广义分数阶算子下非标准Lagrange系统的Noether定理

Noether Theorem for Non-standard Lagrangian System under Generalized Fractional Operators

自然界中几乎不存在简单的线性动力学系统,大多数都是以非保守非线性动力学系统的形式存在,而非标准Lagrange函数可以用于非保守非线性问题的动力学建模.同时,分数阶模型也是研究复杂动力学及物理行为的较好选择.因此本文研究了广义分数阶算子下非标准Lagrange系统的Noether对称性与守恒量.首先,建立广义分数阶算子下非标准Lagrange系统的Lagrange方程,然后基于Hamilton作用量在无限小变换下的不变性,建立广义分数阶算子下非标准Lagrange系统的Noether定理,并给出该系统的对称性及相应的守恒量.在特定条件下广义分数阶算子下非标准Lagrange系统的Noether守恒量可以退化为整数阶非标准Lagrange系统的Noether守恒量,最后举例说明所得结果的具体应用.

There are almost no simple linear dynamic systems in nature,and most of them exist in the form of non-conservative nonlinear dynamic systems.Non-standard Lagrange functions can be used for dynamic modeling of non-conservative nonlinear problems.The fractional model is also a good choice for studying complex dynamics and physical behavior.Therefore,this paper studies the Noether symmetry and conserved quantity of non-standard Lagrange systems under generalized fractional operators.Firstly,the Lagrange equation of non-standard Lagrangian system under generalized operator is established.Then,based on the invariance of Hamilton action under infinitesimal transformation,the Noether theorem of non-standard Lagrangian system under generalized fractional operator is established,and the symmetry and corresponding conserved quantity of the system are given.Under certain conditions,the Noether conserved quantities of non-standard Lagrangian systems under generalized fractional operators can be reduced to the Noether conserved quantities of non-standard Lagrangian systems of integer order.Finally,examples are given to illustrate the specific application of the obtained results.