  We introduce the concept of a variational tricomplex, which is applicable
both to variational and non-variational gauge systems. Assigning this
tricomplex with an appropriate symplectic structure and a Cauchy foliation, we
establish a general correspondence between the Lagrangian and Hamiltonian
pictures of one and the same (not necessarily variational) dynamics. In
practical terms, this correspondence allows one to construct the generating
functional of weak Poisson structure starting from that of Lagrange structure.
As a byproduct, a covariant procedure is proposed for deriving the classical
BRST charge of the BFV formalism by a given BV master action. The general
approach is illustrated by the examples of Maxwell's electrodynamics and chiral
bosons in two dimensions.
