Research on the Symmetry of the Hamiltonian System under Generalized Operators

Generalized operators have recently been proposed with great potential applications. Here, we present research carried out on Noether figury and perturbation to Noether symmetry for Hamiltonian systems within generalized operators. There are four parts, and each part contains two kinds of generalized operator. Firstly, Hamilton equations are established. Secondly, the Noether symmetry method is used for finding the solutions to the differential equations of motion, and conserved quantities are obtained. Thirdly, perturbation to Noether symmetry and adiabatic invariants are further explored. In the end, two examples are given to illustrate the methods and results.