Classical Superintegrable Systems Research Articles

Racah algebra R(n) from coalgebraic structures and chains of R(3) substructures

The recent interest in the study of higher-rank polynomial algebras related to n-dimensional classical and quantum superintegrable systems with coalgebra symmetry and their connection with the generalised Racah algebra R(n), a higher-rank generalisation of the rank one Racah algebra R(3), raises the problem of understanding the role played by the n − 2 quadratic subalgebras generated by the left and right Casimir invariants (sometimes referred as universal quadratic substructures) from this new perspective. Such subalgebra structures play a significant role in the algebraic derivation of spectrum of quantum superintegrable systems. In this work, we tackle this problem and show that the above quadratic subalgebra structures can be understood, at a fixed n > 3, as the images of n − 2 injective morphisms of R(3) into R(n). We show that each of the n − 2 quadratic subalgebras is isomorphic to the rank one Racah algebra R(3). As a byproduct, we also obtain an equivalent presentation for the universal quadratic substructures generated by the partial Casimir invariants of the coalgebra. The construction, which relies on explicit (symplectic or differential) realisations of the generators, is performed in both the classical and the quantum cases.

Superintegrable systems in non-Euclidean plane: Hidden symmetries leading to linearity

Nineteen classical superintegrable systems in two-dimensional non-Euclidean spaces are shown to possess hidden symmetries leading to their linearization. They are the two Perlick systems [Ballesteros et al., Classical Quantum Gravity 25, 165005 (2008)], the Taub–NUT system [Ballesteros et al., SIGMA 7, 048 (2011)], and all the 17 superintegrable systems for the four types of Darboux spaces as determined by Kalnins et al. [J. Math. Phys. 44, 5811–5848 (2003)].

Classical Superintegrable Systems in a Magnetic Field that Separate in Cartesian Coordinates

We consider superintegrability in classical mechanics in the presence of magnetic fields. We focus on three-dimensional systems which are separable in Cartesian coordinates. We construct all possible minimally and maximally superintegrable systems in this class with additional integrals quadratic in the momenta. Together with the results of our previous paper [J. Phys. A: Math. Theor. 50 (2017), 245202, 24 pages], where one of the additional integrals was by assumption linear, we conclude the classification of three-dimensional quadratically minimally and maximally superintegrable systems separable in Cartesian coordinates. We also describe two particular methods for constructing superintegrable systems with higher-order integrals.

Higher order superintegrability, Painlevé transcendents and representations of polynomial algebras

Résumé. In recent years, progress toward the classification of superintegrable systems with higher order integrals of motion has been made. In particular, a complete classification of all exotic potentials with a third or a fourth order integrals, and allowing separation of variables in Cartesian coordinates. All doubly exotic potentials with a fifth order integral have also been completely classified. It has been demonstrated how the Chazy class of third order differential equations plays an important role in solving determining equations. Moreover, taking advantage of various operator algebras defined as Abelian, Heisenberg, Conformal and Ladder case of operator algebras, we re-derived these models. These new techniques also provided further examples of superintegrable Hamiltonian with integrals of arbitrary order. It has been conjectured that all quantum superintegrable potentials that do not satisfy any linear equation satisfy nonlinear equations having the Painlevé property. In addition, it has been discovered that their integrals naturally generate finitely generated polynomial algebras and the representations can be exploited to calculate the energy spectrum. For certain very interesting cases associated with exceptional orthogonal polynomials, these algebraic structures do not allow to calculate the full spectrum and degeneracies. It has been demonstrated that alternative sets of integrals which can be build and used to provide a complete solution. This this allow to make another conjecture i.e. that higher order superintegrable systems can be solved algebraically, they require alternative set of integrals than the one provided by a direct approach.

Higher-order superintegrable systems separating in polar coordinates

Classical and Quantum mechanical superintegrable systems that are separating in polar coordinates are analyzed. The motion is restricted to a Euclidean plane E2 and the additional integral is assumed to be a polynomial of degree N ≥ 3 in momenta. Cases N = 3, 4 and 5 are investigated in detail. This leads to a general and unified description of higher-order superintegrability in the case of potentials allowing separation in polar coordinates.

Are all classical superintegrable systems in two-dimensional space linearizable?

Several examples of classical superintegrable systems in a two-dimensional space are shown to possess hidden symmetries leading to their linearization. They include those determined fifty years ago in the work of Friš et al. [Phys. Lett. 13, 354–356 (1965)], their generalizations, and the more recent Tremblay-Turbiner-Winternitz system [F. Tremblay et al., J. Phys. A: Math. Theor. 42, 242001 (2009)]. We conjecture that all classical superintegrable systems in the two-dimensional space have hidden symmetries that make them linearizable.

