Periodic orbits in the logarithmic potential

TL;DR

This paper develops analytic methods using perturbation theory and Hamiltonian normal forms to accurately approximate periodic orbits in the logarithmic potential, relevant for galactic dynamics.

Contribution

It introduces a resummation technique based on continued fractions to improve the convergence of series solutions for periodic orbits in galactic potentials.

Findings

Normal form truncation yields accurate solutions for normal modes.

Resummation enhances series convergence.

Method applicable to general position periodic orbits.

Abstract

Analytic methods to investigate periodic orbits in galactic potentials. To evaluate the quality of the approximation of periodic orbits in the logarithmic potential constructed using perturbation theory based on Hamiltonian normal forms. The solutions of the equations of motion corresponding to periodic orbits are obtained as series expansions computed by inverting the normalizing canonical transformation. To improve the convergence of the series a resummation based on a continued fraction may be performed. This method is analogous to that looking for approximate rational solutions (Prendergast method). It is shown that with a normal form truncated at the lowest order incorporating the relevant resonance it is possible to construct quite accurate solutions both for normal modes and periodic orbits in general position.