This thesis explores several areas in dynamics which can be viewed as applications of the local state-space form of a mechanical system. The simulation of mechanical systems often involves the solution of differential algebraic equations (DAEs). DAEs occur in every mechanism containing kinematic loops. Such systems can be found in a wide range of areas including the aerospace, automotive, construction, and farm equipment industries. The numerical treatment of DAEs is a topic which is relatively recent and continues to be studied. One can regard DAEs as ordinary differential equations (ODEs) on certain invariant manifolds after index reduction. Thus, the numerical solutions of the DAEs can be obtained through integration of their underlying ODEs. In certain circumstances, difficulties may occur since the numerical solutions of the underlying ODE can drift away from the invariant manifold. In this thesis, the underlying ODEs are locally transformed into ODEs of minimal dimension via local parameterizations of the invariant manifold. By their nature, such ODEs are local and implicit, but their solutions do not suffer from the drift phenomenon. Since the states of these minimal ODEs are independent, they are known as a local state-space form of the equations of motion. This work focuses on generalizing the application of the local state-space form and applying it towards problem areas in multibody dynamics and robotics. The first application of the local state-space form is in deriving a formulation of dynamics called the Singularity Robust Null Space Formulation. This formulation utilizes several aspects of the singular value decomposition for an approach which is efficient, does not fail at singularities, and is better suited than most near singularities. The second application area in this work is the study of the linearized mechanical system. Since the linearized model is also useful in optimization and implicit integration problems, an efficient recursive algorithm for its construction is derived. The algorithm appeals to a formulation of the dynamics found in robotics to ease in a coherent derivation.