Perturbation to Noether symmetries and adiabatic invariants for Birkhoffian systems based on El-Nabulsi dynamical models

Abstract

In this paper, we study the problem of perturbation to Noether symmetries and adiabatic invariants for a Birkhoffian system under small disturbance based on the El-Nabulsi dynamical model. First, the dynamical model presented by El-Nabulsi, which is based on the Riemann-Liouville fractional integral under the framework of the fractional calculus, is extended to the Birkhoffian system, and El-Nabulsi-Birkhoff equations for the Birkhoffian system are established. Then, by using the invariance of the El-Nabulsi-Pfaff action under the infinitesimal transformations, the definition and criterion of the Noether quasi-symmetric transformation are given, and the exact invariant caused directly by the Noether symmetry is obtained. Furthermore, by introducing the concept of high-order adiabatic invariant of a mechanical system, the relationship between the perturbation to the Noether symmetry and the adiabatic invariant after the action of small disturbance is studied, the condition that the perturbation of symmetry leads to the adiabatic invariant and its formulation are presented. As a special case, the perturbation to Noether symmetries and corresponding adiabatic invariants mechanics of non-conservative systems in phase space under El-Nabulsi models and classical Birkhoffian systems are discussed. At the end of the paper, taking the well-known Hojman-Urrutia problem for example, its Noether symmetries under the El-Nabulsi dynamical model is investigated and corresponding exact invariants and adiabatic invariants are presented.