On the validity of the vakonomic model and the chetaev model for constraint dynamical systems

By analyzing the virtual work of reaction forces, we prove the failure of the vakonomic model in obtaining the correct equations of motion for nonholonomic mechanical systems. Only when the constraint is integrable can the actual equation of motion be obtained. We show that the null virtual work condition of reaction forces is a robust criterion on the validity of various models of analytical mechanics. For an illustration, classical examples are discussed.