We consider the classical momentum-or velocity-dependent two-dimensional Hamiltonian given by whe... more 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 H N 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, H N 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 (2N -1)th-order polynomial algebra. As a byproduct, the Hamiltonian H N is interpreted as a family of superintegrable perturbations of the classical Zernike system. Finally, it is shown that H N (and so the Zernike system as well) is endowed with a Poisson sl(2, R)-coalgebra symmetry which would allow for further possible generalizations that are also discussed.
Integrable deformations of a class of Rikitake dynamical systems are constructed by deforming the... more Integrable deformations of a class of Rikitake dynamical systems are constructed by deforming their underlying Lie-Poisson Hamiltonian structures, which are considered linearizations of Poisson-Lie structures on certain (dual) Lie groups. By taking into account that there exists a one-to one correspondence between Poisson-Lie groups and Lie bialgebra structures, a number of deformed Poisson coalgebras can be obtained, which allow the construction of integrable deformations of coupled Rikitake systems. Moreover, the integrals of the motion for these coupled systems can be explicitly obtained by means of the deformed coproduct map. The same procedure can be also applied when the initial system is bi-Hamiltonian with respect to two different Lie-Poisson algebras. In this case, to preserve a bi-Hamiltonian structure under deformation, a common Lie bialgebra structure for the two Lie-Poisson structures has to be found. Coupled dynamical systems arising from this bi-Hamiltonian deformation scheme are also presented, and the use of collective 'cluster variables', turns out to be enlightening in order to analyse their dynamical behaviour. As a general feature, the approach here presented provides a novel connection between Lie bialgebras and integrable dynamical systems.
We consider the classical momentum-or velocity-dependent two-dimensional Hamiltonian given by H N... more We consider the classical momentum-or velocity-dependent two-dimensional Hamiltonian given by H N = p 2 1 + p 2 2 + N n=1 γ n (q 1 p 1 + q 2 p 2) n , 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 H N 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, H N 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 (2N − 1)th-order polynomial algebra. As a byproduct, the Hamiltonian H N is interpreted as a family of superintegrable perturbations of the classical Zernike system. Finally, it is shown that H N (and so the Zernike system as well) is endowed with a Poisson sl(2, R)-coalgebra symmetry which would allow for further possible generalizations that are also discussed.
Integrable perturbations of Hénon-Heiles systems from Poisson coalgebras
Nucleation and Atmospheric Aerosols, 2012
ABSTRACT The integrable perturbations of the two-dimensional integrable Hénon-Heiles Hamiltonians... more ABSTRACT The integrable perturbations of the two-dimensional integrable Hénon-Heiles Hamiltonians of KdV type are revisited by making use of their underlying sl(2,R)⊕h3 Poisson symmetry (co)algebra. As an application, a straightforward N-dimensional integrable generalization of all these systems is constructed by using N-dimensional symplectic realizations of sl(2,R)⊕h3. Finally, a new quantum biparametric deformation of the sl(2,R)⊕h3 Poisson coalgebra is presented, and a method based on this q-deformed symmetry is proposed in order to find new integrable deformations of the Hénon-Heiles systems.
Journal of Physics: Conference Series, Apr 13, 2015
The constant curvature analogue on the two-dimensional sphere and the hyperbolic space of the int... more The constant curvature analogue on the two-dimensional sphere and the hyperbolic space of the integrable Hénon-Heiles Hamiltonian H given by 1 Based on the contribution presented at "The 30th International Colloquium on Group Theoretical Methods in
A method to construct integrable deformations of Hamiltonian systems of ODEs endowed with Lie-Poi... more 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 Rössler 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 generalization to the 2D sphere S 2 and to the hyperbolic space H 2 of the 2D Eu... more A new integrable generalization to the 2D sphere S 2 and to the hyperbolic space H 2 of the 2D Euclidean anisotropic oscillator Hamiltonian with Rosochatius (centrifugal) terms is presented, and its curved integral of the motion is shown to be quadratic in the momenta. In order to construct such a new integrable Hamiltonian H κ , we will make use of a group theoretical approach in which the curvature κ 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. It turns out that when the oscillator parameters Ω 1 and Ω 2 are such that Ω 2 = 4Ω 1 , the system turns out to be the well-known superintegrable 1 : 2 oscillator on S 2 and H 2. Nevertheless, numerical integration of the trajectories of H κ suggests that for other values of the parameters Ω 1 and Ω 2 the system is not superintegrable. In this way, we support the conjecture that for each commensurate (and thus superintegrable) m : n Euclidean oscillator there exists a two-parametric family of curved integrable (but not superintegrable) oscillators that turns out to be superintegrable only when the parameters are tuned to the m : n commensurability condition.
