Lobachevsky Space Research Articles

Уравнения Максвелла в пространстве Лобачевского и моделирование среды со специальными свойствами

Lobachevsky geometry simulates a medium with special constitutive relations Di = ϵ0ϵikEk, Bi = μ0μikHk, where two matrices coincide: ϵik(x) = μik(x). The situation is specified in quasi-Cartesian coordinates (x, y, z) in Lobachevsky space, they are appropriate for modeling a medium nonuniform along the axis z. Exact solutions of the Maxwell equations in complex form of Majorana-Oppenheimer have been constructed. The problem reduces to a second-order differential equation for a certain primary function which can be associated with the one-dimensional Schrödinger problem for a particle in external potential field U(z) = U0e2z. In the frames of the quantum mechanics, Lobachevsky geometry acts as an effective potential barrier with reflection coefficient R = 1; in electrodynamic context, this geometry simulates a medium that effectively acts as an ideal mirror distributed in space. Penetration of the electromagnetic field into the effective medium along the axis z depends on the parameters of an electromagnetic waves ω, k2 1 + k2 2 and the curvature radius ρ of the used Lobachevsky model. The generalized quasi-plane wave solutions f(t, x, y, z) = E + iB and the relevant system of equations are transformed into the real form, which permit us to relate geometry characteristics with expressions for effective tensors of electric and magnetic permittivities.

Nonrelativistic Quantum Mechanical Problem for the Cornell Potential in Lobachevsky Space

In Friedmann–Lobachevsky space-time with a radius of curvature slowly varying over time, we study numerically the problem of motion of a particle moving in the Cornell potential. The mass of the particle is taken to be a reduced mass of the charmonium system. In contrast to the similar problem in flat space, in Lobachevsky space the Cornell potential has a finite depth and, as a consequence, the number of bound states of the system is finite and motion with a continuum energy spectrum is also possible. In this paper, we study the bound states as well as the scattering states of the system.

The Hall effect in Lobachevsky space

In this paper, we consider the problem of the classical and quantum movement of a charged particle in a two-dimensional Lobachevsky space in the presence of analogues of uniform magnetic and electric fields. Based on this consideration, equations for the conductivity for the classical and quantum Hall effect are obtained. It is shown that in Lobachevsky space the presence of a small electrical field leads to a shift of the stair structure of the quantum Hall conductivity.

Corrections for regular identification of high energy positive particles in experimental data using lobachevsky space

In this work, high-energy positive charged particles are distinguished using the Lobachevsky space or Hyperbolic space, which is defined as the total rapidity multiplied by hyperbolic cosines of the transverse and longitudinal rapidity of the particles. Experimental data from eight different types of interactions detected in the bubble chambers accumulated in the high-energy sector were used in the calculations. The weights used to construct the proton and positive pion distributions for each of the interacting secondary particles have been eliminated, allowing such studies to be performed such as particle counting and clustering.These weights do not include calculated weights at azimuth angles, near the center of the star, or without momentum measurements. We now have the opportunity to study positive pions and protons. The percentage of confused particles increases with the beam energy.
 After the reconstruction, we conducted a study of the temperature of the charged particles produced by the p + p interaction of 205 GeV, where Tsallis temperatures are close to Hagedorn . On the other hand, Hagedor and temperatures are higher than Tsallis, which means that the unstable states exchange heat as they move to equilibrium.

К динамике твердого стержня в пространстве Лобачевского

An absolutely solid infinitely thin rectilinear immaterial finite length rod helical motion problem is considered. Motion is heical relative to the inertia center. Located at rod ends point masses are identiall. The rod movement occurs in the constant negative curvature space (Lobachevsky space) under the gyroscopic forces system influence and a constant tracking force. The gyroscopic forces structure is set with using special conditions, which contain six characteristic constant parameters (gyroscopic coefficients). The characteristic motion properties are given. Components analytical dependense on the kinematic rod screw, rod orientation parameters is founded. Expressions of these parameters as time function, are obtained.

Quantum-Mechanical Scattering Problem in Lobachevsky Space at Low Energies

Based on the use of the asymptotics for the wave function of a scattered particle in a Lobachevsky space in a form close to the asymptotics in flat space, general formulas for the theory of quantum mechanical scattering in this space are derived. This approach makes it possible to represent the basic formulas of the theory of scattering in the Lobachevsky space in the form that coincides with the corresponding expressions in three-dimensional Euclidean space. We o.er quantities (length of scattering, effective scattering radius), that are used in describing scattering at short-range potentials and are convenient as phenomenological parameters in describing nuclear interactions at low energies. Numerical estimates of these quantities and cross sections at low energies, that are characteristic of nuclear physics, are given.

