Fractional time-scales Noether theorem with Caputo [formula omitted] derivatives for Hamiltonian systems

Abstract

This paper examines a new Noether theorem for Hamiltonian systems with Caputo Δ derivatives based on fractional time-scales calculus, which overcomes the difficulties unified to study the Noether theorems of fractional continuous systems and fractional discrete systems. To begin with, the fractional time-scales definitions and properties with Caputo Δ derivatives are introduced. Next, the fractional time-scales Hamilton canonical equations with Caputo Δ derivatives are formulated. For the fractional time-scales Hamiltonian system, the definitions and criteria of Noether symmetries without transforming time and with transforming time are given, respectively. Furthermore, the corresponding Noether theorem without transforming time and its Noether theorem with transforming time are obtained. The latter one can reduce to the time-scales Noether theorem with Δ derivatives or the fractional Noether theorem with Caputo derivatives for Hamiltonian systems. Finally, the fractional time-scales damped oscillator and Kepler problem are taken as examples to verify the correctness of the results.