基于非标准Lagrange函数的分数阶Lagrange系统的Noether对称性与守恒量

Noether symmetry and conserved quantities of fractional Lagrange systems based on non-standard Lagrangians

研究基于两类非标准Lagrange函数(指数Lagrange函数和幂律Lagrange函数)的分数阶Lagrange系统的Noether对称性与守恒量.首先,分别导出Caputo分数阶导数下两类非标准Lagrange系统的运动微分方程;其次,根据作用量在无穷小变换下的不变性,给出了分数阶非标准Lagrange系统的Noether对称变换的定义和判据;最后,建立系统的Noether定理并举例说明结果的应用.

The Noether symmetry and conserved quantities of fractional Lagrange systems based on two kinds of non-standard Lagrangians(i.e.,exponential Lagrangian and power-law Lagrangian)are studied.Firstly,the differential equations of motion for two kinds of non-standard Lagrange systems with Caputo fractional derivatives are derived respectively.Secondly,according to the invariance of the action under the infinitesimal transformations,the definitions and criterions of Noether symmetric transformations for fractional non-standard Lagrange systems are given.Finally,Noether’s theorems are established and two examples are given to illustrate the application of the results.