  The Hamiltonian description of classical gauge theories is a very well
studied subject. The two best known approaches, namely the covariant and
canonical Hamiltonian formalisms have received a lot of attention in the
literature. However, a full understanding of the relation between them is not
available, specially when the gauge theories are defined over regions with
boundaries. Here we consider this issue, by first making precise what we mean
by equivalence between the two formalisms. Then we explore several first order
gauge theories, and assess whether their corresponding descriptions satisfy the
notion of equivalence. We shall show that, even when in several cases the two
formalisms are indeed equivalent, there are counterexamples that signal that
this is not always the case. Thus, non-equivalence is a generic feature for
gauge field theories. These results call for a deeper understanding of the
subject.
