Variational problems defined by local data 1

We study, in the framework of the global inverse problem of the calculus of variations, the first order variational problems defined by a family of local Lagrangian densities. We will show that those local variational problems for which the differential of the Poincaré–Cartan form is globally defined, admit a geometrical formulation which apart from some cohomological obstructions is closely related to the one given in the ordinary case. In particular, we will give a criterion for deciding when a local variational problem is indeed a global one. We also give in this setup the proper definition of an infinitesimal symmetry and its associated Noether invariants. We will show that although every infinitesimal symmetry has a virtual Noether invariant, there is a cohomological obstruction to the existence of a global Noether invariant for a given infinitesimal symmetry. We end with a discussion of the associated Poisson algebras.