Inšthis paper wešinvestigate various kinematic properties ofšrolling ofšone rigid body onšanother both for the classical model ofšrolling without slipping (the velocities ofšbodies atšthe point ofšcontact coincide) and for the model ofšrubber-rolling (with the additional condition that the spinning ofšthe bodies relative tošeach other bešexcluded). Furthermore, inšthe case where both bodies are bounded byšspherical surfaces and one ofšthem isšfixed, the equations ofšmotion for ašmoving ball are represented inšthe form ofšthe Chaplygin system. When the center ofšmass ofšthe moving ball coincides with its geometric center, the equations ofšmotion are represented inšconformally Hamiltonian form, and inšthe case where the radii ofšthe moving and fixed spheres coincides, they are written inšHamiltonian form.

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.

In this paper we investigate various kinematic properties of rolling of one rigid body on another both for the classical model of rolling without slipping (the velocities of bodies at the point of contact coincide) and for the model of rubber-rolling (with the additional condition that the spinning of the bodies relative to each other be excluded). Furthermore, in the case where both bodies are bounded by spherical surfaces and one of them is fixed, the equations of motion for a moving ball are represented in the form of the Chaplygin system. When the center of mass of the moving ball coincides with its geometric center, the equations of motion are represented in conformally Hamiltonian form, and in the case where the radii of the moving and fixed spheres coincides, they are written in Hamiltonian form.

The paper considers two two-dimensional dynamical problems for anšabsolutely rigid cylinder interacting with ašdeformable flat base (the motion ofšanšabsolutely rigid disk onšašbase which inšnon-deformed condition isšašstraight line). The base isšašsufficiently stiff viscoelastic medium that creates ašnormal pressure $p(x) = kY(x)+?\dot{Y}(x)$, where $x$šisšašcoordinate onšthe straight line, $Y(x)$ isšašnormal displacement ofšthe pointš$x$, and $k$šand $?$šare elasticity and viscosity coefficients (the Kelvin?Voigt medium). Wešare also ofšthe opinion that during deformation the base generates friction forces, which are subject tošCoulomb?s law. Wešconsider the phenomenon ofšimpact that arises during anšarbitrary fall ofšthe disk onto the straight line and investigate the disk?s motion ?along the straight line? including the stages ofšsliding and rolling.