  Using the example of the harmonic oscillator, we illustrate the use of hybrid
dynamical brackets in analyzing quantum-classical interaction. We only assume
that a hybrid dynamical bracket exists, is bilinear, and reduces to the pure
quantum/classical bracket when acting on pure quantum/classical variables. Any
hybrid bracket obeying these natural requirements will produce the same
dynamics for pure classical or quantum variables, given a hybrid Hamiltonian.
Backreaction is manifested in the evolution of a nonvanishing commutator
between classical variables. The more massive the classical system is, the less
it is affected by backreaction. Interestingly, we show that while pure
variables evolve to violate the pure canonical relations, they always obey the
hybrid canonical relations. The dynamics of hybrid variables, on the other
hand, is shown to require a fully specified and consistent hybrid bracket,
otherwise evolution cannot be defined.
