327 research outputs found
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 degrees of freedom; starting with a general Hamiltonian , 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 () and on the
hyperbolic plane () 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 (, \IR^2, ) and with
the the transition from the classical -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 -dependent Schr\"odinger
equation. First the characterization of the -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 then a discrete
spectrum is obtained. The wavefunctions, that are related with a
-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 ,
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, and , in such a way that for
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 , with a
-dependent non-polynomial rational potential and with an additional
isotonic term. The Schr\"odinger equation is exactly solved and the
-dependent wave functions and bound state energies are explicitly
obtained for both .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 , but of the Noether momenta 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
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