86 research outputs found
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
Superintegrable deformations of superintegrable systems: Quadratic superintegrability and higher-order superintegrability
The superintegrability of four Hamiltonians Hr = ¿ H r, r = a, b, c, d, where Hr are known Hamiltonians and ¿ is a certain function defined on the configuration space and depended on a parameter ¿, is studied. The new Hamiltonians, and the associated constants of motion Jri, i = 1, 2, 3, are continous functions of the parameter ¿. The first part is concerned with separability and quadratic superintegrability (the integrals of motion are quadratic in the momenta) and the second part is devoted to the existence of higher-order superintegrability. The results obtained in the second part are related with the Tremblay-Turbiner-Winternitz and the Post-Winternitz systems
On the harmonic oscillator on the Lobachevsky plane
We introduce the harmonic oscillator on the Lobachevsky plane with the aid of
the potential where is the curvature
radius and is the geodesic distance from a fixed center. Thus the potential
is rotationally symmetric and unbounded likewise as in the Euclidean case. The
eigenvalue equation leads to the differential equation of spheroidal functions.
We provide a basic numerical analysis of eigenvalues and eigenfunctions in the
case when the value of the angular momentum, , equals 0.Comment: to appear in Russian Journal of Mathematical Physics (memorial volume
in honor of Vladimir Geyler
Linked and knotted beams of light, conservation of helicity and the flow of null electromagnetic fields
Maxwell's equations allow for some remarkable solutions consisting of pulsed
beams of light which have linked and knotted field lines. The preservation of
the topological structure of the field lines in these solutions has previously
been ascribed to the fact that the electric and magnetic helicity, a measure of
the degree of linking and knotting between field lines, are conserved. Here we
show that the elegant evolution of the field is due to the stricter condition
that the electric and magnetic fields be everywhere orthogonal. The field lines
then satisfy a `frozen field' condition and evolve as if they were unbreakable
filaments embedded in a fluid. The preservation of the orthogonality of the
electric and magnetic field lines is guaranteed for null, shear-free fields
such as the ones considered here by a theorem of Robinson. We calculate the
flow field of a particular solution and find it to have the form of a Hopf
fibration moving at the speed of light in a direction opposite to the
propagation of the pulsed light beam, a familiar structure in this type of
solution. The difference between smooth evolution of individual field lines and
conservation of electric and magnetic helicity is illustrated by considering a
further example in which the helicities are conserved, but the field lines are
not everywhere orthogonal. The field line configuration at time t=0 corresponds
to a nested family of torus knots but unravels upon evolution
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
A new look at the Feynman 'hodograph' approach to the Kepler first law
Hodographs for the Kepler problem are circles. This fact, known for almost two centuries, still provides the simplest path to derive the Kepler first law. Through Feynman's 'lost lecture', this derivation has now reached a wider audience. Here we look again at Feynman's approach to this problem, as well as the recently suggested modification by van Haandel and Heckman (vHH), with two aims in mind, both of which extend the scope of the approach. First we review the geometric constructions of the Feynman and vHH approaches (that prove the existence of elliptic orbits without making use of integral calculus or differential equations) and then extend the geometric approach to also cover the hyperbolic orbits (corresponding to E > 0). In the second part we analyse the properties of the director circles of the conics, which are used to simplify the approach, and we relate with the properties of the hodographs and Laplace–Runge–Lenz vector the constant of motion specific to the Kepler problem. Finally, we briefly discuss the generalisation of the geometric method to the Kepler problem in configuration spaces of constant curvature, i.e. in the sphere and the hyperbolic plane
Unusual formations of the free electromagnetic field in vacuum
It is shown that there are exact solutions of the free Maxwell equations
(FME) in vacuum allowing an existence of stable spherical formations of the
free magnetic field and ring-like formations of the free electric field. It is
detected that a form of these spheres and rings does not change with time in
vacuum. It is shown that these convergent solutions are the result of an
interference of some divergent solutions of FME. One can surmise that these
electromagnetic formations correspond to Kapitsa's hypothesis about
interference origin and a structure of fireball.Comment: Revtex-file, without figures. To get lournal-pdf-copy with figures
contact with [email protected]
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