开普勒问题的极倒数轨道
Orbits of the Kepler problem via polar reciprocals
摘要：
有人认为，在一个中心力场的运动，极倒数的轨迹是一个优雅的替代速端。极倒数的主要优点是，从轨迹到极倒数的变换是其自身的逆变换。建立了开普勒问题轨道的极倒数$k_*$的形式，并利用$k$是$k_*$的极倒数的事实，证明了轨道本身是圆锥曲线.本文从开普勒问题的极倒数出发，给出了开普勒问题的轨道的几何构造。不需要模糊的二次曲线知识来证明该方法的有效性。与费曼提出的图形化方法不同（并由德贝斯修正），基于极倒数的算法对开普勒问题中的所有三种轨迹（椭圆、抛物线和双曲线）都没有改变。
英文摘要：
It is argued that, for motion in a central force field, polar reciprocals of trajectories are an elegant alternative to hodographs. The principal advantage of polar reciprocals is that the transformation from a trajectory to its polar reciprocal is its own inverse. The form of polar reciprocals $k_*$ of Kepler problem orbits is established, and then the orbits $k$ themselves are shown to be conic sections using the fact that $k$ is the polar reciprocal of $k_*$. A geometrical construction is presented for the orbits of the Kepler problem starting from their polar reciprocals. No obscure knowledge of conics is required to demonstrate the validity of the method. Unlike a graphical procedure suggested by Feynman (and amended by Derbes), the algorithm based on polar reciprocals works without alteration for all three kinds of trajectories in the Kepler problem (elliptical, parabolic, and hyperbolic).
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