The life and career of the great French mathematician and politician Paul Painlevé is described. His contribution to the analytical theory of nonlinear differential equations was significant. The paper outlines the achievements of Paul Painlevé and his students in the investigation of an interesting class of nonlinear second-order equations and new equations defining a completely new class of special functions, now called the Painlevé transcendents. The contribution of Paul Painlevé to the study of algebraic nonintegrability of the $N$-body problem, his remarkable observations in mechanics, in particular, paradoxes arising in the dynamics of systems with friction, his attempt to create the axiomatics of mechanics and his contribution to gravitation theory are discussed.

We consider two impact mappings, the Brach impact mapping and an energetic impact mapping, for rigid-body mechanisms with impacts and friction. The two impact mappings represent the opposite end of the spectrum from basic to advanced impact mappings. Both impact mappings are briefly derived and described. For the Brach impact mapping we will introduce the concept of impulse ratio and discuss how the kinetic energy changes during an impact as the impulse ratio is varied. This analysis is used to further extend the Brach impact mapping to cover situations that were previously omitted. Finally, we make comparisons between the two impact mappings and show how the Painlevé paradox appears in the two impact mappings. The conclusion of the comparisons is that while the basic impact mapping seems easy to implement in a computer simulator it may in the end be more complex and also introduce unnecessary complications that are completely artificial.

We study second-order, second-degree systems related to the Painlevé equations which possess one and two parameters. In every case we show that by introducing a quantity related to the canonical Hamiltonian variables it is possible to derive such a second-degree equation. We investigate also the contiguity relations of the solutions of these higher-degree equations. In most cases these relations have the form of correspondences, which would make them non-integrable in general. However, as we show, in our case these contiguity relations are indeed integrable mappings, with a single ambiguity in their evolution (due to the sign of a square root).

It is well known that the self-similar solutions of the Korteweg–de Vries equation and the modified Korteweg–de Vries equation are expressed via the solutions of the first and second Painlevé equations. In this paper we solve this problem for all equations from the Korteveg–de Vries, modified Korteweg–de Vries, Kaup–Kupershmidt, Caudrey–Dodd–Gibbon and Fordy–Gibbons hierarchies. We show that the self-similar solutions of equations corresponding to hierarchies mentioned above can be found by means of the general solutions of higher-order Painlevé hierarchies introduced more than ten years ago.

Painlevé's paradox occurs in the rigid-body dynamics of mechanical systems with frictional contacts at configurations where the instantaneous solution is either indeterminate or inconsistent. Dynamic jamming is a scenario where the solution starts with consistent slippage and then converges in finite time to a configuration of inconsistency, while the contact force grows unbounded. The goal of this paper is to demonstrate that these two phenomena are also relevant to the field of robotic walking, and can occur in two classical theoretical models of passive dynamic walking — the rimless wheel and the compass biped. These models typically assume sticking contact and ignore the possibility of foot slippage, an assumption which requires sufficiently large ground friction. Nevertheless, even for large friction, a perturbation that involves foot slippage can be kinematically enforced due to external forces, vibrations, or loose gravel on the surface. In this work, the rimless wheel and compass biped models are revisited, and it is shown that the periodic solutions under sticking contact can suffer from both Painlevé's paradox and dynamic jamming when given a perturbation of foot slippage. Thus, avoidance of these phenomena and analysis of orbital stability with respect to perturbations that include slippage are of crucial importance for robotic legged locomotion.

An effective equation of motion of a rattleback is obtained from the basic equation of motion with viscous friction depending on slip velocity. This effective equation of motion is used to estimate the number of spin reversals and the rattleback’s shape that causes the maximum number of spin reversals. These estimates are compared with numerical simulations based on the basic equation of motion.

We discuss the basic problem of the dynamics of mechanical systems with constraints, namely, the problem of finding accelerations as a function of the phase variables. It is shown that in the case of Coulomb friction, this problem is equivalent to solving a variational inequality. The general conditions for the existence and uniqueness of solutions are obtained. A number of examples are considered.
For systems with ideal constraints the problem under discussion was solved by Lagrange in his "Analytical Dynamics" (1788), which became a turning point in the mathematization of mechanics. In 1829, Gauss gave his principle, which allows one to obtain the solution as the minimum of a quadratic function of acceleration, called the constraint. In 1872 Jellett gave examples of non-uniqueness of solutions in systems with static friction, and in 1895 Painlevé showed that in the presence of friction, the absence of solutions is possible along with the nonuniqueness. Such situations were a serious obstacle to the development of theories, mathematical models and the practical use of systems with dry friction. An elegant, and unexpected, advance can be found in the work [1] by Pozharitskii, where the author extended the Gauss principle to the special case where the normal reaction can be determined from the dynamic equations regardless of the values of the coefficients of friction. However, for systems with Coulomb friction, where the normal reaction is a priori unknown, there are still only partial results on the existence and uniqueness of solutions [2–4].
The approach proposed here is based on a combination of the Gauss principle in the form of reactions with the representation of the nonlinear algebraic system of equations for the normal reactions in the form of a variational inequality. The theory of such inequalities [5] includes results on the existence and uniqueness, as well as the developed methods of solution.

In this paper we consider the dynamics of a rigid body with a sharp edge in contact with a rough plane. The body can move so that its contact point is fixed or slips or loses contact with the support. In this paper, the dynamics of the system is considered within three mechanical models which describe different regimes of motion. The boundaries of the domain of definition of each model are given, the possibility of transitions from one regime to another and their consistency with different coefficients of friction on the horizontal and inclined surfaces are discussed.

In this paper we consider the control of a dynamically asymmetric balanced ball on a plane in the case of slipping at the contact point. Necessary conditions under which a control is possible are obtained. Specific algorithms of control along a given trajectory are constructed.