Perturbation to symmetries and adiabatic invariant for nonholonomic controllable mechanical system in phase place

Abstract

This paper studies the perturbation to symmetries and the adiabatic invariant for nonholonomic controllable mechanical system in the phase space. The exact invariants introduced by the form invariance of the nonholonomic controllable mechanical system in phase space without perturbation are given. Based on the definition of high-order adiabatic invariants of the mechanical system, the perturbation to symmetries and the adiabatic invariant for nonholonomic controllable mechanical system in phase space under the action of small disturbance is investigated, then the form of high-order adiabatic invariants and the conditions for their existence are presented. An example is finally given to illustrate the application of the results.