Stationary configurations of a vortex filament in background flows

We investigate the three-dimensional configurations of a thin vortex filament embedded in background flows of an inviscid incompressible fluid, based on the localized induction approximation. An analogy is found between stationary configurations of a vortex filament in a steady flow and trajectories of a charged particle in a steady magnetic field. An analogy with trajectories of a sound ray in a steady low-Mach number flow is drawn as well. These analogies allow us to use the Lagrangian and Hamiltonian formalism of classical mechanics for calculating fully nonlinear forms of a vortex filament. The equations for the filament curve are integrable when the background flow has two spatial symmetries. Only integrable cases are explored in some detail. To illustrate the advantages of use of the analogy, we re-examine the invariant shapes of a vortex filament moving through a still fluid obtained by Kida (1981). Analogies of the Kida class with the motions of a heavy symmetrical top and a charged spherical pendulum in the field of a magnetic monopole are discussed. As next examples, we consider a point source (sink) and a line source (sink) flows. A filament takes the form of a non-uniform helix wound around a cone. The last example is a circular jet of a parabolic velocity profile. We construct a closed filament with azimuthal waves, lying on a plane orthogonal to the flow direction. Among them, an elliptic vortex ring is found to be distinct from higher waves.