Noether Quasi-symmetry Of Fractional Variational Problems

In this paper,the Noether theorems for the non-conservative mechanical systems based on fractional derivatives are studied.The fractional Noether quasi-symmetry and conserved quantities for non-conservative systems based on Riemann-Liouville derivatives and in terms of combined Caputo derivatives are discussed,the dynamic equations of the mechanical systems on fractional derivatives are established,the definitions and the criterions of the fractional Noether quasi-symmetry for the systems,which from the basic fractional variational formulas of the Lagrange systems on the fractional derivatives,are derived,the fractional Noether theorems are derived.The main content of the paper is divided into:Noether quasi-symmetry for non-conservative systems based on fractional derivatives: the fractional Lagrange equations of the non-conservative systems based on fractional derivatives are established,the definitions and the criterions of the fractional Noether quasi-symmetry for the systems,which from the basic fractional variational formulas of the Lagrange systems on the fractional derivatives,are derived,the Noether theorems of the non-conservative systems on fractional derivatives are derived.Finally,the two special cases,which the generalized nonpotential forces are not exist or the gauge functions are equal to zero,are discussed;Noether quasi-symmetry for nonconservative mechanical systems in phase space based on fractional derivatives: the fractional dynamics equations of the non-conservative systems in phase space are established,the definitions and the criterions of the fractional Noether quasi-symmetry for the systems,which from the basic fractional variational formulas of the Hamilton systems on the fractional derivatives,are derived,the Noether theorems of the non-conservative systems in phase space on fractional derivatives are derived;Finally,the two special cases,which the generalized nonpotential forces are not exist or the gauge functions are equal to zero,are discussed;Noether quasi-symmetry for non-conservative systems in terms of combined Caputo fractional derivatives: the fractional Lagrange equations of the nonconservative systems in terms of combined Caputo derivatives are established,the definitions and the criterions of the fractional Noether quasi-symmetry for the systems,which from the basic fractional variational formulas of the Lagrange systems in terms of combined Caputo derivatives,are derived,the Noether theorems of thenon-conservative systems in terms of combined Caputo derivatives are derived;And the two special cases of the non-conservative systems in terms of combined Caputo derivatives,which the generalized nonpotential forces are not exist or the gauge functions are equal to zero,are discussed;Noether quasi-symmetry for non-conservative systems in phase space based on combined Caputo derivatives: the fractional dynamics equations of the nonconservative systems in phase space in terms of combined Caputo derivatives are established,the definitions and the criterions of the fractional Noether quasi-symmetry for the systems,which from the basic fractional variational formulas of the Hamilton systems in terms of combined Caputo derivatives,are derived,the Noether theorems of the non-conservative systems in phase space in terms of combined Caputo derivatives are derived;And the two special cases of the non-conservative systems in phase space in terms of combined Caputo derivatives,which the generalized nonpotential forces are not exist or the gauge functions are equal to zero,are discussed.