Alfonso Blasco

(Super)integrability from coalgebra symmetry: Formalism and applications

The coalgebra approach to the construction of classical integrable systems from Poisson coalgebras is reviewed, and the essential role played by symplectic realizations in this framework is emphasized. Many examples of Hamiltonians with either undeformed or q-deformed coalgebra symmetry are given, and their Liouville superintegrability is discussed. Among them, (quasi-maximally) superintegrable systems on N-dimensional curved spaces of nonconstant curvature are analysed in detail. Further generalizations of the coalgebra approach that make use of comodule and loop algebras are presented. The generalization of such a coalgebra symmetry framework to quantum mechanical systems is straightforward.

Lie–Hamilton systems on the plane: Properties, classification and applications

We study Lie-Hamilton systems on the plane, i.e. systems of first-order differential equations describing the integral curves of a t-dependent vector field taking values in a finite-dimensional real Lie algebra of planar Hamiltonian vector fields with respect to a Poisson structure. We start with the local classification of finite-dimensional real Lie algebras of vector fields on the plane obtained in (A. Gonzalez-Lopez, N. Kamran and P.J. Olver, Proc. London Math. Soc. 64, 339 (1992)) and we interpret their results as a local classification of Lie systems. Moreover, by determining which of these real Lie algebras consist of Hamiltonian vector fields with respect to a Poisson structure, we provide the complete local classification of Lie-Hamilton systems on the plane. We present and study through our results new Lie-Hamilton systems of interest which are used to investigate relevant non-autonomous differential equations, e.g. we get explicit local diffeomorphisms between such systems. In particular, the Milne-Pinney, second-order Kummer-Schwarz, complex Riccati and Buchdahl equations as well as some Lotka-Volterra and nonlinear biomathematical models are analysed from this Lie-Hamilton approach.

Integrable deformations of Rössler and Lorenz systems from Poisson–Lie groups

Abstract A method to construct integrable deformations of Hamiltonian systems of ODEs endowed with Lie–Poisson symmetries is proposed by considering Poisson–Lie groups as deformations of Lie–Poisson (co)algebras. Moreover, the underlying Lie–Poisson symmetry of the initial system of ODEs is used to construct integrable coupled systems, whose integrable deformations can be obtained through the construction of the appropriate Poisson–Lie groups that deform the initial symmetry. The approach is applied in order to construct integrable deformations of both uncoupled and coupled versions of certain integrable types of Rossler and Lorenz systems. It is worth stressing that such deformations are of non-polynomial type since they are obtained through an exponentiation process that gives rise to the Poisson–Lie group from its infinitesimal Lie bialgebra structure. The full deformation procedure is essentially algorithmic and can be computerized to a large extent.

A new integrable anisotropic oscillator on the two-dimensional sphere and the hyperbolic plane

A new integrable generalization to the two-dimensional (2D) sphere, , and to the hyperbolic space, , of the 2D Euclidean anisotropic oscillator Hamiltonian with Rosochatius (centrifugal) terms is presented, and its curved integral of motion is shown to be quadratic in the momenta. To construct such a new integrable Hamiltonian, , we will use a group theoretical approach in which the curvature, , of the underlying space will be treated as an additional (contraction) parameter, and we will make extensive use of projective coordinates and their associated phase spaces. When the oscillator parameters ?1 and ?2 are such that , the system turns out to be the well-known superintegrable oscillator on and . Nevertheless, numerical integration of the trajectories of suggests that for other values of the parameters ?1 and ?2, the system is not superintegrable. In this way, we support the conjecture that for each commensurate, and thus superintegrable, Euclidean oscillator there exists a two-parametric family of curved integrable oscillators that turns out to be superintegrable only when the parameters are tuned to the commensurability condition.

Integrable Hénon–Heiles Hamiltonians: A Poisson algebra approach

Abstract The three integrable two-dimensional Henon–Heiles systems and their integrable perturbations are revisited. A family of new integrable perturbations is found, and N-dimensional completely integrable generalizations of all these systems are constructed by making use of sl ( 2 , R ) ⊕ h 3 as their underlying Poisson symmetry algebra. In general, the procedure here introduced can be applied in order to obtain N-dimensional integrable generalizations of any 2D integrable potential of the form V q 1 2 , q 2 , and the formalism gives the explicit form of all the integrals of the motion. Further applications of this algebraic approach in different contexts are suggested.

N-dimensional integrability from two-photon coalgebra symmetry

A wide class of Hamiltonian systems with N degrees of freedom and endowed with, at least, (N − 2) functionally independent integrals of motion in involution is constructed by making use of the two-photon Lie–Poisson coalgebra (h6, Δ). The set of (N − 2) constants of the motion is shown to be a universal one for all these Hamiltonians, irrespective of the dependence of the latter on several arbitrary functions and N free parameters. Within this large class of quasi-integrable N-dimensional Hamiltonians, new families of completely integrable systems are identified by finding explicitly a new independent integral through the analysis of the sub-coalgebra structure of h6. In particular, new completely integrable N-dimensional Hamiltonians describing natural systems, geodesic flows and static electromagnetic Hamiltonians are presented.

N-dimensional superintegrable systems from symplectic realizations of Lie coalgebras

The construction of N-dimensional (ND) integrable systems from coalgebras is reviewed. In the case of Poisson coalgebras, a necessary condition for the integrability of the ND coalgebra Hamiltonian coming from a given coalgebra is obtained in terms of the dimension of the symplectic realization and the number of nonlinear Casimir functions. From this viewpoint, the full set of three-, four- and five-dimensional Lie–Poisson coalgebras is analysed, and many new families of multiparametric ND integrable systems coming from the cases that fulfil the integrability condition are obtained, including the explicit form of the integrals of the motion. The superintegrability of these Hamiltonians is also emphasized, and the generalization of the whole construction to the quantum mechanical case is straightforward.

An integrable Hénon–Heiles system on the sphere and the hyperbolic plane

We construct a constant curvature analogue on the two-dimensional sphere

Non-coboundary Poisson?Lie structures on the book group

{\mathbf S}^2

Integrable perturbations of the N-dimensional isotropic oscillator

and the hyperbolic space