  Cassini states correspond to the equilibria of the spin axis of a body when
its orbit is perturbed. They were initially described for planetary satellites,
but the spin axes of black hole binaries also present this kind of equilibria.
In previous works, Cassini states were reported as spin-orbit resonances, but
actually the spin of black hole binaries is in circulation and there is no
resonant motion. Here we provide a general description of the spin dynamics of
black hole binary systems based on a Hamiltonian formalism. In absence of
dissipation the problem is integrable and it is easy to identify all possible
trajectories for the spin for a given value of the total angular momentum. As
the system collapses due to radiation reaction, the Cassini states are shifted
to different positions, which modifies the dynamics around them. This is why
the final spin distribution may differ from the initial one. Our method
provides a simple way of predicting the distribution of the spin of black hole
binaries at the end of the inspiral phase.
