Relationships among the three symmetries——Noether symmetry,Lie symmetry and M ei symmetry of a relativistic Birkhoff system have been obtained. Some stabilit y criteria are given by using Liapunov method. Finally the conservation quanti ty and the stability of the relativistic Birkhoff system are correlated by the Energy－Casimir functional, and an example is given, to illustrate the applicat ion of the method.