  We derive and study stochastic dissipative dynamics on coadjoint orbits by
incorporating noise and dissipation into mechanical systems arising from the
theory of reduction by symmetry, including a semidirect-product extension.
Random attractors are found for this general class of systems when the Lie
algebra is semi- simple, provided the top Lyapunov exponent is positive. We
study two canonical examples, the free rigid body and the heavy top, whose
stochastic integrable reductions are found and numerical simulations of their
random attractors are shown.
