非保守动力学系统Noether对称性的摄动与绝热不变量

Perturbation to Noether symmetries and adiabatic invariants for nonconservative dynamic systems

基于非保守系统的El-Nabulsi动力学模型,研究了非保守动力学系统Noether对称性的摄动与绝热不变量问题.首先,引入El-Nabulsi在分数阶微积分框架下基于Riemann-Liouville分数阶积分提出的类分数阶变分问题,列出非保守系统的Euler-Lagrange方程;其次,给出了Noether准对称变换的定义和判据,建立了Noether对称性与不变量之间的关系,得到了精确不变量;最后,提出并研究了该系统受小扰动作用后Noether对称性的摄动与绝热不变量问题,证明了绝热不变量存在的条件及形式,并举例证明结果的应用.

The problem of perturbation to Noether symmetry and adiabatic invariant for a nonconservative dynamic system is studied under a dynamic model presented by E1-Nabulsi. First of all, the fractional action-like variational problem proposed by E1-Nabulsi under the framework of the fractional calculus and based on the definition of the Riemann-Liouville fractional integral is introduced, and the Euler-Lagrange equations of the nonconservative system are given. Secondly, the definition and criterion of the Noether quasisymmetric transformation are given, the relationship between the Noether symmetry and the invariant is established, and the exact invariant is obtained. Finally, the perturbation to the Noether symmetry of the system after the action of a small disturbance and corresponding adiabatic invariant are proposed and studied, the conditions for the existence of adiabatic invariant and the formulation are given. An example is given to illustrate the application of results.