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Noether’s theorem for fractional Birkhoffian system of Herglotz type with time delay

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  • Ding, Juan-Juan
  • Zhang, Yi

Abstract

Noether symmetry theorem of Herglotz type for time-delayed fractional Birkhoffian system are studied. Firstly, based on the fractional derivative of Riemann-Liouville, the Herglotz variational principle of time-delayed fractional Birkhoffian system is established, and the time-delayed Birkhoff′s equation of Herglotz type is derived. Secondly, the definition and criterion of Herglotz type Noether symmetric transformation of time-delayed fractional Birkhoffian system are established. Thirdly, the Noether′s theorem of the system is proposed and proved, in addition, the inner relationship between Noether symmetries and conservation is accurately explored. Next, the special case of the theorem is discussed, in other words, when the Herglotz generalized variational principle is reduced to the classical variational principle, the result of this paper is degraded into the Noether symmetry theorem of the time-delayed fractional Birkhoffian system. Finally, an example is given.

Suggested Citation

  • 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).
    • Handle: RePEc:eee:chsofr:v:138:y:2020:i:c:s0960077920303131
    DOI: 10.1016/j.chaos.2020.109913
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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.
    3. 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.
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    Cited by:

    1. Jin, Shi-Xin & Chen, Xiang-Wei & Li, Yan-Min, 2024. "Approximate Noether theorem and its inverse for nonlinear dynamical systems with approximate nonstandard Lagrangian," Chaos, Solitons & Fractals, Elsevier, vol. 182(C).
    2. Zhang, Yi & Jia, Yun-Die, 2023. "Generalization of Mei symmetry approach to fractional Birkhoffian mechanics," Chaos, Solitons & Fractals, Elsevier, vol. 166(C).
    3. Huang, Li-Qin & Zhang, Yi, 2024. "Herglotz-type vakonomic dynamics and Noether theory of nonholonomic systems with delayed arguments," Chaos, Solitons & Fractals, Elsevier, vol. 182(C).

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