Winternitz System Research Articles

On higher-dimensional superintegrable systems: a new family of classical and quantum Hamiltonian models

We introduce a family of n-dimensional Hamiltonian systems which, contain, as special reductions, several superintegrable systems as the Tremblay–Turbiner–Winternitz system, a generalized Kepler potential and the anisotropic harmonic oscillator with Rosochatius terms. We conjecture that there exist special values in the space of parameters, apart from those leading to known cases, for which this new Hamiltonian family is superintegrable.

N-dimensional Smorodinsky–Winternitz model and related higher rank quadratic algebra

The N-dimensional Smorodinsky–Winternitz system is a maximally superintegrable and exactly solvable model, being subject of study from different approaches. The model has been demonstrated to be multiseparable with wavefunctions given by Laguerre and Jacobi polynomials. In this paper we present the complete symmetry algebra of the system, which it is a higher-rank quadratic one containing the recently discovered Racah algebra as subalgebra. The substructures of distinct quadratic algebras and their related Casimirs are also studied. In this way, from the constraints on the oscillator realizations of these substructures, the energy spectrum of the N-dimensional Smorodinsky–Winternitz system is obtained. We show that allows different set of substructures based on the Racah algebra which can be applied independently to algebraically derive the spectrum of the system.

Superintegrability on the three-dimensional spaces with curvature. Oscillator-related and Kepler-related systems on the sphere S 3 and on the hyperbolic space H 3

The superintegrability of several Hamiltonian systems defined on three-dimensional configuration spaces of constant curvature is studied. We first analyze the properties of the Killing vector fields, Noether symmetries and Noether momenta. Then we study the superintegrability of the harmonic oscillator, the Smorodinsky–Winternitz system and the harmonic oscillator with ratio of frequencies 1:1:2 and additional nonlinear terms on the three-dimensional sphere S 3 (κ > 0) and on the hyperbolic space H 3 (κ < 0). In the second part we present a study first of the Kepler problem and then of the Kepler problem with additional nonlinear terms in these two curved spaces, S 3 (κ > 0) and H 3 (κ < 0). We prove their superintegrability and we obtain, in all the cases, the maximal number of functionally independent integrals of motion. All the mathematical expressions are presented using the curvature κ as a parameter, in such a way that particularizing for κ > 0, κ = 0, or κ < 0, the corresponding properties are obtained for the system on the sphere S 3, the Euclidean space , or the hyperbolic space H 3, respectively.

Haantjes algebras of classical integrable systems

A tensorial approach to the theory of classical Hamiltonian integrable systems is proposed, based on the geometry of Haantjes tensors. We introduce the class of symplectic-Haantjes manifolds (or $$\omega {\mathscr {H}}$$ manifolds), as a natural setting where the notion of integrability can be formulated. We prove that the existence of suitable Haantjes algebras of (1,1) tensor fields with vanishing Haantjes torsion is a necessary and sufficient condition for a Hamiltonian system to be integrable in the Liouville–Arnold sense. We also show that new integrable models arise from the Haantjes geometry. Finally, we present an application of our approach to the study of the Post–Winternitz system and of a stationary flow of the KdV hierarchy.

Embedding of the Racah algebra R(n) and superintegrability

The rank-1 Racah algebra R(3) plays a pivotal role in the theory of superintegrable systems. It appears as the symmetry algebra of the 3-parameter system on the 2-sphere from which all second-order conformally flat superintegrable models in 2D can be obtained by means of suitable limits and contractions. A higher rank generalisation of R(3), the so-called rank n−2 Racah algebra R(n), has been considered recently and showed to be the symmetry algebra of the general superintegrable model on the (n−1)-sphere. In the present work, we show that such an algebraic structure naturally arises as embedded inside a larger quadratic algebra characterising nD superintegrable models with non-central terms. This is shown both in classical and quantum mechanics through suitable (symplectic or differential) realisations of the Racah and additional generators. Among the main results, we present an explicit construction of the complete symmetry algebras for two families of n-dimensional maximally superintegrable models, the Smorodinsky–Winternitz system and the generalised Kepler–Coulomb system. For both families, the underlying symmetry algebras are higher-rank quadratic algebras containing the Racah algebra R(n) as subalgebra. High-order algebraic relations among the generators of the full quadratic algebras are also obtained both in the classical and quantum frameworks. These results should shed new light to the further understanding of the structures of quadratic algebras in the context of superintegrable systems.

Formula omitted]-Smorodinsky–Winternitz system in a constant magnetic field

We propose the superintegrable generalization of Smorodinsky–Winternitz system on the N-dimensional complex Euclidian space which is specified by the presence of constant magnetic field. We find out that in addition to 2N Liouville integrals the system has additional functionally independent constants of motion, and compute their symmetry algebra. We perform the Kustaanheimo–Stiefel transformation of C2-Smorodinsky–Winternitz system to the (three-dimensional) generalized MICZ-Kepler problem and find the symmetry algebra of the latter one. We observe that constant magnetic field appearing in the initial system has no qualitative impact on the resulting system.

