Symmetry and conserved quantities of Hamilton system with comfortable fractional derivatives

In this paper, we present the Noether symmetry and Lie symmetry of fractional Hamilton system by the comfortable fractional derivative. Firstly, we introduce the fractional Noether symmetrical transformations and quasi-symmetrical transformations of the fractional Hamilton system, then, the accordingly criteria for these symmetrical transformations are obtained. Secondly, we investigate the fractional Noether theorems with the comfortable fractional derivative and the fractional Noether conserved quantities are found under the Noether symmetrical transformations and quasi-symmetrical transformations. Thirdly, we further study the fractional Lie symmetry of fractional Hamilton system and the corresponding conserved quantity is given. Finally, an example is given to illustrate the results.