Closed-form normalizations of perturbed two-body problems

We study small perturbations of two- and three-dimensional Kepler problems through analytical techniques. Based on Lie transformation procedures, the normal form of the initial system is calculated up to a certain order of approximation. After truncation the transformation introduces a symmetry; therefore the number of degrees of freedom of the original system is reduced by one. The normalization process leads in all the cases to the calculation of primitives and definite integrals of some functions with respect to one of the variables of the problem. The corresponding integrals are computed in closed form to make the theory valid for all types of elliptic trajectories. This is achieved by using two special functions: a generalization of the (complex) generalized incomplete gamma function and the polylogarithm of complex argument. We apply the method in three situations: (i) the simplification of a Hamiltonian system modelling the trapping mechanisms for the electron of an ionized hydrogen atom; (ii) the closed-form calculation of some integrals appearing in the analysis of strong and resonant perturbations in gravity fields; and (iii) the normal form computations of an artificial satellite orbiting the Earth at low altitude.