 used the inverse map to obtain a canonical variational principle, following Holm and Kupershmidt (1983). As noted in Hydon (2005), a multisymplectic formulation can be obtained by choosing variables such that the Lagrangian at most linear in the first-order derivatives, and contains no higher-order derivatives. We have shown how to construct the multisymplectic formulation for the Euler-Poincaré equations for diffeomorphisms, using the example of the EPDiff($H^1$) equations, and how to extend the method to the Euler-Poincaré equations with advected quantities. These equations encompass many fluid systems, including incompressible Euler, shallow-water, Euler-alpha, Green-Naghdi, perfect complex fluids, inviscid magnetohydrodynamics, *etc.*

The techniques of Hydon (2005) have led to conservation laws for these systems, including the usual multisymplectic conservation laws for energy and momentum plus an infinite set of conservation laws which arise from the particle-relabelling symmetry of fluid dynamics. We have highlighted the connection between these latter conservation laws and Kelvin’s circulation theorem, and showed that multisymplectic integrators based on this formulation will have discrete conservation laws associated with this symmetry.

Outlook
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In the last section of this paper we have discussed the possibility of developing multisymplectic integrators for fluids using this framework. It is undoubtedly simple to construct such integrators, but the issue of accuracy with time arises whenever the flow is strongly mixing and numerical errors make the label field $\MM{l}$ very noisy. A discretisation of the relabelling map discussed in this paper could provide a way to prevent this problem whilst retaining some of the geometric properties of the method. These ideas may aid the future development of integrators that have conservation laws for vorticity and circulation, which are desirable for numerical weather prediction and other applications.

In a different direction, we believe that multisy