  We present a consistent formalism to describe the dynamics of hybrid systems
with mixed classical and quantum degrees of freedom. The probability function
of the system, which, in general, will be a combination of the classical
distribution function and the quantum density matrix, is described in terms of
its corresponding moments. We then define a hybrid Poisson bracket, such that
the dynamics of the moments is ruled by an effective Hamiltonian. In
particular, a closed formula for the Poisson brackets between any two moments
for an arbitrary number of degrees of freedom is presented, which corrects
previous expressions derived in the literature for the purely quantum case.
This formula is of special relevance for practical applications of the
formalism. Finally, we study the dynamics of a particular hybrid system given
by two coupled oscillators, one being quantum and the other classical. Due to
the coupling, specific quantum and classical properties are transferred between
different sectors. In particular, the quantum sector is allowed to violate the
uncertainty relation, though we explicitly show that there exists a minimum
positive bound of the total uncertainty of the hybrid system.
