This paper can be regarded as a continuation of our previous work [70,71] on the hierarchy of the dynamical behavior of nonholonomic systems. We consider different mechanical systems with nonholonomic constraints; in particular, we examine the existence of tensor invariants (laws of conservation) and their connection with the behavior of a system. Considerable attention is given to the possibility of conformally Hamiltonian representation of the equations of motion, which is mainly used for the integration of the considered systems.

The paper generalizes the classical Gauss principle for non-constrained dynamical systems. For large anisotropic external forces of viscous friction our statement transforms into the common Gauss principle for systems with constraints.

The classical mechanical problem about the motion of a heavy rigid body on a horizontal plane is considered within the framework of theory of systems with unilateral constraints. Under general assumptions about the character of friction, we examine the question on the possibility of detachment of the body fromthe plane under the action of reaction of the plane and forces of inertia. For systems with rolling, we find new scenarios of the appearing of motions with jumps and impacts. The results obtained are applied to the study of stationary motions of a disk.We have showed the following.
1) In the absence of friction, the detachment conditions on stationary motions do not hold. However, if the angle $θ$ between the symmetry axis and the vertical decreases to zero, motions close to stationary motions are necessarily accompanied by detachments.
2) The same conclusion holds for a thin disk that rolls on the support without sliding.
3)For a disk of nonzero thickness in the absence of sliding, the detachment conditions hold on stationary motions in some domain in the space of parameters; in this case, the angle $θ$ is not less than 49 degrees. For small values of $θ$ the contact between the body and the support does not break in a neighborhood of stationary motions.

Mechanical systems with unilateral constraints that can be represented in the contact mode on the phase plane are considered. On the phase plane we construct domains that satisfy the following conditions 1) a detachment from the constraint is impossible; 2) the sign of the constraint reaction corresponds to its unilateral character. These conditions are equivalent for an ideal constraint [1, 2], but they can differ in the presence of friction [3]. Trajectories without detachments belong to intersections of these domains. A circular disc moving on a horizontal support with viscous friction and a disc with the sharp edge moving on an icy surface [4, 5] are considered as examples.

Usually for the control of contact conservation one uses only the second condition from above, which can lead to invalid qualitative conclusions.

We examine simultaneous diagonalization of pairs of commuting Hamiltonians of two degrees of freedom, quadratic in momenta, and their reduction to canonical form. A real partial separation of variables for the Clebsch top is carried out.

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.