Caputo \(\Delta\)-type fractional time-scales Noether theorem of Birkhoffian systems
Birkhoff systems extend Hamiltonian and Lagrangian formalism, including dissipative systems. They are characterized by invariance of time-dependent symplectic forms on the integral curves. A time-scale is a non-empty closed subset \(\mathbb T\) of \(\mathbb R\), possibly discrete, and represents the ``points'' in time with respect to which the dynamics is analyzed. A calculus is developed for functions \(\mathbb T\rightarrow \mathbb R\). In this article, fractional derivatives and integrals, of Riemann-Liouville and Caputo types, are extended to functions on time-scales. Fractional calculus naturally describes systems with memory, its extension to time-scales allows the inclusion of time-discrete systems in the formalism. Both the above formalisms are employed to define fractional time-scale Birkhoff mechanics. The authors determine Noether-type theorems for this extended Birkhoff mechanics, in particular for Hamiltonian and Lagrangian systems. The Hojman-Urrutia model is studied as an example.
scientific article; zbMATH DE number 7612090
scientific article