Noether's theorems for dynamical systems of two kinds of non-standard Hamiltonians

This paper focuses on studying Noether's theorems for dynamical systems with two kinds of non-standard Hamiltonians, respectively, namely exponential Hamiltonian and power-law Hamiltonian. Firstly, the differential equations of motion for dynamical systems with exponential Hamiltonian and power-law Hamiltonian are established. Secondly, according to the invariance of the action under the infinitesimal transformations, the definitions and criteria of Noether symmetric transformations and Noether quasi-symmetric transformations are given. Then, Noether's theorems for dynamical systems with exponential Hamiltonian and power-law Hamiltonian are obtained, respectively. Finally, two examples are given to illustrate the applications of the results.