### Constants of motion for non-differentiable quantum variational problems

#### Abstract

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.

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.

#### Keywords

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

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