  It is well known that the Lagrangian and the Hamiltonian formalisms can be
combined and lead to "covariant symplectic" methods. For that purpose a
"pre-symplectic form" has been constructed from the Lagrangian using the
so-called Noether form. However, analogously to the standard Noether currents,
this symplectic form is only determined up to total divergences which are
however essential ingredients in gauge theories.
  We propose a new definition of the symplectic form which is covariant and
free of ambiguities in a general first order formulation. Indeed, our
construction depends on the equations of motion but not on the Lagrangian. We
then define a generalized Hamiltonian which generates the equations of motions
in a covariant way. Applications to Yang-Mills, general relativity,
Chern-Simons and supergravity theories are given. We also consider nice sets of
possible boundary conditions that imply the closure and conservation of the
total symplectic form.
  We finally revisit the construction of conserved charges associated with
gauge symmetries, from both the "covariant symplectic" and the "covariantized
Regge-Teitelboim" points of view. We find that both constructions coincide when
the ambiguity in the Noetherian pre-symplectic form is fixed using our new
prescription. We also present a condition of integrability of the equations
that lead to these quantities.
