Evolving force-free magnetic fields. IV. A variational formulation of the problem

An alternative formulation is presented of the theoretical model of evolving force-free magnetic fields treated in the first paper of this series. Beginning with the distribution of the magnetic footpoints, we derive the equations for the force-free fields above the photosphere from a variational principle. The new formulation presents a nonlinear boundary value problem which appears to be intractable at the present but suggests a quantitative program that may allow us to predict from a given photospheric motion when a magnetic field will transit from a state of quasi-static evolution to a state of non-equilibrium