Aspects of variational noncommutative Poisson geometry

We outline the notions and concepts of the calculus of variational
multivectors within the Poisson formalism over the spaces of infinite
jets of mappings from commutative (non-)graded smooth manifolds to the
factors of noncommutative associative algebras over the invariance under
cyclic permutations of the letters in the associative words. We outline
the basic properties of the variational Schouten bracket and derive an
interesting criterion for noncommutative differential operators to be
Hamiltonian (and thus determine the noncommutative Poisson structures).
We place the noncommutative jet-bundle construction at hand in the
context of quantum string theory.