Geometry of the Madelung transform
| Autor: | Khesin, Boris, Misiolek, Gerard, Modin, Klas |
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| Rok vydání: | 2018 |
| Předmět: | |
| Zdroj: | Arch. Ration. Mech. Anal., 234(2):549-573, 2019 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1007/s00205-019-01397-2 |
| Popis: | The Madelung transform is known to relate Schr\"odinger-type equations in quantum mechanics and the Euler equations for barotropic-type fluids. We prove that, more generally, the Madelung transform is a K\"ahler map (i.e. a symplectomorphism and an isometry) between the space of wave functions and the cotangent bundle to the density space equipped with the Fubini-Study metric and the Fisher-Rao information metric, respectively. We also show that Fusca's momentum map property of the Madelung transform is a manifestation of the general approach via reduction for semi-direct product groups. Furthermore, the Hasimoto transform for the binormal equation turns out to be the 1D case of the Madelung transform, while its higher-dimensional version is related to the problem of conservation of the Willmore energy in binormal flows. Comment: 27 pages, 2 figures |
| Databáze: | arXiv |
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