Physics

Maxwell's electrodynamics postulates the finite propagation speed of electromagnetic (EM) action and the notion of EM fields, but it only satisfies the requirement of the covariance in Minkowski metric (Lorentz invariance). Darwin's force law of moving charges, which originates from Maxwell's field theory complies the Lorentz invariance as well. Poincaré's principle stating that physical laws can be formulated with identical meaning on different geometries suggest, that the retarded EM interaction of moving charges might be covariant even in Euclidean geometry (Galilean invariance). Keeping the propagation speed finite, but breaking with Maxwell's field theory in this study an attempt is made to find a Galilean invariant force law. Through the altering of the Liénard-Wiechert potential (LWP) a new retarded potential of two moving charges, the Common Retarded Electric Potential (CREP) is introduced which depends on the velocities of both interacting charges. The sought after force law is determined by means of the second order approximation of CREP. The law obtained is the Galilean invariant Weber's force law, surprisingly. Its rediscovery from the second order approximation of a retarded electric potential confirms the significance of Weber's force law and proves it to be a retarded and approximative law. The fact that Weber's force law implies even the magnetic forces tells us that magnetic phenomena are a manifestation of the retarded electric interaction exclusively. The third order approximation of the CREP opens the possibility of EM waves, and the creation of a complete, Euclidean electrodynamics.

We aim to create deterministic collisions between orbiting bodies by applying a time-dependent external force to one or both bodies, whether the bodies are mutually repulsive, as in the two- or multi-electron atomic case or mutually attractive, as in the planetary-orbit case. Specifically, we have devised a mathematical framework for causing deterministic collisions by launching an inner orbiting body to a higher energy such that this inner body is guaranteed to collide with the outer body. Our method first expresses the problem mathematically as coupled nonlinear differential equations with a time-dependent driving force and solves to find a feasible solution for the force function. Although our calculation is based strictly on classical physics, our approach is suitable for the case of helium with two highly excited electrons and is also valid for creating collisions in the gravitational case such as for our solar system.

In the classroom demonstration of the two-ball drop, some conditions lead to a "delayed rebound effect," with the second bounce of the upper ball higher than the first. This paper uses two models to explore the causes of this phenomenon. The classic independent contact model (ICM) is reviewed for the first bounce, and extended semi-analytically to the second bounce in the perfectly elastic case. A dynamical model based on a linear dashpot force is studied numerically. The delayed rebound effect is found for a range of parameters, most commonly in cases where the first bounce is lower than the ICM prediction.

The Lagrangian formulation of classical mechanics is extremely useful for a vast array of physics problems encountered in the undergraduate and graduate physics curriculum. Unfortunately, many treatments of this topic lack explanations of the most basic details that make Lagrangian mechanics so practical. In this paper, we detail the steps taken to arrive at the principle of stationary action, the Euler-Lagrange equations, and the Lagrangian of classical mechanics. These steps are: 1) the calculation of the minimal distance between two points in a plane, to introduce the variational principle and to derive the Euler-Lagrange equation; 2) proving the Euler-Lagrange equations are independent of arbitrary coordinate transformations and motivating that this independence is desirable for classical mechanics; and 3) a straightforward reformulation of Newton's second law in the form of Euler-Lagrange equations and formulation of the principle of stationary action. This paper is targeted toward the advanced undergraduate student who, like our own experiences, struggles with details which are not seen as crucial to the utilization of the tools developed by Lagrangian mechanics, and is especially frustrated by the question "\textit{why} is the Lagrangian always kinetic minus potential energy?" We answer this question in a simple and approachable manner.

In classical mechanics, the motion of an object is described with Newton's three laws of motion, which means that the motion of the material elements composing a continuum can be described with the particle model. However, this viewpoint is not objective, since the existence of transverse wave cannot be predicted by the theory of elasticity based on the particle model. In this paper, the material element of an elastomer is regarded as a rigid body, and the traditional elastic wave equation is derived based on it. In the derivation, the constitutive relations and strain-displacement relations are correspondingly modified. The study reveals that the longitudinal and transverse waves in elastomer correspond to the translational and rotational motion of the material element, respectively. Besides, the reciprocity of shear stress and shear strain is no longer requisite in continuum mechanics, and the local rigid body rotation contributes stress.

