  The EHP and the MCAP provide new rigorous weak variational formalism for a
broad range of initial boundary value problems in mathematical physics and
mechanics. Both approaches utilize the mixed formulation and lead to the
development of various space-time finite element methods. In this paper, their
potential when adopting temporally higher order approximations is investigated.
The classical single-degree-of-freedom dynamical systems are primarily
considered to validate and to investigate the performance of the numerical
algorithms developed from both formulations. For the undamped system, all the
algorithms are found to be symplectic and unconditionally stable with respect
to the time step. On the other hand, for the damped system, the approach is
shown to be robust and to be accurate with good convergence characteristics.
