  In field theory, as well as in mechanics, the substitution of some fields in
terms of other fields at the level of the action raises an issue of consistency
with respect to the equations of motion. We discuss this issue and give an
expression which neatly displays the difference between doing the substitution
at the level of the Lagrangian or at the level of the equations of motion. Both
operations do not commute in general. A very relevant exception is the case of
auxiliary variables, which are discussed in detail together with some of their
relevant applications. We discuss the conditions for the preservation of
symmetries - Noether as well as non-Noether - under the reduction of degrees of
freedom provided by the mechanism of substitution. We also examine how the
gauge fixing procedures fit in our framework and give simple examples on the
issue of consistency in this case.
