A note on some superintegrable Hamiltonian systems

We consider some examples of superintegrable system which were recently isolated through a differential Galois group analysis. The identity of these systems is clarified and the corresponding Poisson algebras derived

First integrals from conformal symmetries: Darboux–Koenigs metrics and beyond

On spaces of constant curvature, the geodesic equations automatically have higher order integrals, which are just built out of first order integrals, corresponding to the abundance of Killing vectors. This is no longer true for general conformally flat spaces, but in this case there is a large algebra of conformal symmetries. In this paper we use these conformal symmetries to build higher order integrals for the geodesic equations. We use this approach to give a new derivation of the Darboux–Koenigs metrics, which have only one Killing vector, but two quadratic integrals. We also consider the case of possessing one Killing vector and two cubic integrals. The approach allows the quantum analogue to be constructed in a simpler manner

Poisson Algebras and 3D Superintegrable Hamiltonian Systems

Using a Poisson bracket representation, in 3D, of the Lie algebra sl (2), we first use highest weight representations to embed this into larger Lie algebras. These are then interpreted as symmetry and conformal symmetry algebras of the “kinetic energy”, related to the quadratic Casimir function. We then consider the potentials which can be added, whilst remaining integrable, leading to families of separable systems, depending upon arbitrary functions of a single variable. Adding further integrals, in the superintegrable case, restricts these functions to specific forms, depending upon a finite number of arbitrary parameters. The Poisson algebras of these superintegrable systems are studied. The automorphisms of the symmetry algebra of the kinetic energy are extended to the full Poisson algebra, enabling us to build the full set of Poisson relations

Classical and Quantum Super-integrability: From Lissajous Figures to Exact Solvability

The first part of this paper explains what super-integrability is and how it differs in the classical and quantum cases. This is illustrated with an elementary example of the resonant harmonic oscillator. For Hamiltonians in “natural form”, the kinetic energy has geometric origins and, in the flat and constant curvature cases, the large isometry group plays a vital role. We explain how to use the corresponding first integrals to build separable and super-integrable systems. We also show how to use the automorphisms of the symmetry algebra to help build the Poisson relations of the corresponding non–Abelian Poisson algebra. Finally, we take both the classical and quantum Zernike system, recently discussed by Pogosyan et al., and show how the algebraic structure of its super-integrability can be understood in this framework

A Kaluza–Klein reduction of super-integrable systems

Given a super-integrable system in n degrees of freedom, possessing an integral which is linear in momenta, we use the “Kaluza–Klein construction” in reverse to reduce to a lower dimensional super-integrable system. We give two examples of a reduction from 3 to 2 dimensions. The constant curvature metric (associated with the kinetic energy) is the same in both cases, but with two different super-integrable extensions. For these, we use different elements of the algebra of isometries of the kinetic energy to reduce to 2-dimensions. Remarkably, the isometries of the reduced space can be derived from those of the 3-dimensional space, even though it requires the use of quadratic expressions in momenta

Eisenhart lift of 2-dimensional mechanics

The Eisenhart lift is a variant of geometrization of classical mechanics with d degrees of freedom in which the equations of motion are embedded into the geodesic equations of a Brinkmann-type metric defined on (d+2) (d+2) -dimensional spacetime of Lorentzian signature. In this work, the Eisenhart lift of 2-dimensional mechanics on curved background is studied. The corresponding 4-dimensional metric is governed by two scalar functions which are just the conformal factor and the potential of the original dynamical system. We derive a conformal symmetry and a corresponding quadratic integral, associated with the Eisenhart lift. The energy–momentum tensor is constructed which, along with the metric, provides a solution to the Einstein equations. Uplifts of 2-dimensional superintegrable models are discussed with a particular emphasis on the issue of hidden symmetries. It is shown that for the 2-dimensional Darboux–Koenigs metrics, only type I can result in Eisenhart lifts which satisfy the weak energy condition. However, some physically viable metrics with hidden symmetries are presented

Symmetries of ℤN graded discrete integrable systems

We recently introduced a class of ℤN graded discrete Lax pairs and studied the associated discrete integrable systems (lattice equations). We discuss differential–difference equations which then we interpret as symmetries of the discrete systems. In particular, we present nonlocal symmetries which are associated with the 2D Toda lattice

Superintegrable systems on 3 dimensional conformally flat spaces

We consider Hamiltonians associated with 3 dimensional conformally flat spaces, possessing 2, 3 and 4 dimensional isometry algebras. We use the conformal algebra to build additional quadratic first integrals, thus constructing a large class of superintegrable systems and the complete Poisson algebra of first integrals. We then use the isometries to reduce our systems to 2 degrees of freedom. For each isometry algebra we give a universal reduction of the corresponding general Hamiltonian. The superintegrable specialisations reduce, in this way, to systems of Darboux–Koenigs type, whose integrals are reductions of those of the 3 dimensional system

Generalised Darboux-Koenigs Metrics and 3-Dimensional Superintegrable Systems

The Darboux-Koenigs metrics in 2D are an important class of conformally flat, non-constant curvature metrics with a single Killing vector and a pair of quadratic Killing tensors. In [arXiv:1804.06904] it was shown how to derive these by using the conformal symmetries of the 2D Euclidean metric. In this paper we consider the conformal symmetries of the 3D Euclidean metric and similarly derive a large family of conformally flat metrics possessing between 1 and 3 Killing vectors (and therefore not constant curvature), together with a number of quadratic Killing tensors. We refer to these as generalised Darboux-Koenigs metrics. We thus construct multi-parameter families of super-integrable systems in 3 degrees of freedom. Restricting the parameters increases the isometry algebra, which enables us to fully determine the Poisson algebra of first integrals. This larger algebra of isometries is then used to reduce from 3 to 2 degrees of freedom, obtaining Darboux-Koenigs kinetic energies with potential functions, which are specific cases of the known super-integrable potentials

The role of commuting operators in quantum superintegrable systems

We discuss the role of commuting operators for quantum superintegrable systems, showing how they are used to build eigenfunctions. These ideas are illustrated in the context of resonant harmonic oscillators, the Krall–Sheffer operators, with polynomial eigenfunctions, and the Calogero–Moser system with additional harmonic potential. The construction is purely algebraic, avoiding the use of separation of variables and differential equation theory