Jacobi最后乘子在二阶微分方程Lagrange结构中的应用  

Application of Jacobi Last Multipliers in Lagrangian Structures of Second-Order Differential Equations

作　　者：彭新龙 章海 PENG Xinong;ZHANG Hai(School of Mathematics and Physics,Anqing Normal University,Anqing 246133,China)

机构地区：[1]安庆师范大学数理学院,安徽安庆246133

出　　处：《安庆师范大学学报（自然科学版）》2023年第3期20-24,共5页Journal of Anqing Normal University(Natural Science Edition)

基　　金：国家自然科学基金项目(11701009)。

摘　　要：寻找微分方程的变分结构即Lagrange结构对于理解方程的性质非常重要。本文针对一些二阶常微分方程,通过求解它们的Jacobi最后乘子并结合方程的Lie对称,获得了这些方程的Lagrange函数,从而将其转化为某些变分问题的运动方程。此外,我们推导了这些微分方程的多个Jacobi最后乘子和多个Lagrange函数,因而证明了常见微分方程有很多个不等价的Lagrange结构。Finding the variational structure,also known as the Lagrangian structure,is crucial for understanding the prop-erties of differential equations.For certain second-order ordinary differential equations,their Lagrangian functions are ob-tained by working out their Jacobi last multipliers and employing the Lie symmetries of the equations.Therefore,theses equa-tions are transformed into the equations of motion of some variational problems.We have derived several Jacobi last multipli-ers and Lagrangian functions for these differential equations,thus proving that usual differential equations have many non-equivalent Lagrangian structures.

关 键 词：Jacobi最后乘子 二阶常微分方程 LAGRANGE函数 LIE对称 

分 类 号：O411.1[理学—理论物理]