Chaos in the Kepler problem with quadrupole perturbations

TL;DR

This paper proves that the Kepler problem with quadrupole perturbations exhibits chaos in the Hamiltonian flow, clarifying conflicting numerical results in the physical literature.

Contribution

It provides a rigorous mathematical proof of chaos in the perturbed Kepler problem using the Melnikov integral method.

Findings

Chaotic dynamics occur in the zero-energy manifold for quadrupole-perturbed Kepler problem.

Chaos is proven for both prolate and oblate quadrupole perturbations.

The results reconcile previous conflicting numerical studies.

Abstract

We use the Melnikov integral method to prove that the Hamiltonian flow on the zero-energy manifold for the Kepler problem perturbed by a quadrupole moment is chaotic, irrespective of the perturbation being of prolate or oblate type. This result helps to elucidate some recent conflicting works in the physical literature based on numerical simulations.