  In this paper we use a gradient flow to deform closed planar curves to curves
with least variation of geodesic curvature in the $L^2$ sense. Given a smooth
initial curve we show that the solution to the flow exists for all time and,
provided the length of the evolving curve remains bounded, smoothly converges
to a multiply-covered circle. Moreover, we show that curves in any homotopy
class with initially small $L^3\lVert k_s\rVert_2^2$ enjoy a uniform length
bound under the flow, yielding the convergence result in these cases.
