  Matched pairs of Lie groupoids and Lie algebroids are studied. Discrete
Euler-Lagrange equations are written for the matched pairs of Lie groupoids. As
such, a geometric framework to analyse a discrete system by decomposing it into
two mutually interacting subsystems is established. Two examples are provided
to illustrate this strategy; the discrete dynamics on the trivial groupoid, and
the discrete dynamics on the special linear group.
