Geometric Hydrodynamics via Madelung Transform

31 Oct 2018 · Khesin Boris, Misiolek Gerard, Modin Klas ·

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.

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Geometric Hydrodynamics via Madelung Transform

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