The kinematic representations of general open-loop chains in many robotic applications are based on the Denavit-Hartenberg (DH) notation. However, when the DH representation is used for kinematic modeling, the relative joint constraints cannot be described explicitly using the common formulation methods. In this paper, we propose a new formulation of solving a system of differential-algebraic equations (DAEs) where the method of Lagrange multipliers is incorporated into the optimization problem for optimal motion planning of redundant systems. In particular, a set of fictitious joints is modeled to solve for the joint constraint forces and moments, as well as the optimal dynamic motion and the required actuator torques, of redundant manipulators described in DH representation. The proposed method is formulated within the framework of our earlier study on the generation of load-effective optimal dynamic motions of redundant manipulators that guarantee successful execution of given tasks, in which the Lagrangian dynamics for general external loads are incorporated. Some example tasks of a simple planar manipulator and a high-degree-of-freedom digital human model are illustrated, and the results show accurate calculation of joint constraint loads without altering the original planned motion. The proposed optimization formulation is equivalent to solving DAEs without integration.