Historical Studies on the systematization of classical mechanics in the 18^<th> century

经典力学通常被认为是由艾萨克·牛顿在17世纪末创立的，但他在《原理》一书中的数学论点是几何的，与我们的非常不同。在这项研究中，我们讨论了18世纪经典力学解析和系统体系的建立过程，特别是在这一过程中起主要作用的莱昂哈德·欧拉的力学理论。欧拉首先使用了系统的方法，即确定在空间中固定的坐标轴，并推导出运动方程作为每个坐标的二次微分方程。由此，他还得到了“Vis viva”(动能)、线动量和角动量守恒定律。他进一步将他的方法的对象从质点扩展到刚体，并导出了所谓的刚体欧拉方程。这种从力学到建立在解析微积分基础上的新力学的转变发生在17世纪40年代下半叶，欧拉对行星和刚体的运动进行了研究。非常重要的是，欧拉运动方程的数学公式在这一时期发生了很大的变化。在他年轻的时候，他从伽利略的自由落体定律推导出了运动方程，它具有空间一阶微分方程的特殊公式。后来，它的公式变成了二阶坐标微分方程，伽利略定律的影响减弱了，但我们可以从他的运动方程的因子2和他的力学单位系来区分它的影响。

Classical mechanics is generally regarded as having been founded by Isaac Newton in the late 17th century, but his mathematical arguments in Principia are geometrical and very different from those of ours. In this study, we discuss the process of constructing the analytic and systematic system of classical mechanics in the 18th century, particularly the mechanical theory of Leonhard Euler who played the principal role in this process.Euler firstly used the systematic method of setting the coordinate axes fixed in space and deriving the equation of motion as the second-differential equation for each coordinate. From it, he also the conservation laws of "vis viva" (kinetic energy), linear momentum and angular momentum. Further he extended the object of his method from mass points to rigid bodies, and derived the so-called Euler equation of rigid bodies.This transition of mechanics to new one founded on the analytical calculus happed in the second part of 1740s by Euler's study of motion of planets and rigid bodies. It is very important that in this period the mathematical formula of Euler's equation of motion has changed largely.In his younger period, he derived the equation of motion from Galileo's law of free fall, and it had the particular formula of the first-order differential equation of space. Later, its formula has changed to the second-order differential equation of coordinate and the influence of Galileo's law has decreased, but we can distinguish its influence in the factor 2 of his equation of motion and his unit system of mechanics.