  For the free probability analogue of Euclidean space endowed with the
Gaussian measure we apply the approach of Arnold to derive Euler equations for
a Lie algebra of non-commutative vector fields which preserve a certain trace.
We extend the equations to vector fields satisfying non-commutative smoothness
requirements. We introduce a cyclic vorticity and show that it satisfies a
vorticity equation and that it produces a family of conserved quantities.
