The Euler equations for a flow which preserves the Gaussian measure on Euclidean space can be translated in terms of Gaussian random variables, which raises the question about an analogue in free probability. We derive these free Euler equations by applying the approach of Arnold for Euler equations to a Lie algebra of infinitesimal automorphisms of the von Neumann algebra of a free group. We then extend the equations to noncommutative vector fields satisfying only certain weaker noncommutative smoothness conditions. We also introduce a cyclic vorticity and show that it satisfies appropriate vorticity equations and that it gives rise to a family of conserved quantities.