牛顿求解最小阻力体问题的思想辨析

Newton’s Ideas for Solving the Problem of the Solid of Least Resistance

牛顿在撰写《原理》时曾提出和研究了历史上第一个变分法问题——最小阻力体问题,并先后给出两种解法(1685年和1694年),在此过程中萌生了变分法的一些重要原始思想.以《原理》相关内容和《牛顿数学手稿》提供的信息为基础,重点对牛顿关于此问题的两种解法中所蕴含的数学思想和技术进行分析与比较.结果表明,两种解法在思想上,特别是在表现形式和曲线变分方式上存有差异:1685年解法立足极值局部化和泛函稳定性等新思想,采取了通过曲线上个别点处某个坐标变化的曲线局部变分技术,这些思想和技术对于求解变分问题具有一定的普遍性,是构成后来变分法中欧拉方法的核心思想与基本技术;1694年的解法则完全局限于普通极值思想和方法的框架之内,表现出了较强的技巧性.

The problem of the solid of least resistance,being the first problem of the calculus of variations,was formulated by Newton at a time when he was writing his Principia.Newton gave two solutions to the problem in 1685 and 1694 respectively,and in doing so,he conceived some important original ideas of the calculus of variations.Based on Principia and The Mathematical Papers of Isaac Newton,the two solutions mentioned above are comparatively investigated and some differences between them are recognized in this paper.The conclusion is that the 1685 solution is more general because it involves the following new ideas and technique: the localization of the extremum,the stability of the functional and the local variation of the curve,by means of changing an ordinate of one point in it,which constituted the core elements of the later Euler’s method in the calculus of variations while the 1694 solution is entirely confined to the framework of the ordinary function extremum.