On Geodesic Hydrodynamic Motions. Heisenberg Spin Connections

Abstract

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 Schrödinger equation is obtained. Corresponding results hold "mutatis mutandis" in magneto-hydrostatics.

Nested toroidal flux surfaces in magnetohydrostatics. Generation via soliton theory

“…Remarkably, the geometric significance of this observation has been revealed only recently by Rogers (2000) and Rogers and Schief (2000). Thus, in the remainder of this paper, we are concerned with the magnetohydrostatic system (2.1) subject to the constraint (2.3).…”

The Euler Equations of Spatial Gasdynamics and the Integrable Heisenberg Spin Equation

“…The latter were subsequently investigated by Howard [4], Wasserman [5], and Marris [6]. In [7,8], it was established that, remarkably, constant speed flows are governed by the integrable Heisenberg spin equation [9] subject to a geometric constraint. The significance of the mathematically equivalent analog of constant speed flows in magnetohydrostatics was discussed in [10,11].…”

Novel integrable reductions in nonlinear continuum mechanics via geometric constraints

“…17 In Ref. 18, it has been established that, remarkably, this classical problem may be decomposed into a study of the solitonic Heisenberg spin equation ͑2.2͒ with S ϭt subject to the condition ͑2.4͒. The results carry over, mutatis mutandis, to the steady motion of an inviscid and thermally nonconducting gas with arbitrary equation of state as well as to magnetohydrostatics.…”

“…18, it has been established that, remarkably, this classical problem may be decomposed into a study of the solitonic Heisenberg spin equation ͑2.2͒ with S ϭt subject to the condition ͑2.4͒. 12,18,19 In the sequel, an account of the derivation of the Heisenberg spin equation in hydrodynamics and magnetohydrostatics is presented. 12,18,19 In the sequel, an account of the derivation of the Heisenberg spin equation in hydrodynamics and magnetohydrostatics is presented.…”