Application of Lie-Series to Numerical Integration in Celestial Mechanics

Abstract

W. Gröbner gave in his book “Lie-series and Their Applications” the definition of this series. With a linear differential operator D we can evaluate the terms of a convergent series. For testing, this method is applied to the two body problem. It could be shown that the evaluation of the first five terms are sufficient to get good results. Then, Lie series were used to integrate a restricted three body system, a three body system, and a restricted four body system. Because the terms are very complex, much computing time is required. The advantage of solving these problems with Lie series lies in the fact that the same procedure may be used for solving non-conservative systems; for example, a two body system with variable masses. For testing, the method of Lie series is also applied to solve non-autonomous systems.