning for earth satellites.