The Madelung equations are Euler equations for quantum mechanics. What is the Lagrangian description of quantum hydrodynamics? The Madelung equations of quantum mechanics suggest the hydrodynamic model of quantum mechanics (quantum mechanics is described as a fluid of universes in a multiverse, which, in the non-relativistic setting, is described by configuration space). However, the Madelung equations are Euler hydrodynamic equations, and Euler equations only describe bulk flow of a fluid. To get the full dynamics of a fluid, one can use a Lagrange description (among other solutions).
This suggests that the Madelung equations give an incomplete description of the hydrodynamic model of quantum mechanics, and a proper description of the hydrodynamic model would be (or be equivalent to) a Lagrange description.
Has anyone developed such a description?
(Note: I'm aware that the Bohmian mechanics an be derived from the Madelung equations by assuming that the universe fluid's particle velocity is equal to the bulk flow velocity. However, I've never seen any motivation for this assumption. I'd also be satisfied with a justification of why this works.)
 A: In the original Madelung  paper, as far as I know, the quantum mechanical potential depended only on the position of a given particle in the fluid, and was independent of other particles that might also be guided by the fluid.  The Bohm quantum mechanical potential depends in general on all the coordinates of all the particles, ie. on the full configuration space.  This difference has been pointed out by Sheldon Goldstein in some of his papers, for example. I think that this can be remedied though, and Madelung's theory modified in the following way.  If the particles in the Madelung fluid are solitons in the fluid, then each particle creates a disturbance in the velocity field of the fluid that can be felt by the other particles.  In this way they interact, the quantum mechanical potential might then depend on the full configuration space.  I don't know of any fluid model which exists that illustrates this idea though.  The bouncing drop models of Yves Coudet might yield such a configuration space dependent fluidic interaction, but how close it is to quantum theory I don't know, and the bouncing drop Lagrangians are quite different from the Madelung Eulerian models.
