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Conserved Quantities and Adiabatic Invariants for El-Nabulsi’s Fractional Birkhoff System

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Abstract

Based on El-Nabulsi-Birkhoff fractional equations, Lie symmetry and the Hojman conserved quantity, the Noether conserved quantity deduced indirectly by the Lie symmetry and adiabatic invariants of Lie symmetrical perturbation are studied under the framework of El-Nabulsi’s fractional model. Firstly, Lie symmetry and the Hojman conserved quantity are obtained, including the equations of motion of EI-Nabulsi’s fractional Birkhoff system, the determining equations of Lie symmetry for the system and the generalization of the Hojman theorem. Secondly, the Noether conserved quantity deduced indirectly by the Lie symmetry is obtained. Thirdly, the adiabatic invariants of Lie symmetrical perturbation for disturbed EI-Nabulsi’s fractional Birkhoff system is achieved, including the disturbed El-Nabulsi-Birkhoff fractional equations, the determining equations of Lie symmetrical perturbation and adiabatic invariants for disturbed El-Nabulsi’s fractional Birkhoff system. Fourthly, adiabatic invariants and exact invariants under the special ifinitesimal transformations are presented. Finally, the Hojman-Urrutia problem is discussed to illustrate the application of these methods and results.

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Acknowledgments

This work is supported by the National Natural Science Foundation of China (grant Nos.10972151 and 11272227), and the Innovation Program for Scientific Research of Nanjing University of Science and Technology.

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Authors and Affiliations

  1. College of Science, Nanjing University of Science and Technology, Nanjing, 210094, People’s Republic of China

    Chuan-Jing Song

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

    Yi Zhang

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  1. Chuan-Jing Song
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Correspondence to Yi Zhang.

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Song, CJ., Zhang, Y. Conserved Quantities and Adiabatic Invariants for El-Nabulsi’s Fractional Birkhoff System. Int J Theor Phys 54, 2481–2493 (2015). https://doi.org/10.1007/s10773-014-2475-0

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  • DOI: https://doi.org/10.1007/s10773-014-2475-0

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