Consider the flow of an inviscid, incompressible fluid with non-zero vorticity governed by the Euler pdes in \(\mathcal{D}=R^{3}\), let the vorticity be specified initially as a smooth vector field \(\Omega_{\alpha}(0, \mathbf{X}) \in L_{\mathcal{D}}^{2}\), then establish the equation for the Lagrangean position field \(\Phi_{\alpha}(\tau, \mathbf{X})\) in terms of vorticity \(\Omega_{\alpha}(0, \mathbf{X})\) at the reference time.