Geometry of the Madelung Transform
Journal article, 2019

The Madelung transform is known to relate Schrödinger-type equations in quantum mechanics and the Euler equations for barotropic-type fluids. We prove that, more generally, the Madelung transform is a Kähler 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.


Boris Khesin

University of Toronto

Gerard Misiołek

University of Notre Dame

Klas Modin

Chalmers, Mathematical Sciences, Applied Mathematics and Statistics

Archive for Rational Mechanics and Analysis

0003-9527 (ISSN) 1432-0673 (eISSN)

Vol. 234 2 549-573

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