This paper presents a series expansion that describes the evolution of a mechanical system starting at rest and subject to a time-varying external force. Mechanical systems are presented as second-order systems on a configuration manifold via the notion of a ne connections. The series expansion is derived by exploiting the homogeneity property of mechanical systems and the variations of constant formula. A convergence analysis is obtained using some analytic functions and combinatorial analysis results. This expansion provides a rigorous means of analyzing locomotion gaits in robotics and lays the foundation for the design of motion control algorithms for a large class of underactuated mechanical systems.