Helmholtz theorem for Hamiltonian systems on time scales

Abstract

We derive the Helmholtz theorem for Hamiltonian systems defined on time scales in the context of nonshifted calculus of variations which encompass the discrete and continuous case. Precisely, we give a theorem characterizing first order equation on time scales, admitting a Hamiltonian formulation which is defined with non-shifted calculus of variation. Moreover, in the affirmative case, we give the associated Hamiltonian.

Continuous versus discrete structures II -- Discrete Hamiltonian systems and Helmholtz conditions

“…Contrary to what happens in the Lagrangian case where technical difficulties related to the composition of operators seem to cancel such a generalization, the Hamiltonian case is possible due to its linear nature in the discrete differential operators. We refer to [13] for more details.…”

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

“…For the Hamiltonian case it has been done for the discrete calculus of variations in [1] using the framework of [32] and in [20] using a discrete embedding procedure derived in [19]. In the case of time-scale calculus it has been done in [36]; for the Stratonovich stochastic calculus see [37].…”