  We extend the DuBois-Reymond necessary optimality condition and Noether's
symmetry theorem to the scale relativity theory setting. Both Lagrangian and
Hamiltonian versions of Noether's theorem are proved, covering problems of the
calculus of variations with functionals defined on sets of non-differentiable
functions, as well as more general non-differentiable problems of optimal
control. As an application we obtain constants of motion for some linear and
nonlinear variants of the Schrodinger equation.
