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

Abstract

We derive the Helmholtz theorem for nondifferentiable Hamiltonian systems in the framework of Cresson's quantum calculus. Precisely, we give a theorem characterizing nondifferentiable equations, admitting a Hamiltonian formulation. Moreover, in the affirmative case, we give the associated Hamiltonian. 2010 Mathematics Subject Classification. 49N45, 70S05. Key words and phrases. Cresson's quantum calculus, nondifferentiable calculus of variations, nondifferentiable Hamiltonian systems, inverse problem of the calculus of variations.

About the structure of the discrete and continuous Eringen’s nonlocal elastica

“…Pierret [19] introduced a notion of Hamiltonian systems on time scales adapted to the framework of the non-shifted calculus of variations. Precisely, we have the following: Definition 7.5.…”