Noether Symmetry Method for Hamiltonian Mechanics Involving Generalized Operators

Based on the generalized operators, Hamilton equation, Noether symmetry, and perturbation to Noether symmetry are studied. The main contents are divided into four parts, and every part includes two generalized operators. Firstly, Hamilton equations within generalized operators are established. Secondly, the Noether symmetry method and conserved quantity are studied. Thirdly, perturbation to the Noether symmetry and adiabatic invariant are presented. And finally, two applications are presented to illustrate the methods and results.