  It is well known that the Lagrangian and Hamiltonian descriptions of field
theories are equivalent at the discrete time level when variational integrators
are used. Besides the symplectic Hamiltonian structure, many physical systems
exhibit a Hamiltonian structure when written in mixed form. In this
contribution, the discrete equivalence of Lagrangian, symplectic Hamiltonian
and mixed formulations is investigated for linear wave propagation phenomena.
Under compatibility conditions between the finite elements, the Lagrangian and
mixed formulations are indeed equivalent. For the time discretization the
leapfrog scheme and the implicit midpoint rule are considered. In mixed methods
applied to wave problems the primal variable (e.g. the displacement in
mechanics or the magnetic potential in electromagnetism) is not an unknown of
the problem and is reconstructed a posteriori from its time derivative. When
this reconstruction is performed via the trapezoidal rule, then these
time-discretization methods lead to equivalent formulations.
