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Noether’s theorem for fractional Herglotz variational principle in phase space

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  • Tian, Xue
  • Zhang, Yi

Abstract

The aim of this paper is to bring together two approaches, Heglotz variational principle and fractional calculus, to deal with non-conservative systems in phase space. Namely, we study the functional of Herglotz type whose extremum is sought, by the differential equation that involves Caputo fractional derivatives in phase space. Firstly, Herglotz variational principle under fractional Hamilton action in phase space is presented, and its Hamilton canonical equations are derived. Secondly, two basic formulae for the variation of the fractional Hamilton–Herglotz action in phase space are obtained. Furthermore, the definition and the criterion of Noether symmetry for fractional Herglotz variational principle are given, and the corresponding Noether’s theorem is established. Under appropriate conditions, the Noether’s theorem can reduce to the classical one of Herglotz type in phase space. Finally, two examples are given to illustrate the application of the results.

Suggested Citation

  • Tian, Xue & Zhang, Yi, 2019. "Noether’s theorem for fractional Herglotz variational principle in phase space," Chaos, Solitons & Fractals, Elsevier, vol. 119(C), pages 50-54.
    • Handle: RePEc:eee:chsofr:v:119:y:2019:i:c:p:50-54
    DOI: 10.1016/j.chaos.2018.12.005
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    References listed on IDEAS

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    1. Garra, Roberto & Taverna, Giorgio S. & Torres, Delfim F.M., 2017. "Fractional Herglotz variational principles with generalized Caputo derivatives," Chaos, Solitons & Fractals, Elsevier, vol. 102(C), pages 94-98.
    2. Alkahtani, B.S.T. & Atangana, A., 2016. "Controlling the wave movement on the surface of shallow water with the Caputo–Fabrizio derivative with fractional order," Chaos, Solitons & Fractals, Elsevier, vol. 89(C), pages 539-546.
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    Cited by:

    1. Tian, Xue & Zhang, Yi, 2021. "Fractional time-scales Noether theorem with Caputo Δ derivatives for Hamiltonian systems," Applied Mathematics and Computation, Elsevier, vol. 393(C).
    2. Wang, Jieyang & Mou, Jun & Xiong, Li & Zhang, Yingqian & Cao, Yinghong, 2021. "Fractional-order design of a novel non-autonomous laser chaotic system with compound nonlinearity and its circuit realization," Chaos, Solitons & Fractals, Elsevier, vol. 152(C).
    3. Ding, Juan-Juan & Zhang, Yi, 2020. "Noether’s theorem for fractional Birkhoffian system of Herglotz type with time delay," Chaos, Solitons & Fractals, Elsevier, vol. 138(C).

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