Noether's Theorem with Momentum and Energy Terms for Cresson's Quantum Variational Problems

Abstract

We prove a DuBois-Reymond necessary optimality condition and a Noether symmetry theorem to the recent quantum variational calculus of Cresson. The results are valid for problems of the calculus of variations with functionals defined on sets of nondifferentiable functions. As an application, we obtain a constant of motion for a linear Schrödinger equation.

Helmholtz Theorem for Nondifferentiable Hamiltonian Systems in the Framework of Cresson’s Quantum Calculus

“…In [18], a Noether type theorem is proved but only with the momentum term. This result is further extended in [24] by considering invariance transformations that also change the time variable, thus obtaining not only the generalized momentum term of [18] but also a new energy term. In [3], nondifferentiable variational problems with a free terminal point, with or without constraints, of first and higher-order, are investigated.…”