The variational bicomplex seams to provide a modern formulation of the variational problem in terms of modern differential geometry. In particular the bigraded complex of differential forms $\Omega^{p,q} (J^\infty E)$ with $D = \delta + d$ on the infinite jet bundle of some vector bundle $E$ encodes all the information from the Lagrangian view: If $L \in \Omega^{0, n}$ is the Lagrangian density and $\theta \in \Omega^{1, n-1}$ the variational form than $\delta L + d \theta$ equals the Euler-Lagrange equation.

Futhermore as $\omega = \delta \theta \in \Omega^{2, n-1}$ is $\delta$-closed (and also $d$-closed on solutions) it seams natural to regard it as a generalized symplectic form. But I couldn't find any account on the Hamiltion viewpoint (Legendre transformation, Hamiltion equation, Hamiltion Jacobi, Poisson structure, ect.). What are the possible routes to generalize symplectic geometry to this regime and what are the problems one is faced with?

The standard procedure seams to be to choose an Cauchy surface of the base manifold and integrate $\omega$ over it to get a (pre)symplectic form. But there seams to be issues for non-compact Cauchy surfaces and the independence wrt. to the choosen surface does not work off-shell. Also this approach is not really "in the spirit" of the variational bicomplex, so I search for alternatives.