Superintegrability of the Post-Winternitz system on spherical and hyperbolic spaces

TL;DR

This paper extends the study of superintegrability to the Post-Winternitz system on spherical and hyperbolic spaces, providing explicit constants of motion and proving higher-order superintegrability for all curvatures.

Contribution

It introduces a curvature-dependent formalism for the Post-Winternitz system, demonstrating its higher-order superintegrability on curved spaces with explicit constants of motion.

Findings

Proves higher-order superintegrability of the Post-Winternitz system on curved spaces.

Provides explicit expressions for constants of motion depending on curvature.

Establishes the correct formulation of the PW system on spherical and hyperbolic geometries.

Abstract

The properties of the Tremblay-Turbiner-Winternitz system (related to the harmonic oscillator) were recently studied on the two-dimensional spherical $S_{\kappa}^2$ ($\kappa>0$) and hiperbolic $H_{\kappa}^2$ ($\kappa<0$) spaces (J. Phys. A : Math. Theor. 47, 165203, 2014). In particular, it was proved the higher-order superintegrability of the TTW system by making use of (i) a curvature-dependent formalism, and (ii) existence of a complex factorization for the additional constant of motion. Now a similar study is presented for the Post-Winternitz system (related to the Kepler problem). The curvature $\kappa$ is considered as a parameter and all the results are formulated in explicit dependence of $\kappa$. This technique leads to a correct definition of the Post-Winternitz (PW) system on spaces with curvature $\kappa$, to a proof of the existence of higher-order superintegrability (in both cases, $\kappa>0$ and $\kappa<0$), and to the explicit expression of the constants of motion.