欧拉变分法基本方程不变性思想及其探源

Euler′s thought on the invariance of the fundamental equation of the calculus of variations and its origin

目的廓清欧拉变分法基本方程不变性思想及其产生的历史根源。方法历史分析和文献考证。结果欧拉基本方程不变性思想与18世纪分析学研究对象变革的大背景密切相关。随着分析学逐渐脱离几何传统,抽象的公式或作为解析表达式的函数逐步取代几何曲线,成为分析学研究的基本对象,欧拉基本方程不变性思想是当时正在发生的这种变革的真实写照。结论欧拉基本方程不变性思想,是18世纪分析学研究对象变革的产物,为其后拉格朗日在纯分析基础上创立形式的变分法以及更进一步拓展变分法的力学应用,提供了认识上的先导。

Aim To explore Euler＇s thought on the invariance of the fundamental equation of the calculus of variations and its background. Methods Historic analysis and literatural review. Results Euler＇s thought on the invariance of the fundamental equation of the calculus of variations was closely connected with the change of the subject of analysis in the 18th century. With the separation of analysis subject from geometry, the formula involving letters and numbers or the functions as analytical expressions came the fundamental concept of analysis subject. The gradually replaced the geometric curves and eventually bethought of Euler reflected the change at that time. Conclusion Euler＇s thought of the invariance of the fundamental equation was the result of the change of the subject of analysis in the 18th century. Following this congnitive guidance, Lagrange created his formal method of variation based on pure analysis and further expanded the employments of the calculus of variations into dynamical problems.