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

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.

Keywords:

• Fractional Noether-type theorem
• fractional calculus of variation
• nonlinear and chaotic systems

2010 Mathematics Subject Classifications:

• 49K21
• 26A33
• 49S05
• 70K99

Acknowledgments

The authors are grateful to the Brazilian foundations CNPq and Capes for financial support.

Disclosure statement

The authors declare that they have no conflict of interest.