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By means of the Schouten calculus for contravariant antisymmetric tensorfields, we apply the Lie transform method to investigate smooth deformations oftensor fields and, in particular, to perturbations of Hamiltonian systemsgenerated by deformations of the Poisson bracket. Using results by Karasev andVorobiev on the computation of Poisson cohomology we describe infinitesimalgenerators for the Lie transformations. We give applications to perturbed Eulerequations on six dimensional Lie coalgebras and to Hamiltonian systems onPoisson manifolds equipped with Dirac brackets.

The Lie Transform Method for perturbations of contravariant antisymmetric tensor fields and its applications to Hamiltonian dynamics