  In this paper we explore the nonholonomic Lagrangian setting of mechanical
systems in local coordinates on finite-dimensional configuration manifolds. We
prove existence and uniqueness of solutions by reducing the basic equations of
motion to a set of ordinary differential equations on the underlying
distribution manifold $D$. Moreover, we show that any $D-$preserving
discretization may be understood as beeing generated by the exact evolution map
of a time-periodic non-autonomous perturbation of the original continuous-time
nonholonomic system. By means of discretizing the corresponding
Lagrange-d'Alembert principle, we construct geometric integrators for the
original nonholonomic system. We give precise conditions under which these
integrators generate a discrete flow preserving the distribution $D$. Also, we
derive corresponding consistency estimates. Finally, we carefully treat the
example of the nonholonomic particle, showing how to discretize the equations
of motion in a reasonable way, particularly regarding the nonholonomic
constraints. The exploration in this paper lays the ground to analyze the
dynamics of appropriate discretizations of nonholonomic mechnical systems in
the Lagrangian framework and to relate that dynamics to the dynamics of the
original nonholonomic systems. We postpone this analysis to a series of
forthcoming papers.
