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

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 differ...

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Bibliographic Details
Published in:Applicable analysis 2021-06, Vol.100 (8), p.1727-1743
Main Authors: Lazo, M. J., Frederico, G. S. F., Carvalho-Neto, P. M.
Format: Article
Language:English
Subjects:
Calculus of variations
Conservation laws
Derivatives
Euler-Lagrange equation
Fractional calculus
fractional calculus of variation
Fractional Noether-type theorem
Lagrangian function
nonlinear and chaotic systems
Nonlinear systems
Theorems
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Summary: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.
ISSN:0003-6811
1563-504X
DOI:10.1080/00036811.2019.1659958