Extension of the Lie Transform Theory Depending on a Small Parameter for Multi-parametric Dynamical Systems

Lie transform method derived by Deprit and Hori in the 1960s allowed researchers to solve the perturbation problems depending on a small parameter. But actually the real systems are very complicated. In the astrodynamics, one usually encounters problems involving several perturbations which in turn yields dynamical system with several small parameters, e.g., oblateness of the massive objects, radiation pressure, mass loss, relativistic effects, drag perturbations, etc. To involve as many perturbations as the system requires, the theory of canonical Lie transform depending on a small parameter is extended to N-small parameters. Lie transform based on one small parameter is briefly surveyed. Some useful lemmas are proved. Then, the generalized Lie transformation method is developed.