  We study the relationship between the equations of first order Lagrangian
field theory on fiber bundles and the covariant Hamilton equations on the
finite-dimensional polysymplectic phase space of covariant Hamiltonian field
theory. The main peculiarity of these Hamilton equations lies in the fact that,
for degenerate systems, they contain additional gauge fixing conditions. We
develop the BRST extension of the covariant Hamiltonian formalism,
characterized by a Lie superalgebra of BRST and anti-BRST symmetries.
