Constants of motion for non-differentiable quantum variational problems

Jacky Cresson, Gastão S. F. Frederico, Delfim F. M. Torres


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 Schrödinger equation.


Non-differentiability; scale calculus of variations; symmetries; constants of motion; DuBois-Reymond necessary condition; Noether's theorem; Schrödinger equations

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