On a new generalized local fractal derivative operator

• Published in: Chaos, solitons and fractals Vol. 161; p. 112329
• Main Authors: El-Nabulsi, Rami Ahmad, Khalili Golmankhaneh, Alireza, Agarwal, Praveen
• Format: Journal Article
• Language: English
• Published: Elsevier Ltd 01.08.2022
• Subjects: Fractal calculus; Fractal time; Generalized derivative operator; Generalized local fractal derivative operator; The nonlocal classical mechanics; The nonlocal classical mechanics; Fractal time; 81T20; 37N05; 28A80; Fractal calculus; Generalized derivative operator; Generalized local fractal derivative operator
• ISSN: 0960-0779; 1873-2887
• DOI: 10.1016/j.chaos.2022.112329

In this study, a new generalized local fractal derivative operator is introduced and we discuss its implications in classical systems through the Lagrangian and Hamiltonian formalisms. The variational approach has been proved to be practical to describe dissipative dynamical systems. Besides, the Hamiltonian formalism is characterized by the emergence of auxiliary constraints free from Dirac auxiliary functions. In field theory, it was found that both damped Klein-Gordon and Dirac equations are generalized, and for specific parameters, a field equation comparable to the Barut equation describing the electromagnetic interactions between N spin-1/2 particles in lepton physics is obtained. A Hamiltonian formulation of higher-order Lagrangian has been constructed and discussed as well. The reformulation of the problem based on fractal calculus has been also addressed in details and compared with the basic approach. •Classical mechanics equation involving fractal generalized derivative operator is given.•Fractal generalized derivative is applied in classical field theory.•Classical mechanics and classical field theory are reformulated by using generalized derivatives.