The Kepler Problem and the Laplace–Runge–Lenz Vector on Spaces of Constant Curvature and Arbitrary Signature

Abstract

The superintegrability property is used to derive, by purely algebraic means, the equation of the orbits of the Kepler potential in any 2d space with any constant curvature and signature type, and we show that they are in all \({\mathbf S}^2_{\kappa_1[\kappa_2]}\) a CK conic with a focus at the potential origin. If the motion is not seen in configuration space but in the momentum \({\mathcal{P}}_{1}, {\mathcal{P}}_{2}\) plane, then the orbits are always ‘cycles’ in the natural geometry of this space. This result provides a generalization of an earlier result by Hamilton on the circular character for the hodographs of the Kepler problem and leans on the existence of a constant of motion, a ‘Laplace–Runge–Lenz’ conserved vector for the Kepler motion in any 2d space with any constant curvature and signature type.