Perturbation of symmetries and Hojman adiabatic invariants of discrete mechanical systems in the phase space

Abstract

The perturbation of symmetries and adiabatic invariants of discrete mechanical systems in the phase space are studied. The Hojman exact invariants introduced by the special Lie symmetries of discrete mechanical systems in the phase space without perturbation are given. Based on the definition of high-order adiabatic invariants of a mechanical system in the phase space, the perturbation of Lie symmetries of the system by the action of small disturbance is investigated, and a type of new adiabatic invariants of the system are obtained, which can be called the Hojman adiabatic invariants. An example is given to illustrate the application of the results.