  We propose in this paper a Proper Generalized Decomposition (PGD) approach
for the solution of problems in linear elastodynamics. The novelty of the work
lies in the development of weak formulations of the PGD problems based on the
Lagrangian and Hamiltonian Mechanics, the main objective being to devise
numerical methods that are numerically stable and energy conservative. We show
that the methodology allows one to consider the Galerkin-based version of the
PGD and numerically demonstrate that the PGD solver based on the Hamiltonian
formulation offers better stability and energy conservation properties than the
Lagrangian formulation. The performance of the two formulations is illustrated
and compared on several numerical examples describing the dynamical behavior of
a one-dimensional bar.
