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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