Abstract:

Several aspects of the inverse problem of the calculus of variations and a short overview of the domestic development of this problem are presented. The main topic is a new method to construct Lagrangians from the first integral of mechanical system, and the equivalent Lagrangians and family of Lagrangians can be obtained by the new method. Some examples are given to illustrate the theoretical significance and the application value of this method. Finally, it is pointed out that great attention should be paid to the study of the inverse problem of the calculus of variations.

Key words: the inverse problem of the calculus of variations, Lagrangian, differential equations of motion, first integral, family of Lagrangians

摘要：

概述变分法逆问题的基本内容和国内在此领域的若干进展。重点阐述由力学系统第一积分构造Lagrange 函数的新方法, 指出利用此方法能够得到等价的 Lagrange 函数和函数族。举例说明该方法的理论意义和应用价值。最后指出, 应当重视变分法逆问题的研究。

关键词: 变分法逆问题, Lagrange 函数, 运动微分方程, 第一积分, Lagrange 函数族