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The Madelung transform as a momentum map
Department of Mathematics, University of Toronto, Toronto, ON M5S 2E4, Canada |
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.
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. |
| [2] |
B. Khesin, G. Misiolek and K. Modin, Geometry of Newton's equation on diffeomorphisms and densities, work in progress. |
| [3] |
B. Kolev,
Poisson brackets in hydrodynamics, Discrete and Continuous Dynamical Systems, 19 (2007), 555-574.
doi: 10.3934/dcds.2007.19.555. |
| [4] |
E. Madelung,
Quantentheorie in hydrodynamischer Form, Zeitschrift für Physik, 40 (1927), 322-326.
doi: 10.1007/BF01400372. |
| [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. |
| [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. |
| [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. |
| [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. |
| [9] |
A. Weinstein,
The local structure of Poisson manifolds, Journal of Differential Geometry, 18 (1983), 523-557.
doi: 10.4310/jdg/1214437787. |
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. |
| [2] |
B. Khesin, G. Misiolek and K. Modin, Geometry of Newton's equation on diffeomorphisms and densities, work in progress. |
| [3] |
B. Kolev,
Poisson brackets in hydrodynamics, Discrete and Continuous Dynamical Systems, 19 (2007), 555-574.
doi: 10.3934/dcds.2007.19.555. |
| [4] |
E. Madelung,
Quantentheorie in hydrodynamischer Form, Zeitschrift für Physik, 40 (1927), 322-326.
doi: 10.1007/BF01400372. |
| [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. |
| [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. |
| [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. |
| [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. |
| [9] |
A. Weinstein,
The local structure of Poisson manifolds, Journal of Differential Geometry, 18 (1983), 523-557.
doi: 10.4310/jdg/1214437787. |
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