We study Lie-Hamilton systems on the plane, i.e. systems of first-order differential equations de... more 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. González-López, 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.
Lotka-Volterra systems as Poisson-Lie dynamics on solvable groups
Nucleation and Atmospheric Aerosols, 2012
ABSTRACT A class of integrable 3D Lotka-Volterra (LV) equations is shown to be a particular insta... more ABSTRACT A class of integrable 3D Lotka-Volterra (LV) equations is shown to be a particular instance of Poisson-Lie dynamics on a family of solvable 3D Lie groups. As a consequence, the classification of all possible Poisson-Lie structures on these groups is shown to provide a systematic approach to obtain multiparametric integrable deformations of this LV system. Moreover, by making use of the coproduct map induced by the group multiplication, a twisted set of 3N-dimensional integrable Lotka-Volterra equations can be constructed. Finally, the quantization of one of the Poisson-Lie LV structures is performed, and is shown to give rise to a quantum euclidean algebra.
Integrable systems from maps between Poisson manifolds
Nucleation and Atmospheric Aerosols, 2012
ABSTRACT The loop coproduct approach to integrable systems introduced in [1, 2] is reformulated i... more ABSTRACT The loop coproduct approach to integrable systems introduced in [1, 2] is reformulated in terms of maps between Poisson manifolds. This reformulation is a generalization of that of the coalgebra symmnetry method given in [3] by using Poisson maps.
A Lie-Hamilton system is a nonautonomous system of first-order ordinary differential equations de... more A Lie-Hamilton system is a nonautonomous system of first-order ordinary differential equations describing the integral curves of a t-dependent vector field taking values in a finitedimensional real Lie algebra of Hamiltonian vector fields with respect to a Poisson structure. We provide new algebraic/geometric techniques to easily determine the properties of such Lie algebras on the plane, e.g., their associated Poisson bivectors. We study new and known Lie-Hamilton systems on R 2 with physical, biological and mathematical applications. New results cover Cayley-Klein Riccati equations, the here defined planar diffusion Riccati systems, complex Bernoulli differential equations and projective Schrödinger equations. Constants of motion for planar Lie-Hamilton systems are explicitly obtained which, in turn, allow us to derive superposition rules through a coalgebra approach.
The exact analytical solution in closed form of a modified SIR system where recovered individuals... more The exact analytical solution in closed form of a modified SIR system where recovered individuals are removed from the population is presented. In this dynamical system the populations S(t) and R(t) of susceptible and recovered individuals are found to be generalized logistic functions, while infective ones I(t) are given by a generalized logistic function times an exponential, all of them with the same characteristic time. The dynamics of this modified SIR system is analyzed and the exact computation of some epidemiologically relevant quantities is performed. The main differences between this modified SIR model and original SIR one are presented and explained in terms of the zeroes of their respective conserved quantities. Moreover, it is shown that the modified SIR model with timedependent transmission rate can be also solved in closed form for certain realistic transmission rate functions.
All possible Poisson-Lie (PL) structures on the 3D real Lie group generated by a dilation and two... more All possible Poisson-Lie (PL) structures on the 3D real Lie group generated by a dilation and two commuting translations are obtained. Their classification is fully performed by relating these PL groups with the corresponding Lie bialgebra structures on the corresponding 'book' Lie algebra. By construction, all these Poisson structures are quadratic Poisson-Hopf algebras for which the group multiplication is a Poisson map. In contrast to the case of simple Lie groups, it turns out that most of the PL structures on the book group are non-coboundary ones. Moreover, from the viewpoint of Poisson dynamics, the most interesting PL book structures are just some of these non-coboundaries, which are explicitly analysed. In particular, we show that the two different q-deformed Poisson versions of the sl(2, R) algebra appear as two distinguished cases in this classification, as well as the quadratic Poisson structure that underlies the integrability of a large class of 3D Lotka-Volterra equations. Finally, the quantization problem for these PL groups is sketched.
