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Caputo \(\Delta \)-type fractional time-scales Noether theorem of Birkhoffian systems

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Abstract

A new type of Noether theorem with Caputo \(\Delta \)-derivative of Birkhoffian systems is the focus of this paper based upon the fractional time-scales calculus, which constructs a model to unify continuous fractional systems and discrete fractional systems. The Caputo \(\Delta \)-type fractional time-scales Pfaff-Birkhoff principle is given, and its Birkhoff equations are deduced. According to the definition of fractional time-scales Noether symmetry of Birkhoffian systems, its criterion is proved and its Noether theorem is formulated by the generalized Jost method. Furthermore, the corresponding transformation conditions are given to study the relationship among fractional time-scales Birkhoffian systems, Hamiltonian systems and Lagrangian systems. Taking the fractional time-scales Hojman-Urrutia model as an example, the simulation results show the validity of the new Noether theorem.

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (11972241, 11572212); the Postgraduate Research & Practice Innovation Program of Jiangsu Province (KYCX20_0251); and the Natural Science Foundation of Jiangsu Province (BK20191454).

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Authors and Affiliations

  1. School of Science, Nanjing University of Science and Technology, Xiaolingwei Street No. 200, Nanjing, 210094, Jiangsu, China

    Xue Tian

  2. College of Civil Engineering, Suzhou University of Science and Technology, Binhe Road No. 1701, Suzhou, 215011, Jiangsu, China

    Yi Zhang

  3. School of Mathematics and Big Data, Anhui University of Science and Technology, Taifeng Street No. 168, Huainan, 232001, Anhui, China

    Xue Tian

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Tian, X., Zhang, Y. Caputo \(\Delta \)-type fractional time-scales Noether theorem of Birkhoffian systems. Acta Mech 233, 4487–4503 (2022). https://doi.org/10.1007/s00707-022-03338-9

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  • DOI: https://doi.org/10.1007/s00707-022-03338-9

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