A procedure for finding first integrals of mechanical systems with gauge-variant Lagrangians

In this paper a procedure is given for finding first integrals of non-conservative mechanical systems in which the Lagrangian is gauge-variant under infinitesimal transformations as functions of time, position and generalized velocities. The procedure is founded on the generalized Noetherian theorem. The existence of first integrals depends on the existence of solutions of the system of partial differential equations. Two examples are analyzed in detail using this theory. The considerations are based on mechanical systems but the results obtained can be used on all problems in physics, engineering and mathematics for which one can construct an action integral in Hamilton's sense.