arXiv subjects

Manuel F. Rañada

Publications and source records attributed to Manuel F. Rañada.

At least 19 recordsLinked to original sources

arXiv2021

Superintegrability on the 3-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 (S-W) system and the Harmonic Oscillator with ratio of frequencies 1:1:2 and additional nonlinear terms on the 3-dimensional sphere $S^3$ ($\kp>0)$ and on the hyperbolic space $H^3$ ($\kp<0$). 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, $S^3$ ($\kp>0)$ and $H^3$ ($\kp<0$). 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 $\kp$ as a parameter, in such a way that particularizing for $\kp>0$, $\kp=0$, or $\kp<0$, the corresponding properties are obtained for the system on the sphere $S^3$, the Euclidean space $\IE^3$, or the hyperbolic space $H^3$, respectively.

math-ph↗

arXiv2021

Jacobi multipliers and Hojman symmetry

The geometric intrinsic approach to Hojman symmetry is developed and use is made of the theory of the Jacobi last multipliers to find the corresponding conserved quantity for non divergence-free vector fields. The particular cases of autonomous Lagrangian and Hamiltonian systems are studied as well as the generalization of these results to normalizer vector fields of the dynamics. The nonautonomous cases, where normalizer vector fields play a relevant role, are also developed.

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arXiv2020

Complex functions and geometric structures associated to the superintegrable Kepler-related family of systems endowed with generalized Runge-Lenz integrals of motion

The existence of quasi-bi-Hamiltonian structures for a two-dimensional superintegrable $(k_1,k_2,k_3)$-dependent Kepler-related problem is studied. We make use of an approach that is related with the existence of some complex functions which satisfy interesting Poisson bracket relations and that was previously applied to the standard Kepler problem as well as to some particular superintegrable systems as the Smorodinsky-Winternitz (SW) system, the Tremblay-Turbiner-Winternitz (TTW) and Post-Winternitz (PW) systems. We prove that these complex functions are important for two reasons: first, they determine the integrals of motion, and second they determine the existence of some geometric structures (in this particular case, quasi-bi-Hamiltonian structures). All the results depend of three parameters ($k_1, k_2, k_3$) in such a way that in the particular case $k_1\ne 0$, $k_2= k_3= 0$, we recover the results of the original Kepler problem (previously studied in SIGMA 12, 010 (2016)). This paper can be considered as divided in two parts and every part present a different approach (different complex functions and different quasi-bi-Hamiltonian structures).

math-ph↗

arXiv2017

Quantization of Hamiltonian systems with a position dependent mass: Killing vector fields and Noether momenta approach

The quantization of systems with a position dependent mass (PDM) is studied. We present a method that starts with the study of the existence of Killing vector fields for the PDM geodesic motion (Lagrangian with a PDM kinetic term but without any potential) and the construction of the associated Noether momenta. Then the method considers, as the appropriate Hilbert space, the space of functions that are square integrable with respect to a measure related with the PDM and, after that, it establishes the quantization, not of the canonical momenta $p$, but of the Noether momenta $P$ instead. The quantum Hamiltonian, that depends on the Noether momenta, is obtained as an Hermitian operator defined on the PDM Hilbert space. In the second part several systems with position-dependent mass, most of them related with nonlinear oscillators, are quantized by making use of the method proposed in the first part.

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arXiv2017

Superintegrable systems on 3-dimensional curved spaces: Eisenhart formalism and separability

The Eisenhart geometric formalism, which transforms an Euclidean natural Hamiltonian $H=T+V$ into a geodesic Hamiltonian ${\cal T}$ with one additional degree of freedom, is applied to the four families of quadratically superintegrable systems with multiple separability in the Euclidean plane. Firstly, the separability and superintegrability of such four geodesic Hamiltonians ${\cal T}_r$ ($r=a,b,c,d$) in a three-dimensional curved space are studied and then these four systems are modified with the addition of a potential ${\cal U}_r$ leading to ${\cal H}_r={\cal T}_r +{\cal U}_r$. Secondly, we study the superintegrability of the four Hamiltonians $\widetilde{\cal H}_r= {\cal H}_r/ μ_r$, where $μ_r$ is a certain position-dependent mass, that enjoys the same separability as the original system ${\cal H}_r$. All the Hamiltonians here studied describe superintegrable systems on non-Euclidean three-dimensional manifolds with a broken spherically symmetry.

