The [variational bicomplex](http://ncatlab.org/nlab/show/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.



Extending the variational bicomplex to Hamiltion or Hamiltion-Jacobi formalism
I'm not sure that there is a right answer to your question, since what is and what is not a satisfactory generalization of symplectic geometry can be quite subjective and in the end only can judge what you find satisfactory. I myself find what you call the "standard approach" (at least in my understanding after your clarifying comment) rather satisfactory.


One of the drawbacks thereof that you refer to, that the integrated presymplectic form depends on the integration surface, is I think irrelevant. The reason is that the actual phase space where the presymplectic structure is to be defined is precisely the space of solutions, which is usually identified with the space of initial data (provided the equations of motion are well posed). The second drawback, the ambiguities associated with non-compact Cauchy surfaces, are in a sense unavoidable. These ambiguities are essentially the same as the ambiguities in the choice of asymptotic boundary conditions for the equations of motion, which also enter into the definition of the (infinite dimensional) phase space (in the guise of the space of solutions or of initial data). So they cannot be avoided if one is concerned with the global behavior of solutions.


On the other hand, people have considered some generalizations of symplectic geometry to field theories, whose goal is to stay as much as possible within a local, finite-dimensional context. One fairly popular generalization is called *multisymplectic geometry* (I think there are other ones, but I know much less about them). A very nice article that connects multisymplectic geometry with the "standard approach" is


*Covariant Poisson Brackets in Geometric Field Theory*, by
Michael Forger, Sandro V. Romero
<http://arxiv.org/abs/math-ph/0408008>


Unfortunately, I don't know enough about appropriate generalizations of Hamilton-Jacobi theory to say something intelligent about it here.