Family of N-dimensional superintegrable systems and quadratic algebra structures

Classical and quantum superintegrable systems have a long history and they possess more integrals of motion than degrees of freedom. They have many attractive properties, wide applications in modern physics and connection to many domains in pure and applied mathematics. We overview two new families of superintegrable Kepler-Coulomb systems with non-central terms and superintegrable Hamiltonians with double singular oscillators of type (n, N — n) in N-dimensional Euclidean space. We present their quadratic and polynomial algebras involving Casimir operators of so(N + 1) Lie algebras that exhibit very interesting decompositions Q(3) ⊕ so(N — 1), Q(3) ⊕ so(n) ⊕ so(N — n) and the cubic Casimir operators. The realization of these algebras in terms of deformed oscillator enables the determination of a finite dimensional unitary representation. We present algebraic derivations of the degenerate energy spectra of these systems and relate them with the physical spectra obtained from the separation of variables.

Examples of Complete Solvability of 2D Classical Superintegrable Systems

Classical (maximal) superintegrable systems in $n$ dimensions are Hamiltonian systems with $2n-1$ independent constants of the motion, globally defined, the maximum number possible. They are very special because they can be solved algebraically. In this paper we show explicitly, mostly through examples of 2nd order superintegrable systems in 2 dimensions, how the trajectories can be determined in detail using rather elementary algebraic, geometric and analytic methods applied to the closed quadratic algebra of symmetries of the system. We treat a family of 2nd order degenerate systems: oscillator analogies on Darboux, nonzero constant curvature, and flat spaces, related to one another via contractions, and obeying Kepler's laws. Then we treat two 2nd order nondegenerate systems, an analogy of a caged Coulomb problem on the 2-sphere and its contraction to a Euclidean space caged Coulomb problem. In all cases the symmetry algebra structure provides detailed information about the trajectories. An interesting example is the occurrence of ''metronome orbits'', trajectories confined to an arc rather than a loop, which are indicated clearly from the structure equations but might be overlooked using more traditional methods. We also treat the Post-Winternitz system, an example of a classical 4th order superintegrable system that cannot be solved using separation of variables. Finally we treat a superintegrable system, related to the addition theorem for elliptic functions, whose constants of the motion are only rational in the momenta, a system of special interest because its constants of the motion generate a closed polynomial algebra. This paper contains many new results but we have tried to present most of the materials in a fashion that is easily accessible to nonexperts, in order to provide entr\'ee to superintegrablity theory.

Classical and quantum higher order superintegrable systems from coalgebra symmetry

The N-dimensional generalization of Bertrand spaces as families of maximally superintegrable (M.S.) systems on spaces with a nonconstant curvature is analyzed. Considering the classification of two-dimensional radial systems admitting three constants of motion at most quadratic in momenta, we will be able to generate a new class of spherically symmetric M.S. systems by using a technique based on coalgebra. The three-dimensional realization of these systems provides the entire classification of classical spherically symmetric M.S. systems admitting periodic trajectories. We show that in dimension N > 2, these systems (classical and quantum) admit, in general, higher order constants of motion and turn out to be exactly solvable. Furthermore, it is possible to obtain non-radial M.S. systems by introducing the projection of the original radial system to a suitable lower dimensional space.

Invariants in 3D for Classical Superintegrable Systems in Complex Phase Space

Physical dynamical systems in higher dimensions are always interesting. In this context we present here the possibility of its three-dimensional complex dynamical invariant in extended complex phase space(ECPS). Lie algebraic method is used to study three-dimensional classical superintegrable system on the extended complex phase space. Such complex invariants play an important role in the analysis of complex trajectories, also study of non-hermitian Hamiltonian systems.

Infinite families of superintegrable systems separable in subgroup coordinates

A method is presented that makes it possible to embed a subgroup separable superintegrable system into an infinite family of systems that are integrable and exactly-solvable. It is shown that in two dimensional Euclidean or pseudo-Euclidean spaces the method also preserves superintegrability. Two infinite families of classical and quantum superintegrable systems are obtained in two-dimensional pseudo-Euclidean space whose classical trajectories and quantum eigenfunctions are investigated. In particular, the wave-functions are expressed in terms of Laguerre and generalized Bessel polynomials.

