A numerical investigation of 2n first-order Hamilton's equations, which describe the motion of a dynamical system, has been conducted using Galerkin's approximations and a derivative-free analogue of Newton's iteration method. Furthermore, the motion stability of a dynamical system in the neighborhood of the approximate periodic solutions due to the effect of the extraneous forces, introduced by the process of using the approximate solutions rather than the actual solutions, has been studied by solving the nonlinear nonhomogeneous differential systems of the perturbed motion. The perturbation solutions are obtained to determine the motion stability. An example, using the van der Pol equation, illustrates the accuracy and error bounds between the approximate solutions and the actual solutions. Furthermore, the example also illustrates the motion stability of perturbation solutions. A computer program for numerical computions has been developed for solving the van der Pol equation with a harmonic forcing term. / Ph. D.