Homoclinic orbits around spinning black holes. II. The phase space portrait

In paper I in this series, we found exact expressions for the equatorial homoclinic orbits: the separatrix between bound and plunging, whirling and not whirling motion. As a companion to that physical space study, in this paper we paint a phase space portrait of the homoclinic orbits that includes exact expressions for the actions and fundamental frequencies. Additionally, we develop a reduced Hamiltonian description of Kerr motion that allows us to track groups of trajectories with a single global clock. This facilitates a variational analysis, whose stability exponents and eigenvectors could potentially be useful for future studies of families of black hole orbits and their associated gravitational waveforms.