  In recent works, the authors considered various Lagrangians, which are
invariant under a Lie group action, in the case where the independent variables
are themselves invariant. Using a moving frame for the Lie group action, they
showed how to obtain the invariantized Euler-Lagrange equations and the space
of conservation laws in terms of vectors of invariants and the adjoint
representation of a moving frame.
  In this paper, we show how these calculations extend to the general case
where the independent variables may participate in the action. We take for our
main expository example the standard linear action of SL(2) on the two
independent variables. This choice is motivated by applications to variational
fluid problems which conserve potential vorticity. We also give the results for
Lagrangians invariant under the standard linear action of SL(3) on the three
independent variables.
