  This paper presents a geometric-variational approach to continuous and
discrete {\it second-order} field theories following the methodology of
\cite{MPS}. Staying entirely in the Lagrangian framework and letting $Y$ denote
the configuration fiber bundle, we show that both the multisymplectic structure
on $J^3Y$ as well as the Noether theorem arise from the first variation of the
action function. We generalize the multisymplectic form formula derived for
first order field theories in \cite{MPS}, to the case of second-order field
theories, and we apply our theory to the Camassa-Holm (CH) equation in both the
continuous and discrete settings. Our discretization produces a
multisymplectic-momentum integrator, a generalization of the Moser-Veselov
rigid body algorithm to the setting of nonlinear PDEs with second order
Lagrangians.
