On the linear integrals of the chaplygin equations

Abstract

A geometrical property is established of the lines of congruence determined by the first integral of the dynamic equations of the nonholonomic Chaplygin system, with the integral linear with respect to the velocities. In the case of two degrees of freedom and active nonzero forms applied to the system, the property yields explicitly the necessary and sufficient conditions of existence of a linear integral. The result is illustrated by an example.