Quasi-bi-Hamiltonian structures, complex functions and superintegrability: the Tremblay–Turbiner–Winternitz (TTW) and the Post–Winternitz (PW) systems

The existence of quasi-bi-Hamiltonian structures for the Tremblay–Turbiner–Winternitz (TTW) and the Post–Winternitz (PW) systems is studied. We first recall that the superintegrability of these two systems is related with the existence of certain complex functions endowed with interesting Poisson bracket properties, and then we prove the existence of several quasi-bi-Hamiltonian structures making use of these complex functions. This is done in two steps: first with complex 2-forms (wedge product of the differentials of the complex functions) and then with several real 2-forms. The properties of these geometric structures and the associated recursion operators are also analyzed.The paper can be considered as divided in two parts. In the first part a we study the TTW system (related to the harmonic oscillator) and in the second part we study the PW system (related to the Kepler problem).

The Post–Winternitz system on spherical and hyperbolic spaces: A proof of the superintegrability making use of complex functions and a curvature-dependent formalism

Two important advances in integrability have been the recent discovery of the higher-order superintegrability of the Tremblay–Turbiner–Winternitz system (related to the harmonic oscillator) and the Post–Winternitz system (related to the Kepler problem). The properties of the TTW system have been recently studied on the two-dimensional spherical Sκ2 (κ>0) and hyperbolic Hκ2 (κ<0) spaces by making use of a curvature-dependent formalism and the existence of a complex factorization for the higher-order constant of motion. Now in this Letter we prove that a similar technique can also be applied for the study of the PW system.

Superintegrable deformations of superintegrable systems: Quadratic superintegrability and higher-order superintegrability

The superintegrability of four Hamiltonians Hr˜=λ Hr, r = a, b, c, d, where Hr are known Hamiltonians and λ is a certain function defined on the configuration space and depended on a parameter κ, is studied. The new Hamiltonians, and the associated constants of motion Jri, i = 1, 2, 3, are continous functions of the parameter κ. The first part is concerned with separability and quadratic superintegrability (the integrals of motion are quadratic in the momenta) and the second part is devoted to the existence of higher-order superintegrability. The results obtained in the second part are related with the Tremblay-Turbiner-Winternitz and the Post–Winternitz systems.

The Tremblay–Turbiner–Winternitz system on spherical and hyperbolic spaces: superintegrability, curvature-dependent formalism and complex factorization

The higher order superintegrability of the Tremblay–Turbiner–Winternitz system (related to the harmonic oscillator) is studied on the two-dimensional spherical and hyperbolic spaces, (κ > 0) and (κ < 0). The curvature κ is considered as a parameter and all the results are formulated in explicit dependence on κ. The idea is that the additional constant of motion can be factorized as the product of powers of two particular rather simple complex functions (here denoted by Mr and Nϕ). This technique leads to a proof of the superintegrability of the Tremblay–Turbiner–Winternitz system on (κ > 0) and (κ < 0), and to the explicit expression of the constants of motion.

Generalized Lenard chains and multi-separability of the Smorodinsky–Winternitz system

We show that the notion of generalized Lenard chains allows to formulate in a natural way the theory of multi-separable systems in the context of bi-Hamiltonian geometry. We prove that the existence of generalized Lenard chains generated by a Hamiltonian function and by a Nijenhuis tensor defined on a symplectic manifold guarantees the separation of variables. As an application, we construct such a chain for the case I of the classical Smorodinsky–Winternitz model.

From constants of motion to superposition rules for Lie–Hamilton systems

A Lie system is a non-autonomous system of first-order differential equations possessing a superposition rule, i.e. a map expressing its general solution in terms of a generic finite family of particular solutions and some constants. Lie–Hamilton systems form a subclass of Lie systems whose dynamics is governed by a curve in a finite-dimensional real Lie algebra of functions on a Poisson manifold. It is shown that Lie–Hamilton systems are naturally endowed with a Poisson coalgebra structure. This allows us to devise methods for deriving in an algebraic way their constants of motion and superposition rules. We illustrate our methods by studying Kummer–Schwarz equations, Riccati equations, Ermakov systems and Smorodinsky–Winternitz systems with time-dependent frequency.

Higher order superintegrability of separable potentials with a new approach to the Post–Winternitz system

The higher order superintegrability of separable potentials is studied. It is proved that these potentials possess (in addition to the two quadratic integrals) a third integral of higher order in the momenta that can be obtained as the product of powers of two particular rather simple complex functions. Some systems related to the harmonic oscillator, such as the generalized SW system and the TTW system, were studied in previous papers; now a similar analysis is presented for superintegrable systems related to the Kepler problem. In this way, a new proof of the superintegrability of the Post–Winternitz system is presented and the explicit expression for the integral is obtained. Finally, the relations between the superintegrable systems with quadratic constants of motion (separable in several different coordinate systems) and the superintegrable systems with higher order constants of motion are analyzed.

