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 conserved quantities of first-order Lagrange systems, the perturbation to the symmetries and adiabatic invariants of first-order Lagrange systems are discussed. Firstly, the concept of higher-order adiabatic invariants of the first-order Lagrange system is proposed. Then, conditions for the existence of the exact and adiabatic invariants are proved, and their forms are given. Finally, an example is presented to illustrate these results.