Second-order multisymplectic field theory: A variational approach to second-order multisymplectic field theory

Journal article

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.

Authors: Kouranbaeva, S; Shkoller, S

• Journal: J.Geom.Phys.
• Volume: 35
• Pages: 333-366
• Publication date: 1999-09-17
• Keywords: math-ph; math.MP; 58B20; 58B05; 76E99; math.DG