American Institute of Mathematical Sciences

June  2017, 9(2): 157-165. doi: 10.3934/jgm.2017006

The Madelung transform as a momentum map

 Department of Mathematics, University of Toronto, Toronto, ON M5S 2E4, Canada

Received  December 2015 Revised  May 2016 Published  May 2017

The Madelung transform relates the non-linear Schrödinger equation and a compressible Euler equation known as the quantum hydrodynamical system. We prove that the Madelung transform is a momentum map associated with an action of the semidirect product group $\mathrm{Diff}(\mathbb{R}^{n}) \ltimes H^∞(\mathbb{R}^n; \mathbb{R})$, which is the configuration space of compressible fluids, on the space $Ψ = H^∞(\mathbb{R}^{n}; \mathbb{C})$ of wave functions. In particular, this implies that the Madelung transform is a Poisson map taking the natural Poisson bracket on $Ψ$ to the compressible fluid Poisson bracket. Moreover, the Madelung transform provides an example of "Clebsch variables" for the hydrodynamical system.

Citation: Daniel Fusca. The Madelung transform as a momentum map. Journal of Geometric Mechanics, 2017, 9 (2) : 157-165. doi: 10.3934/jgm.2017006
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