Automated Perturbation Theory for Hamiltonian Systems

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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.