Perturbation to Mei symmetry and Mei adiabatic invariants for mechanical systems in phase space

For a perturbed mechanical system in phase space, considering${tilde{{rm d}}}/{ {{rm d}t}}$ in the structure equation andprocess of proof including infinitesimal parameter $varepsilon $obviously, this paper studies the perturbation to Mei symmetry andadiabatic invariants. Firstly, the exact invariant induced directlyfrom the Mei symmetry of the system without perturbation is given.Secondly, based on the concept of high-order adiabatic invariant,the determining equations of the perturbation to Mei symmetry areestablished, the condition of existence of the Mei adiabaticinvariant led by the perturbation to Mei symmetry is obtained, andits form is presented. Lastly, an example is given to illustrate theapplication of the results.