We describe 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 so-called Madelung transform between the Schr¨odinger-type equations on wave functions and Newton’s equations on densities turns out to be a K¨ahler map between the corresponding phase spaces, equipped with the Fubini-Study and Fisher-Rao information metrics. This is a joint work with G. Misiolek and K. Modin.