A variational approach to second-order multisymplectic field theory

This paper presents a geometric-variational approach to continuous and discrete second-order field theories following the methodology of [Marsden, Patrick, Shkoller, Comm. Math. Phys. 199 (1998) 351-395]. Staying entirely in the Lagrangian framework and letting Y denote the configuration fiber bundle, we show that both the multisymplectic structure on J3Y 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 [Marsden, Patrick, Shkoller, Comm. Math. Phys. 199 (1998) 351-395], 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. © 2000 Elsevier Science B.V