The Lobachevsky geometry method in the relativistic kinematics of particle collisions: special frame of reference

The use of the geometry of the Lobachevsky momentum space in the relativistic kinematics of particle collisions is demonstrated by the example of the problem of a special reference system. That system complements the geometric image of the process of elastic scattering of two particles of unequal masses. The speed of a special reference system relative to the center of mass and the angle of scattering of particles in it are determined. The conditions for the existence of such a reference system are analyzed. It is shown that in the case of a process with equal masses, the point corresponding to such a system goes into the ideal region of the extended Lobachevsky space - beyond the cone, and the lines intersecting in it become diverging lines in the sense of Lobachevsky geometry. In this case, the angle between the divergent straight lines (geodesics) of the geometric image is purely imaginary and connected to the minimum length of the segment perpendicular to the diverging straight lines (geodesics).

Covariant formulation of the generalized uncertainty principle

We present a formulation of the generalised uncertainty principle based on commutator $\left[ {\hat x}^i, {\hat p}_j \right]$ between position and momentum operators defined in a covariant manner using normal coordinates. We show how any such commutator can acquire corrections if the momentum space is curved. The correction is completely determined by the extrinsic curvature of the surface $p^2=$ constant in the momentum space, and results in non-commutativity of normal position coordinates $\left[ {\hat x}^i, {\hat x}^j \right] \neq 0$. We then provide a construction for the momentum space geometry as a suitable four dimensional extension of a geometry conformal to the three dimensional relativistic velocity space - the Lobachevsky space - whose curvature is determined by the dispersion relation $F(p^2)=-m^2$, with $F(x)=x$ yielding the standard Heisenberg algebra.

Some problems of convex analysis in the Lobachevsky space

Some problems of convex analysis in the Lobachevsky space

Задача Александрова и метод Погорелова для гиперболических многообразий малой размерности

For hyperbolic manifolds a single the relation of a broken line with vertices on the given rays and a given rotation volume, the existence of a broken line with vertices on the given rays and given turns, etc. are proving. One affirms that if in the three-dimensional Lobachevsky space an orientable polyhedral surface with smooth metric and edge, which consists of the geodesic cycles ℎ1, ℎ2, ..., ℎ𝑘, then this surface can be smoothly extended by tapes 𝑣1, 𝑣2, ..., 𝑣𝑘, for which vertex curvaturesof of the outer edges are given.

Dirac Particle in the Coulomb Field on the Background of Hyperbolic Lobachevsky Model

The known systems of radial equations describing the relativistic hydrogen atom on the base of the Dirac equation in Lobachevsky hyperbolic space is solved. The relevant 2-nd order differential equation has six regular singular points, its solutions of Frobenius type are constructed explicitly. To produce the quantization rule for energy values we have used the known condition for determination of the transcendental Frobenius solutions. This defines the energy spectrum which is physically interpretable and similar to the spectrum arising for the scalar Klein-Fock-Gordon equation in Lobachevsky space. In the present paper, exact analytical solutions referring to this spectrum are constructed. Convergence of the series involved is proved analytically and numerically. Squared integrability of the solutions is demonstrated numerically. It is shown that the spectrum coincides precisely with that previously found within the semi-classical approximation.

Some Generalizations of the Shadow Problem in the Lobachevsky Space

Some Generalizations of the Shadow Problem in the Lobachevsky Space

Деякi узагальнення задачi про тiнь в просторi Лобачевського

УДК 514.13, 515.12, 513.83, 517.5 Розглянуто узагальнення задачі про тінь у гіперболічному просторі. Цю задачу можна розглядати як задачу про знаходження умов, які забезпечують належність точок до узагальнено опуклої оболонки сім'ї множин. Визначено граничні значення параметрів, при яких одні й ті ж конфігурації куль забезпечують належність точки до узагальнено опуклої оболонки куль в евклідовому й гіперболічному просторах. Крім куль розглянуто сім'ї орикуль, а також комбінації куль і орикуль.

