Exact and adiabatic invariants of first-order Lagrange systems

A system of first-order differential equations is expressed in the form of first-order Lagrange equations. Based on the theory of symmetries and conse rved quantities of first-order Lagrange systems, the perturbation to the sy mmetries and adiabatic invariants of first-order Lagrange systems are discu ssed. Firstly, the concept of higher-order adiabatic invariants of the firs t-order Lagrange system is proposed. Then, conditions for the existence of the exact and adiabatic invariants are proved, and their forms are given. F inally, an example is presented to illustrate these results.