A model of an underactuated mechanical system consists of a system of coupled nonlinear differential equations of second order, where the right hand side describes the effect of the external (generalized) forces, i.âe., the system input. Its components are assigned to the scalar differential equations via a (general) configuration dependent matrix. However, in the literature, it is often assumed that the configuration coordinates and the input can be chosen such that a decomposition of the equations of motion into a nonactuated and a fully actuated subsystem is possible, which significantly facilitates further steps of systems analysis and controller design. The present contribution proposes a criterion whether or not such a choice of coordinates is possible. As mathematical tools, differential forms and the Frobenius Theorem are used. The application of the criterion is demonstrated by means of three examples.