Alexander Kuleshov

The problem of the motion, in the free molecular flow of particles, of a rigid body with a fixed point bounded by the surface of an ellipsoid of revolution is considered. This problem is similar in many aspects to the classical problem of the motion of a heavy rigid body about a fixed point. In particular, this problem possesses the integrable cases corresponding to the classical Euler – Poinsot, Lagrange and Hess cases of integrability of the equations of motion of a heavy rigid body with a fixed point. A natural question arises about the existence of analogues of other integrable cases that exist in the problem of motion of a heavy rigid body with a fixed point (Kovalevskaya case, Goryachev – Chaplygin case, etc) for the system considered. Using the standard Euler angles as generalized coordinates, the Hamiltonian function of the system is derived. Equations of motion of the body in the flow of particles are presented in Hamiltonian form. Using the theorem on the Liouville-type nonintegrability of Hamiltonian systems near elliptic equilibrium positions, which has been proved by V. V. Kozlov, necessary conditions for the existence in the problem under consideration of an additional analytic first integral independent of the energy integral are presented. We have proved that the necessary conditions obtained are not fulfilled for a rigid body with a mass distribution corresponding to the classical Kovalevskaya integrable case in the problem of the motion of a heavy rigid body with a fixed point. Thus, we can conclude that this system does not possess an integrable case similar to the Kovalevskaya integrable case in the problem of the motion of a heavy rigid body with a fixed point.

We present a kinematic analysis and numerical simulation of the toy known as the oloid. The oloid is defined by the convex hull of two equal radius disks whose symmetry planes are at right angles with the distance between their centers equal to their radius. The no-slip constraints of the oloid are integrable, hence the system is essentially holonomic. In this paper we present analytic expressions for the trajectories of the ground contact points, basic dynamic analysis, and observations on the unique behavior of this system.

The paper considers two two-dimensional dynamical problems for an absolutely rigid cylinder interacting with a deformable flat base (the motion of an absolutely rigid disk on a base which in non-deformed condition is a straight line). The base is a sufficiently stiff viscoelastic medium that creates a normal pressure $p(x) = kY(x)+ν\dot{Y}(x)$, where $x$ is a coordinate on the straight line, $Y(x)$ is a normal displacement of the point $x$, and $k$ and $ν$ are elasticity and viscosity coefficients (the Kelvin—Voigt medium). We are also of the opinion that during deformation the base generates friction forces, which are subject to Coulomb’s law. We consider the phenomenon of impact that arises during an arbitrary fall of the disk onto the straight line and investigate the disk’s motion «along the straight line» including the stages of sliding and rolling.

Analysis and simulation are performed for a simplifiedmodel of a skateboard in the absence of rider control. Equations of motion of the model are derived and the problem of integrability of the obtained equations is investigated. The influence of various parameters of the model on its dynamics and stability are studied.

In this paper we continue our investigation of dynamics and stability of motion of a skateboard with a rider. In our previous papers we assumed that the rider, modeled as a rigid body, remains fixed and perpendicular with respect to the board. Hence if the board tilts through γ, the rider tilts through the same angle relative to the vertical, i. e. only one generalized coordinate γ describes the tilt of the board and rider.
Now we make the next step in modeling complexity and we allow the board and rider to have separate degrees of freedom, γ and φ, respectively. Here the rider is assumed to be connected to the board with a pin along the central line of the board through a torsional spring which exerts a torque on the rider and board proportional to the difference in their tilts relative to the vertical. Equations of motion of the model are derived and the problem of integrability of the obtained equations is investigated. The influence of various parameters of the model on its dynamics and stability is studied.