Geometric Hydrodynamics via Madelung Transform

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 PDEs 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.

17 pages, 2 figures

References in corpus (1)

• The curvature of semidirect product groups associated with two-component Hunter-Saxton systems

Cited by in corpus (14)

• Madelung transform and probability densities in hybrid classical-quantum dynamics
• Geometric hydrodynamics and infinite-dimensional Newton's equations
• Geometry of nonadiabatic quantum hydrodynamics
• Geometry of the Madelung transform
• Towards a mathematical Theory of the Madelung Equations
• Semi-invariant Riemannian metrics in hydrodynamics
• Geometric Hydrodynamics in Open Problems
• Non-Hamiltonian Kelvin wave generation on vortices in Bose-Einstein condensates
• Deep Learning: Hydrodynamics, and Lie-Poisson Hamilton-Jacobi Theory
• Zero absolute vorticity plane Couette flow as an hydrodynamic representation of quantum energy states under perpendicular magnetic field
• Holonomy and vortex structures in quantum hydrodynamics
• Geometric Hydrodynamics: from Euler, to Poincaré, to Arnold
• Geometry of quantum hydrodynamics in theoretical chemistry
• Areas on the space of smooth probability density functions on $S^2$