Normal forms for the epicyclic approximations of the perturbed Kepler problem

TL;DR

This paper derives normal forms for Hamiltonians related to epicyclic approximations of the perturbed Kepler problem, facilitating simplified solutions in the Hill synodic system with potential applications to spatial and non-autonomous cases.

Contribution

It provides a systematic method to compute normal forms for the perturbed Kepler problem, extending epicyclic approximations within a Hamiltonian framework.

Findings

Normal forms match Keplerian solutions in unperturbed cases

Perturbed solutions are obtained as perturbation series

Method extends to spatial and non-autonomous problems

Abstract

We compute the normal forms for the Hamiltonian leading to the epicyclic approximations of the (perturbed) Kepler problem in the plane. The Hamiltonian setting corresponds to the dynamics in the Hill synodic system where, by means of the tidal expansion of the potential, the equations of motion take the form of perturbed harmonic oscillators in a rotating frame. In the unperturbed, purely Keplerian case, the post-epicyclic solutions produced with the normal form coincide with those obtained from the expansion of the solution of the Kepler equation. In all cases where the perturbed problem can be cast in autonomous form, the solution is easily obtained as a perturbation series. The generalization to the spatial problem and/or the non-autonomous case is straightforward.