Symplectic mechanics of relativistic spinning compact bodies: Canonical formalism in the Schwarzschild spacetime

This work constitutes the second part of a series of studies that aim to utilize tools from Hamiltonian mechanics to investigate the motion of an extended body in general relativity. The first part of this work [On the integrability of extended test body dynamics around black holes, Classical Quantum Gravity.] established the ten-dimensional, covariant Hamiltonian framework, incorporating all quadratic-in-spin corrections from the secondary spin to the geodesic motion in arbitrary spacetime. Notably, this framework yields an integrable Hamiltonian system (up to quadratic terms in spin) when both the background spacetime and secondary compact objects are rotating Kerr black holes. 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 formulas 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.