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

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.

arXiv comment27 pages, 4 figures. Minor corrections

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