  In this paper, we prove necessary and sufficient conditions for a
hybridizable discontinuous Galerkin (HDG) method to satisfy a multisymplectic
conservation law, when applied to a canonical Hamiltonian system of partial
differential equations. We show that these conditions are satisfied by the
"hybridized" versions of several of the most commonly-used finite element
methods, including mixed, nonconforming, and discontinuous Galerkin methods.
(Interestingly, for the continuous Galerkin method in dimension greater than
one, we show that multisymplecticity only holds in a weaker sense.)
Consequently, these general-purpose finite element methods may be used for
structure-preserving discretization (or semidiscretization) of canonical
Hamiltonian systems of ODEs or PDEs. This establishes multisymplecticity for a
large class of arbitrarily-high-order methods on unstructured meshes.
