Geometric Hydrodynamics via Madelung Transform

We introduce a geometric framework to study Newton's equations on infinite-dimensional configuration spaces of diffeomorphisms and smooth probability densities. It turns out that several important PDEs of hydrodynamical origin can be described in this framework in a natural way. In particular, the Madelung transform between the Schr\"odinger equation and Newton's equations is a symplectomorphism of the corresponding phase spaces. Furthermore, the Madelung transform turns out to be a K\"ahler map when the space of densities is equipped with the Fisher-Rao information metric. We describe several dynamical applications of these results.