周颖, 张毅. 基于Caputo导数的分数阶广义Birkhoff系统的Noether定理[J]. 云南大学学报(自然科学版), 2019, 41(5): 953-959. doi: 10.7540/j.ynu.20190130
引用本文: 周颖, 张毅. 基于Caputo导数的分数阶广义Birkhoff系统的Noether定理[J]. 云南大学学报(自然科学版), 2019, 41(5): 953-959. doi: 10.7540/j.ynu.20190130
ZHOU Ying, ZHANG Yi. Noether symmetries for fractional generalized Birkhoffian systems in terms of Caputo derivatives[J]. Journal of Yunnan University: Natural Sciences Edition, 2019, 41(5): 953-959. DOI: 10.7540/j.ynu.20190130
Citation: ZHOU Ying, ZHANG Yi. Noether symmetries for fractional generalized Birkhoffian systems in terms of Caputo derivatives[J]. Journal of Yunnan University: Natural Sciences Edition, 2019, 41(5): 953-959. DOI: 10.7540/j.ynu.20190130

基于Caputo导数的分数阶广义Birkhoff系统的Noether定理

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

  • 摘要: 研究基于Caputo导数的分数阶广义Birkhoff系统的Noether定理. 首先,建立分数阶广义Pfaff−Birkhoff原理,导出分数阶广义Birkhoff方程. 其次,研究时间不变的特殊无限小变换下的分数阶Noether对称性与分数阶守恒量,建立分数阶广义Birkhoff系统的Noether定理. 再次,研究时间变化的一般无限小变换下的分数阶Noether对称性与分数阶守恒量,建立分数阶广义Birkhoff系统的Noether定理,并利用时间重参数方法给出其证明. 最后,给出了一个算例以说明其应用.

     

    Abstract: Noether’s theorems for a fractional generalized Birkhoffian system in terms of Caputo derivatives are studied. Firstly, the generalized Pfaff–Birkhoff principle based on Caputo fractional derivatives is established, the fractional generalized Birkhoffian equations are derived. Then, the fractional Noether symmetry and the fractional conserved quantity under special infinitesimal transformations without transforming the time are studied. Noether’s theorem for the fractional generalized Birkhoffian system is established. Once more, the fractional Noether symmetry and the fractional conserved quantity under general infinitesimal transformations with transforming the time are studied, Noether’s theorem for the fractional generalized Birkhoffian system is established. The proof is given by using the time reparametric method. Finally, an example is given to illustrate its application.

     

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