Scientific Journal of KubSAU

The article discusses various examples of dynamical systems in which the motion is determined by the logarithmic law - quark systems, hydrodynamic systems, galaxies. Set the general nature of angular motion on a hypersphere in a space of arbitrary dimension and radial movement 6D in the metric of a logarithmic potential. We investigate the 6D metric describing the case of motion with two centers of symmetry. It is shown that in such a metric exists a class of exact solutions, logarithmically dependent on the gravity center coordinates. It was established that in spiral galaxies the orbital motion is due to the logarithmic potential, which is the exact solution of the field equations of Einstein's theory of gravity. The most well-known and widespread in nature case is turbulent flow over a smooth or rough surface, in which the mean velocity depends logarithmically on the distance from the wall. We derivate the logarithmic velocity profile in turbulent flow from the NavierStokes equations. An analogy of the logarithmic velocity profile and the logarithmic law in the case of erosion of materials under impacts been proposed. In electrodynamics, Ampere's law, which describes the interaction of current-carrying conductors, is a consequence of the logarithmic dependence of the vector potential of the distance from the conductor axis. There is, however, an alternative derivation of Ampere law of the Riemann hypothesis about the currents due to the motion of charges

The work discusses various examples of physical systems which state is determined by the logarithmic law - quantum and classical statistical systems and relativistic motion in multidimensional spaces. It was established that the Fermi-Dirac statistics and BoseEinstein-Maxwell-Boltzmann distribution could be described by a single equation, which follows from Einstein's equations for systems with central symmetry. We have built the rate of emergence of classical and quantum systems. The interrelation between statistical and dynamic parameters in supergravity theory in spaces of arbitrary dimension was established. It is shown that the description of the motion of a large number of particles can be reduced to the problem of motion on a hypersphere. Radial motion in this model is reduced to the known distributions of quantum and classical statistics. The model of angular movement is reduced to a system of nonlinear equations describing the interaction of a test particle with sources logarithmic type. The HamiltonJacobi equation was integrated under the most general assumptions in the case of centrally-symmetric metric. The dependence of actions on the system parameters and metrics was found out. It is shown that in the case of fermions the action reaches extremum in fourdimensional space. In the case of bosons there is a local extremum of action in spaces of any dimension

In this work, we investigate the problem of collisions of particles linked to the singularities of the gravitational field in the Ricci flow. A system of non-linear parabolic equations describing the evolution of the axially symmetric metrics proposed. We consider the metric having axial symmetry and comprising two singularities simulating particles of finite mass. There was numerically investigated the change of the metric in the collision of particles. Two formulations of the problem have been considered, one of which scatter particles after the collision, and the other as a result of the merger of two particles, a new stable static system, which can be interpreted as a new particle. The initial and boundary conditions using the exact solution of the static problem, so the collision persist particularly metrics caused by the presence of particles. In numerical experiments determined that the collision of the particles in the Ricci flow leads to the formation of gravitational waves, similar in structure to the waves, registered in the LIGO experiment. Consequently, we can assume that the observed gravity waves caused mainly by transients associated with the change in the metric system. A model describing the emission of gravitational waves in the collision of particles in the Ricci flow proposed. The influence of the parameters of the problem - the speed and mass of the particles, on the amplitude and intensity of the emission of gravitational waves was numerically simulated

In this study, we investigate the problem of the emission of gravitational waves produced in collisions of particles submitted to the singularities of the gravitational field. A system of non-linear parabolic equations describing the evolution of the axially symmetric metrics in the Ricci flow derived. A model describing the emission of gravitational waves in the collision and merger of the particles in the Ricci flow proposed. It is shown that the theory of the Ricci flow describes the problem of black holes merge, consistent with Einstein-Infeld theory, which describes the dynamics of the material particles provided by the singularities of the gravitational field. As an example, we consider the metric having axial symmetry and comprising two singularities simulating particles of finite mass. We have numerically investigated the change of the metric in the collision and merger of the particles. The initial and boundary conditions using the exact solution of the static problem, so the collision persist particularly metrics caused by the presence of particles. In numerical experiments determined that the collision of the particles in the Ricci flow leads to the formation of gravitational waves, similar in structure to the waves, registered in the LIGO experiment. Consequently, we can assume that the observed gravity waves caused mainly by transients associated with the change in the metric of a system. The influence of the parameters of the problem - the speed and mass of the particles, on the amplitude and intensity of the emission of gravitational waves was numerically simulated. We have found chaotic behavior of gravitational potentials at the merger of the singularities in the Ricci flow

The article deals with the problem of changing the polarity of the geomagnetic field in the satellite model. It is assumed that the central core of the earth magnetized and surrounded by a number of satellites, each of which has a magnetic moment. Satellites interact with a central core and one another by means of gravity and through a magnetic field. It is shown that satellites distributed in orbit around a central core in such a system. It displays two models, one of which on the outer orbit satellites interact with each other and with a central body - the core and satellites, located on the inner orbit. The central body can make sudden upheavals in the fall at the core of one or more satellites, which leads to the excitation of vibrations in the satellite system, located on the outer orbit. It is shown that the duration of phase with constant polarity and upheaval time depends on the magnitude of the disturbance torque and core asymmetry. The second model contains two magnets subsystems and the central core. The rapid change of the geomagnetic field polarity detected on the basis of paleomagnetic data is modeled based on the Euler theory describing the rigid body rotation. In this model, there are modes with a quick flip of the body while maintaining the angular momentum. If the body has a magnetic moment, when there is a change coup magnetic field polarity. This leads to the excitation of vibrations in the satellite subsystems that are on the inner and outer orbits. Numerical simulation of the dynamics of the system consisting of the core and 10-13 satellites was run to determine the period of constant polarity magnetic field

