Physics

Solutions are obtained for the dual form of the Schrödinger equation got from the transformation of Poisson equation for the vector and the scalar potential, in dielectric and magnetic materials, having into account homogeneous isotropic linear mechanisms. We study and apply these dual equation solutions in some specific potentials.

The paper presents an analysis of the dynamic behaviour of discrete flexural systems composed of Euler--Bernoulli beams. The canonical object of study is the discrete Green's function, from which information regarding the dynamic response of the lattice under point loading by forces and moments can be obtained. Special attention is devoted to the interaction between flexural and torsional waves in a square lattice of Euler--Bernoulli beams, which is shown to yield a range of novel effects, including extreme dynamic anisotropy, non-reciprocity, wave-guiding, filtering, and the ability to create localised defect modes, all without the need for additional resonant elements or interfaces. The analytical study is complimented by numerical computations and finite element simulations, both of which are used to illustrate the effects predicted. A general algorithm is provided for constructing Green's functions as well as defect modes. This algorithm allows the tuning of the lattice to produce pass bands, band gaps, resonant modes, wave-guides, and defect modes, over any desired frequency range.

The dynamical analysis of vibrational systems of masses interconnected by restitution elements each with a single degree of freedom, and different configurations between masses and spring constants, is presented. Finite circular and linear arrays are studied using classical arguments, and their proper solution is given using methods often found in quantum optical systems. We further study some more complicated arrays where the solutions are given by using Lie algebras.

We study the dynamics of a particle in a space that is non-differentiable. Non-smooth geometrical objects have an inherently probabilistic nature and, consequently, introduce stochasticity in the motion of a body that lives in their realm. We use the mathematical concept of fiber bundle to characterize the multivalued nature of geodesic trajectories going through a point that is non-differentiable. Then, we generalize our concepts to everywhere non-smooth structures. The resulting theoretical framework can be considered a hybridization of the theory of surfaces and the theory of stochastic processes. We keep the concepts as general as possible, in order to allow for the introduction of other fundamental processes capable of modeling the fractality or the fluctuations of any conceivable continuous, but non-differentiable space.

We consider the 1-D motion of an electron under a periodic force and taking into account the effect of radiation reaction dissipation force on its motion, using the formulation of the radiation reaction force as a function of the external force. Two cases are considered: a simple sinusoidal time depending force, and sinusoidal electromagnetic force with position and time dependence. We found that the difference of the normalized (with respect the speed of light) velocities, with and without radiation reaction, are quite small between 10 −31 to 10 −14 for intensities on the electric field of 10 −8 to 1 Dynes/ues , which may represent some concern to measure experimentally.

The study of the evolution of the dynamics of a massive or massless particle shows that in special relativity theory, the energy is not conserved. From the law of evolution of the velocity over time of a particle subjected to a constant acceleration, it is possible to calculate the total energy acquired by this particle during its movement when its velocity tends towards the celerity of light. The energy transferred to the particle in relativistic mechanics overestimates the theoretical value. Discrete mechanics applied to this same problem makes it possible to show that the movement reflects that of Newtonian mechanics at low velocity, to obtain a velocity which tends well towards the celerity of the medium when the time increases, but also to conserve the energy at its theoretical value. This consistent behavior is due to the proposed physical analysis based on the compressible nature of light propagation.

We analyze planar electromagnetic waves confined by a slab waveguide formed by two perfect electrical conductors. Remarkably, 2D Maxwell equations describing transverse electromagnetic modes in such waveguides are exactly mapped onto equations for acoustic waves in fluids or gases. We show that interfaces between two slab waveguides with opposite-sign permeabilities support 1D edge modes, analogous to surface acoustic plasmons at interfaces with opposite-sign mass densities. We analyze this novel type of edge modes for the cases of isotropic media and anisotropic media with tensor permeabilities (including hyperbolic media). We also take into account `non-Hermitian' edge modes with imaginary frequencies or/and propagation constants. Our theoretical predictions are feasible for optical and microwave experiments involving 2D metamaterials.

We realize an elastic second-order topological insulator hosting both one-dimensional gapped edge states and zero-dimensional in-gap corner modes in the double-sided pillared phononic crystal plates with square lattice. Changing the width of two neighbor pillars breaks the inversion symmetry and induces the band inversion to emulate the quantum spin Hall effect where the gapless edge states are obtained. Further breaking the space-symmetry at interface, the gapless edge states are gapped and inducing the edge topological transitions and then giving rise to the zero-dimensional in-gap corner modes. Our work offers a novel way for elastic wave trapping and robustly guiding.

A relativistic particle undergoing successive boosts which are non collinear will experience a rotation of its coordinate axes with respect to the boosted frame. This rotation of coordinate axes is caused by a relativistic phenomenon called Thomas Rotation. We assess the importance of Thomas rotation in the calculation of physical quantities like electromagnetic fields in the relativistic regime. We calculate the electromagnetic field tensor for general three dimensional successive boosts in the particle's rest frame as well as the laboratory frame. We then compare the electromagnetic field tensors obtained by a direct boost β → +δ β → and successive boosts β → and Δ β → and check their consistency with Thomas rotation. This framework might be important to situations such as the calculation of frequency shifts for relativistic spin-1/2 particles undergoing Larmor precession in electromagnetic fields with small field non-uniformities.

A local permittivity model is proposed to accurately characterize spatial dispersion in non-local wire-medium (WM) structures with arbitrary terminations. A closed-form expression for the local thickness-dependent permittivity is derived for a general case of a bounded WM with lumped impedance insertions and terminated with impedance surfaces, which takes into account the effects of spatial dispersion and loads/terminations in the averaged sense per length of the wire medium. The proposed approach results in a local model formalism and accurately predicts the response of WM structures for near-field and far-field excitation. It is also shown that a traditional transmission network and circuit model can be effectively used to quantify the interaction of propagating and evanescent waves with WM structures. In addition, the derived analytical expression for the local thickness-dependent permittivity has been used in the full-wave numerical solver (CST Microwave Studio) demonstrating a drastic reduction in the computation time and memory in the solution of near-field and far-field problems involving wire media.