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Noether Symmetry and Conserved Quantity for FractiOnal Birkhoffian Mechanics and Its Applications

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Abstract

Noether theorem is an important aspect to study in dynamical systems. Noether symmetry and conserved quantity for the fractional Birkhoffian system are investigated. Firstly, fractional Pfaff actions and fractional Birkhoff equations in terms of combined Riemann-Liouville derivative, Riesz-Riemann-Liouville derivative, combined Caputo derivative and Riesz-Caputo derivative are reviewed. Secondly, the criteria of Noether symmetry within combined Riemann-Liouville derivative, Riesz-Riemann-Liouville derivative, combined Caputo derivative and Riesz-Caputo derivative are presented for the fractional Birkhoffian system, respectively. Thirdly, four corresponding conserved quantities are obtained. The classical Noether identity and conserved quantity are special cases of this paper. Finally, four fractional models, such as the fractional Whittaker model, the fractional Lotka biochemical oscillator model, the fractional Hénon-Heiles model and the fractional Hojman-Urrutia model are discussed as examples to illustrate the results.

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Authors and Affiliations

  1. College of Science, Nanjing University of Science and Technology, No. 200 Xiaolingwei Street, Xuanwu District Nanjing, 210094, Jiangsu Province, China

    Chuan-Jing Song

  2. School of Mathematics and Physics, Suzhou University of Science and Technology, No. 1 Kerui Road, Gaoxin District Suzhou, 215009, Jiangsu Province, China

    Chuan-Jing Song

  3. College of Civil Engineering, Suzhou University of Science and Technology, No. 1 Kerui Road, Gaoxin District Suzhou, 215009, Jiangsu Province, China

    Yi Zhang

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  1. Chuan-Jing Song
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  2. Yi Zhang
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Song, CJ., Zhang, Y. Noether Symmetry and Conserved Quantity for FractiOnal Birkhoffian Mechanics and Its Applications. FCAA 21, 509–526 (2018). https://doi.org/10.1515/fca-2018-0028

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