Superintegrability in a non-conformally-flat space

Superintegrable systems in two- and three-dimensional spaces of constant curvature have been extensively studied. From these, superintegrable systems in conformally flat spaces can be constructed by Stäckel transform. In this paper a method developed to establish the superintegrability of the Tremblay–Turbiner–Winternitz system in two dimensions is extended to higher dimensions and a superintegrable system on a non-conformally-flat four-dimensional space is found. In doing so, curvature corrections to the corresponding classical potential are found to be necessary. It is found that some subalgebras of the symmetry algebra close polynomially.

A new approach to the higher order superintegrability of the Tremblay–Turbiner–Winternitz system

The higher order superintegrability of systems separable in polar coordinates is studied using an approach that was previously applied for the study of the superintegrability of a generalized Smorodinsky–Winternitz system. The idea is that the additional constant of motion can be factorized as the product of powers of two particular rather simple complex functions (here denoted by M and N). This technique leads to a proof of the superintegrability of the Tremblay–Turbiner–Winternitz system and to the explicit expression of the constants of motion. A second family (related with the first one) of superintegrable systems is also studied.

Lie-algebraic description of the quantum superintegrable Smorodinsky–Winternitz system in n dimensions

We apply the potential group method to a family of n-dimensional quantum Smorodinsky–Winternitz systems. The Hamiltonians of the systems are associated with first-order Casimir operators of the unitary group U(3n) restricted to certain subspaces of carrier space of the symmetric representation. Hence, the group U(3n) describes fixed energy states of a family of Smorodinsky–Winternitz systems with different potential strength. Moreover, it is shown that 2n − 1 integrals of motions (including the Hamiltonian) are related to Casimir operators of U(3n) and its subgroups.

Master symmetries, non-Hamiltonian symmetries and superintegrability of the generalized Smorodinsky–Winternitz system

The first part of the paper is devoted to the theory of master symmetries using the geometric formalism as an approach. It is shown that certain superintegrable systems are endowed with this property as a consequence of the existence of a family of master symmetries. In the second part, the properties of dynamical but non-Hamiltonian symmetries are studied. It is proved that the higher order superintegrability of the generalized Smorodinsky–Winternitz system is a consequence of the existence of symplectic symmetries not preserving the Hamiltonian function.

Three and four-body systems in one dimension: Integrability, superintegrability and discrete symmetries

Families of three-body Hamiltonian systems in one dimension have been recently proved to be maximally superintegrable by interpreting them as one-body systems in the three-dimensional Euclidean space, examples are the Calogero, Wolfes and Tramblay Turbiner Winternitz systems. For some of these systems, we show in a new way how the superintegrability is associated with their dihedral symmetry in the three-dimensional space, the order of the dihedral symmetries being associated with the degree of the polynomial in the momenta first integrals. As a generalization, we introduce the analysis of integrability and superintegrability of four-body systems in one dimension by interpreting them as one-body systems with the symmetries of the Platonic polyhedra in the four-dimensional Euclidean space. The paper is intended as a short review of recent results in the sector, emphasizing the relevance of discrete symmetries for the superintegrability of the systems considered.

Quadratic algebra approach to relativistic quantum Smorodinsky–Winternitz systems

There exists a relation between the Klein–Gordon and the Dirac equations with scalar and vector potentials of equal magnitude and the Schrödinger equation. We obtain the relativistic energy spectrum for the four relativistic quantum Smorodinsky–Winternitz systems from their quasi-Hamiltonian and the quadratic algebras studied by Daskaloyannis in the nonrelativistic context. We also apply the quadratic algebra approach directly to the initial Dirac equation for these four systems and show that the quadratic algebras obtained are the same than those obtained from the quasi-Hamiltonians. We point out how results obtained in context of quantum superintegrable systems and their polynomial algebras can be applied to the quantum relativistic case.

A Recurrence Relation Approach to Higher Order Quantum Superintegrability

We develop our method to prove quantum superintegrability of an integrable 2D system, based on recurrence relations obeyed by the eigenfunctions of the system with respect to separable coordinates. We show that the method provides rigorous proofs of superintegrability and explicit constructions of higher order generators for the symmetry algebra. We apply the method to 5 families of systems, each depending on a parameter $k$, including most notably the caged anisotropic oscillator, the Tremblay, Turbiner and Winternitz system and a deformed Kepler-Coulomb system, and we give proofs of quantum superintegrability for all rational values of $k$, new for 4 of these systems. In addition, we show that the explicit information supplied by the special function recurrence relations allows us to prove, for the first time in 4 cases, that the symmetry algebra generated by our lowest order symmetries closes and to determine the associated structure equations of the algebras for each $k$. We have no proof that our generating symmetries are of lowest possible order, but we have no counterexamples, and we are confident we can can always find any missing generators from our raising and lowering operator recurrences. We also get for free, one variable models of the action of the symmetry algebra in terms of difference operators. We describe how the St\"ackel transform acts and show that it preserves the structure equations.