Intrinsic geometry of curves and the Lorentz equation

We show that the trajectory of a point charge in a uniform electromagnetic field is a helix if the Lorentz equation governs its motion. Our approach is totally relativistic, and it is based on the use of the Frenet-Serret formulae which describe the intrinsic geometry of world lines in Minkowski spacetime.     

On the geometry of null hypersurfaces in Minkowski space

The present work is divided into three parts. First we study the null hypersurfaces of the Minkowski space ${\mathbb{R}}_1^{n+2}$, classifying all rotation null hypersurfaces in ${\mathbb{R}}_1^{n+2}$. In the second part we start our analysis of the submanifold geometry of the null hypersurfaces. In the particular case of the $(n+1)$-dimensional light cone, we characterize its totally umbilical spacelike hypersurfaces, show the existence of non-totally umbilical ones and give a uniqueness result for the minimal spacelike rotation surfaces in the $3$-dimensional light cone. In the third and final part we consider an isolated umbilical point on a spacelike surface immersed in the 3-dimensional light cone of ${\mathbb{R}}_1^4$ and obtain the differential equation of the principal configuration associated to this point, showing that every classical generic Darbouxian principal configuration appears in this context.     

Algebrodynamics over complex space and phase extension of the Minkowski geometry

First principles should predetermine physical geometry and dynamics both together. In the “algebrodynamics” they follow solely from the properties of biquaternion algebra and the analysis over . We briefly present the algebrodynamics over Minkowski background based on a nonlinear generalization to of the Cauchi-Riemann analyticity conditions. Further, we consider the effective real geometry uniquely resulting from the structure of multiplication and found it to be of the Minkowski type, with an additional phase invariant. Then we pass to study the primordial dynamics that takes place in the complex space and brings into consideration a number of remarkable structures: an ensemble of identical correlated matter pre-elements (“duplicons”), caustic-like signals (interaction carriers), a concept of random complex time resulting in irreversibility of physical time at macrolevel, etc. In partucular, the concept of “dimerous electron” naturally arises in the framework of complex algebrodynamics and, together with the above-mentioned phase invariant, allows for a novel approach to explanation of quantum interference phenomena alternative to recently accepted wave—particle dualism paradigm. The text was submitted by the author in English.     

Motion of a rigidly rotating circle in terms of Minkowski geometry

It is my great pleasure to dedicate this article to Josh Goldberg, a long-standing friend of mine. In terms of special theory of relativity, a discussion of a geometric model representing a rigidly rotating circle of material particles is given. It is shown that the ratio of the proper length of the spatial circumference of the circle to its radius is the same as in the case of a circle that is at rest relative to a global inertial frame of reference when the circumference is measured in terms of invariants determined by a set of local inertial observers who are momentarily comoving with the particles at equal values of their proper times which are synchronized in turn by every pair of neighbouring observers one after the other.     

Relativistic particles and the geometry of 4-D null curves

We study actions in (d+1)$(d+1)$-dimensions associated with null curves, mainly when d=3$d=3$, whose Lagrangian is a linear function on the curvature of the particle path, showing that null helices are always possible trajectories of the particles. We find Killing vector fields along critical curves of the action which correspond to the linear and the angular momenta of the particle. They provide two constants of the motion which can be interpreted in terms of the mass and the spin of the system. Moreover, we are able to integrate both the Euler–Lagrange equations and the Cartan equations in cylindrical coordinates around a certain plane.     

Quantum anticentrifugal force for wormhole geometry

We show the existence of an anticentrifugal force in a wormhole geometry in R³. This counterintuitive force was shown to exist in a flat R² space. The role the geometry plays in the appearance of this force is discussed.     

Influence of the normal force and contact geometry on the static force of friction of an oscillating sample

The paper is devoted to an experimental and theoretical investigation of the static friction force between a rapidly oscillating sample and a steel plate. The static frictional force is studied experimentally as function of the oscillating amplitude, the normal force and the contact geometry. A simplest model of tangent contact with a constant friction coefficient is proposed and shows a good agreement with experiment. The static friction force is proved to be a universal function of the ratio of the oscillation amplitude, the indentation depth and to the friction coefficient.     

Null Cartan Bertrand curves of AW(k)-type in Minkowski 4-space

This Letter considers the curvature conditions of AW(k)-type (k=1,2,3)$(k=1,2,3)$ null Cartan curves, and investigates null Cartan Bertrand curves. We show that null Cartan Bertrand curves are AW(k)-type (k=1,2,3)$(k=1,2,3)$ curves in Minkowski 4-space.     

