基于经典和Riesz导数的分数阶广义Birkhoff系统的Noether定理  

Noether’s theorems for fractional generalized Birkhoffian systems in terms of classical and Riesz derivatives

作　　者：周颖[1] 张毅[2] Zhou Ying;Zhang Yi(School of Mathematical Science,Suzhou University of Science and Technology,Suzhou 215009,China;School of Civil Engineering,Suzhou University of Science and Technology,Suzhou 215011,China)

机构地区：[1]苏州科技大学数学科学学院,江苏苏州215009 [2]苏州科技大学土木工程学院,江苏苏州215011

出　　处：《南京理工大学学报》2021年第5期621-628,共8页Journal of Nanjing University of Science and Technology

基　　金：国家自然科学基金(11972241,11572212,11272227);江苏省自然科学基金(BK20191454)。

摘　　要：为了研究分数阶模型下Birkhoff系统的对称性与守恒量之间的内在联系,该文提出并证明含经典和Riesz导数(包括Riesz-Riemann-Liouville导数和Riesz-Caputo导数)的分数阶广义Birkhoff系统的Noether定理。基于经典和Riesz导数的分数阶广义Pfaff-Birkhoff原理,导出相应的分数阶广义Birkhoff方程。分析系统的Noether对称性与守恒量,采用时间重新参数化方法证明分数阶Noether定理,并利用“传递公式”给出了分数阶守恒量的显形式。最后给出一个算例以说明其应用。In order to study the internal relation between symmetry and conserved quantity of Birkhoffian systems under fractional models,this paper proposes and proves Noether’s theorems for fractional generalized Birkhoffian systems in terms of classical and Riesz derivatives(including Riesz-Riemann-Liouville derivatives and Riesz-Caputo derivatives).The fractional generalized Pfaff-Birkhoff principles based on classical and Riesz derivatives are established,and the corresponding fractional generalized Birkhoff equations are derived.The Noether symmetry and conserved quantities of the systems are analyzed,and the fractional Noether’s theorems are proved by using the time re-parameterization method,and the explicit form of fractional conserved quantity is given by using"transfer formula".Finally,an example is given to illustrate its application.

关 键 词：广义BIRKHOFF系统 NOETHER对称性 分数阶守恒量 分数阶微积分 Riesz导数 

分 类 号：O316[理学—一般力学与力学基础]