ABSTRACT
To Madelung, just like to Schrödinger himself, the stationary Schrödinger equation was an eigenvalue equation. Likewise, the nonstationary equation was worth considering just as long as it provided this eigenvalue equation. In the beginning, though, the connection was not quite as direct as it seems today. To wit, the stationary equation was instrumental in guiding the reason towards a nonstationary equation, for both, Madelung as well as Schrödinger himself. The main difference in the two approaches is that Madelung assumes a Born-type of interpretation for the wave function in connection with density of a fluid, and this, just like the actual probabilistic interpretation of the wave function, leaves the phase of the wave function undecided. In view of the identification of the phase with the classical action, this lack of decision might not be quite out of line: after all, the classical action itself was always undecided in theoretical mechanics. However, the wave mechanics, with its quantum amendment, brings an important point to fore: a specific intervention of the geometry, in fact, of the differential geometry.
