Ptolemaic Transformation in Keplerian Problem

Abstract

We propose the Ptolemaic transformation: a canonical change of variables reducing the Keplerian motion to the form of a perturbed Hamiltonian problem. As a solution of the unperturbed case, the Ptolemaic variables define an intermediary orbit, accurate up to the first power of eccentricity, like in the kinematic model of Claudius Ptolemy. In order to normalize the perturbed Hamiltonian we modify the recurrent Lie series algorithm of HoriuuMersman. The modified algorithm accounts for the loss of a term's order during the evaluation of a Poisson bracket, and thus can be also applied in resonance problems. The normalized Hamiltonian consists of a single Keplerian term; the mean Ptolemaic variables occur to be trivial, linear functions of the Delaunay actions and angles. The generator of the transformation may serve to expand various functions in Poisson series of eccentricity and mean anomaly.