  This paper presents the geometric setting of quantum variational principles
and extends it to comprise the interaction between classical and quantum
degrees of freedom. Euler-Poincar\'e reduction theory is applied to the
Schr\"odinger, Heisenberg and Wigner-Moyal dynamics of pure states. This
construction leads to new variational principles for the description of mixed
quantum states. The corresponding momentum map properties are presented as they
arise from the underlying unitary symmetries. Finally, certain
semidirect-product group structures are shown to produce new variational
principles for Dirac's interaction picture and the equations of hybrid
classical-quantum dynamics.
