Lie symmetrical perturbation and adiabatic invariants of generalized Hojman type for Lagrange systems

Based on the invariance of differential equations underinfinitesimal transformations of group, Lie symmetries, exactinvariants, perturbation to the symmetries and adiabatic invariantsin form of non-Noether for a Lagrange system are presented. Firstly,the exact invariants of generalized Hojman type led directly by Liesymmetries for a Lagrange system without perturbations are given.Then, on the basis of the concepts of Lie symmetries and higherorder adiabatic invariants of a mechanical system, the perturbationof Lie symmetries for the system with the action of smalldisturbance is investigated, the adiabatic invariants of generalizedHojman type for the system are directly obtained, the conditions forexistence of the adiabatic invariants and their forms are proved.Finally an example is presented to illustrate these results.