Geometry of the Madelung transform

Autor: Khesin, Boris, Misiolek, Gerard, Modin, Klas
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