  This work constitutes the second part of a series of studies that aim to
utilise tools from Hamiltonian mechanics to investigate the motion of an
extended body in general relativity. The first part of this work [Refs. [1, 2]]
constructed a ten-dimensional, covariant Hamiltonian framework encompassing all
the linear-in-spin corrections to the geodesic motion in arbitrary spacetime.
This framework was proven to be integrable in the Schwarzschild and Kerr
spacetimes, specifically. The present work translates this abstract
integrability result into tangible applications for linear-in-spin Hamiltonian
dynamics of a compact object in a Schwarzschild spacetime. In particular, a
canonical system of coordinates is constructed explicitly, which exploits the
spherical symmetry of the Schwarzschild spacetime. These coordinates are based
on a relativistic generalization of the classical Andoyer variables of
Newtonian rigid body motion. This canonical setup allows us to derive
ready-to-use formulae for action-angle coordinates and gauge-invariant
Hamiltonian frequencies, which automatically include all linear-in-spin
effects. No external parameters or ad hoc choices are necessary, and the
framework can be used to find complete solutions by quadrature of generic
(bound or unbound), linear-in-spin orbits, including orbital inclination,
precession and eccentricity, as well as spin precession. The efficacy of the
formalism is demonstrated here in the context of circular orbits with arbitrary
spin and orbital precession, with the results validated against known results
in the literature.
