Duality of the Lagrangian and Eulerian representations of collective motion—a connection built around vorticity

To allow a non-zero 'vorticity' associated with the generalized momentum, the Lagrangian describing general fluid-mechanical collective motions must incorporate a non-canonical structure. The canonical formalism, symbolized by the basic Hamilton–Jacobi equation P = ∇S relating the momentum 'P' with the action 'S', does not permit finite vorticity. The Lagrangian in the Eulerian view (suited for coupling with other fields such as the electromagnetic) must include 'topological constraints' embodying this non-canonical feature. Analyzing the role of the abstract fields (introduced as Lagrange multipliers) constituting the constraints, we may unify the Lagrangians in both Eulerian and Lagrangian views. Relativistic (Lorentz-invariant) formulation reveals the natural meaning of the Clebsch parametrization.