Construction of classical superintegrable systems with higher order integrals of motion from ladder operators

TL;DR

This paper presents a method to construct higher order integrals of motion for classical superintegrable systems using ladder operators, leading to new systems with polynomial Poisson algebras and closed bounded trajectories.

Contribution

It introduces a novel approach to generate superintegrable systems with higher order integrals from ladder operators, expanding the class of known integrable models.

Findings

Constructed new superintegrable systems with higher order integrals

Derived polynomial Poisson algebra for these systems

Numerically confirmed all bounded trajectories are closed

Abstract

We construct integrals of motion for multidimensional classical systems from ladder operators of one-dimensional systems. This method can be used to obtain new systems with higher order integrals. We show how these integrals generate a polynomial Poisson algebra. We consider a one-dimensional system with third order ladders operators and found a family of superintegrable systems with higher order integrals of motion. We obtain also the polynomial algebra generated by these integrals. We calculate numerically the trajectories and show that all bounded trajectories are closed.