We have discovered a new first integral in the problem of motion of a dynamically symmetric ball, subject to gravity, on the surface of a paraboloid. Using this integral, we have obtained conditions for stability (in the Lyapunov sense) of steady rotations of the ball at the upmost, downmost and saddle point.

We constructed Hirota–Kimura type discretization of the classical nonholonomic Suslov problem of motion of rigid body fixed at a point. We found a first integral proving integrability. Also, we have shown that discrete trajectories asymptotically tend to a line of discrete analogies of so-called steady-state rotations. The last property completely corresponds to well-known property of the continuous Suslov case. The explicite formulae for solutions are given. In $n$-dimensional case we give discrete equations.

In the 1950s and 1960s, Charles F. Taylor designed and tested various prototype one-wheeled vehicles. These machines were stabilized and steered using gyroscopes. In this paper, a simple model of a one-wheeled vehicle is presented and analyzed. This analysis explains the ability of these machines to exhibit stable steady motions.

We study turning strategies in low-dimensional models of legged locomotion in the horizontal plane. Since the constraints due to foot placement switch from stride to stride, these models are piecewise-holonomic, and this can cause stride-to-stride changes in angular momentum and in the ratio of rotational to translational kinetic energy. Using phase plane analyses and parameter studies based on experimental observations of insects, we investigate how these changes can be harnessed to produce rapid turns, and compare the results with dynamical cockroach data. Qualitative similarities between the model and insect data suggest general strategies that could be implemented in legged robots.

The basic notions of the dynamics of nonholonomic systems are revisited in order to give a general and simple method for writing the dynamical equations for linear as well as nonlinear kinematical constraints. The method is based on the representation of the constraints by parametric equations, which are interpreted as dynamical equations, and leads to firstorder differential equations in normal form, involving the Lagrangian coordinates and auxiliary variables (the use of Lagrangian multipliers is avoided). Various examples are illustrated.

A fast rotating tippe top (TT) defies our intuition because, when it is launched on its bottom, it flips over to spin on its handle. The existing understanding of the flipping motion of TT is based on analysis of stability of asymptotic solutions for different values of TT parameters: the eccentricity of the center of mass $0 \leqslant \alpha \leqslant 1$ and the quotient of main moments of inertia $\gamma=I_1/I_3$. These results provide conditions for flipping of TT but they say little about dynamics of inversion.
I propose here a new approach to study the equations of TT and introduce a Main Equation for the tippe top. This equation enables analysis of dynamics of TT and explains how the axis of symmetry $\hat{3}$ of TT moves on the unit sphere $S^2$. This approach also makes possible to study the relationship between behavior of TT and the law of friction.

An everyday example of a nonholonomic mechanical system is a pencil rolling on an inclined plane. Aspects of the motion are discussed in various approximations, of which the most realistic assumes rolling without sliding and that a constant fraction of the pencil’s kinetic energy is retained after each collision with the plane.

In the paper the analytical solution of the partially linearized equations of motion of nearly horizontally rolling of a thick rigid disk on a perfectly rough horizontal plane under the action of gravity is given in terms of Whittaker functions. The solution is used to obtain the asymptotic solutions for a very small inclination angle, the study of unilateral contact between a disk and a plane and the study of disk colliding motion.

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 from the 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 $\theta$ 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 $\theta$ is not less than 49 degrees. For small values of $\theta$, the contact between the body and the support does not break in a neighborhood of stationary motions.

This classical paper by S.A. Chaplygin presents a part of his research in nonholonomic mechanics. In this paper, Chaplygin suggests a general method for integration of the equations of motion for non-holonomic systems, which he himself called the "reducing-multiplier method". The method is illustrated on two concrete problems from non-holonomic mechanics.
This paper produced a considerable effect on the further development of the Russian nonholonomic community. With the help of Chaplygin’s reducing-multiplier theory the equations for quite a number of non-holonomic systems were solved (such systems are known as Chaplygin systems). First published about a hundred years ago, this work has not lost its scientific significance and is hoped to be estimated at its true worth by the English-speaking world.
This publication contributes to the series of RCD translations of Chaplygin’s scientific heritage. In 2002 we published two of his works (both cited in this one) in the special issue dedicated to non-holonomic mechanics (RCD, Vol. 7, no. 2). These translations along with translations of his other two papers on hydrodynamics (RCD, Vol. 12, nos. 1,2) aroused considerable interest and are broadly cited by modern researches.