Fractional Birkhoffian Dynamics Based on Quasi-fractional Dynamics Models and Its Lie Symmetry  

基于准分数阶模型的分数阶Birkhoff动力学及其Lie对称性

作　　者：JIA Yundie ZHANG Yi 贾云碟;张毅(苏州科技大学数学科学学院,苏州215009;苏州科技大学土木工程学院,苏州215011)

机构地区：[1]College of Mathematical Sciences,Suzhou University of Science and Technology,Suzhou 215009,P.R.China [2]College of Civil Engineering,Suzhou University of Science and Technology,Suzhou 215011,P.R.China

出　　处：《Transactions of Nanjing University of Aeronautics and Astronautics》2021年第1期84-95,共12页南京航空航天大学学报（英文版）

基　　金：supported by the National Natural Science Foundation of China (Nos.11972241,11572212 and 11272227);the Natural Science Foundation of Jiangsu Province(No. BK20191454)。

摘　　要：In order to investigate the dynamic behavior of non-conservative systems,the Lie symmetries and conserved quantities of fractional Birkhoffian dynamics based on quasi-fractional dynamics model are proposed and studied.The quasi-fractional dynamics model here refers to the variational problem based on the definition of RiemannLiouville fractional integral(RLFI),the variational problem based on the definition of extended exponentially fractional integral(EEFI),and the variational problem based on the definition of fractional integral extended by periodic laws(FIEPL).First,the fractional Pfaff-Birkhoff principles based on quasi-fractional dynamics models are established,and the corresponding Birkhoff’s equations and the determining equations of Lie symmetry are obtained.Second,for fractional Birkhoffian systems based on quasi-fractional models,the conditions and forms of conserved quantities are given,and Lie symmetry theorems are proved.The Pfaff-Birkhoff principles,Birkhoff’s equations and Lie symmetry theorems of quasi-fractional Birkhoffian systems and classical Birkhoffian systems are special cases of this article.Finally,some examples are given.为了探究非保守系统的动力学行为,该文提出并研究基于准分数阶动力学模型的分数阶Birkhoff动力学的Lie对称性和守恒量。准分数阶动力学模型是指基于Riemann-Liouville分数阶积分定义的变分问题、基于按指数律扩展的分数阶积分定义的变分问题和基于按周期函数律拓展的分数阶积分定义的变分问题。首先,建立了基于准分数阶模型的分数阶Pfaff-Birkhoff原理,得到了相应的Birkhoff方程和Lie对称性确定方程。其次,对于基于准分数阶模型的分数阶Birkhoff系统,给出了守恒量的条件和形式,并证明了Lie对称性定理。准分数阶Birkhoff系统与经典Birkhoff系统的Pfaff-Birkhoff原理、Birkhoff方程和Lie对称定理均是该文的特例。最后,给出了若干算例。

关 键 词：quasi-fractional dynamics model Lie symmetry conserved quantity fractional Birkhoffian system Riemann-Liouville derivative 

分 类 号：O316[理学—一般力学与力学基础]