质点在与距离成正比的有心引力作用下做椭圆轨道运动的三种证明方法

Three Proof Methods of Elliptical Orbit Motion of Particles under the Action of Gravity Proportional to Distance

首先导出正椭圆在以中心为极点的极坐标系中的方程;然后从直角坐标系中的牛顿第二定律、比奈公式和能量守恒这三个角度证明了质点在与距离成正比的有心引力作用下的轨道是椭圆,给出椭圆轨道的具体参量、能量、角动量的表达式;最后对这三种证法进行了比较,并对计算过程中某些符号的选择进行了解释。尽管简单,研究所给出的较详细的推理过程与结论或可为相关内容的教学提供一些参考。

In this paper,the equation of the positive ellipse in polar coordinate system with the center as the pole is derived first.Then,from Newton′s second law in rectangular coordinate system,Binet formula and energy conservation,it is proved that the orbit of a particle under the action of gravity proportional to distance is an ellipse,and the expressions of the specific parameters,energy and angular momentum of the ellipse orbit are given.Finally,the comparison of these three methods is made,and the choice of some symbols in the calculation process is explained.Although simple,the detailed reasoning process and conclusion given in this paper may provide some reference for the teaching of relevant content.