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Noether symmetries for fractional generalized Birkhoffian systems in terms of classical and combined Caputo derivatives

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Abstract

In this paper, we research Noether’s theorems of fractional generalized Birkhoffian systems in terms of classical and combined Caputo derivatives. First, the generalized Pfaff–Birkhoff principle and Birkhoff’s equations with classical and combined Caputo derivatives are given. Then, in the case of without and with transforming time, respectively, we obtain two kinds of Noether symmetry and their conserved quantities by the method of time re-parameterization. Finally, by taking case of generalized Birkhoffian systems, we study the relationship between Noether symmetry and Mei symmetry. It is concluded that the generalized Birkhoff equations, Noether identity and Noether conserved quantity obtained by using the generalized Pfaff–Birkhoff principle based on the action functional with dynamical functions after infinitesimal transformation are completely consistent with the criterion equation, structural equation and Mei conserved quantity of the fractional generalized Birkhoffian system, respectively.

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (Nos.11972241, 11572212, 11272227), the Natural Science Foundation of Jiangsu Province (No. BK20191454).

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  1. College of Mathematics and Physics, Suzhou University of Science and Technology, Suzhou, 215009, People’s Republic of China

    Ying Zhou

  2. College of Civil Engineering, Suzhou University of Science and Technology, Suzhou, 215011, People’s Republic of China

    Yi Zhang

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Zhou, Y., Zhang, Y. Noether symmetries for fractional generalized Birkhoffian systems in terms of classical and combined Caputo derivatives. Acta Mech 231, 3017–3029 (2020). https://doi.org/10.1007/s00707-020-02690-y

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