Who was Painlevé and why his paradoxes are so important for the study of friction

There is a Russian anecdote about a mathematics professor (Nikolai Luzin) who, when asked “which historical event happened in October (November) 1917?” answered “that autumn we have started studying trigonometric series”. The point of the joke was of course that the Soviet propaganda taught that the Russian October (November) 1917 Revolution was the most important historical event of the 20th century. Beyond the Russian revolution, other events happened at the same time. The Balfour declaration was issued in London, while Paul Painlevé became a Prime Minister of France in Fall 1917.

However, for us Painlevé is of interest not as a politician but as a mechanician and mathematitian, after whom the frictional Painlevé paradoxes were called. There is a family of such paradoxes, and one of the simplest is shown in the figure below. To solve the equations of statics, an assumption should be made about the direction of the friction force. However, after the solution is obtained, it may turn out that the assumed direction of the friction force contradicts the direction of the velocities dx/dt in the system, therefore resulting in a paradox.


Figure 1. A setup for the Painlevé paradox: two sliders (one frictional and the other is frictionless) connected by a rigid bar; the paradox is resolved if the bar is assumed to be elastically deformable with the stiffness k.

In the system shown in Fig. 1, two sliders both having the mass m are connected by a link with a constant length l forming the angle alpha with the sliding surface. The upper slider is frictionless while the lower slider is frictional with the coefficient of friction μ. An external force P is applied to the upper slider. The motion of such a system is governed by the equation 2md2x/dt2 = P – μ|R|sign(dx/̇dt), where R is the normal force acting at the first slider (Rcosφ is the compression force in the link). From the balance of forces acting on the second slider, md2x/dt2 = P + R/tan(φ).

To find the unknown acceleration d2x/dt2 and force R, we should assume the value of sign(dx/̇dt). However, if μ tanφ>2 then two solutions exists for a positive velocity dx/dt>0 satisfying md2x/dt2 = P(1±μ tanφ )/(2±μtanφ), while no solution exists for the negative velocity dx/dt<0.

The Painlevé paradox indicates that the Coulomb friction is not always logically compatible with the rest of the equations of mechanics. However, there is an interesting relationship between the Painlevé paradoxes and the friction-induced instabilities. If an elastically deformable link is considered instead of the rigid link, the sliding system has an additional degree of freedom. In that case, the paradox corresponds to the unstable solution with the reaction force growing until the value of φ decreases so that the paradox condition μ tan(φ)>2 will not be satisfied anymore. Thus the static paradox of a non-existent solution, when studied in dynamics, corresponds to an unstable solution. The Painlevé paradoxes turn out to be related to very central and universal aspects of friction.