Abstract: Paradoxes of dry friction were discovered by Painlevé in 1895 and caused a controversy on whether the Coulomb-Amontons laws of dry friction are compatible with the Newtonian mechanics of the rigid bodies. Various resolutions of the paradoxes have been suggested including the abandonment of the model of rigid bodies and modifications of the law of friction. For compliant (elastic) bodies, the Painlevé paradoxes may correspond to the friction-induced instabilities. Here we investigate another possibility to resolve the paradoxes: the introduction of the three-value logic. We interpret the three states of a frictional system as either rest-motion-paradox or as rest-stable motion-unstable motion depending on whether a rigid or compliant system is investigated. We further relate the ternary logic approach with the entropic stability criteria for a frictional system and with the study of ultraslow sliding friction (intermediate between the rest and motion or between stick and slip).
5.1. Note on the history of mechanics
Throughout the history of pre-modern mechanics, the state of rest and the state of motion were considered two opposite states of a mechanical system, rather than rest being a special case of motion. This is because in Aristotle’s physics, no motion by inertia was possible, and motion always implied the presence of a moving force or an effective cause of motion .
Several paradoxes have emerged accompanying the opposition of rest vs. motion, including the classical Zeno’s arrow paradox, formulated by Aristotle as “If everything when it occupies an equal space is at rest, and if that which is in locomotion is always occupying such a space at any moment, the flying arrow is therefore motionless.” [25, 33]
Furthermore, it was a matter of discussion, whether rest can be obtained as a combination of two uniform motions, until the so-called Tusi couple was discovered in the 13th century. The Tusi couple is an imaginary device used for the Copernican astronomical model. The Tusi couple consists of two spheres with a smaller sphere rolling inside a larger sphere having twice the same diameter. A point on a smaller sphere performs an oscillatory motion. At an extreme point of its trajectory, the point changes its direction for an opposite one, so that the instantaneous velocity is zero. While the mechanism is similar to the crank-slider linkage, which converts rotation into the reciprocating motion, and it was known from ancient times, the Tusi couple demonstrated that continuous rotation can produce motion with instantaneous zero velocity, which was not obvious until the concept of instantaneous velocity was suggested .
It took significant efforts, until the motion by inertia (without cause) was discovered by Galileo in the early 1600s (in fact, Giuseppe Moletti had already established that objects of different weight fall with the same acceleration ). This required the realization that friction is what prevents moving objects from continuous motion by inertia. Therefore, friction and inertia were in a complimentary relationship: without identifying friction, inertia could not be recognized.
H. A. Wiltsche pointed out that pre-Galilean Aristotelian mechanics studied natural occurrences as opposed to the study of phenomena (“the invariant forms that allegedly underline natural occurrences”) introduced by Galileo. The latter systematically excluded causal accidents as impediments, and friction largely fell as a victim in the search of refined and purified phenomena . Even today, despite almost universal occurrence of friction, it is studied by materials scientists and engineers much more often than by physicists.
After calculus was also created in the 17th century by Newton and Leibnitz, it became possible to introduce the concept of instantaneous velocity as a derivative of coordinate, and consider rest as a special zero-velocity case of motion. Since then, it has been established that the positions and velocities corresponding to degrees of freedom of a mechanical system characterize a state of that system. The approach was extended in a formal way by the Lagrange mechanics, which views the law of motion as a local extremum of a functional depending on the generalized coordinates in the configurational space of the system, and further by the Hamilton mechanics, which defines the law of motion in the phase space of a mechanical system (a cotangential bundle of the configurational space) given by coordinates and momenta.
Галилей предложил изучать идеальные модели природы (они же явления природы, феномены), а не проявления (occurrences). Для идеализации нужно что? Правильно - исключить трение из рассмотрения. Вот с тех пор трение и выпало из рассмотрения физики, его изучают материаловеды и инженерА.
Однако модели систем с трением могут включать в себя противоречия. Когда не существует решения при определенных исходных условиях, или более чем одно решение. Это проблема модели. Но как к ней относиться? Можно по-разному. Можно улучшать модель. А можно сказать, что механическая система характеризуется не только координатами и импульсами, но еще и логическим предикатом.