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

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.

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 γ=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.