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

摘要: 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.

Abstract: 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.