Sliding motion between two rough solids under light normal loading involves myriad micro-impacts between antagonist micro-asperities. Those micro-impacts are at the origin of many emerging macroscopic phenomena, including the friction force, the slider's vibrations and the noise radiated in the surroundings. However, the individual properties of the micro-impacts (e.g. maximum force, position along the interface, duration) are essentially elusive to measurement. Here, we introduce an instrumented slider aimed at measuring the position and the normal component of the micro-impact forces during sliding against a rough track. It is based on an array of piezoelectric sensors, each placed under a single model asperity. Its dynamical characteristics are established experimentally and compared to a finite elements model. We then validate its measurement capabilities by using it against a track bearing simple, well-defined topographical features. The measurements are interpreted thanks to a simple multi-asperity contact model. Our slider is expected to be useful in future studies to provide local insights into a variety of tribological questions involving dry rough sliding interfaces.

A convex triangular obstacle forms a vital part of a periodic echellete grating. A triangular grating is characterized by three parameters like period, depth and flare angle. Knowledge of groove field is essential for precise designing of triangular corrugated structures for studying the blazing effect of propagating EM wave. In the present paper, an attempt has been made to determine the power of Groove fields belonging to a pair of groove regions adjacent to a convex triangular prism. Groove fields and their associated powers based on Dirichlet conditions on the groove surfaces have been determined. The governing Helmholtz wave equation has been solved for determining the free surface field and the groove field. Fourier-Bessel series, oblique coordinate transformations and Lommel's integral are used as tools.

We present simple expressions for load required to indent a layer of arbitrary thickness with a conical, paraboloidal or cylindrical punch. A rigid substrate underneath the sample leads to an increase of load required for indentation. This effect has to be corrected for to prevent overestimation of Young's modulus from indentation measurements, such as force - distance curves recorded with the Atomic Force Microscope (AFM). The problems of the frictionless contact of an axisymmetric punch and an isotropic, linear-elastic layer are reducible to Fredholm integral equations of the second kind. We solved them numerically and used the Remez algorithm to obtain piecewise polynomial approximations of the load - indentation relation for samples that are either in frictionless contact with the rigid substrate or bonded to it. Their relative error due to approximation is negligible and uniformly spread. Combining the numerical approximations with asymptotic solutions for very thin layers, we obtained equations appropriate for samples of arbitrary thickness. They were implemented in a new version of AtomicJ, our free, open source application for analysis of AFM recordings.

We solve the problem of determining basic topological properties of flat samples by performing measurements on their outer edge. The global maximum of four probe resistances shows a characteristic behaviour, which is dependent on the genus (i.e., the number of holes) of the domain. An extension of the van der Pauw method on domains having zero, one, or two holes is presented and discussed. A possibility of measuring topological properties of condensed matter is demonstrated. Experimental results for triply connected domains are presented and explained by continuous symmetry breaking caused by the presence of two holes. The results are consistent with the topological theorem of Hurwitz on the number of automorphisms of Riemann surfaces.

In this paper, one of the major shortcomings of the conventional numerical approaches is alleviated by introducing the probabilistic nature of molecular transitions into the framework of classical computational electrodynamics. The main aim is to develop a numerical method, which is capable of capturing the statistical attributes caused by the interactions between a group of spontaneous as well as stimulated emitters and the surrounding electromagnetic field. The electromagnetic field is governed by classical Maxwell's equations, while energy is absorbed from and emitted to the (surrounding) field according to the transitions occurring for the emitters, which are governed by time-dependent probability functions. These probabilities are principally consistent with quantum mechanics. In order to validate the proposed method, it is applied to three different test-cases; directionality of fluorescent emission in a corrugated single-hole gold nano-disk, spatial and temporal coherence of fluorescent emission in a hybrid photonic-plasmonic crystal, and stimulated emission of a core-shell SPASER. The results are shown to be closely comparable to the experimental results reported in the literature.