  We discuss a field theoretical extension of the basic structures of classical
analytical mechanics within the framework of the De Donder--Weyl (DW) covariant
Hamiltonian formulation. The analogue of the symplectic form is argued to be
the {\em polysymplectic} form of degree $(n+1)$, where $n$ is the dimension of
space-time, which defines a map between multivector fields or, more generally,
graded derivation operators on exterior algebra, and forms of various degrees
which play a role of dynamical variables. The Schouten-Nijenhuis bracket on
multivector fields induces the graded analogue of the Poisson bracket on forms,
which turns the exterior algebra of (horizontal) forms to a Gerstenhaber
algebra. The equations of motion are written in terms of the Poisson bracket on
forms and it is argued that the bracket with $H\vol$, where $H$ is the DW
Hamiltonian function and $\vol$ is the horizontal (i.e. space-time) volume
form, is related to the operation of exterior differentiation of forms.
