On the integrability of spinning-body dynamics around black holes

In general relativity, the trajectory of a celestial body in a given spacetime is influenced by its proper rotation, or \textit{spin}. We present a covariant and physically self-consistent Hamiltonian framework to study this motion, at linear order in the body's spin and in an arbitrary fixed spacetime. The choice of center-of-mass and degeneracies coming from Lorentz invariance are treated rigorously with adapted tools from Hamiltonian mechanics. Applying the formalism to a background space-time described by the Kerr metric, we prove that the motion of a spinning body around a generic rotating black hole is an \textit{integrable} Hamiltonian system. In particular, linear-in-spin effects do not break the integrability of Kerr geodesics, and induce no \textit{chaos} within the associated phase space. Our findings suggest a natural way to improve current gravitational waveform modelling for asymmetric binary systems, and provide a mean to extend classical features of Kerr geodesics to linear-in-spin trajectories.