General fractional classical mechanics: Action principle, Euler-Lagrange equations and Noether theorem

Standard action principle and first Noether theorem are generalized to take into account general form of nonlocality in time (memory) and to describe dissipative and non-Lagrangian nonlinear systems. General fractional calculus (GFC) of the Luchko form is used to take into account a wide class of nonlocalities time compared to the usual fractional calculus. Nonlocality is described by a pair of operator kernels belonging to the Luchko set. Non-holonomic variation equations of Sedov type is used to take into account equations of motion for wide class of dissipative and non-Lagrangian systems. In addition, within the framework of the proposed approach, equations of motion are considered not only with general fractional derivatives, but also for general fractional integrals. Example of application is considered.

Automated summary

This paper extends the standard action principle and the first Noether theorem to consider the general form of nonlocality in time and describes dissipative and non-Lagrangian nonlinear systems. The general fractional calculus is used to handle a wide class of nonlocalities in time compared to the usual fractional calculus. The nonlocality is described by a pair of operator kernels belonging to the Luchko set. The non-holonomic variation equations of the Sedov type are used to describe the motion equations of a wide class of dissipative and non-Lagrangian systems. Additionally, the equations of motion are considered not only with general fractional derivatives but also with general fractional integrals. An application example is presented.