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arXiv2017

Revisiting Lie integrability by quadratures from a geometric perspective

After a short review of the classical Lie theorem, a finite dimensional Lie algebra of vector fields is considered and the most general conditions under which the integral curves of one of the fields can be obtained by quadratures in a prescribed way will be discussed, determining also the number of quadratures needed to integrate the system. The theory will be illustrated with examples andbn an extension of the theorem where the Lie algebras are replaced by some distributions will also be presented.

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arXiv2016

Superintegrable systems with a position dependent mass : Kepler-related and Oscillator-related systems

The superintegrability of two-dimensional Hamiltonians with a position dependent mass (pdm) is studied (the kinetic term contains a factor $m$ that depends of the radial coordinate). First, the properties of Killing vectors are studied and the associated Noether momenta are obtained. Then the existence of several families of superintegrable Hamiltonians is proved and the quadratic integrals of motion are explicitly obtained. These families include, as particular cases, some systems previously obtained making use of different approaches. We also relate the superintegrability of some of these pdm systems with the existence of complex functions endowed with interesting Poisson bracket properties. Finally the relation of these pdm Hamiltonians with the Euclidean Kepler problem and with the Euclidean harmonic oscillator is analyzed.

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arXiv2016

Quasi-Bi-Hamiltonian Structures of the 2-Dimensional Kepler Problem

The existence of quasi-bi-Hamiltonian structures for the Kepler problem is studied. We first relate the superintegrability of the system with the existence of two complex functions endowed with very interesting Poisson bracket properties and then we prove the existence of a quasi-bi-Hamiltonian structure by making use of these two functions. The paper can be considered as divided in two parts. In the first part a quasi-bi-Hamiltonian structure is obtained by making use of polar coordinates and in the second part a new quasi-bi-Hamiltonian structure is obtained by making use of the separability of the system in parabolic coordinates.

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arXiv2015

Geometrical and Dynamical Aspects of Nonlinear Higher-Order Riccati Systems

We study a geometrical formulation of the nonlinear second-order Riccati equation (SORE) in terms of the projective vector field equation on $S^1$, whichn in turn is related to the stability algebra of Virasoro orbit. Using Darboux integrability method we obtain the first-integral of SORE and the results are applied to the study of its Lagrangian and Hamiltonian description. We unveil the relation between the Darboux polynomials and master symmetries associated to second-order Riccati. Using these results we show the existence of a Lagrangian description for the related system, and the Painlevé II equation is analysed.

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arXiv2014

A Quantum Quasi-Harmonic Nonlinear Oscillator with an Isotonic Term

The properties of a nonlinear oscillator with an additional term $k_g/x^2$, characterizing the isotonic oscillator, are studied. The nonlinearity affects to both the kinetic term and the potential and combines two nonlinearities associated to two parameters, $κ$ and $k_g$, in such a way that for $κ=0$ all the characteristics of of the standard isotonic system are recovered. The first part is devoted to the classical system and the second part to the quantum system. This is a problem of quantization of a system with position-dependent mass of the form $m(x)=1/(1 - κ x^2)$, with a $κ$-dependent non-polynomial rational potential and with an additional isotonic term. The Schrödinger equation is exactly solved and the $(κ,k_g)$-dependent wave functions and bound state energies are explicitly obtained for both $κ<0$ and $κ>0$.

math-ph↗

arXiv2013

Canonoid transformations and master symmetries

Different types of transformations of a dynamical system, that are compatible with the Hamiltonian structure, are discussed making use of a geometric formalism. Firstly, the case of canonoid transformations is studied with great detail and then the properties of master symmetries are also analyzed. The relations between the existence of constants of motion and the properties of canonoid symmetries is discussed making use of a family of boundary and coboundary operators.

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arXiv2013

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 with the harmonic oscillator, as the generalized SW system and the TTW system, were studied in previous papers; now a similar analysis is presented for superintegrable systems related with the Kepler problem. In this way, a new proof of the superintegrability of the Post-Winternitz system is presented and the explicit expression of the integral is obtained. Finally, the relations between the superintegrable systems with quadratic constants of the motion (separable in several different coordinate systems) and the superintegrable systems with higher-order constants of the motion are analyzed.

