  This paper presents a variational and multisymplectic formulation of both
compressible and incompressible models of continuum mechanics on general
Riemannian manifolds. A general formalism is developed for non-relativistic
first-order multisymplectic field theories with constraints, such as the
incompressibility constraint. The results obtained in this paper set the stage
for multisymplectic reduction and for the further development of Veselov-type
multisymplectic discretizations and numerical algorithms. The latter will be
the subject of a companion paper.
