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Fractional Noether’s theorem with classical and Caputo derivatives: constants of motion for non-conservative systems

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Abstract

Since the seminal work of Emmy Noether, it is well know that all conservations laws in physics, e.g., conservation of energy or conservation of momentum, are directly related to the invariance of the action under a family of transformations. However, the classical Noether’s theorem cannot yield information about constants of motion for non-conservative systems since it is not possible to formulate physically meaningful Lagrangians for this kind of systems in classical calculus of variation. On the other hand, in recent years the fractional calculus of variation within Lagrangians depending on fractional derivatives has emerged as an elegant alternative to study non-conservative systems. In the present work, we obtained a generalization of the Noether’s theorem for Lagrangians depending on mixed classical and Caputo derivatives that can be used to obtain constants of motion for dissipative systems. In addition, we also obtained Noether’s conditions for the fractional optimal control problem.

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Acknowledgments

This work was partially supported by CNPq and CAPES (Brazilian research funding agencies).

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  1. G. S. F. Frederico

    Present address: Department of Mathematics, Federal University of Santa Catarina, Florianópilis, SC, Brazil

Authors and Affiliations

  1. Department of Science and Technology, University of Cape Verde, Praia, Santiago, Cabo Verde

    G. S. F. Frederico

  2. Institute of Mathematics, Statistics and Physics, Federal University of Rio Grande, Rio Grande, RS, Brazil

    M. J. Lazo

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Frederico, G.S.F., Lazo, M.J. Fractional Noether’s theorem with classical and Caputo derivatives: constants of motion for non-conservative systems . Nonlinear Dyn 85, 839–851 (2016). https://doi.org/10.1007/s11071-016-2727-z

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