  The standard Hamiltonian machinery, being applied to field theory, leads to
infinite-dimensional phase spaces. It is not covariant. In this article, we
present covariant finite-dimensional multimomentum Hamiltonian formalism for
field theory. This is the multisymplectic generalization of the Hamiltonian
formalism in mechanics. In field theory, multimomentum canonical variables are
field functions and momenta corresponding to derivatives of fields with respect
all world coordinates, not only the time. In case of regular Lagrangian
densities, the multimomentum Hamiltonian formalism is equivalent to the
Lagrangian formalism, otherwise for degenerate Lagrangian densities. In this
case, the Euler-Lagrange equations become undetermined and require additional
conditions which remain elusive. In the framework of the multimomentum
Hamiltonian machinery, one obtaines them automatically as a part of Hamilton
equations. The key point consists in the fact that, given a degenerate
Lagrangian density, one must consider a family of associated multimomentum
Hamiltonian forms in order to exaust solutions of the Euler-Lagrange equations.
We spell out degenerate quadratic and affine Lagrangian densities. The most of
field models are of these types. As a result, we get the general procedure of
describing constraint field systems.
