质点沿最速降线和首尾固定的两相连线段下落问题的研究

Study of the problems of particles falling along the fastest down line and two connected line segments with fixed ends

目前关于最速降线即圆滚线的由来原理的文章有不少,但多数是直接套用光学折射定律进行类比推理,缺少对方法原委的详细介绍和对运动细节的探讨,或者仅是局部的数值计算而缺乏物理内涵.至于与圆滚线存在关联的质点沿首尾固定的两相连直线段降落的时间这样更为基础的问题,却少有人撰文研究.本文首先推导出质点速降问题中的“折射定律”,然后利用它推导出最速降线即圆滚线的方程.证明了质点沿圆滚线的速降运动等价于匀速滚动圆周上对应点的绝对运动.给出最速降运动方程和计算圆滚线具体参量及质点降落时间的方法,并举例说明.指出当降落终点处在圆滚线左右半拱时,圆滚线参量的计算公式有所区别.简单地论证了圆滚线是唯一的捷线.最后,通过求极值的方法,计算出首尾固定的两相连直线段的最速降折点位置,以及在纵、横坐标给定时的最速降折点位置.通过列表和作图,分析出降落时间随折点的变化规律.

Although there are many articles about the origin principle of the brachistochrone(i.e.,the cycloid)at present,most of them directly apply the law of optical refraction to analogical reasoning,lacking of detailed introduction to the source of method and discussion on the details of motion,or just do local numerical calculations and lack of physical connotation.As for the more basic problem associated with the cycloid that the time when the particle falls along two connected straight lines fixed at the beginning and end,few papers have been written to study it.This paper first derives the"refraction law"in the problem of prompt drop of particle,and then uses it to deduce the equation of the brachistochrone(i.e.the cycloid).It is proved that the dropping motion of a particle along a cycloid is equivalent to the absolute motion of the corresponding point on a uniform rolling circle.The equation of the steepest descent motion and the methods of calculating the specific parameters of the cycloid and the descent time of the particle are given,and some examples are given to illustrate them.It is pointed out that there are different formulas for calculating the parameters of cycloid when the landing end is at the left and right half arch of the cycloid.It is simply proved that the cycloid is only shortcut.Finally,through the method of finding extremum,the position of the fastest descending break point of two connected straight lines with fixed ends,and the positions of the fastest descending break points with given longitudinal or abscissa coordinates are calculated.Through listing and drawing,the change rule of landing time with breaking point is analyzed.