On a new generalized local fractal derivative operator

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.