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

On the harmonic oscillator on the Lobachevsky plane

We introduce the harmonic oscillator on the Lobachevsky plane with the aid of the potential $V(r)=(a^2\omega^2/4)sinh(r/a)^2$ where $a$ is the curvature radius and $r$ 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, $m$, 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

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]

The dynamical nature of time

It is usually assumed that the "$t$" parameter in the equations of dynamics can be identified with the indication of the pointer of a clock. Things are not so easy, however. In fact, since the equations of motion can be written in terms of $t$ but also of $t'=f(t)$, $f$ being any well behaved function, each one of those infinite parametric times $t'$ is as good as the Newtonian one to study classical dynamics. Here we show that the relation between the mathematical parametric time $t$ in the equations of dynamics and the physical dynamical time $\sigma$ that is measured with clocks is more complex and subtle than usually assumed. These two times, therefore, must be carefully distinguished since their difference may have significant consequences. Furthermore, we show that not all the dynamical clock-times are necessarily equivalent and that the observational fingerprint of this non-equivalence has the same form as that of the Pioneer anomaly.Comment: 13 pages, no figure

Hamiltonians separable in cartesian coordinates and third-order integrals of motion

We present in this article all Hamiltonian systems in E(2) that are separable in cartesian coordinates and that admit a third-order integral, both in quantum and in classical mechanics. Many of these superintegrable systems are new, and it is seen that there exists a relation between quantum superintegrable potentials, invariant solutions of the Korteweg-De Vries equation and the Painlev\'e transcendents.Comment: 19 pages, Will be published in J. Math. Phy

A non-linear Oscillator with quasi-Harmonic behaviour: two- and nn-dimensional Oscillators

A nonlinear two-dimensional system is studied by making use of both the Lagrangian and the Hamiltonian formalisms. The present model is obtained as a two-dimensional version of a one-dimensional oscillator previously studied at the classical and also at the quantum level. First, it is proved that it is a super-integrable system, and then the nonlinear equations are solved and the solutions are explicitly obtained. All the bounded motions are quasiperiodic oscillations and the unbounded (scattering) motions are represented by hyperbolic functions. In the second part the system is generalized to the case of $n$ degrees of freedom. Finally, the relation of this nonlinear system with the harmonic oscillator on spaces of constant curvature, two-dimensional sphere $S^2$ and hyperbolic plane $H^2$, is discussed.Comment: 30 pages, 4 figures, submitted to Nonlinearit