Higher-order superintegrable momentum-dependent Hamiltonians on curved spaces from the classical Zernike system

TL;DR

This paper introduces a family of higher-order superintegrable Hamiltonians on curved spaces, generalizing the classical Zernike system, and explores their algebraic structures and symmetries.

Contribution

It explicitly constructs higher-order constants of motion for these Hamiltonians on various curved geometries, extending superintegrability beyond Euclidean space.

Findings

All Hamiltonians are superintegrable with explicit higher-order constants of motion.

The Hamiltonians form a polynomial algebra of order (2N-1).

They possess a Poisson rak{sl}(2,\,\mathbb{R})-coalgebra symmetry.

Abstract

We consider the classical momentum- or velocity-dependent two-dimensional Hamiltonian given by $$\mathcal H_N = p_1^2 + p_2^2 +\sum_{n=1}^N \gamma_n(q_1 p_1 + q_2 p_2)^n ,$$ where $q_i$ and $p_i$ are generic canonical variables, $\gamma_n$ are arbitrary coefficients, and $N\in \mathbb N$. For $N=2$, being both $\gamma_1,\gamma_2$ different from zero, this reduces to the classical Zernike system. We prove that $\mathcal H_N$ always provides a superintegrable system (for any value of $\gamma_n$ and $N$) by obtaining the corresponding constants of the motion explicitly, which turn out to be of higher-order in the momenta. Such generic results are not only applied to the Euclidean plane, but also to the sphere and the hyperbolic plane. In the latter curved spaces, $\mathcal H_N $ is expressed in geodesic polar coordinates showing that such a new superintegrable Hamiltonian can be regarded as a superposition of the isotropic 1:1 curved (Higgs) oscillator with even-order anharmonic curved oscillators plus another superposition of higher-order momentum-dependent potentials. Furthermore, the symmetry algebra determined by the constants of the motion is also studied, giving rise to a $(2N-1)$th-order polynomial algebra. As a byproduct, the Hamiltonian $\mathcal H_N $ is interpreted as a family of superintegrable perturbations of the classical Zernike system. Finally, it is shown that $\mathcal H_N$ (and so the Zernike system as well) is endowed with a Poisson $\mathfrak{sl}(2,\mathbb R)$-coalgebra symmetry which would allow for further possible generalizations that are also discussed.