José L. Trueba

ANALYTICAL ESTIMATES OF THE EFFECT OF NONLINEAR DAMPING IN SOME NONLINEAR OSCILLATORS

This paper reports on the effect of nonlinear damping on certain nonlinear oscillators, where analytical estimates provided by the Melnikov theory are obtained. We assume general nonlinear damping terms proportional to the power of velocity. General and useful expressions for the nonlinearly damped Duffing oscillator and for the nonlinearly damped simple pendulum are computed. They provide the critical parameters in terms of the damping coefficient and damping exponent, that is, the power of the velocity, for which complicated behavior is expected. We also consider generalized nonlinear damped systems, which may contain several nonlinear damping terms. Using the idea of Melnikov equivalence, we show that the effect of nonlinear dissipation can be equivalent to a linearly damped nonlinear oscillator with a modified damping coefficient.

A generalized perturbed pendulum

Abstract The simple pendulum is a paradigm in the study of oscillations and other phenomena in physics and nonlinear dynamics. This explains why it has deserved much attention, from many viewpoints, for a long time. Here, we attempt to describe what we call a generalized perturbed pendulum, which comprises, in a single model, many known situations related to pendula, including different forcing and nonlinear damping terms. Melnikov analysis is applied to this model, with the result of general formulae for the appearance of chaotic motions that incorporate several particular cases. In this sense, we give a unified view of the pendulum.

High-speed intensified video recordings of sprites and elves over the western Mediterranean Sea during winter thunderstorms.

We report the first intensified high‐speed video images of elves, sprites, and halos observed in Europe. All the events corresponded to winter season thunderstorms over the Mediterranean Sea. The observations comprise many elves generated by both cloud‐to‐ground lightning current polarities. In 8 of the 14 sprite observations we observed an elve previous to the sprite. In three cases we observed also an elve quickly followed by a halo and a sprite. In several observations we observed lightning light before the mesospheric transient luminous event. We present a case where the lightning from cloud tops was visible during the entire event. Thanks to the high‐speed videos and their resolution and low intensifier phosphor persistence we analyzed the timing distribution of the development phase of sprite elements, the persisting luminosity phase, and the total duration. Finally, we summarize one particular observation where a streamer collides and bounces with a previous formed column; it may be a new phenomenon of collision between an existing column body that interacts with a later streamer point‐like tip which is not merged and attached.

The electromagnetic helicity

It is shown that the electromagnetic helicity, defined as the sum of the helicities of the electric and magnetic fields of an electromagnetic field in vacuum, is a constant of the motion. Furthermore, it is equal to twice the classical limit of the difference between the numbers of right- and left-handed photons, which establishes a close relationship between the wave and the particle meanings of the word helicity. Resumen. Se demuestra que la helicidad electromagnetica, definida como la suma de las helicidades electrica y magnetica de un campo electromagnetico en el vacio, es una constante del movimiento. Ademas, es igual a dos veces el limite clasico de la diferencia entre los numeros de fotones polarizados a la derecha y a la izquierda, lo cual establece una relacion cercana entre los significados ondulatorio y corpuscular de la palabra helicidad.

Mechanism of branching in negative ionization fronts

When a strong electric field is applied to nonconducting matter, narrow channels of plasma called streamers may form. Branchlike patterns of streamers have been observed in anode directed discharges. We explain a mechanism for branching as the result of a balance between the destabilizing effect of impact ionization and the stabilizing effect of electron diffusion on ionization fronts. The dispersion relation for transversal perturbation of a planar negative front is obtained analytically when the ratio D between the electron diffusion coefficient and the intensity of the externally imposed electric field is small. We estimate the spacing lambda between streamers and deduce a scaling law lambda approximately D(1/3).

Energy dissipation in a nonlinearly damped Duffing oscillator

In this paper, we study the effect of including a nonlinear damping term proportional to the power of the velocity in the dynamics of a double-well Duffing oscillator. In particular, we focus our attention in understanding how the energy dissipates over a cycle and along the time, by the use of different tools of analysis. Analytical and numerical results for different damping terms are shown, and the presence of a discontinuity and an inversion of behavior depending on the initial energy are discussed. An averaged power loss in a period is defined, showing similar characteristics as the energy dissipation over a cycle, although no discontinuity is present. The discontinuity gap which appears for the energy dissipation at a certain value of the initial energy decreases as the power of the damping term increases and an associated scaling law is found.

Exchange of helicity in a knotted electromagnetic field

There are solutions of Maxwell equations in vacuum in which the magnetic and the electric lines have a nontrivial topology. This behaviour has physical consequences since it is related to classical expressions indicating aspects of the photon content of the 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. c

A class of non-null toroidal electromagnetic fields and its relation to the model of electromagnetic knots

An electromagnetic knot is an electromagnetic field in vacuum in which the magnetic lines and the electric lines coincide with the level curves of a pair of complex scalar fields ϕ and θ (see equations (A.1), (A.2)). When electromagnetism is expressed in terms of electromagnetic knots, it includes mechanisms for the topological quantization of the electromagnetic helicity, the electric charge, the electromagnetic energy inside a cavity and the magnetic flux through a superconducting ring. In the case of electromagnetic helicity, its topological quantization depends on the linking number of the field lines, both electric and magnetic. Consequently, to find solutions of the electromagnetic knot equations with nontrivial topology of the field lines has important physical consequences. We study a new class of solutions of Maxwellʼs equations in vacuum Arrayas and Trueba (2011 arXiv:1106.1122) obtained from complex scalar fields that can be interpreted as maps , in which the topology of the field lines is that of the whole torus-knot set. Thus this class of solutions is built as electromagnetic knots at initial time. We study some properties of those fields and consider if detection based on the energy and momentum observables is possible.

Motion of charged particles in a knotted electromagnetic field

In this paper we consider the classical relativistic motion of charged particles in a knotted electromagnetic field. After reviewing how to construct electromagnetic knots from maps between the three-sphere and the two-sphere, we introduce a mean quadratic radius of the energy density distribution in order to study some properties of this field. We study the classical relativistic motion of electrons in the electromagnetic field of the Hopf map, and compute their trajectories. It is observed that these electrons initially at rest are strongly accelerated by the electromagnetic force, becoming ultrarelativistic in a period of time that depends on the knot energy and size.

A topological mechanism of discretization for the electric charge

We present a topological mechanism of discretization, which gives for the fundamental electric charge the value q0 = √ ¯ = 5.29 ×10 19 C, about 3.3 times the electron charge. Its basis is the following recently proved property of the standard linear classical Maxwell equations: they can be obtained by change of variables from an underlying