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.

Newton's equations

Hydrodynamics

Quantum information

Fisher-Rao

Infinite-dimensional geometry

Author

Boris Khesin

University of Toronto

Gerard Misiolek

University of Notre Dame

Klas Modin

University of Gothenburg

Chalmers, Mathematical Sciences, Applied Mathematics and Statistics

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

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

Vol. 115 24 6165-6170

Numerical methods for computational anatomy

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

Geometry and Computational Anatomy (GEOCA)

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

European Commission (Horizon 2020), 2015-03-01 -- 2017-06-30.

Subject Categories

Computational Mathematics

Geometry

Mathematical Analysis

DOI

10.1073/pnas.1719346115

More information

Latest update

7/2/2018 2