On Geodesic Hydrodynamic Motions. Heisenberg Spin Connections

It is established that, in steady circulation-preserving hydrodynamic motions with velocity q = qt subject to either of the geometric constraints t · curl t = 0 or div t = 0, the geodesic unit tangent t-field is constrained by privileged Heisenberg spin-type equations. In the case div t = 0, remarkably, the integrable Heisenberg spin equation related to the solitonic nonlinear Schr¨odinger equation is obtained. Corresponding results hold “mutatis mutandis” in magneto-hydrostatics