We study the class of mechanical control systems whose Lagrangian is ``kinetic energy minus potential energy.'' With these systems it is useful and interesting to formulate the control problem in terms of the configuration of the system rather than the state, the latter including velocity. If the system is underactuated (i.e., the number of inputs is less than the number of degrees of freedom) then it is a nontrivial problem to ascertain which configurations are accessible under the application of a given class of inputs. We provide a description of these accessible points in terms of the system geometry. The affine connection associated with the kinetic energy's natural Riemannian metric plays an essential role in this description. In particular, we define a product, which we call the symmetric product, for vector fields on a manifold with an affine connection. We give a geometric interpretation of the symmetric product and explain its appearance in our computations for the mechanical control problem.

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