Modified Lagrange Method to Analyze Problems of Sliding and Rolling

A modified Lagrange method to analyze problems of sliding and rolling is presented. The method is based on modeling the friction as a process of collisions between the sliding and rolling body, and particles of the surface on which it slips and rolls. The process of collisions does not need to describe the exact friction process. Instead it can represent another equivalent mechanism of loss of energy of the body due to sliding and rolling. The function that describes the rate of increase of the kinetic energy of the particles, as a result of the collisions, plays a major roll in the modified Lagrange equations. Cases of isotropic and anisotropic friction can be modeled. Three examples of using the method are presented. It is shown that when an infinitely rough surface is assumed, the classical equations for nonholonomic constraints of rolling without sliding are obtained. Lagrange multipliers that appear in these equations obtain direct physical meaning and the mechanism behind the constraint becomes clear. [S0021-8936(00)01404-5]