Optimizing phase imaging via dynamic force curves

Tapping mode (TM, also called intermittent contact mode) atomic force microscopy (AFM) has been routinely used in many laboratories. However, consistent or deliberate control of measuring conditions and interpretation of results are often difficult. In this article, we demonstrate how measurement parameters (drive frequency, cantilever stiffness and oscillation amplitude) affect the tapping tip's state. This has been done by systematic dynamic force measurements performed on mica and polystyrene surfaces together with computer simulations. Our study shows the following results. (1) Weaker cantilevers, smaller amplitude and higher drive frequency (around the resonance) lead to an extension of the attractive region (greater phase lag) in amplitude–phase–distance curves and thus can help to achieve stable high-setpoint TM imaging with minimal tip–sample pressure. (2) Bistability of tapping tips often exists and may cause height artefacts if the setpoint falls in the bistable region. (3) Tapping tips with high vibrating energy (stiff cantilevers and large amplitude) driven at resonance are only slightly perturbed by tip–sample interactions and usually remain monostable during the sweep of the scanner position. This can help to achieve good phase contrast without significant artefacts when the setpoint falls in a continuous negative–positive phase shift transition region. (4) Low energy cantilevers (compliant cantilevers and small amplitude) usually result in large phase shift and can be used to acquire large phase contrast images. However, height artefacts will occur when the setpoint falls in the bistable region usually existing for such cantilevers. (5) Computer simulations are useful in understanding the bistability in dynamic force curves and determining either material properties or the optimal imaging parameters.     

Sheared force networks: anisotropies, yielding, and geometry

A scenario for the yielding of granular matter is presented by considering the ensemble of force networks for a given contact network and applied shear stress tau. As tau is increased, the probability distribution of contact forces becomes highly anisotropic, the difference between average contact forces along minor and major axes grows, and the allowed networks span a shrinking subspace of all force networks. Eventually, contacts start to break, and at the maximal shear stress the packing becomes effectively isostatic. The size of the allowed subspace exhibits simple scaling properties, which lead to a prediction for the yield stress for packings of an arbitrary contact number.     

A high-precision Ab initio determination of the equilibrium geometry and force field of HOC+

Ab initio molecular orbital theory is used to determine if the molecular ion HOC⁺ is linear (as are the isoelectronic species HCN, HNC, HCO⁺, etc.), or if it is quasilinear. Near the Hartree-Fock limit the molecule is either linear, or very close to linear. Electron correlation favors the linear geometry, leading to the unequivocal prediction of a linear molecule. Detailed comparisons between HOC⁺ and isoelectronic HNC show an apparent lack of convergence in the bending potential for the former which is remedied by the addition of f functions to the basis set. The HOC⁺ potential energy surface is computed in the bending and bend-stretch coordinates and fit to an analytical function. Use of this function to compute the rotation-vibration energies results in improved agreement with experiment relative to previous potentials by nearly two orders of magnitude, as documented in the accompanying paper.     

Near-field optical imaging using force detection with new tip-electrode geometry

We propose a new tip-electrode geometry to detect an (optical) evanescent field using noncontact atomic force microscopy. Using a semi-transparent metal electrode on the prism surface, the force sensitivity due to evanescent field in new tip-electrode geometry was enhanced by a factor of about 1000, comparing with that in old tip-electrode geometry where electrode was located behind the prism. Furthermore, this tip-electrode geometry avoids the electrostatic field caused by the residual charges and contact-electrified charges near the prism surface, which affects the force sensitivity due to evanescent field. We demonstrated the high resolution imaging of the evanescent field on the Au film with 15-nm (λ/33) lateral resolution.     

Increasing the adhesion force of electrostatic adhesives using optimized electrode geometry and a novel manufacturing process

This paper presents a method to increase the adhesion level of electrostatic adhesives by optimizing the electrode geometry and using a novel manufacturing technique. Simulation software, Comsol Multiphysics, was used to find the average electric field strength generated by a specific electrode geometry. The geometry was then optimized based on a gradient descent algorithm that changed each individual electrode width. Four different electrode patterns were simulated: concentric circles, comb (inter-digital), square spiral, and Hilbert curve (a fractal space-filling geometry). Among these designs the concentric circle pattern was the most effective. The optimized concentric circle pattern had varying electrode widths and the smallest allowable gap between the electrodes. These results were experimentally validated on a variety of materials with varying roughness: drywall, wood, tile, glass and steel. Overall, the experimental data closely matched the simulation results. Utilization of a new fabrication process also allowed for a significant increase in shear adhesion capability. With the optimized electrode geometry and the new fabrication process, we are able to achieve between a 2.2 and 15× improvement in shear pressure compared to previously published values, depending on the substrate material.     

On the Reality of Minkowski Space

Should physicists deal with the question of the reality of Minkowski space (or any relativistic spacetime)? It is argued that they should since this is a question about the dimensionality of the world at the macroscopic level and it is physics that should answer it.     

Phonon dispersion curves of transition metals using modified general tensor force model

The general tensor force model has been modified by incorporating separately the electron-ion interactions. The model satisfies the translational symmetry requirements of the lattice and is used to obtain the phonon dispersion curves of chromium, molybdenum and tungsten. The agreement between the theoretical and experimental frequencies is very good.     

Mach's Principle and Minkowski Spacetime

It is shown that when the Minkowski metric is approached by a limiting process using two different static, spherically-symmetric, closed cosmological models, that although the energy-stress tensors for the Einstein-Friedmann field equations vanishes, their integral does not. Since part of this integral consists of the mass of the incoherent dust background, which is the same in both models, the Minkowski metric obtained by this limiting process cannot be regarded as anti-Machian, since there is an infinite amount of ponderable matter in the background, albeit at vanishing density. One of the models is the Einstein static universe with its cosmological term. The other model does not employ this term, but instead uses a tensor that has vanishing trace, negative energy density and negative pressure. Gravitational energy is also studied, and it is pointed out that for both models, this energy becomes infinitely negative in the Minkowski limit.