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Geometry of the Madelung Transform

Journal article
Authors B. Khesin
G. Misiolek
Klas Modin
Published in Archive for Rational Mechanics and Analysis
Volume 234
Issue 2
Pages 549-573
ISSN 0003-9527
Publication year 2019
Published at Department of Mathematical Sciences
Pages 549-573
Language en
Links dx.doi.org/10.1007/s00205-019-01397...
Keywords quantum-mechanics, Mathematics, Mechanics
Subject categories Mathematics

Abstract

The Madelung transform is known to relate Schrodinger-type equations in quantum mechanics and the Euler equations for barotropic-type fluids. We prove that, more generally, the Madelung transform is a Kahler map (that is a symplectomorphism and an isometry) between the space of wave functions and the cotangent bundle to the density space equipped with the Fubini-Study metric and the Fisher-Rao information metric, respectively. We also show that Fusca's momentum map property of the Madelung transform is a manifestation of the general approach via reduction for semi-direct product groups. Furthermore, the Hasimoto transform for the binormal equation turns out to be the 1D case of the Madelung transform, while its higher-dimensional version is related to the Willmore energy in binormal flows.

Page Manager: Webmaster|Last update: 9/11/2012
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