Generalized fractional operators for nonstandard Lagrangians

Abstract

In this note we study the application of generalized fractional operators to a particular class of nonstandard Lagrangians. These are typical of dissipative systems and the corresponding Euler-Lagrange and Hamilton equations are analyzed. The dependence of the equation of motion on the generalized kernel permits to obtain a wide range of different configurations of motion. Some examples are discussed and analyzed.Comment: This is a preprint of a paper whose final and definite form will appear in Mathematical Methods in the Applied Sciences, ISSN 0170-4214. Paper submitted 31/Jan/2014; revised 23/Apr/2014; accepted for publication 25/Apr/201

Complex-Valued Fractional Derivatives on Time Scales

“…The subject is very rich and several different definitions and approaches are available, either in discrete [1], continuous [11], and time-scale settings [2]. In continuous time, i.e., for the time scale T = R, several definitions are based on the classical Euler Gamma function Γ .…”

Fractional variational approach with non-standard power-law degenerate Lagrangians and a generalized derivative operator

“…nonlinear differential equations [7,8,9,10] and dissipative dynamical systems [12,13,14,15,16,17,21,37,38,41,42,44] as well as in theoretical physics [18,19,20,22]. In general, NSL are characterized by a deformed kinetic term and a deformed potential function, yet the Euler-Lagrange equation that results from the standard calculus of variations lead to equations of motion that correspond to physically attractive nonlinear dynamical systems.…”

“…The resulting equations of motion are based on the standard calculus of variations and it was observed that both types exhibit interesting dynamical properties. More recently, in [44], the application of generalized fractional operators was applied to NSL and the consequential approach has proved to be useful to understand dissipative systems. More recently degenerate NSL approach was also applied to nonlinear oscillators [21] where it was observed that the theory describes correctly oscillators characterized by a positiondependent mass which represent an interesting class of problem in different field of sciences and engineering.…”

A necessary condition of Pontryagin type for fuzzy fractional optimal control problems

“…The two subjects have recently been put together and a theory of the calculus of variations and optimal control that deals with more general systems containing noninteger order derivatives is now available: see the books [2,21,29]. In particular, the fractional Hamiltonian perspective is a very active subject, being investigated in a series of publications: see, e.g., [4,7,13,24,25,33,39,40].…”