Lie-series for orbital elements -- II. The spatial case

If one has to attain high accuracy over long timescales during the numerical computation of the N-body problem, the method called Lie-integration is one of the most effective algorithms. In this paper we present a set of recurrence relations with which the coefficients needed by the Lie-integration of the orbital elements related to the spatial N-body problem can be derived up to arbitrary order. Similarly to the planar case, these formulae yields identically zero series in the case of no perturbations. In addition, the derivation of the formulae has two stages, analogously to the planar problem. Namely, the formulae are obtained to the first order, and then, higher order relations are expanded by involving directly the multilinear and fractional properties of the Lie-operator.