Symmetry And Conserved Quantity Of Fractional Birkhoffian System Based On Quasi-fractional Model

Almost all observable physical phenomena or dynamic processes in nature are non-conservative.In order to study the non-conservative mechanics better,El-Nabulsi proposed three kinds of non-conservative dynamic models,i.e.quasi-fractional model.As a generalization of integer order Birkhoffian system,fractional Birkhoffian system can more accurately describe the physical behavior of the corresponding system.Therefore,based on the quasi-fractional dynamics model,this paper presents and studies the fractional symmetry and conservation quantity of the quasi-fractional dynamics model in Birkhoffian system.In this paper,the quasi-fractional models based on Riemann-Liouville integral definition,exponential law integral definition and periodic function law integral definition given by El-Nabulsi are studied respectively.The fractional Noether symmetry,fractional Lie symmetry and fractional Mei symmetry of Birkhoffian system are studied respectively,and their conservation is derived.1.The variational problems and Birkhoff's equations of fractional Birkhoffian system are studied in the quasi-fractional model.Based on Riemann-Liouville fractional derivative,the fractional Pfaff-Birkhoff principles based on quasi-fractional model are established,and the differential equations of motion of fractional Birkhoffian system are obtained.2.The Noether symmetries of fractional Birkhoffian system are studied under quasi-fractional model.Based on the invariance of the fractional Pfaff action in the quasi-fractional model under infinitesimal transformation,the definitions of Noether symmetry transformation and quasi-symmetric transformation of the fractional Birkhoffian system are established.The fractional Noether symmetry theorems of the system are given.3.The Lie symmetries of fractional Birkhoffian system are studied under quasi-fractional model.Based on the invariance of the motion differential equation of fractional Birkhoffian system based on the quasi-fractional model under infinitesimal transformation,the Lie symmetry determination equations of the fractional Birkhoffian system are established,the definitions and criterions of the corresponding fractional Lie symmetry are given,and the Lie symmetry theorems of the system are derived.4.The Mei symmetries of fractional Birkhoffian system are studied under quasi-fractional model.Based on the form invariance of fractional Birkhoff's equation with quasi-fractional model after its dynamic function has undergone infinitely small transformation,the definitions and criterions of Mei symmetry of the fractional Birkhoffian system are established,and the fractional Mei symmetry theorems of the system are given.5.The Noether symmetries of fractional generalized Birkhoffian system are studied under quasi-fractional model.Based on Riemann-Liouville fractional derivative,the fractional generalized Pfaff-Birkhoff principles based on quasi-fractional model are established,and the motion differential equations of fractional generalized Birkhoffian system are obtained.Secondly,the definitions and criterions of Noether symmetric transformation and quasi-symmetric transformation of the fractional generalized Birkhoffian system are given.Finally,the fractional Noether symmetry theorems of the system are obtained.