  We introduce new Langevin-type equations describing the rotational and
translational motion of rigid bodies interacting through conservative and
non-conservative forces, and hydrodynamic coupling. In the absence of
non-conservative forces the Langevin-type equations sample from the canonical
ensemble. The rotational degrees of freedom are described using quaternions,
the lengths of which are exactly preserved by the stochastic dynamics. For the
proposed Langevin-type equations, we construct a weak 2nd order geometric
integrator which preserves the main geometric features of the continuous
dynamics. The integrator uses Verlet-type splitting for the deterministic part
of Langevin equations appropriately combined with an exactly integrated
Ornstein-Uhlenbeck process. Numerical experiments are presented to illustrate
both the new Langevin model and the numerical method for it, as well as to
demonstrate how inertia and the coupling of rotational and translational motion
can introduce qualitatively distinct behaviours.