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arXiv2012

The quantum free particle on spherical and hyperbolic spaces: A curvature dependent approach II

This paper is the second part of a study of the quantum free particle on spherical and hyperbolic spaces by making use of a curvature-dependent formalism. Here we study the analogues, on the three-dimensional spherical and hyperbolic spaces, $S_\k^3$ ($κ>0$) and $H_\k^3$ ($κ<0$), to the standard {\itshape spherical waves} in $E^3$. The curvature $\k$ is considered as a parameter and for any $\k$ we show how the radial Schrödinger equation can be transformed into a $\k$-dependent Gauss hypergeometric equation that can be considered as a $\k$-deformation of the (spherical) Bessel equation. The specific properties of the spherical waves in the spherical case are studied with great detail. These have a discrete spectrum and their wave functions, which are related with families of orthogonal polynomials (both $\k$-dependent and $\k$-independent), and are explicitly obtained.

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arXiv2012

Curvature-dependent formalism, Schrödinger equation and energy levels for the harmonic oscillator on three-dimensional spherical and hyperbolic spaces

A nonlinear model representing the quantum harmonic oscillator on the three-dimensional spherical and hyperbolic spaces, $S_\k^3$ ($κ>0$) and $H_k^3$ ($κ<0$), is studied. The curvature $\k$ is considered as a parameter and then the radial Schrödinger equation becomes a $\k$-dependent Gauss hypergeometric equation that can be considered as a $\k$-deformation of the confluent hypergeometric equation that appears in the Euclidean case. The energy spectrum and the wavefunctions are exactly obtained in both the three-dimensional sphere $S_\k^3$ ($κ>0$) and the hyperbolic space $H_k^3$ ($κ<0$). A comparative study between the spherical and the hyperbolic quantum results is presented.

math-ph↗

arXiv2012

A geometric approach to a generalized virial theorem

The virial theorem, introduced by Clausius in statistical mechanics, and later applied in both classical mechanics and quantum mechanics, is studied by making use of symplectic formalism as an approach in the case of both the Hamiltonian and Lagrangian systems. The possibility of establishing virial's like theorems from one-parameter groups of non-strictly canonical transformations is analysed; and the case of systems with a position dependent mass is also discussed. Using the modern symplectic approach to quantum mechanics we arrive at the quantum virial theorem in full analogy with the classical case.

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arXiv2012

The quantum free particle on spherical and hyperbolic spaces: A curvature dependent approach

The quantum free particle on the sphere $S_κ^2$ ($κ>0$) and on the hyperbolic plane $H_κ^2$ ($κ<0$) is studied using a formalism that considers the curvature $\k$ as a parameter. The first part is mainly concerned with the analysis of some geometric formalisms appropriate for the description of the dynamics on the spaces ($S_κ^2$, $\IR^2$, $H_κ^2$) and with the the transition from the classical $κ$-dependent system to the quantum one using the quantization of the Noether momenta. The Schrödinger separability and the quantum superintegrability are also discussed. The second part is devoted to the resolution of the $κ$-dependent Schrödinger equation. First the characterization of the $κ$-dependent `curved' plane waves is analyzed and then the specific properties of the spherical case are studied with great detail. It is proved that if $κ>0$ then a discrete spectrum is obtained. The wavefunctions, that are related with a $κ$-dependent family of orthogonal polynomials, are explicitly obtained.

math-ph↗

arXiv2010

A geometric approach to integrability of Abel differential equations

A geometric approach is used to study the Abel first order differential equation of the first kind. The approach is based on the recently developed theory of quasi-Lie systems which allows us to characterise some particular examples of integrable Abel equations. Second order Abel equations will be discussed and the inverse problem of the Lagrangian dynamics is analysed: the existence of two alternative Lagrangian formulations is proved, both Lagrangians being of a non-natural class. The study is carried out by means of the Darboux polynomials and Jacobi multipliers.

math-ph↗

arXiv2010

A new proof of the higher-order superintegrability of a noncentral oscillator with inversely quadratic nonlinearities

The superintegrability of a rational harmonic oscillator (non-central harmonic oscillator with rational ratio of frequencies) with non-linear "centrifugal" terms is studied. In the first part, the system is directly studied in the Euclidean plane; the existence of higher-order superintegrability (integrals of motion of higher order than 2 in the momenta) is proved by introducing a deformation in the quadratic complex equation of the linear system. The constants of motion of the nonlinear system are explicitly obtained. In the second part, the inverse problem is analyzed in the general case of $n$ degrees of freedom; starting with a general Hamiltonian $H$, and introducing appropriate conditions for obtaining superintegrability, the particular "centrifugal" nonlinearities are obtained.

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