  We obtain a weak formulation of the stationarity condition for the half
Dirichlet energy, which can be expressed in terms of a fractional analogous to
the Hopf differential. As an application we show that conformal harmonic maps
from the disc are precisely the harmonic extensions of stationary points of the
half Dirichlet energy on the circle. We also derive a Noether theorem and a
Pohozaev identity for stationary points of the half Dirichlet energy.
