Rasetti–Regge Dirac bracket formulation of Lagrangian fluid dynamics of vortex filaments Original Research Article

We review the Rasetti–Regge Dirac bracket (RRDB) for determining the constrained Hamiltonian dynamics of vortex filaments moving with an incompressible potential flow of superfluid helium-II in the Lagrangian fluid picture. We compare the equations for Lagrangian vortex filaments with their corresponding Eulerian vorticity dynamics in the examples of the Euler fluid, superfluid vortices, the local induction approximation (LIA), the Rosenhead regularization and a new class of alternative regularized theories including the Euler-alpha model. The RRDB formulation generalizes the Betchov–Da Rios equation for the transverse self-induction velocity of a vortex filament from LIA to the case of an incompressible fluid whose energy may expressed as an arbitrary functional of spatial vorticity. We also discuss the relation of RRDB to the Marsden–Weinstein bracket for vortex filaments and its implications under the Hasimoto transformation for physically meaningful nonlocal nonlinear Schrödinger (NLNLS) equations.