Geometric hydrodynamics via Madelung transform
Journal article, 2018

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 Schrödinger equation and Newton's equations is a symplectomorphism of the corresponding phase spaces. Furthermore, the Madelung transform turns out to be a Kähler map when the space of densities is equipped with the Fisher-Rao information metric. We describe several dynamical applications of these results.


Quantum information

Newton's equations

Infinite-dimensional geometry



Boris Khesin

University of Toronto

Gerard Misiolek

University of Notre Dame

Klas Modin

Chalmers, Mathematical Sciences, Applied Mathematics and Statistics

University of Gothenburg

Proceedings of the National Academy of Sciences of the United States of America

0027-8424 (ISSN) 1091-6490 (eISSN)

Vol. 115 24 6165-6170

Geometry and Computational Anatomy (GEOCA)

The Swedish Foundation for International Cooperation in Research and Higher Education (STINT) (11253452-STINT K Modin 15-18), 2015-04-01 -- 2018-06-30.

European Commission (EC) (EC/H2020/661482), 2015-03-01 -- 2017-06-30.

Numerical methods for computational anatomy

Swedish Foundation for Strategic Research (SSF), 2013-11-01 -- 2018-01-31.

Subject Categories

Computational Mathematics


Mathematical Analysis



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