Orbits of the Kepler problem via polar reciprocals

Abstract

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).Comment: 11 pages, 5 figures, accepted for publication by American Journal of Physic

Elementary Solution of Kepler Problem (and a few other problems)

“…There is a handful of non trivial problems in classical mechanics one can solve exactly, such as the onedimensional harmonic oscillator (1dHO) and the Kepler problem [1][2][3][4][5][6][7][8][9]. The latter one, that is, to find the orbit of a point mass under the action of a inverse square force, is particularly important for its historical role as well as applications in celestial mechanics.…”

Elementary Solution of Kepler Problem (and a few other problems)

“…There is a handful of non trivial problems in classical mechanics one can solve exactly, such as the one-dimensional harmonic oscillator (1dHO) and the Kepler problem [1][2][3][4][5][6][7][8][9] . The latter one, that is, to find the orbit of a point mass under the action of a inverse square force, is particularly important for its historical role as well as applications in celestial mechanics.…”