rigid-body dynamics In the physical science of dynamics, rigid-body dynamics studies the movement of systems of interconnected bodies under the action of external forces. The assumption that the bodies are ''rigid'' (i.e. they do not deform under the action of a ...

, the Painlevé paradox (also called frictional paroxysms by

Jean Jacques Moreau Jean Jacques Moreau (31 July 1923 – 9 January 2014) was a French mathematician and mechanician. He normally published under the name J. J. Moreau. Moreau was born in Blaye. He received his doctorate in mathematics from the University of Paris, ...

) is the paradox that results from inconsistencies between the

contact Contact may refer to: Interaction Physical interaction * Contact (geology), a common geological feature * Contact lens or contact, a lens placed on the eye * Contact sport, a sport in which players make contact with other players or objects * ...

and

Coulomb The coulomb (symbol: C) is the unit of electric charge in the International System of Units (SI). In the present version of the SI it is equal to the electric charge delivered by a 1 ampere constant current in 1 second and to elementary char ...

models of

friction Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. There are several types of friction: *Dry friction is a force that opposes the relative lateral motion of t ...

. It is named for former French prime minister and mathematician

Paul Painlevé Paul Painlevé (; 5 December 1863 – 29 October 1933) was a French mathematician and statesman. He served twice as Prime Minister of the Third Republic: 12 September – 13 November 1917 and 17 April – 22 November 1925. His entry into politic ...

. To demonstrate the paradox, a hypothetical system is constructed where analysis of the system requires assuming the direction of the frictional force. Using that assumption, the system is solved. However, once the solution is obtained, the final direction of motion is determined to contradict the assumed direction of the friction force, leading to a paradox. This result is due to a number of discontinuities in the behavior of rigid bodies and the discontinuities inherent in the Coulomb friction law, especially when dealing with large coefficients of friction. There exist, however, simple examples which prove that the Painlevé paradoxes can appear even for small, realistic friction.

Explanations

Simplified models of friction applied to fully rigid bodies are extremely useful for a basic understanding of physical principles, or when modelling systems for applications such as animation, robotics and bio-mechanics. However, they are is only an approximation to a full elastic model requiring complex systems of

partial differential equations In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to ...

. Several resolutions of the paradox have been published. A mathematical resolution was published in the 1990s by David E. Stewart. In the same decade, Franck Génot and Bernard Brogliato, published an explanation of the paradox from a more mechanical point of view. Génot and Brogliato have studied in great detail the rod dynamics in the neighborhood of a singular point of the phase space, when the rod is sliding. The dynamical equations are then a particular singular ordinary differential equation with vector field ''f''(''x'')/''g''(''x''), where both ''f'' and ''g'' may vanish at a certain point (angle and angular velocity). One of the results is that at this singular point the contact force may grow unbounded, however its impulse remains always bounded (this may explain why time-stepping numerical methods like Moreau's scheme can well handle such situations since they estimate the impulse, not the force). Hence the infinite contact force is not at all an obstacle to the integration. Another situation (different from the first one) is that the trajectories may attain a zone in the phase space, where the linear complementarity problem (LCP) that gives the contact force, has no solution. Then the solution (i.e. the angular velocity of the rod) has to jump to an area where the LCP has a solution. This creates indeed a sort of "impact" with velocity discontinuity. It is noteworthy that J. J. Moreau has shown in his seminal paper through numerical simulation with his time-stepping scheme (afterwards called Moreau's scheme) that Painlevé paradoxes can be simulated with suitable time-stepping methods, for the above reasons given later by Génot and Brogliato.

Physical realizations

A common demonstration of the paradox is the "bouncing" of chalk when forced to slide across a blackboard. Since the Painlevé paradoxes are based on a mechanical model of Coulomb friction, where the calculated friction force can have multiple values when the contact point has no tangential velocity, this is a simplified model of contact. It does, nevertheless, encapsulate the main dynamical effects of friction, such as sticking and slipping zones. In addition to this simple example, more complex realizations of Painlevé paradoxes have been demonstrated.

References

{{DEFAULTSORT:Painleve paradox Physical paradoxes Articles containing video clips