Superintegrable dynamics on $H^2$ generated by coupling the Morse and Rosen-Morse potentials

TL;DR

This paper studies a Hamiltonian system on the hyperbolic plane combining Morse and Rosen-Morse potentials, showing that bounded orbits are closed when a specific parameter condition is met, indicating superintegrability.

Contribution

It demonstrates the superintegrability of the coupled Morse and Rosen-Morse potentials on the hyperbolic plane and explicitly constructs the associated polynomial constants of motion.

Findings

Bounded orbits are closed iff a parameter condition is satisfied.

The system exhibits maximal superintegrability.

Explicit polynomial constants of motion are constructed.

Abstract

A Hamiltonian dynamics defined on the two-dimensional hyperbolic plane by coupling the Morse and Rosen-Morse potentials is analyzed. It is demonstrated that orbits of all bounded motions are closed iff the product of the parameter $\tilde a$ of the Morse potential and the square root of the absolute value of the curvature is a rational number. This property of trajectories equivalent to the maximal superintegrability is confirmed by explicit construction of polynomial superconstant of motion.