  We consider controllability for divergence-free systems that have a conserved
quantity and satisfy a H\"ormander condition. It is shown that such systems are
controllable, provided that the conserved quantity is a proper function. The
proof of the result combines analytic tools with probabilistic arguments. While
this statement is well-known in geometric control theory, the probabilistic
proof given in this note seems to be new. We show that controllability follows
from H\"ormander's condition, together with the a priori knowledge of an
invariant measure with full topological support for a diffusion that
`implements' the control system.
  Examples are given that illustrate the relevance of the assumptions required
for the result to hold. Applications of the result to ergodicity questions for
systems arising from non-equilibrium statistical mechanics and to the
controllability of Galerkin approximations to the Euler equations are also
given.
