Planetary Motion and the Duality of Force Laws

Trajectories of Hooke's law in the complex plane, which are conic sections, are mapped onto trajectories of Newton's law of gravitation via the transformation z → z2. Newton's law of ellipses (objects attracted to a center by a force inversely proportional to the square of the distance travel in conic sections) follows from a geometric analysis of this map. An extension of this approach reveals a similar relation between more general pairs of power laws of centripetal attraction. The implications of these relations are discussed and a Matlab program is provided for their numerical study. This material is suitable for an undergraduate complex analysis class.