Research Of The Fractional Pfaff-Birkhoff Variational Problem And Symmetry Of Variable Order

In this paper,the Noether symmetries and corresponding conserved quantities of the variable order fractional Birkhoffian systems and the variable order fractional generalized Birkhoffian systems are presented and studied in terms of three different variable order fractional models,i.e.,variable order fractional Riemann-Liouville,variable order fractional Caputo,variable order fractional Riesz.The fractional Pfaff-Birkhoff principle of variable order and the fractional generalized Pfaff-Birkhoff principle of variable order are established.Then corresponding variable order fractional Birkhoff's equations and variable order fractional generalized Birkhoff's equations are deduced.Furthermore,the Noether's theorems for the variable order fractional Birkhoffian systems and the variable order fractional generalized Birkhoffian systems are established.In first part,the fractional Pfaff-Birkhoff principle is established in terms of fractional Riemann-Liouville derivatives of variable order,and the variable order fractional Birkhoff's equations are deduced.The fractional Pfaff-Birkhoff variational problems of variable order are studied,and the Noether's theorem for fractional Birkhoffian systems of variable order is established.Then,the fractional generalized Pfaff-Birkhoff principle of variable order is presented,and the variable order fractional generalized Birkhoff's equations are given.Moreover,the Noether's theorem for the variable order fractional generalized Birkhoffian systems is established.In second part,the fractional Pfaff-Birkhoff principle is established in terms of fractional Caputo derivatives of variable order,and the variable order fractional Birkhoff's equations are deduced.The fractional Pfaff-Birkhoff variational problems of variable order are studied,and the Noether's theorem for fractional Birkhoffian systems of variable order is established.Then,the fractional generalized Pfaff-Birkhoff principle of variable order is presented,and variable order fractional generalized Birkhoff's equations are given.Moreover,the Noether's theorem for variable order fractional generalized Birkhoffian systems is established.In third part,the fractional Pfaff-Birkhoff principle is established in terms of fractional Riesz-Riemann-Liouville derivatives of variable order and fractional Riesz-Caputo derivatives of variable order,and the corresponding variable order fractional Birkhoff's equations are deduced.The fractional Pfaff-Birkhoff variational problems of variable order are studied,and the Noether's theorems for fractionalBirkhoffian systems of variable order are established.Then,the fractional generalized Pfaff-Birkhoff principles of variable order are presented,and variable order fractional generalized Birkhoff's equations are given.Moreover,the Noether's theorems for variable order fractional generalized Birkhoffian systems are established.