基于按正弦周期律拓展的分数阶积分的变分问题的Noether定理

Noether's Theorem for Variational Problem Based on Fractional Integral Extended by Sine Periodic Law

基于按正弦周期律拓展的分数阶积分的类分数阶动力学建模方法,研究完整系统的类分数阶Noether对称性和守恒量。首先,基于按正弦周期律拓展的分数阶积分,建立了类分数阶变分问题,导出了类分数阶d'Alembert-Lagrange原理,给出了类分数阶Euler-Lagrange方程;其次,基于类分数阶Hamilton作用量在无限小群变换下的不变性,提出了类分数阶Noether对称变换和Noether准对称变换的定义和判据;最后,建立了类分数阶Noether定理,揭示了系统的Noether对称性与守恒量之间的关系,并举例说明结果的应用。

Based on the fractional integral extended by sine periodic law introduced by EI-Nabulsi, the fractional action-like Noether symmetries and conserved quantities for holonomic systems are studied. First, the fractional action-like variational problem based on the fractional integral extended by sine peri- odic law is established, the fractional action-like d＇Alembert-Lagrange principle is deduced, as well as the fractional action-like Euler-Lagrange equations is obtained. Secondly, the definitions and criteria of the fractional action-like Noether＇s （quasi-） symmetrical transformations are presented in terms of the in- variance of the fractional action-like Hamilton action under the infinitesimal transformations of group. Fi- nally, fractional action-like Noether＇s theorem for holonomic systems is explored, the relationship between the Noether symmetry and the conserved quantity of the system is revealed, and two examples are given to illustrate the application of the results.