  The evolution of a mixed quantum-classical system is expressed in the mapping
formalism where discrete quantum states are mapped onto oscillator states,
resulting in a phase space description of the quantum degrees of freedom. By
defining projection operators onto the mapping states corresponding to the
physical quantum states, it is shown that the mapping quantum-classical
Liouville operator commutes with the projection operator so that the dynamics
is confined to the physical space. It is also shown that a trajectory-based
solution of this equation can be constructed that requires the simulation of an
ensemble of entangled trajectories. An approximation to this evolution equation
which retains only the Poisson bracket contribution to the evolution operator
does admit a solution in an ensemble of independent trajectories but it is
shown that this operator does not commute with the projection operators and the
dynamics may take the system outside the physical space. The dynamical
instabilities, utility and domain of validity of this approximate dynamics are
discussed. The effects are illustrated by simulations on several quantum
systems.
