Perturbation To Symmetries And Adiabatic Invariants For Nonholonomic Mechanical Systems

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. There exists intimate relation between the integrability of the system and the variations of its symmetries as well as the adiabatic invariants under the action of small disturbance. Therefore, the studies on perturbation to symmetries and adiabatic invariants, which baced on studies on symmetries and exact invariants, are of great significance. In this paper, we study the perturbation to symmetries and adiabatic invariants for nonholonomic mechanical systems. First, perturbation to Noether symmetry, Mei symmetry and Lie symmetry as well as their adiabatic invariants for nonholonomic mechanical systems are studied. The exact invariants induced respectively from the three symmetries of the system without perturbation are given. Based on the concept of adiabatic invariant, perturbations to these symmetries are discussed and the adiabatic invariants of the perturbed system are obtained. Then, perturbation to Noether-Mei symmetry and Noether-Lie symmetry as well as their adiabatic invariants for nonholonomic mechanical systems are studied. The exact invariants induced respectively from the two united symmetries of the system without perturbation are given. Perturbations to these symmetries are discussed and the adiabatic invariants of the perturbed system are obtained. Finally, Lie symmetries and conserved quantities of the nonholonomic Vacco dynamical systems are studied; perturbation to Lie symmetry and Hojman adiabatic invariants for nonholonomic Vacco dynamical systems are discussed.