  We consider the calculation of Euler--Lagrange systems of ordinary difference
equations, including the difference Noether's Theorem, in the light of the
recently-developed calculus of difference invariants and discrete moving
frames. We introduce the difference moving frame, a natural discrete moving
frame that is adapted to difference equations by prolongation conditions.
  For any Lagrangian that is invariant under a Lie group action on the space of
dependent variables, we show that the Euler--Lagrange equations can be
calculated directly in terms of the invariants of the group action.
Furthermore, Noether's conservation laws can be written in terms of a
difference moving frame and the invariants. We show that this form of the laws
can significantly ease the problem of solving the Euler--Lagrange equations,
and we also show how to use a difference frame to integrate Lie group invariant
difference equations. In this Part I, we illustrate the theory by applications
to Lagrangians invariant under various solvable Lie groups. The theory is also
generalized to deal with variational symmetries that do not leave the
Lagrangian invariant.
  Apart from the study of systems that are inherently discrete, one significant
application is to obtain geometric (variational) integrators that have finite
difference approximations of the continuous conservation laws embedded
\textit{a priori}. This is achieved by taking an invariant finite difference
Lagrangian in which the discrete invariants have the correct continuum limit to
their smooth counterparts. We show the calculations for a discretization of the
Lagrangian for Euler's elastica, and compare our discrete solution to that of
its smooth continuum limit.
