从第一积分构造Lagrange函数的直接方法

提出力学系统Lagrange函数和第一积分之间存在一种新关联,在此基础上给出变分法逆问题的一种新的直接解法.证明系统Lagrange函数可以由带修正因子的第一积分构成,导出修正因子应满足的偏微分方程,运用此解法构建不同系统的Lagrange函数和函数族,并讨论新解法的特点.

A direct approach to construction of the lagrangian from the first integral

A new relation between the Lagranian and the first integral of a mechanical system was presented. On the basis of the relation, a new direct method to solve the inverse problem of the calculus of variations was given. It is proved that a first integral of differential equations with modified factors can be constructed a Lagrangian of the system. The partial differential equations to determine the modified factors were obtained. The new method was used to construct the Lagrangians and the families of Lagrangians of some different systems. The characteristics of the new solution were disscused.