Closed vortex filament in a cylindrical domain: Circulation quantization

Summary: This article investigates the small oscillations of a vortex ring with zero thickness that evolves under the Local Induction Equation (LIE). The dynamics of these oscillations are described by a deduced differential equation. A new approach to the Hamiltonian description of this dynamic system is suggested, which extends the set of dynamical variables by adding the circulation gamma. The constructed theory is invariant under the transformations of the Galilei group, allowing for a new viewpoint on the energy of a vortex filament with zero thickness. The quantization of this dynamical system and the calculation of the energy spectrum and acceptable circulation values are performed. The physical states of the theory are constructed using coherent states for the Heisenberg-Weyl group.

Summary: Velocity circulation is a fundamental observable in classical and quantum flows, with different characteristics in each. In classical fluids, it is a Lagrangian invariant, while in quantum flows, it is quantized. The dependence of flow structure on scale can be measured by varying the size of closed loops enclosing vortex filaments.