Perturbation To Symmetry And Adiabatic Invariants For Dynamical Stystems With Nonstandard Lagrangians

Nonlinear and nonconservative dynamical systems are one of the most common problems in the natural world.The nonstandard Lagrangians has some properties that the standard Lagrangians doesn't have.And it can be used to describe the nonlinear and nonconservative problems.So it has important theoretical significance and applicable value to study.In this paper,perturbation to Noether symmetry and adiabatic invariants,perturbation to Lie symmetry and adiabatic invariants and perturbation to Mei symmetry and adiabatic invariants for dynamical systems based on exponential Lagrangians and power-law Lagrangians are studied.In the first part,firstly,based on exponential Lagrangians and power-law Lagrangians,and exponential Lagrangians and power-law Lagrangians under the El-Nabulsi models,the exact invariants of Noether type are given.Secondly,started from the definition of high-order adiabatic invariants,the relationship between the perturbation of Noether symmetry and the adiabatic invariants of Noether type of the system under a small disturbance are studied.Finally,the corresponding theorems of adiabatic invariants based on the conditions and forms of the existence of the higher order adiabatic invariants are established.In the second part,firstly,based on exponential Lagrangians and power-law Lagrangians,the exact invariants of the Noether and the Hojman types are given.Secondly,started from the definition of high-order adiabatic invariants,the relationship between the perturbation of Lie symmetry and the adiabatic invariants of the Noether and the Hojman types of the system under a small disturbance are studied.Finally,the corresponding theorems of adiabatic invariants based on the conditions and forms of the existence of the higher order adiabatic invariants are established.In the third part,firstly,based on exponential Lagrangians and power-law Lagrangians,the exact invariants of Mei type are given.Secondly,started from the definition of high-order adiabatic invariants,the relationship between the perturbation of Mei symmetry and the adiabatic invariants of Mei type of the system under a small disturbance are studied.Finally,the corresponding theorems of adiabatic invariants based on the conditions and forms of the existence of the higher order adiabatic invariants are established.