# Multisymplectic Geometry, Variational Integrators, and Nonlinear PDEs

**Marsden, J. E., G. W. Patrick and S. Shkoller
**

*Comm. Math. Phys.*, **199**, 351-395

### Abstract:

This paper presents a geometric-variational approach to
continuous and discrete mechanics and field theories. Using
multisymplectic geometry, we show that the existence of the
fundamental geometric structures as well as their preservation
along solutions can be obtained directly from the variational
principle. In particular, we prove that a unique
multisymplectic structure is obtained by taking the derivative
of an action function, and use this structure to prove
covariant generalizations of conservation of symplecticity and
Noether's theorem. Natural discretization schemes for PDEs,
which have these important preservation properties, then follow
by choosing a discrete action functional. In the case of
mechanics, we recover the variational symplectic integrators of
Veselov type, while for PDEs we obtain covariant spacetime
integrators which conserve the corresponding discrete
multisymplectic form as well as the discrete momentum mappings
corresponding to symmetries. We show that the usual notion of
symplecticity along an infinite-dimensional space of fields can
be naturally obtained by making a spacetime split. All of the
aspects of our method are demonstrated with a nonlinear
sine-Gordon equation, including computational results and a
comparison with other discretization schemes.