Lie-series for orbital elements -- I. The planar case

Lie-integration is one of the most efficient algorithms for numerical integration of ordinary differential equations if high precision is needed for longer terms. The method is based on the computation of the Taylor-coefficients of the solution as a set of recurrence relations. In this paper we present these recurrence formulae for orbital elements and other integrals of motion for the planar $N$-body problem. We show that if the reference frame is fixed to one of the bodies -- for instance to the Sun in the case of the Solar System --, the higher order coefficients for all orbital elements and integrals of motion depend only on the mutual terms corresponding to the orbiting bodies.