Related papers: Safe domain and elementary geometry
We prove well-posedness and regularity results for elliptic boundary value problems on certain domains with a smooth set of singular points. Our class of domains contains the class of domains with isolated oscillating conical singularities,…
Systems of identical particles with equal charge are studied under a special type of confinement. These classical particles are free to move inside some convex region S and on the boundary of it $\Omega$ (the $S^{d-1}-$ sphere, in our…
The derivation of the equation of one-dimensional movement of a solitary shock wave is given. This derivation shows, that the differential equation of movement of a solitary plane shock wave in the channel with variable area, is exact, if…
We establish that many fundamental concepts and techniques in quantum field theory and collider physics can be naturally understood and unified through a simple new geometric language. The idea is to equip the space of collider events with…
Cosmological domain walls can be formed as a result of symmetry breaking at any epoch during the evolution of our universe. We study their interaction with a classical macroscopic object, like Earth or a satellite in Earth's orbit. We set…
This work proposes an algorithm to bound the minimum distance between points on trajectories of a dynamical system and points on an unsafe set. Prior work on certifying safety of trajectories includes barrier and density methods, which do…
We propose geometric tools that are suitable for studying the behavior of a billiard trajectory in a homogeneous force field. Two examples are considered: a vertical plane with an open top and with a parabolic or right angle boundary at the…
Stable topological defects of light (pseudo)scalar fields can contribute to the Universe's dark energy and dark matter. Currently the combination of gravitational and cosmological constraints provides the best limits on such a possibility.…
In this paper, we apply the one dimensional quantum law of motion, that we recently formulated in the context of the trajectory representation of quantum mechanics, to the constant potential, the linear potential and the harmonic…
Non-relativistic particles that are effectively confined to two dimensions can in general move on curved surfaces, allowing dynamical phenomena beyond what can be described with scalar potentials or even vector gauge fields. Here we…
The elliptical stadium is a plane region bounded by a curve constructed by joining two half-ellipses by two parallel segments of equal length. The billiard inside it, as a map, generates a two parameters family of dynamical systems. It is…
The control problem of the working tool movement along a predefined trajectory is considered. The integral of kinetic energy and weighted inertia forces for the whole period of motion is considered as a cost functional. The trajectory is…
The problem of determining the angle at which a point mass launched from ground level with a given speed is a standard exercise in mechanics. Similar, yet conceptually and calculationally more difficult problems have been suggested to…
We study the motion of a particle in a particular magnetic field configuration both classically and quantum mechanically. For flux-free radially symmetric magnetic fields defined on circular regions, we establish that particle escape speeds…
The curve joining the points of maximum height in the parabolas of ideal projectile motion is shown to be an ellipse. Some features of the motion are illustrated with the help of such ellipse.
We study the influence of geometry on semilinear elliptic equations of bistable or nonlinear-field type in unbounded domains. We discover a surprising dichotomy between epigraphs that are bounded from below and those that contain a cone of…
The notion of microscopic state of the system at a given moment of time as a point in the phase space as well as a notion of trajectory is widely used in classical mechanics. However, it does not have an immediate physical meaning, since…
Conventionally while we talk about geometry associated with a simple harmonic oscillator, we draw a circle with a radius equal to the amplitude of Oscillator and imagine a particle moving along the perimeter with a frequency same as that of…
This paper investigates positive harmonic functions on a domain which contains an infinite cylinder, and whose boundary is contained in the union of parallel hyperplanes. (In the plane its boundary consists of two sets of vertical…
Both the physical picture of the dynamics of atoms and molecules in intense infrared fields and its theoretical description use the concept of electron trajectories. Here we address a key question which arises in this context: Are…