基于联合Caputo导数的分数阶Lagrange系统的Noether对称性

Noether symmetries for fractional Lagrange systems in terms of combined Caputo derivatives

提出并研究了基于联合Caputo导数的分数阶Lagrange系统的Noether对称性与守恒量。首先,建立了力学系统的分数阶Hamilton原理,并进一步得到了分数阶Lagrange方程;其次,根据系统的分数阶Hamilton作用量在无限小群变换下的不变性,给出了分数阶Noether对称变换和分数阶Noether准对称变换的定义和判据;最后,建立了分数阶Noether对称性与守恒量之间的联系;文末,举例说明结果的应用。

This paper has proposed and studied the Noether symmetries and the conserved quantities for fractional Lagrange systems based on combined Caputo fractional derivatives. Firstly, the fractional Hamilton variational principle for the Lagrange systems was established, and the fractional Lagrange equations were obtained. Secondly, based upon the invariance of the fractional Hamilton action under the infinitesimal transformation of groups, we provided the definition and criterion of the fractional Noether symmetry transformation and the fractional Noether quasi-symmetry transformation for the system. Finally, the relation between the fractional Noether symmetries and the conserved quantities was established. At the end of the paper, two examples were given to illustrate the application of the results.