中文摘要：

通过变分原理的方法研究力电耦合场中带电小球的最速降线问题,导出其轨迹控制方程,即Euler-Lagrange方程。对此强非线性的两点边值问题,通过打靶法编制相应程序进行求解,得到带电小球在电场强度和电场方向变化时的最速降落轨迹。同时讨论最速降线为直线时电场强度与电场方向应满足的条件,并求得此时降落所需要的总时间。结果表明,最速降线是可以通过外场进行定量调控的。

英文摘要：

The brachistochrone of an electric ball in the electromechanical coupling field was investigated according to variational method, and the corresponding governing equation （Euler-Lagrange equation） was derived. For the two-point value problem with strong nonlinearity, the shooting method was adopted, and the fastest dropping lines of the electric ball were obtained when the direction and intensity of the electric field were changed. At the same time, when the brachistoehrone is a linear line, the relation between the direction and intensity of the electric field was discussed in detail. Furthermore, the total consuming time of the fast dropping in this case was calculated. The results indicate that the braehistochrone of an electric ball can be modulated quantitatively by the external field.