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General Noncommuting Curvilinear Coordinates and Fluid Mechanics

We show that restricting the states of a charged particle to the lowest Landau level introduces noncommutativity between general curvilinear coordinate operators. The Cartesian, circular cylindrical and spherical polar coordinates are three special cases of our quite general method. The connection between U(1) gauge fields defined on a general noncommuting curvilinear coordinates and fluid mechanics is explained. We also recognize the Seiberg–Witten map from general noncommuting to commuting variables as the quantum correspondence of the Lagrange-to-Euler map in fluid mechanics.