Automated Perturbation Theory for Hamiltonian Systems

哈密​​顿系统的自动摄动理论

• 批准号: 9712410
• 负责人: Katherine Yelick
• 金额: $ 9.72万
• 依托单位: University of California-Berkeley
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-09-01 至 2000-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9712410&HistoricalAwards=false
• 关键词: Automated; Perturbation; Theory; Hamiltonian; Systems

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.

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，通过光谱学识别分子，以及加速器中光束的稳定性。这些多维系统是不可积的，但扰动方法可以应用于将它们转换为具有较少自由度的近似。 大多数情况下，这种近似是由没有数学复杂性的基本程序来执行的，结果是计算变得巨大。这个项目的目标是建立一个工具箱来解决动力系统，特别是针对哈密顿系统的扰动方法。 该工具箱将足够灵活，可以有效地处理不同类别的系统，使研究人员能够探索新的问题，尝试新的转换，并验证已发表的结果。 作为实用性、灵活性和效率的证明，将为低高度的月球卫星开发一种分析理论。月球探测任务的计划，如1994年的克莱门汀使命，寻找适合侦察的轨道。 他们需要预测卫星的长期行为，以选择一个低偏心率（恒定的传感器高度）和固定的近心点（节省燃料的机动）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值选择可能的轨道，然后通过数值积分进行检查。如果没有一个长期动态的全局视图，使命规划者只能希望他们明智地采样。从分析模型中生成长期相空间的“地图”将消除猜测。这样的地图已经被证明在地球卫星的使命规划中是有用的。