  Though ubiquitous as first-principles models for conservative phenomena,
Hamiltonian systems present numerous challenges for model reduction even in
relatively simple, linear cases. Here, we present a method for the
projection-based model reduction of canonical Hamiltonian systems that is
variationally consistent for any choice of linear reduced basis: Hamiltonian
models project to Hamiltonian models. Applicable in both intrusive and
nonintrusive settings, the proposed method is energy-conserving and symplectic,
with error provably decomposable into a data projection term and a term
measuring deviation from canonical form. Examples from linear elasticity with
realistic material parameters are used to demonstrate the advantages of a
variationally consistent approach, highlighting the steady convergence
exhibited by consistent models where previous methods reliant on inconsistent
techniques or specially designed bases exhibit unacceptably large errors.
