丁金凤, 张毅. 事件空间中一类拟分数阶Birkhoff系统的Noether定理[J]. 云南大学学报(自然科学版), 2020, 42(4): 673-678. doi: 10.7540/j.ynu.20190329
引用本文: 丁金凤, 张毅. 事件空间中一类拟分数阶Birkhoff系统的Noether定理[J]. 云南大学学报(自然科学版), 2020, 42(4): 673-678. doi: 10.7540/j.ynu.20190329
DING Jin-feng, ZHANG Yi. Noether’s theorems for a type of quasi–fractional Birkhoffian systems in even space[J]. Journal of Yunnan University: Natural Sciences Edition, 2020, 42(4): 673-678. DOI: 10.7540/j.ynu.20190329
Citation: DING Jin-feng, ZHANG Yi. Noether’s theorems for a type of quasi–fractional Birkhoffian systems in even space[J]. Journal of Yunnan University: Natural Sciences Edition, 2020, 42(4): 673-678. DOI: 10.7540/j.ynu.20190329

事件空间中一类拟分数阶Birkhoff系统的Noether定理

Noether’s theorems for a type of quasi–fractional Birkhoffian systems in even space

  • 摘要: 基于按指数律拓展的分数阶积分,研究事件空间中拟分数阶Birkhoff系统的Noether对称性与守恒量. 首先,基于按指数律拓展的分数阶积分定义,给出事件空间中拟分数阶Pfaff作用量,建立事件空间中拟分数阶Pfaff–Birkhoff原理,并导出Pfaff–Birkhoff–d’Alembert原理,得到事件空间中拟分数阶Birkhoff系统的运动微分方程. 其次,计算Pfaff作用量的全变分,给出事件空间中拟分数阶Pfaff作用量的两个变分公式. 建立事件空间中拟分数阶Birkhoff系统的Noether对称性的定义和判据. 最后,建立事件空间中拟分数阶Birkhoff系统的Noether定理,揭示了系统的Noether对称性与守恒量之间的内在联系. 如果分数阶时间积分参数γ=1,则该定理退化为经典的事件空间中Birkhoff系统的Noether定理. 文末举例说明结果的应用.

     

    Abstract: Based on the fractional integral extended by exponential law, Noether symmetries and conserved quantities of quasi−fractional Birkhoffian systems in event space have been studied. Firstly, based on the definition of fractional integral extended by exponential law, the quasi−fractional Pfaff action in event space is presented, the quasi−fractional Pfaff−Birkhoff principle in event space is established, and the Pfaff−Birkhoff−d'Alembert principle is derived, and the differential equations of motion for quasi−fractional Birkhoffian systems in event space are obtained. Secondly, the total variation of Pfaff action is calculated, and two variational formulas of quasi−fractional Pfaff action in event space are given. The definition and criterion of Noether symmetry of quasi−fractional Birkhoffian system in event space are established. Finally, Noether’s theorems for quasi−fractional Birkhoffian systems in event space are established which reveal the internal relationship between Noether symmetries and conserved quantities. If the fractional time integral parameter \gamma = 1, the theorems degrade to Noether’s theorems of classical Birkhoffian systems in event space. At the end of the paper, an example is given to illustrate the application of the results.

     

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