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

The Madelung transform as a momentum map

Daniel Fusca

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.

Keywords: Madelung transform, moment map, quantum fluid, semidirect product group, nonlinear Schrödinger.
Mathematics Subject Classification: Primary: 53D20, 76Y05; Secondary: 37K65.
Citation: Daniel Fusca. The Madelung transform as a momentum map. Journal of Geometric Mechanics, 2017, 9 (2) : 157-165. doi: 10.3934/jgm.2017006
References:
[1]

R. Carles, R. Danchin and J.-C. Saut, Madelung, Gross-Pitaevskii and Korteweg, Nonlinearity, 25 (2012), 2843-2873.  doi: 10.1088/0951-7715/25/10/2843.  Google Scholar

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B. Kolev, Poisson brackets in hydrodynamics, Discrete and Continuous Dynamical Systems, 19 (2007), 555-574.  doi: 10.3934/dcds.2007.19.555.  Google Scholar

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J. E. Marsden and T. Ratiu, Introduction to Mechanics and Symmetry, 2nd edition, Springer-Verlag, New York, 1999. doi: 10.1007/978-0-387-21792-5.  Google Scholar

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J. E. Marsden and A. Weinstein, Coadjoint orbits, vortices, and Clebsch variables for incompressible fluids, Physica D: Nonlinear Phenomena, 7 (1983), 305-323.  doi: 10.1016/0167-2789(83)90134-3.  Google Scholar

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M.-K. von Renesse, An optimal transport view of Schrödinger's equation, Canadian Mathematical Bulletin, 55 (2012), 858-869.  doi: 10.4153/CMB-2011-121-9.  Google Scholar

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show all references

References:
[1]

R. Carles, R. Danchin and J.-C. Saut, Madelung, Gross-Pitaevskii and Korteweg, Nonlinearity, 25 (2012), 2843-2873.  doi: 10.1088/0951-7715/25/10/2843.  Google Scholar

[2]

B. Khesin, G. Misiolek and K. Modin, Geometry of Newton's equation on diffeomorphisms and densities, work in progress. Google Scholar

[3]

B. Kolev, Poisson brackets in hydrodynamics, Discrete and Continuous Dynamical Systems, 19 (2007), 555-574.  doi: 10.3934/dcds.2007.19.555.  Google Scholar

[4]

E. Madelung, Quantentheorie in hydrodynamischer Form, Zeitschrift für Physik, 40 (1927), 322-326.  doi: 10.1007/BF01400372.  Google Scholar

[5]

J. E. Marsden, T. Ratiu and A. Weinstein, Semidirect products and reduction in mechanics, Transactions of the American Mathematical Society, 281 (1984), 147-177.  doi: 10.2307/1999527.  Google Scholar

[6]

J. E. Marsden and T. Ratiu, Introduction to Mechanics and Symmetry, 2nd edition, Springer-Verlag, New York, 1999. doi: 10.1007/978-0-387-21792-5.  Google Scholar

[7]

J. E. Marsden and A. Weinstein, Coadjoint orbits, vortices, and Clebsch variables for incompressible fluids, Physica D: Nonlinear Phenomena, 7 (1983), 305-323.  doi: 10.1016/0167-2789(83)90134-3.  Google Scholar

[8]

M.-K. von Renesse, An optimal transport view of Schrödinger's equation, Canadian Mathematical Bulletin, 55 (2012), 858-869.  doi: 10.4153/CMB-2011-121-9.  Google Scholar

[9]

A. Weinstein, The local structure of Poisson manifolds, Journal of Differential Geometry, 18 (1983), 523-557.  doi: 10.4310/jdg/1214437787.  Google Scholar

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