On bi-Hamiltonian formulation of the perturbed Kepler problem

TL;DR

This paper demonstrates that the perturbed Kepler problem is a bi-Hamiltonian system despite not meeting the Fernandes theorem conditions, due to its isochronous nature.

Contribution

It reveals the bi-Hamiltonian structure of the perturbed Kepler problem, challenging existing theorems that link hypersurface conditions to bi-Hamiltonian systems.

Findings

Both initial and perturbed Kepler systems are isochronous.

The Fernandes theorem does not apply to these systems.

The perturbed Kepler problem admits a bi-Hamiltonian formulation.

Abstract

The perturbed Kepler problem is shown to be a bi-Hamiltonian system in spite of the fact that the graph of the Hamilton function is not a hypersurface of translation, which is against a necessary condition for the existence of the bi-Hamiltonian structure according to the Fernandes theorem. In fact, both the initial and perturbed Kepler systems are isochronous systems and, therefore, the Fernandes theorem cannot be applied to them.