A variational approach to second-order multisymplectic field theory

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.