  By means of the Schouten calculus for contravariant antisymmetric tensor
fields, we apply the Lie transform method to investigate smooth deformations of
tensor fields and, in particular, to perturbations of Hamiltonian systems
generated by deformations of the Poisson bracket. Using results by Karasev and
Vorobiev on the computation of Poisson cohomology we describe infinitesimal
generators for the Lie transformations. We give applications to perturbed Euler
equations on six dimensional Lie coalgebras and to Hamiltonian systems on
Poisson manifolds equipped with Dirac brackets.
