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I construct a global version of the local polysymplectic approach tocovariant Hamiltonian field theory pioneered by C. Gunther. Beginning with thegeometric framework of the theory, I specialize to vertical vector fields toconstruct the (poly)symplectic structures, derive Hamilton's field equations,and construct a more-or-less natural Poisson bracket. I then examine a few keyexamples to determine the nature of the necessary vertical projections and findthat the theory provides the geometric analog of the canonical transformationapproach to covariant Hamiltonian field theory advanced by Struckmeier andRedelbach. I conclude with a few remarks about possible applications of thisframework to the geometric quantization of classical field theories.

A global version of Gunther's polysymplectic formalism using vertical projections