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This paper presents a geometric-variational approach to continuous anddiscrete mechanics and field theories. Using multisymplectic geometry, we showthat the existence of the fundamental geometric structures as well as theirpreservation along solutions can be obtained directly from the variationalprinciple. In particular, we prove that a unique multisymplectic structure isobtained by taking the derivative of an action function, and use this structureto prove covariant generalizations of conservation of symplecticity andNoether's theorem. Natural discretization schemes for PDEs, which have theseimportant preservation properties, then follow by choosing a discrete actionfunctional. In the case of mechanics, we recover the variational symplecticintegrators of Veselov type, while for PDEs we obtain covariant spacetimeintegrators which conserve the corresponding discrete multisymplectic form aswell as the discrete momentum mappings corresponding to symmetries. We showthat the usual notion of symplecticity along an infinite-dimensional space offields can be naturally obtained by making a spacetime split. All of theaspects of our method are demonstrated with a nonlinear sine-Gordon equation,including computational results and a comparison with other discretizationschemes.Comment: LaTeX2E, 52 pages, 11 figures, to appear in Comm. Math. Phy

Multisymplectic Geometry, Variational Integrators, and Nonlinear PDEs