El-Nabulsi动力学模型下Birkhoff系统Noether对称性的摄动与绝热不变量

基于El-Nabulsi动力学模型,研究了小扰动作用下Birkhoff系统Noether对称性的摄动与绝热不变量问题.首先,将El-Nabulsi提出的在分数阶微积分框架下基于Riemann-Liouville分数阶积分的非保守系统动力学模型拓展到Birkhoff系统,建立El-Nabulsi-Birkhoff方程;其次,基于在无限小变换下El-Nabulsi-Pfaff作用量的不变性,给出Noether准对称性的定义和判据,得到了Noether对称性导致的精确不变量;再次,引入力学系统的绝热不变量概念,研究El-Nabulsi动力学模型下受小扰动作用的Birkhoff系统Noether对称性的摄动与绝热不变量之间的关系,得到了对称性摄动导致的绝热不变量的条件及其形式.作为特例,给出了El-Nabulsi动力学模型下相空间中非保守系统和经典Birkhoff系统的Noether对称性的摄动与绝热不变量.以著名的Hojman-Urrutia问题为例,研究其在El-Nabulsi动力学模型下的Noether对称性,得到了相应的精确不变量和绝热不变量.

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

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.
Keywords： Birkhoffian system El-Nabulsi dynamical model perturbation of Noether symmetry adiabatic invariant