Noether-type theorem for fractional variational problems depending on fractional derivatives of functions

TL;DR

This paper develops a generalized fractional calculus of variation framework that accounts for the unique properties of fractional derivatives, leading to a Noether-type theorem for nonlinear systems and applications to chaotic dynamics.

Contribution

It introduces a new fractional variational calculus that overcomes limitations of fractional derivatives and formulates a Noether-type theorem for nonlinear fractional systems.

Findings

Derived generalized Euler-Lagrange equations for fractional derivatives.

Established a Noether-type theorem for fractional variational problems.

Applied results to nonlinear chaotic dynamical systems.

Abstract

In the present work, by taking advantage of a so-called practical limitation of fractional derivatives, namely, the absence of a simple chain and Leibniz's rules, we proposed a generalized fractional calculus of variation where the Lagrangian function depends on fractional derivatives of differentiable functions. The Euler-Lagrange equation we obtained generalizes previously results and enables us to construct simple Lagrangians for nonlinear systems. Furthermore, in our main result, we formulate a Noether-type theorem for these problems that provides us with a means to obtain conservative quantities for nonlinear systems. In order to illustrate the potential of the applications of our results, we obtain Lagrangians for some nonlinear chaotic dynamical systems, and we analyze the conservation laws related to time translations and internal symmetries.