Lie symmetrical perturbation and a new type of non-Noether adiabatic invariants for disturbed generalized Birkhoffian systems

For a generalized Birkhoffian system with the action of small disturbance, the Lie symmetrical perturbation and a new type of non-Noether adiabatic invariants are presented. On the basis of the invariance of disturbed generalized Birkhoffian system under general infinitesimal transformation of group, the determining equation of Lie symmetrical perturbation of the system is constructed. Based on the definition of higher-order adiabatic invariants of a mechanical system, a new type of non-Noether adiabatic invariants of a disturbed generalized Birkhoffian system is obtained by investigating the Lie symmetrical perturbation. Then, a new type of exact invariants of non-Noether type is given, furthermore adiabatic invariants and exact invariants of non-Noether type are obtained under the special infinitesimal transformation of group. Finally, an example is given to illustrate the application of the method and results.