Lagrangian variational principle in stochastic mechanics: gauge structure and stability

The Lagrangian variational principle with the classical action leads, in stochastic mechanics, to Madelung’s fluid equations, if only irrotational velocity fields are allowed, while new dynamical equations arise if rotational velocity fields are also taken into account. The new equations are shown to be equivalent to the (gauge invariant) system of a Schrödinger equation involving a four‐vector potential (A,Φ) and the coupled evolution equation (of magnetohydrodynamical type) for the vector field A. A general energy theorem can be proved and the stability properties of irrotational and rotational solutions investigated.