Geometric Hydrodynamics via Madelung Transform

TL;DR

This paper develops a geometric framework linking hydrodynamical PDEs, Schrödinger equations, and probability densities through the Madelung transform, revealing symplectic and Kähler structures in infinite-dimensional spaces.

Contribution

It introduces a geometric approach unifying hydrodynamical PDEs and quantum mechanics via the Madelung transform, highlighting symplectic and Kähler properties.

Findings

Madelung transform is a symplectomorphism between Schrödinger and Newton's equations.

The transform is a Kähler map with the Fisher-Rao metric on densities.

Several dynamical applications demonstrate the framework's utility.

Abstract

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.