In this project is presented a general structure fot the explicit equation of motion for constrained mechanical systems, developed by Udwadia and Kalaba, which are subjected to holonomic and non-holonomic constraints. These new equations lead to a simple and new fundamental view of lagrangian mechanics. The Udwadia-Kalaba equation can be derived via the Gauss s principle, d Alembert s principle or extended d Alembert s principle. Once the dynamic equations are explained, shall be demonstrated the applicability of the method in some examples of mechanical systems in engineering problems, as a pendulum with a moving support. Another example will be a particle that is constrained to move along an elliptical trajectory. The results provide deeper insights into the fundamental character of constrained motion in general mechanical systems. Furthermore, a comparison of Udwadia–Kalaba Equation with Newton–Euler Equation is made.