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We introduce the concept of a variational tricomplex, which is applicableboth to variational and non-variational gauge systems. Assigning thistricomplex with an appropriate symplectic structure and a Cauchy foliation, weestablish a general correspondence between the Lagrangian and Hamiltonianpictures of one and the same (not necessarily variational) dynamics. Inpractical terms, this correspondence allows one to construct the generatingfunctional of weak Poisson structure starting from that of Lagrange structure.As a byproduct, a covariant procedure is proposed for deriving the classicalBRST charge of the BFV formalism by a given BV master action. The generalapproach is illustrated by the examples of Maxwell's electrodynamics and chiralbosons in two dimensions.

Variational tricomplex of a local gauge system, Lagrange structure and weak Poisson bracket