  We consider a toy model of rate independent droplet motion on a surface with
contact angle hysteresis based on the one-phase Bernoulli free boundary
problem. Taking advantage of two notions of weak solutions, energy-based and
comparison-principle-based, we study the dynamic contact angle of moving
contact lines and the geometry of de-pinning. We show that these two notions of
solutions coincide in a star-shaped setting, where we show (almost) optimal
regularity of the contact line and the convergence of a minimizing movements
scheme. In a general setting, the notions differ essentially in how they handle
jumps, but both are shown to satisfy a weak motion law.
