Perturbation to Noether Symmetry and Adiabatic Invariants for Dynamical Systems with Nonstandard Lagrangians

The perturbation to Noether symmetry and adiabatic invariants for dynamical systems with nonstandard Lagrangians are studied. Based on two kinds of nonstandard Lagrangians( i. e.exponential Lagrangians and power-law Lagrangians),the exact invariants of Noether type are given. Based on the definition of highorder adiabatic invariants,the relationship between the perturbation of Noether symmetry and the adiabatic invariants of the system under a small disturbance is studied,and then the corresponding theorems of adiabatic invariants are established. Finally, two examples are given to illustrate the methods and results appear in this paper.