Annotation of the results obtained in 2023
The problem of a dynamically asymmetric unbalanced disk (omnidisk) rolling on a plane is considered under the assumption that the point of contact of the disk cannot slip in the direction of the horizontal diameter of the disk. The constraints limit the motion of the system so that the motion of the disk occurs without loss of contact with the plane, the horizontal projection of the center of mass onto the supporting plane is fixed, and the projection of the angular velocity of the disk onto the normal to its plane is zero (Suslov’s constraint). Within the framework of the project, in 2023 we examined the dynamics of the above-mentioned disk, taking into account the action of forces and moments of forces of viscous friction. We found partial solutions which correspond to the upper or lower vertical rotations of the disk. The stability of these solutions is analyzed and it is shown that the lower rotation is always unstable and the upper rotation is stable at sufficiently large angular velocities of rotation of the disk. The region of possible motions of the system is analyzed and some assumptions about restrictions on the initial conditions are made under which a flip-over of the disk will be observed. The problem of a sphere with axisymmetric mass distribution rolling with partial slipping on a horizontal plane is investigated. In the model under study, the sphere can slide in the direction of the projection of the symmetry axis onto the supporting plane and can roll without slipping in the direction perpendicular to the above-mentioned direction. It is shown that in the general case the system is nonintegrable and does not admit the existence of an invariant measure with smooth density. Some partial cases of the existence of an additional integral of motion are found and investigated. In addition, the limiting case is found in which the system is integrable by the Euler-Jacobi theorem.

 

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Annotation of the results obtained in 2022
Equations of motion that describe the dynamics of an inhomogeneous disk on a plane, subject to additional constraints, both in the general case and in a particular case have been obtained. The problem was considered under the assumption that the disk does not lose contact with the supporting plane and does not slip in the direction of the plane of the disk and that the projections of the velocity of the center of mass onto the horizontal plane are equal to zero. It is shown that, in addition to the total energy, the resulting system preserves the projection of the angular velocity onto the axis perpendicular to the plane of the disk and that this projection is equal to zero. Furthermore, the resulting system possesses involution owing to which all trajectories of the system are symmetric relative to a plane in phase space that corresponds to the vertical position of the disk. On the fixed level set of the first integrals of this system, reduced equations that are a system of three ordinary differential first-order equations were obtained. For the system obtained, a two-parameter family of permanent rotations (rotations with a constant inclination angle of the plane of the disk and a constant angular velocity) was found. They are rotations of the disk about the vertical axis with a constant angular velocity under which the plane of the disk is also vertical and the center of mass is offset by some constant angle relative to the vertical. To these rotations there corresponds a two-parameter family of fixed points of the reduced system. A detailed analysis was made of the stability of permanent rotations and it was shown that they are unstable at small angular velocities and stable at large angular velocities. In addition, an analysis was made of the dependence of the curve separating stable and unstable rotations on the mass-geometric parameters of the disk and the parameters of the family. Several scenarios were described for the case where permanent rotations lose stability as the angular velocity of rotation decreases. It was shown that the entire phase space of the reduced system splits into separate regions with different types of dynamics: integrable, superintegrable or chaotic. These regions are separated from each other by two invariant manifolds. On the basis of numerical modeling, the nonintegrability of the problem under consideration was proved. A detailed analysis of dynamics for a particular integrable case, that of a symmetric balanced disk, was carried out. The problem of a sphere having an axisymmetric mass distribution and rolling on a horizontal plane has been addressed. It was assumed that, when the sphere rolls, it moves without loss of contact with the supporting plane and without slipping in the direction of the projection of the symmetry axis onto the supporting plane. It was also assumed that, in the direction perpendicular to the above-mentioned one, the sphere can slide relative to the plane. Examples of realization of the described nonholonomic constraint were given. Equations of motion of the system were obtained and it was shown that it admits a number of first integrals: a geometric integral, an energy integral, two constraints imposed on the system (when there is no loss of contact with the plane and no slipping in either direction), Jellett’s integral, and another additional integral linear in velocities. As a result of existence of these integrals of motion, the system under consideration reduces to a system with one degree of freedom.

 

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