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Geometric hydrodynamics via Madelung transform

Journal article
Authors B. Khesin
G. Misiolek
Klas Modin
Published in Proceedings of the National Academy of Sciences of the United States of America
Volume 115
Issue 24
Pages 6165-6170
ISSN 0027-8424
Publication year 2018
Published at Department of Mathematical Sciences
Pages 6165-6170
Language en
Links https://doi.org/10.1073/pnas.171934...
Keywords hydrodynamics, infinite-dimensional geometry, quantum information, Fisher-Rao, Newton's, quantum-mechanics, optimal transport, equation, fluid, diffeomorphisms, theorem, motion, moser, Science & Technology - Other Topics
Subject categories Geometry

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 partial differential equations of hydrodynamical origin can be described in this framework in a natural way. In particular, the Madelung transform between the Schrodinger equation and Newton's equations is a symplectomorphism of the corresponding phase spaces. Furthermore, the Madelung transform turns out to be a Kahler 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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