最速降线求解和摩擦力影响的研究

最速降线在数学史上名声显赫,不考虑摩擦力时为圆滚线(摆线).不过,较低点在摆线顶点另一侧时,Euler方程求解必须根据一阶导数的正负分别进行,所得摆线实际上由两部分组成.因而方程的平凡解———水平直线可以插到这两部分中间,得到一族极值曲线,但仍以摆线为最优解.摩擦对最速降线的影响可以通过坐标系旋转来解决.由于实际路径在水平方向应是单调的,因而最速降线由竖直方向的直线和与之相切的摆线共同构成,摆线底线与水平方向成摩擦角.给出了直线下滑和最速降线下滑的等时线.

Study on the solution of brachistochrone and the effect of friction

Brachistochrone is a curve connecting two points, along which shortest time is spent for a ball to slip from the upper point to the lower one under gravity. As a cycloid, not considering the effect of friction it is famous in the history of mathematics. However, if the lower point is at the another side of the cycloid top, the solution of Euler equation must be done in two steps according to the negative or positive of the first derivative, and the result is constructed with two parts. In this case, the line solution of derivative zero can be interposed between these two parts. So the results of Euler equation satisfing the boundary conditions are a series of curves, among which cycloid is the brachistochrone. By the coordinate rotation with friction angle, effect of friction on the velocity can be calculated, and brachistochrone is a perpendicular line and partial cycloid which is tangent to the perpendicular line. The cycloid base declines with the friction angle to the horizontal line. The equi_chrone curves are given for dropping along direct line and brachistochrone.