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

General structure of the solutions of the Hamiltonian constraints of gravity

A general framework for the solutions of the constraints of pure gravity is constructed. It provides with well defined mathematical criteria to classify their solutions in four classes. Complete families of solutions are obtained in some cases. A starting point for the systematic study of the solutions of Einstein gravity is suggested.Comment: 17 pages, LaTeX, submitted to International J. of Geom. Meth. in Modern Physics. Added comments in the last sectio

The Pioneer riddle, the quantum vacuum and the variation of the light velocity

It is shown that the same phenomenological Newtonian model recently proposed, which accounts for the cosmological evolution of the fine-structure constant, suggests furthermore an explanation of the unmodelled acceleration a(P) similar or equal to 8.5 x 10(-10) m/s(2) of the Pioneer 10/11 spaceships reported by Anderson et al. in 1998. In the view presented here, the permittivity and permeability of the empty space are decreasing adiabatically, and the light is accelerating therefore, as a consequence of the progressive attenuation of the quantum vacuum due to the combined effect of its gravitational interaction with all the expanding universe and the fourth Heisenberg relation. It is argued that the spaceships might not have any extra acceleration (but would follow instead the unchanged Newton laws), the observed effect being due to an adiabatic acceleration of the light equal to a(P), which has the same observational radio signature as the anomalous acceleration of the Pioneers

Lie systems and integrability conditions for t-dependent frequency harmonic oscillators

Time-dependent frequency harmonic oscillators (TDFHO's) are studied through the theory of Lie systems. We show that they are related to a certain kind of equations in the Lie group SL(2,R). Some integrability conditions appear as conditions to be able to transform such equations into simpler ones in a very specific way. As a particular application of our results we find t-dependent constants of the motion for certain one-dimensional TDFHO's. Our approach provides an unifying framework which allows us to apply our developments to all Lie systems associated with equations in SL(2,R) and to generalise our methods to study any Lie system