Any epidemiological compartmental model with constant population is shown to be a Hamiltonian dyn... more Any epidemiological compartmental model with constant population is shown to be a Hamiltonian dynamical system in which the total population plays the role of the Hamiltonian function. Moreover, some particular cases within this large class of models are shown to be bi-Hamiltonian. New interacting compartmental models among different populations, which are endowed with a Hamiltonian structure, are introduced. The Poisson structures underlying the Hamiltonian description of all these dynamical systems are explicitly presented, and their associated Casimir functions are shown to provide an efficient tool in order to find exact analytical solutions for epidemiological models, such as the ones describing the dynamics of the COVID-19 pandemic.
The Hamiltonian structure of a class of three-dimensional (3D) Lotka-Volterra (LV) equations is r... more The Hamiltonian structure of a class of three-dimensional (3D) Lotka-Volterra (LV) equations is revisited from a novel point of view by showing that the quadratic Poisson structure underlying its integrability structure is just a real three-dimensional Poisson-Lie group. As a consequence, the Poisson coalgebra map ∆ (2) that is given by the group multiplication provides the keystone for the explicit construction of a new family of 3N-dimensional integrable systems that, under certain constraints, contain N sets of deformed versions of the 3D LV equations. Moreover, by considering the most generic Poisson-Lie structure on this group, a new two-parametric integrable perturbation of the 3D LV system through polynomial and rational perturbation terms is explicitly found.
The generalization of (super)integrable Euclidean classical Hamiltonian systems to the twodimensi... more The generalization of (super)integrable Euclidean classical Hamiltonian systems to the twodimensional sphere and the hyperbolic space by preserving their (super)integrability properties is reviewed. The constant Gaussian curvature of the underlying spaces is introduced as an explicit deformation parameter, thus allowing the construction of new integrable Hamiltonians in a unified geometric setting in which the Euclidean systems are obtained in the vanishing curvature limit. In particular, the constant curvature analogue of the generic anisotropic oscillator Hamiltonian is presented, and its superintegrability for commensurate frequencies is shown. As a second example, an integrable version of the Hénon-Heiles system on the sphere and the hyperbolic plane is introduced. Projective Beltrami coordinates are shown to be helpful in this construction, and further applications of this approach are sketched. 1
We construct a constant curvature analogue on the two-dimensional sphere S 2 and the hyperbolic s... more We construct a constant curvature analogue on the two-dimensional sphere S 2 and the hyperbolic space H 2 of the integrable Hénon-Heiles Hamiltonian H given by H = 1 2 (p 2 1 + p 2 2) + Ω q 2 1 + 4q 2 2 + α q 2 1 q 2 + 2q 3 2 , where Ω and α are real constants. The curved integrable Hamiltonian H κ so obtained depends on a parameter κ which is just the curvature of the underlying space, and is such that the Euclidean Hénon-Heiles system H is smoothly obtained in the zero-curvature limit κ → 0. On the other hand, the Hamiltonian H κ that we propose can be regarded as an integrable perturbation of a known curved integrable 1 : 2 anisotropic oscillator. We stress that in order to obtain the curved Hénon-Heiles Hamiltonian H κ , the preservation of the full integrability structure of the flat Hamiltonian H under the deformation generated by the curvature will be imposed. In particular, the existence of a curved analogue of the full Ramani-Dorizzi-Grammaticos (RDG) series V n of integrable polynomial potentials, in which the flat Hénon-Heiles potential can be embedded, will be essential in our construction. Such infinite family of curved RDG potentials V κ,n on S 2 and H 2 will be also explicitly presented.
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Papers by Alfonso Blasco