Small Oscillations of a Vortex Ring: Hamiltonian Formalism and Quantization

TL;DR

This paper develops a Hamiltonian formalism for small oscillations of a vortex ring, extends the dynamical variables, and quantizes the system to analyze its energy spectrum and circulation states.

Contribution

It introduces a new Hamiltonian approach including circulation as a dynamical variable and performs quantization to study vortex ring oscillations.

Findings

Derived differential equation for vortex ring oscillations

Extended Hamiltonian formalism with circulation as a variable

Quantized the system and calculated energy spectrum

Abstract

This article investigates small oscillations of a vortex ring with zero thickness that evolves under the Local Induction Equation (LIE). We deduce the differential equation that describes the dynamics of these oscillations. We suggest the new approach to the Hamiltonian description of this dynamic system. This approach is based on the extension of the set of dynamical variables by adding the circulation $\Gamma$ as a dynamical variable. The constructed theory is invariant under the transformations of the Galilei group. The appearance of this group allows for a new viewpoint on the energy of a vortex filament with zero thickness. We quantize this dynamical system and calculate the spectrum of the energy and acceptable circulation values. The physical states of the theory are constructed with help of coherent states for the Heisenberg -Weyl group.