# Extending the variational bicomplex to Hamiltion or Hamiltion-Jacobi formalism

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.

-
Unfortunately, it's a bit unclear to me what aspects of the "Hamiltonian viewpoint" you think are missing. You might be interested in the answer I gave here, where a lot of these details are filled in, though not explicitly including the Hamilton-Jacobi equation: mathoverflow.net/questions/81800/81857#81857 – Igor Khavkine Apr 29 '13 at 17:16
Thank you very much for the link. I know that there exists this way to get the Hamiltonian formulation (which you described in the link and to which I referred as the "standard procedure" above). But as pointed out, there are some shortcomings and the space-time split is not that nice from a physical point of view. So I was looking for an another procedure which generalized symplectic geometry in some way (or/and a good reference which handles this problems). – Tobias Diez Apr 29 '13 at 20:59
For example generalizing the Hamiltion equation in the obvious way $i_X \omega = \delta H$ would mean, that $X$ is a tensor field of type $(-1, 1)$ instead of a normal vector field. But I couldn't find any reference for such or related extensions directly based on the variational bicomplex. – Tobias Diez Apr 29 '13 at 21:00

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.

-