The study performs the relation between the Lagrange multipliers and the constraint reaction forces in the multi-body systems. It helps to establish a simple and effective method to deal with the non-smooth factors including friction, contact and collision in multi-body systems. First, a general expression calculating the ideal constraint reaction forces is described by a form of complete Cartesian reference coordinate in a discrete dynamical system. Then, an explicit expression is given in the form of independent general generalized coordinates for a multi-body system. A condition under which the constraint reaction forces can be one-to-one corresponding to Lagrange multipliers is discussed. An example is used to demonstrate the process to realize the condition and the effectiveness of this method.