A Configuration Counterpart of the Kepler Problem Hodograph

Abstract

This paper presents two peculiar geometrical structures in the configuration space of the classical Kepler problem: the circularU-graph and the circularS-graph. TheS-graph shows up a configuration space counterpart of the well-known velocity space hodograph. Several interesting results are brought out, such as a peculiar description of the mechanical energy. An extension to the three-dimensional space, through theU-sphere and theS-sphere, characterizes the regular Kepler orbits by means of the north pole of the associatedS-sphere. The Minkowskian parameterization introduced in [1] is easily recovered and is shown to fit naturally in theS-sphere description of the Kepler problem.