Fractional variational problems depending on fractional derivatives of differentiable functions with application to nonlinear chaotic systems

TL;DR

This paper develops a generalized Euler-Lagrange equation for fractional variational problems involving derivatives of differentiable functions, enabling the formulation of Lagrangians for nonlinear and chaotic systems.

Contribution

It introduces a necessary condition for fractional variational problems that extends existing results and facilitates modeling of complex nonlinear dynamics.

Findings

Derived a generalized Euler-Lagrange equation for fractional derivatives.

Constructed Lagrangians for specific chaotic systems.

Extended the applicability of fractional calculus in nonlinear dynamics.

Abstract

In the present work, we formulate a necessary condition for functionals with Lagrangians depending on fractional derivatives of differentiable functions to possess an extremum. The Euler-Lagrange equation we obtained generalizes previously known results in the literature and enables us to construct simple Lagrangians for nonlinear systems. As examples of application, we obtain Lagrangians for some chaotic dynamical systems.