CaputoΔ型分数阶时间尺度Noether定理

CAPUTOΔ-TYPE FRACTIONAL TIME-SCALES NOETHER THEOREM

时间尺度理论将微分方程理论和差分方程理论融合于一体,而分数阶微积分可以为实际问题提供更为切合的模型.分数阶时间尺度微积分因能统一研究连续分数阶系统和离散分数阶系统而备受关注.结合时间尺度和分数阶微积分,研究含CaputoΔ导数的分数阶时间尺度Noether定理,为研究复杂系统动力学行为提供了一个新的视角.首先,回顾了分数阶时间尺度积分和导数的定义.其次,根据所提出的CaputoΔ型分数阶时间尺度Hamilton原理,导出了分数阶时间尺度Lagrange方程.在特定条件下,此方程可退化为时间尺度Lagrange方程、Caputo型分数阶Lagrange方程和经典Lagrange方程.进一步地,在特殊无限小变换和一般无限小变换两种情形下,分别给出了CaputoΔ型分数阶时间尺度Noether对称性的定义和判据.继而,提出并证明了特殊无限小变换下的分数阶时间尺度Noether定理(定理1)和一般无限小变换下的分数阶时间尺度Noether定理(定理2).当α=1时,定理1则退化为特殊无限小变换下的经典时间尺度Noether定理,并且定理2成为利用广义Jost方法所得到的时间尺度Noether定理.此外,当T=R时,定理2还可退化为Caputo型分数阶Noether定理.最后,以平面上的分数阶时间尺度Kepler问题和单自由度分数阶时间尺度线性振动系统为例来验证定理的正确性.

The time-scales theory combines differential equations theory with difference equations theory,and fractional calculus can provide more realistic models in practical problems.The fractional time-scales calculus has attracted much attention because it can unify continuous fractional systems and discrete fractional systems.Combining the time-scales calculus and the fractional calculus,we focus on the fractional time-scales Noether theorem with CaputoΔ-derivatives,which provides a new perspective for studying the dynamic behaviors of complex systems.This paper begins with a review of the definitions of fractional time-scales integrals and derivatives.Then,according to the proposed CaputoΔ-type fractional time-scales Hamilton principle,the fractional time-scales Lagrange equation is derived.Under certain conditions,the fractional time-scales Lagrange equation can be reduced to the time-scales Lagrange equation,the Caputo-type fractional Lagrange equation and the classical Lagrange equation.Furthermore,in the two cases of special infinitesimal transformations and general infinitesimal transformations,the definitions and criteria of CaputoΔ-type fractional time-scales Noether symmetries are given.In addition,the fractional time-scales Noether theorem under special infinitesimal transformations(Theorem 1)and the fractional time-scales Noether theorem under general infinitesimal transformations(Theorem 2)are proposed and proved.Whenα=1,Theorem 1 can be reduced to the classical timescales Noether theorem under special infinitesimal transformations,and Theorem 2 becomes the time-scales Noether theorem obtained by the generalized Jost method.Not only that,but Theorem 2 can be reduced to the Caputo-type fractional Noether theorem if T=R.At the end of this paper,the fractional time-scales Kepler problem in the plane and the fractional time-scales single freedom linear vibration system are taken as examples to verify the correctness of theorems.