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Fractional Noether Theorem Based on Extended Exponentially Fractional Integral

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Abstract

Based on the new type of fractional integral definition, namely extended exponentially fractional integral introduced by EI-Nabulsi, we study the fractional Noether symmetries and conserved quantities for both holonomic system and nonholonomic system. First, the fractional variational problem under the sense of extended exponentially fractional integral is established, the fractional d’Alembert-Lagrange principle is deduced, then the fractional Euler-Lagrange equations of holonomic system and the fractional Routh equations of nonholonomic system are given; secondly, the invariance of fractional Hamilton action under infinitesimal transformations of group is also discussed, the corresponding definitions and criteria of fractional Noether symmetric transformations and quasi-symmetric transformations are established; finally, the fractional Noether theorems for both holonomic system and nonholonomic system are explored. What’s more, the relationship between the fractional Noether symmetry and conserved quantity are revealed.

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Acknowledgements

This work is supported by the National Natural Science Foundation of China (grant Nos. 10972151 and 11272227), the Innovation Program for Postgraduate in Higher Education Institutions of Jiangsu Province (No. CXLX11_0961), and the Innovation Program for Postgraduate of Suzhou University of Science and Technology (No. SKCX11S_051).

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Authors and Affiliations

  1. College of Mathematics and Physics, Suzhou University of Science and Technology, Suzhou, 215009, P.R. China

    Zi-Xuan Long

  2. College of Civil Engineering, Suzhou University of Science and Technology, Suzhou, 215009, P.R. China

    Yi Zhang

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  1. Zi-Xuan Long
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  2. Yi Zhang
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Correspondence to Yi Zhang.

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Long, ZX., Zhang, Y. Fractional Noether Theorem Based on Extended Exponentially Fractional Integral. Int J Theor Phys 53, 841–855 (2014). https://doi.org/10.1007/s10773-013-1873-z

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  • DOI: https://doi.org/10.1007/s10773-013-1873-z

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