In this article, we investigate the problem of creation of matter in the collision of particles, presented by singularities of the gravitational field. A system of nonlinear parabolic equations describing the evolution of the axially symmetric metrics in the Ricci flow derived. A model describing the creation of matter in the collision and merger of the particles in the Ricci flow proposed. It is shown that the theory that describes the Ricci flow in the collision of black holes is consistent with EinsteinInfeld theory, which describes the dynamics of the material particles provided by the singularities of the gravitational field. As an example, we consider the metric having axial symmetry and which contains two singularities simulating particles of finite mass. It is shown that the static metric with two singularities corresponding to in Newton's theory of gravity two particles moving around the center of mass in circular orbits in a non-inertial frame of reference, rotating with a period of two-body system rotation. We have numerically investigated the change of the metric in the collision of particles with subsequent expansion. In numerical experiments, we have determined that the collision of the particles in the Ricci flow leads to the formation of two types of matter with positive and negative energy density, respectively. When moving singularities towards each other in the area between the particles the matter is formed with negative energy density, and in the region behind the particles - with positive density. In the recession of the singularities, the matter with positive energy density is formed in the area between the particles. The question of the nature of baryonic matter in the expanding universe is discussed

In this article, the restricted problem of three and more bodies in the Ricci flow in the general theory of relativity considered. A system of non-linear parabolic equations describing the evolution of the axially symmetric metrics in the Ricci flow proposed. A model describing the motion of particles in the Ricci flow derived. It is shown that the theory describing the Ricci flow in the many-body problem is consistent with the Einstein-Infeld theory, which describes the dynamics of the material particles provided by the singularities of the gravitational field. As an example, consider the metric having axial symmetry and contains two singularities simulating particles of finite mass. It is shown that the static metric with two singularities corresponds to Newton's theory of the two centers of gravity, moving around the center of mass in circular orbits in a noninertial frame of reference, rotating with a period of bodies. We consider the statement of the problem of many bodies distributed at the initial time on the axis of symmetry of the system. In numerical calculations, we studied the properties of the gravitational potential in the problem of establishing a static condition in which multiple singularities retain the initial position on the axis of the system. This is achieved due to relativistic effects, which have no analogues in Newton's theory of gravitation. Using the properties of relativistic potentials we have justified transition from the relativistic motion of the particles to the dynamic equations in the classic theory

In this article, we investigate the restricted problem of many bodies with a logarithmic potential in the general theory of relativity. We consider the metric having axial symmetry and containing a logarithmic singularity. In numerical calculations, we studied the properties of the gravitational potential in the problem of establishing a static condition in which multiple singularities retain the initial position on the axis of the system. This is achieved due to relativistic effects, which have no analogues in Newton's theory of gravitation. The motion of relativistic particles in a logarithmic potential sources distributed on the surface of a torus simulated. It is shown that the trajectory of the particles in these systems form a torus covered with needles. It was found, that the Ricci flow in the general theory of relativity could be born three kinds of matter - positive and negative energy density, as well as the color of matter, the gravitational potential of which is complex. It has been shown that this type of material is associated with the manifestation of the quantummechanical properties, which is consistent with the hypothesis of the origin of Schrodinger quantum mechanics. It is assumed that the most likely candidate for the role of the color of matter is the system of quarks as to describe the dynamics of quarks using the logarithmic potential, and the quarks themselves are not observed in the free state

In this article we consider the many-body problem in general relativity in the case of the distribution of N singularities on the circle. It specifies the exact solution of the problem for an arbitrary distribution of singularities. It is shown that the static metric of N singularities corresponds to Newton's theory of N centers of gravity, moving around the central body in a circular orbit in a non-inertial frame of reference, rotating with a period of bodies revolving. We consider the statement of the problem of many bodies distributed at the initial time on the circle. In numerical calculations, we studied the properties of the gravitational potential in the problem of establishing a static condition in which multiple singularities retain the initial position on the circle. This is achieved due to relativistic effects, which have no analogues in Newton's theory of gravitation. Using the properties of relativistic potentials justified transition from the relativistic motion of the particles to the dynamic equations in the classical theory. A system of non-linear parabolic equations describing the evolution of the metric in the Ricci flow proposed. The problem of the calculation of the potentials in the Ricci flow formulated. The application of the theory to describe the ring galaxy, planetary rings and the asteroid belt considered

In this work, we examine the dynamics of relativistic particles in the ring or spiral galaxy metric in general relativity. On the basis of the solution of Einstein's equations we have derived metric having axial symmetry, comprising N centers of gravity and a logarithmic singularity. The application received metrics to describe the motion of particles in a spiral and ring galaxy. On the basis of Einstein's equations solutions for vacuum we are explained rotation of matter in spiral galaxies. An expression for gravitation potential in the inner region of spiral galaxies in agreement with experimental data on the rotation of the CO and hydrogen is described. It is established that in the metric with N centers of gravity which are distributed on the circumference, exist as a local motion near the center of gravity, and motion around N gravity center as well. The transition from one mode of motion to another is determined by the initial distance to the circle on which the distributed centers of gravity. A system of non-linear parabolic equations describing the evolution of the metric in the Ricci flow proposed. The boundary problem for the gravitational potentials in the Ricci flow was formulated. There are applications of the theory to describe a spiral and ring galaxy