This paper presents a geometric-variational approach to continuous and disc
rete second-order field theories following the methodology of [Marsden, Pat
rick, Shkoller, Comm. Math. Phys. 199 (1998) 351-395]. Staying entirely in
the Lagrangian framework and letting Y denote the configuration fiber bundl
e, we show that both the multisymplectic structure on J(3)Y as well as the
Noether theorem arise from the first variation of the action function. We g
eneralize the multisymplectic form formula derived for first-order field th
eories in [Marsden, Patrick, Shkoller, Comm. Math. Phys. 199 (1998) 351-395
], to the case of second-order field theories, and we apply our theory to t
he Camassa-Holm (CH) equation in both the continuous and discrete settings.
Our discretization produces a multisymplectic-momentum integrator, a gener
alization of the Moser-Veselov rigid body algorithm to the setting of nonli
near PDEs with second-order Lagrangians. (C) 2000 Elsevier Science B.V. All
rights reserved.