Superintegrability on curved spaces, orbits and momentum hodographs: revisiting a classical result by Hamilton

Abstract

The equation of the orbits (in the configuration space) and of the hodographs (in the ‘momentum’ plane) for the ‘curved’ Kepler and harmonic oscillator systems, living in a configuration space of any constant curvature and either signature type, are derived by purely algebraic means. This result extends to the ‘curved’ Kepler or harmonic oscillator for the classical Hamilton derivation of the orbits of the Euclidean Kepler problem through its hodographs. In both cases, the fundamental property allowing these derivations to work is the superintegrability of the ‘curved’ Kepler and harmonic oscillator, no matter whether the constant curvature of the configuration space is zero or not, or whether the configuration space metric is Riemannian or Lorentzian. In the ‘curved’ case the basic result does not refer to the ‘velocity hodograph’ but to the ‘momentum hodograph’; both coincide in a Euclidean configuration space, but only the latter is unambiguously defined in all curved spaces.