Perturbation to Lie symmetry and another type of Hojman adiabatic invariants for Birkhoffian systems

The perturbation to Lie symmetry and another type of Hojmanadiabatic invariants induced from the perturbation to Lie symmetryfor Birkhoffian systems are studied. The exact invariants of Liesymmetry for the system without perturbation are given. Based on theconcept of adiabatic invariant, the perturbation to Lie symmetry isdiscussed and another new type of Hojman adiabatic invariants thathave the different form from that in [{em Acta Phys. Sin.} {bf 55}3833] for the perturbed system are obtained.