Exchange of helicity in a knotted electromagnetic field

In this work we present for the first time an exact solution of Maxwell equations in vacuum, having non trivial topology, in which there is an exchange of helicity between the electric and magnetic part of such field. We calculate the temporal evolution of the magnetic and electric helicities, and explain the exchange of helicity making use of the Chern-Simon form. We also have found and explained that, as time goes to infinity, both helicities reach the same value and the exchange between the magnetic and electric part of the field stops.Comment: 9 pages, 6 fi

A System of n=3 Coupled Oscillators with Magnetic Terms: Symmetries and Integrals of Motion

The properties of a system of n = 3 coupled oscillators with linear terms in the velocities (magnetic terms) depending in two parameters are studied. We proved the existence of a bi-Hamiltonian structure arising from a non-symplectic symmetry, as well the existence of master symmetries and additional integrals of motion (weak superintegrability) for certain particular values of the parameters.Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

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.Comment: 16 page

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

The quantum free particle on the sphere $S_\kappa^2$ ($\kappa>0$) and on the hyperbolic plane $H_\kappa^2$ ($\kappa<0$) 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 ($S_\kappa^2$, \IR^2, $H_\kappa^2$) and with the the transition from the classical $\kappa$-dependent system to the quantum one using the quantization of the Noether momenta. The Schr\"odinger separability and the quantum superintegrability are also discussed. The second part is devoted to the resolution of the $\kappa$-dependent Schr\"odinger equation. First the characterization of the $\kappa$-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 $\kappa>0$ then a discrete spectrum is obtained. The wavefunctions, that are related with a $\kappa$-dependent family of orthogonal polynomials, are explicitly obtained

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, $\kappa$ and $k_g$, in such a way that for $\kappa=0$ 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 $m(x)=1/(1 - {\kappa} x^2)$, with a $\kappa$-dependent non-polynomial rational potential and with an additional isotonic term. The Schr\"odinger equation is exactly solved and the $(\kappa,k_g)$-dependent wave functions and bound state energies are explicitly obtained for both $\kappa0$.Comment: two figure

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.Comment: 21 pages, to appear in J.Phys. A:Math. Theo