Comparative analysis of friction models in dynamics of a ball on a plane

Comparative analysis of the dynamics of a homogeneous ball on a plane with dry friction is conducted for two conjectures: 1) single contact point (non-holonomic statement); 2) the normal load is distributed in the circle spot of contact with radius $\varepsilon$. It is assumed that for given active forces and coefficient of friction the non-slip motion is possible. The expression for load distribution function $\phi$ at the contact spot (second statement) is arbitrary, with general mild restrictions, which ensure correctness of the passage to the limit. It is shown that for $\varepsilon\to 0$ the trajectory of the ball with contact spot approaches the trajectory of the ball with single contact point. Previously similar result was obtained by Fufaev [1] in the case $\phi=\mathrm{const}$. The possibility of approximation of reactions of non-holonomic constraints by means of forces of viscous friction was proved [2], [3], as well as by means of forces of dry friction with infinitely large coefficient of friction [4].