| The Mechanical system in terms of quasi-coordinates means that the differential equation of motion for the system is expressed by quasi-coordinates and quasi-velocity. The studies on symmetries and exact invariants (conserved quantities) play a very important role in mechanics and physics. As we know even a small change, that we can call a perturbation, may influence the original symmetries and exact invariants of mechanical systems. Therefore, the studies on perturbation to symmetries and adiabatic invariants, which based on studies on symmetries and exact invariants, are of great significance. In this paper, we study the perturbation to symmetries and adiabatic invariants for mechanical systems in terms of quasi-coodinates. First, perturbation to Noether symmetry, Lie symmetry and Mei symmetry as well as their adiabatic invariants for mechanical systems in terms of quasi-coodinates are studied, the exact invariants induced respectively from three symmetries without perturbation are given, perturbations to these symmetries are discussed and the adiabatic invariants of the perturbed system are obtained. Then, perturbations to united symmetries of the system are studied, the exact invariants induced from Noether-Mei symmetry and Noether-Lie symmetry are given, perturbations to the two united symmetries are discussed and the adiabatic invariants of the perturbed system are obtained. Finally, the new types of adiabatic invariants deduced from Noether symmetry and Mei symmetry of the mechanical systems in terms of quasi-coordinates are studied, the conditions for the new type exact invariants exist and the forms of the new types of exact invariants are given, the new type Noether adiabatic invariant deduced form perturbation to Noether symmetry and the new type Mei adiabatic invariant deduced form perturbation to Mei symmetry of the system are discussed. |
