  For integrable systems in the sense of multidimensional consistency (MDC) we
can consider the Lagrangian as a form, which is closed on solutions of the
equations of motion. For 2-dimensional systems, described by partial difference
equations with two independent variables, MDC allows us to define an action on
arbitrary 2-dimensional surfaces embedded in a higher dimensional space of
independent variables, where the action is not only a functional of the field
variables but also the choice of surface. It is then natural to propose that
the system should be derived from a variational principle which includes not
only variations with respect to the dependent variables, but also with respect
to variations of the surface in the space of independent variables. Here we
derive the resulting system of generalized Euler-Lagrange equations arising
from that principle. We treat the case where the equations are 2 dimensional
(but which due to MDC can be consistently embedded in higher-dimensional
space), and show that they can be integrated to yield relations of
quadrilateral type. We also derive the extended set of Euler-Lagrange equations
for 3-dimensional systems, i.e., those for equations with 3 independent
variables. The emerging point of view from this study is that the variational
principle can be considered as the set of equations not only encoding the
equations of motion but as the defining equations for the Lagrangians
themselves.