Non-Compact Complex Projective Space as a Phase Space

First of all we give a simple example of Kahler manifolds (pseudo)Euclidean complex spaces as well as compact and non-compact complex projective spaces. Moreover we describe The Klein model of Lobachevsky space and mapping a conformal mechanics to itin one dimensional case first. Than we extend Klein model for the N-dimensional. We got interesting results that isometry generators of the phase space, namely Killing potentials are appear to be the constants of motion of the system.

Simple closed geodesics on regular tetrahedra in Lobachevsky space

All simple closed geodesics on regular tetrahedra in Lobachevsky space are described. The asymptotic behaviour is found for the number of simple closed geodesics of length , as tends to infinity. Bibliography: 22 titles.

Lobachevsky space in analysis of relativistic nuclear interactions. New phenomenon—directed nuclear radiation

The relativistic nature of phenomena is illustrated in terms of Lobachevsky geometry. Lobachevsky space is used for description of particle production in relativistic nuclear physics on the basis of experimental data obtained at bubble chambers in π-C, p-C, C-C, n-p reactions in an energy range from units to tens of GeV . The new phenomenon—directed nuclear radiation—is discussed.

Description of a free quantum-mechanical particle in the Lobachevsky space based on the integral equation

The quantum mechanical problem of the motion of a free particle in the three-dimensional Lobachevsky space is interpreted as space scattering. The quantum case is considered on the basis of the integral equation derived from the Schrödinger equation. The work continues the problem considered in [1] studied within the framework of classical mechanics and on the basis of solving the Schrödinger equation in quasi-Cartesian coordinates. The proposed article also uses a quasi-Cartesian coordinate system; however after the separation of variables, the integral equation is derived for the motion along the axis of symmetry horosphere axis coinciding with the z axis. The relationship between the scattering amplitude and the analytical functions is established. The iteration method and finite differences for solution of the integral equation are proposed.

Three-Dimensional Right-Angled Polytopes of Finite Volume in the Lobachevsky Space: Combinatorics and Constructions

Изучаются комбинаторные свойства многогранников, реализуемых в пространстве Лобачевского $\mathbb L^3$ в виде многогранников конечного объема с прямыми двугранными углами. На основе теоремы Е.М. Андреева показано, что срезка бесконечно удаленных вершин прямоугольных многогранников устанавливает взаимно однозначное соответствие с сильно циклически реберно четырехсвязными многогранниками, отличными от куба и пятиугольной призмы. Показано, что любой такой многогранник получается срезкой паросочетания многогранника из этого класса или куба не более чем с двумя срезанными несмежными перпендикулярными ребрами, так что каждый четырехугольник является результатом срезки ребра. Предложено уточнение конструкции Барнетта таких многогранников, и дано ее приложение к прямоугольным многогранникам. Уточнен метод построения идеальных прямоугольных многогранников при помощи операций скручивания ребер, и описана связь этого метода с конструкцией Барнетта при помощи совершенных паросочетаний. Высказана гипотеза об изменении объема многогранника при операциях, и приведены аргументы в ее поддержку.

On the classification of stably reflective hyperbolic Z[√2]-lattices of rank 4

In this paper we prove that the fundamental polyhedron of a ℤ2-arithmetic reflection group in the three-dimensional Lobachevsky space contains an edge such that the distance between its framing faces is small enough. Using this fact we obtain a classification of stably reflective hyperbolic ℤ2-lattices of rank 4.

Three-Dimensional Right-Angled Polytopes of Finite Volume in the Lobachevsky Space: Combinatorics and Constructions

We study combinatorial properties of polytopes realizable in the Lobachevsky space $$\mathbb{L}^{3}$$ as polytopes of finite volume with right dihedral angles. On the basis of E. M. An-dreev’s theorem we prove that cutting off ideal vertices of right-angled polytopes defines a one-to-one correspondence with strongly cyclically four-edge-connected polytopes different from the cube and the pentagonal prism. We show that any polytope of the latter family can be obtained by cutting off a matching of a polytope from the same family or of the cube with at most two nonadjacent orthogonal edges cut, in such a way that each quadrangle results from cutting off an edge. We refine D. Barnette’s construction of this family of polytopes and present its application to right-angled polytopes. We refine the known construction of ideal right-angled polytopes using edge twists and describe its connection with D. Barnette’s construction via perfect matchings. We make a conjecture on the behavior of the volume under operations and give arguments to support it.