  In this paper, we introduce local expressions for discrete Mechanics. To
apply our results simultaneously to several interesting cases, we derive these
local expressions in the framework of Lie groupoids, following the program
proposed by Alan Weinstein in [19]. To do this, we will need some results on
the geometry of Lie groupoids, as, for instance, the construction of symmetric
neighborhoods or the existence of local bisections. These local descriptions
will be particular useful for the explicit construction of geometric
integrators for mechanical systems (reduced or not), in particular, discrete
Euler-Lagrange equations, discrete Euler-Poincar\'e equations, discrete
Lagrange-Poincar\'e equations... The results contained in this paper can be
considered as a local version of the study that we have started in [13], on the
geometry of discrete Mechanics on Lie groupoids.
