Abstract
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 J 3 Y 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.
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