On Integral Invariants for Vortex Motion under the Localized Induction Approximation

Abstract

Integral invariants associated with the motion of a vortex filament in an incompressible inviscid fluid are explored, under the assumption of localized induction. It is shown that the moment of the fluid impulse and the helicity are both invariant with respect to time. A sufficient condition for a closed filament to be unknotted is presented by using the concept of the total curvature of a closed curve.