Based on an analysis of three kinds of non-equivalent nonholonomic variations，i-e-，the Suslovs variation，Hlders variation and vakonomic variation， the method of Lagrange multipliers and stationary action principle are utilized to discuss the differential equations of motion for nonlinear nonholonomic constrained systems with respect to the three kinds of variations- The condition for the three kinds of equations to be equivalent is investigated- The equations for affine nonholonomic constrained systems are also obtained as special cases of the general nonholonomic systems- Two examples are given to illustrated the validity of the result-