In this paper, we consider the dynamics of a heavy homogeneous ball moving under the influence of dry friction on a fixed horizontal plane. We assume the ball to slide without rolling. We demonstrate that the plane may be divided into two regions, each characterized by a distinct coefficient of friction, so that balls with equal initial linear and angular velocity will converge upon the same point from different initial locations along a certain segment. We construct the boundary between the two regions explicitly and discuss possible applications to real physical systems.

In this paper we investigate the dynamics of a body with a flat base (cylinder) sliding on a horizontal rough plane. For analysis we use two approaches. In one of the approaches using a friction machine we determine the dependence of friction force on the velocity of motion of cylinders. In the other approach using a high-speed camera for video filming and the method of presentation of trajectories on a phase plane for analysis of results, we investigate the qualitative and quantitative behavior of the motion of cylinders on a horizontal plane. We compare the results obtained with theoretical and experimental results found earlier. In addition, we give a systematic review of the well-known experimental and theoretical results in this area.

In this paper we investigate the dynamics of a body with a flat base (cylinder) sliding on a horizontal rough plane. For analysis we use two approaches. In one of the approaches using a friction machine we determine the dependence of friction force on the velocity of motion of cylinders. In the other approach using a high-speed camera for video filming and the method of presentation of trajectories on a phase plane for analysis of results, we investigate the qualitative and quantitative behavior of the motion of cylinders on a horizontal plane. We compare the results obtained with theoretical and experimental results found earlier. In addition, we give a systematic review of the well-known experimental and theoretical results in this area.

This paper reviews the results of stability analysis for polygonal configurations of a point vortex system in an annular region depending on the ratio of the inner and outer radii of the annulus. Conditions are found for linear stability of Thomson?s configurations for the case $N<7$. The paper also shows that a system of two vortices between parallel walls is a limiting case of a two-vortex system in an annular region, as the radii of the annulus tend to infinity.

In this paper we investigate the dynamics of a body with a flat base sliding on a horizontal and inclined rough plane under the assumption of linear pressure distribution of the body on the plane as the simplest dynamically consistent friction model. For analysis we use the descriptive function method similar to the methods used in the problems of Hamiltonian dynamics with one degree of freedom and allowing a qualitative analysis of the system to be made without explicit integration of equations of motion. In addition, we give a systematic review of the well-known experimental and theoretical results in this area.

In this paper we investigate the dynamics of a body with a flat base sliding on a inclined plane under the assumption of linear pressure distribution of the body on the plane as the simplest dynamically consistent friction model. Computer-aided analysis of the system?s dynamics on the inclined plane using phase portraits has allowed us to reveal dynamical effects that have not been found earlier.

Inšthis paper wešinvestigate the dynamics ofšašbody with ašflat base sliding onšašhorizontal plane under the assumption ofšlinear pressure distribution ofšthe body onšthe plane asšthe simplest dynamically consistent friction model.

For analysis wešuse the descriptive function method similar tošthe methods used inšthe problems ofšHamiltonian dynamics with one degree ofšfreedom and allowing ašqualitative analysis ofšthe system tošbešmade without explicit integration ofšequations ofšmotion. Inšaddition, wešgive ašsystematic review ofšthe well-known experimental and theoretical results inšthis area.

An algorithm for the investigation of a mathematical model of the impact of a double pendulum against an obstacle is constructed and realized by computer. This algorithm allows calculation of impact loads and coefficients of restitution at the contact point and hinges. The main goal of this investigation is to minimize negative impact conditions in the hingings.

The problem ofšdynamics ofšheavy uniform ball moving onšthe fixed rough plane under its own inertia and forces ofšdry friction isšconsidered. Assuming that friction coefficient isšvariable, the switching curve for change the value ofšfriction coefficient isšconstructed. Using this curve tošchange the value ofšfriction coefficient wešhave shown that the bundle ofšequal balls starting from one interval with equal linear and angular velocities should gather atšone point.

The problem ofšašuniform straight cylinder (disc) sliding onšašhorizontal plane under the action ofšdry friction forces isšconsidered. The contact patch between the cylinder and the plane coincides with the base ofšthe cylinder. Wešconsider axisymmetric discs, i.e. wešassume that the base ofšthe cylinder isšsymmetric with respect tošthe axis lying inšthe plane ofšthe base. The focus isšonšthe qualitative properties ofšthe dynamics ofšdiscs whose circular base, triangular base and three points are inšcontact with ašrough plane.

Inšthis paper, the system ofštwo vortices inšanšannular region isšshown tošbešintegrable inšthe sense ofšLiouville. Ašfew methods for analysis ofšthe dynamics ofšintegrable systems are discussed and these methods are then applied tošthe study ofšpossible motions ofštwo vortices ofšequal inšmagnitude intensities. Using the previously established fact ofšthe existence ofšrelative choreographies, the absolute motions ofšthe vortices are classified inšrespect tošthe corresponding regions inšthe phase portrait ofšthe reduced system.

For the system of two point vortices in anulus the Hamiltonian is expressed in terms of elliptic functions. The stability of the Thomson configuration is studied.

This study isšthe continuation ofšthe computer experiment [1] with particles ofšgas inšašone-dimensional tube, described earlier. Inšthis paper wešgive investigation results for the statistical properties ofšašrelativistic gas inšašone-dimensional tube. Itšisšshown that this system reaches the state ofšthermodynamical equilibrium whose distribution function isšdetermined byšthe relativistic energy ofšparticles. The system ofšparticles inšašone-dimensional tube isšdescribed byšanalogy with the billiards inšašpolygon.

With the help ofšmathematical modeling, wešstudy the behavior ofšašgas ($\sim10^6$ particles) inšašone-dimensional tube. For this dynamical system, wešconsider the following cases:
?šcollisionless gas (with and without gravity) inšaštube with both ends closed, the particles ofšthe gas bounce elastically between the ends,
?šcollisionless gas inšaštube with its left end vibrating harmonically inšašprescribed manner,
?šcollisionless gas inšaštube with ašmoving piston, the piston?s mass isšcomparable tošthe mass ofšašparticle.
The emphasis isšonšthe analysis ofšthe asymptotic ($t??$)) behavior ofšthe system and specifically onšthe transition tošthe state ofšstatistical oršthermal equilibrium. This analysis allows preliminary conclusions onšthe nature ofšrelaxation processes.
Atšthe end ofšthe paper the numerical and theoretical results obtained are discussed. Itšshould bešnoted that not all the results fit well the generally accepted theories and conjectures from the standard texts and modern works onšthe subject.