  A recent article by the first two authors together with B Andrews and V-M
Wheeler considered the so-called `ideal curve flow', a sixth order curvature
flow that seeks to deform closed planar curves to curves with least variation
of total geodesic curvature in the $L^2$ sense. Critical in the analysis there
was a length bound on the evolving curves. It is natural to suspect therefore
that the length-constrained ideal curve flow should permit a more
straightforward analysis, at least in the case of small initial `energy'. In
this article we show this is indeed the case, with suitable initial data
providing a flow that exists for all time and converges smoothly and
exponentially to a multiply-covered round circle of the same length and winding
number as the initial curve.
