Conserved quantities for Hamiltonian systems on time scales

Conserved quantities for Hamiltonian systems on time scales with nabla derivatives and delta derivatives are presented. First, Hamilton principle on time scales with nabla derivatives is established and Hamilton canonical equation with nabla derivatives is obtained. Second, Noether identity and Noether theorem for Hamiltonian systems with nabla derivatives are achieved. Third, Hamilton canonical equation with delta derivatives, Noether identity and Noether theorem for Hamiltonian systems with delta derivatives are gotten through duality principle on the basis of the corresponding results with nabla derivatives. Fourth, some special cases of Noether identity and Noether theorem are given. And finally, two examples are devoted to illustrate the methods and results.