Abstract: This paper argues that it is not necessary to use the unprovable and geometrically ambiguous Chetaev（Четаев） relations to study the motion of nonholonomic constraint systems. In fact, by applying the Gauss variation to the universal equations of dynamics, the Lagrange equation can be derived for a general first-order nonholonomic constraint system using the method of Lagrange multipliers, the first-order linear nonholonomic constraint system beings only its special case; if the latter must be derived from the constraint equation, the Jourdain variation can be used. The key here is to write the equations satisfied by the imaginary acceleration, imaginary velocity and imaginary displacement respectively correctly.

Key words:  Gauss variation, Jourdain variation, nonholonomic constraint system, Chetaev condition