Perturbation to the symmetries of relativistic Birkhoffian systems and the inver se problems
相对论性Birkhoff系统的对称性摄动及其逆问题

In this paper, we have studied the positive problems of perturbation to the symmetries for relativistic Birkhoffian systems under the action of a small force of the disturbance. The fundamental theory, equations of motion and equation of small disturbance of relativistic Birkhoffian systems were established. Lie symmetries and conserved quantity of this systems were given. Perturbation to the symmetries of the systems under infinitesimal transformations, s-order adiabatic invariants, existence conditions and the form of adiabatic invariants were studied. We have also studied the inverse problems of perturbation to the symmetries for relativistic Birkhoffian systems. We obtain the perturbation to the symmetries of the systems under infinitesimal transformations, when the systems possess s-order adiabatic invariants. The relationship between relativistic Birkhoffian systems and classical Birkhoffian systems were studied. Finally an example was presented to illustrate the result.