Symmetry theories of Hamiltonian systems with fractional derivatives

Abstract

The Noether and Lie symmetries as well as the conserved quantities of Hamiltonian system with fractional derivatives are established. The definitions and criteria for the fractional symmetrical transformations and quasi-symmetrical transformations in the Noether sense of Hamiltonian system are first discussed. Then, using the invariance of Hamiltonian action under the infinitesimal transformations with respect to time, generalized coordinates and generalized momentums, the fractional Noether theorem of the system is obtained. Further, the Lie symmetry and conserved quantity of the system are acquired. Two examples are presented to illustrate the application of the results.