  In this work we show that we can obtain dual equivalent actions following the
symplectic formalism with the introduction of extra variables which enlarge the
phase space. We show that the results are equal as the one obtained with the
recently developed gauging iterative Noether dualization method (NDM). We
believe that, with the arbitrariness property of the zero mode, the symplectic
embedding method (SEM) is more profound since it can reveal a whole family of
dual equivalent actions. We illustrate the method demonstrating that the
gauge-invariance of the electromagnetic Maxwell Lagrangian broken by the
introduction of an explicit mass term and a topological term can be restored to
obtain the dual equivalent and gauge-invariant version of the theory.
