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 where q i and p i are generic canonical variables, γ n are arbitrary coefficients, and N ∈ N . For N = 2, being both γ 1 , γ 2 different from zero, this reduces to the classical Zernike system. We prove that always provides a superintegrable system (for any value of γ 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, 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 ( 2 N − 1 ) th-order polynomial algebra. As a byproduct, the Hamiltonian is interpreted as a family of superintegrable perturbations of the classical Zernike system. Finally, it is shown that (and so the Zernike system as well) is endowed with a Poisson s l ( 2 , R ) -coalgebra symmetry which would allow for further possible generalizations that are also discussed.

On the generalization of classical Zernike system

“…The only obstacle is that the additional integral C(⃗ p) is singular somewhere in momentum space. We show that for the particular Hamiltonians (2) one can replace C(⃗ p), by using a very simple identity (cf equations ( 8) and (A2) below), with nonsingular combination of C, H and L. We suspect that similar procedure is possible for more general Hamiltonians H(⃗ p 2 ,⃗ q •⃗ p) as well; − having the explicit form of the superintegral for the Hamiltonian (2) we reproduce the result of [1]: the superintegral is a polynomial in momenta provided F is such a polynomial; − we sketch the argument that (2) continues to be superintegrable if the Euclidean plane is replaced by the surface of constant curvature; − we extend our results to the case of configuration space of arbitrary dimensions.…”

“…In the present paper we discuss a number of new results concerning the Hamiltonians generalizing that considered in [1]. More precisely: − we show that the more general Hamiltonians of the form…”

“…Due to the rotational invariance it is interesting to study the Hamiltonian (2) in polar coordinates. To this end we make the canonical transformation to polar variables [1]:…”

“…In the recent paper [1] the natural generalization of the classical Zernike system [2][3][4] (for the quantum version see [3][4][5][6][7][8][9]) has been proposed. It is defined by the Hamiltonian (1) with ⃗ q = (q 1 , q 2 ), ⃗ p = (p 1 , p 2 ) being canonical variables.…”

“…It has been shown in [1] that (1) is maximally superintegrable and admits, apart from energy and angular momentum, an additional integral of motion which is polynomial of degree N in momenta.…”