Lie symmetries, perturbation to symmetries and adiabatic invariants of Lagrange system

Based on the invariance of differential equations under infinitesimal transformations, Lie symmetries, laws of conservations, perturbation to the symmetries and adiabatic invariants of Lagrange system are presented. Firstly, the concepts of Lie symmetries and higher order adiabatic invariants of the Lagrange system are proposed. Then, the conditions for existence of the exact invariants and adiabatic invariants are proved, and their forms are given. At last, an example is presented to illustrate these results.