A unified approach to quantum and classical TTW systems based on factorizations

A unifying method based on factorization properties is introduced for finding symmetries of quantum and classical superintegrable systems using the example of the Tremblay–Turbiner–Winternitz (TTW) model. It is shown that the symmetries of the quantum system can be implemented in a natural way to its classical version. Besides, by this procedure we get also other type of constants of motion depending explicitly on time that allow to find directly the motion of the system whose corresponding trajectories coincide with those obtained previously by using its symmetries.

Structure relations for the symmetry algebras of quantum superintegrable systems

A quantum superintegrable system is an integrable n-dimensional Hamiltonian system with potential H = Δn + V that admits 2n − 1 algebraically independent partial differential operators commuting with the Hamiltonian, the maximum number possible. Here, n ≥ 2. The system is of order ℓ if the maximum order of the symmetry operators other than the Hamiltonian is ℓ Typically, the algebra generated by the symmetry operators has been shown to close. There is an analogous definition for classical superintegrable systems with the operator commutator replaced by the Poisson bracket. Superintegrability captures what it means for a Hamiltonian system to be explicitly algebraically and analytically solvable, not just solvable numerically. Until recently there were very few examples of superintegrable systems of order ℓ with ℓ > 3 and and virtually no structure results. The situation has changed dramatically in the last two years with the discovery of families of systems depending on a rational parameter κ = p/q that are superintegrable for all κ and of arbitrarily high order, such as ℓ = p + q + 1. We review a method, based on recurrence formulas for special functions, that proves superintegrability of these higher order quantum systems, and allows us to determine the structure of the symmetry algebra. Just a few months ago, these constructions seemed out of reach.

Classical ladder operators, polynomial Poisson algebras, and classification of superintegrable systems

We recall results concerning one-dimensional classical and quantum systems with ladder operators. We obtain the most general one-dimensional classical systems, respectively, with a third and a fourth-order ladder operators satisfying polynomial Heisenberg algebras. These systems are written in terms of the solutions of quartic and quintic equations. They are the classical equivalent of quantum systems involving the fourth and fifth Painlevé transcendents. We use these results to present two new families of superintegrable systems and examples of trajectories that are deformation of Lissajous's figures.

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

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 ladder 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.

Third-order superintegrable systems separating in polar coordinates

A complete classification of quantum and classical superintegrable systems in E2 is presented that allow the separation of variables in polar coordinates and admit an additional integral of motion of order 3 in the momentum. New quantum superintegrable systems are discovered for which the potential is expressed in terms of the sixth Painlevé transcendent or in terms of the Weierstrass elliptic function.

Models for the 3D singular isotropic oscillator quadratic algebra

We give the first explicit construction of the quadratic algebra for a 3D quantum superintegrable system with nondegenerate (4-parameter) potential together with realizations of irreducible representations of the quadratic algebra in terms of differential—differential or differential—difference and difference—difference operators in two variables. The example is the singular isotropic oscillator. We point out that the quantum models arise naturally from models of the Poisson algebras for the corresponding classical superintegrable system. These techniques extend to quadratic algebras for superintegrable systems in n dimensions and are closely related to Hecke algebras and multivariable orthogonal polynomials.

Periodic orbits for an infinite family of classical superintegrable systems

We show that all bounded trajectories in the two-dimensional classical system with the potential are closed for all integer and rational values of k. The period is and does not depend on k. This agrees with our earlier conjecture suggesting that the quantum version of this system is superintegrable.

Models of quadratic quantum algebras and their relation to classical superintegrable systems

We show how to construct realizations (models) of quadratic algebras for 2D second order superintegrable systems in terms of differential or difference operators in one variable. We demonstrate how various models of the quantum algebras arise naturally from models of the Poisson algebras for the corresponding classical superintegrable system. These techniques extend to quadratic algebras related to superintegrable systems in n dimensions and are intimately related to multivariable orthogonal polynomials.

Models for Quadratic Algebras Associated with Second Order Superintegrable Systems in 2D

There are 13 equivalence classes of 2D second order quantum and classical superintegrable systems with nontrivial potential, each associated with a quadratic algebra of hidden symmetries. We study the finite and infinite irreducible representations of the quantum quadratic algebras though the construction of models in which the symmetries act on spaces of functions of a single complex variable via either differential operators or difference operators. In another paper we have already carried out parts of this analysis for the generic nondegenerate superintegrable system on the complex 2-sphere. Here we carry it out for a degenerate superintegrable system on the 2-sphere. We point out the connection between our results and a position dependent mass Hamiltonian studied by Quesne. We also show how to derive simple models of the classical quadratic algebras for superintegrable systems and then obtain the quantum models from the classical models, even though the classical and quantum quadratic algebras are distinct.