We explore the dynamics of mixed classical-quantum Hamiltonians, a cla
ss of systems that frequently appears in the study of physical systems
characterized by both slow and fast modes. Traditionally such systems
are studied within the adiabatic approximation and as a rule they dis
play level crossings. We show that level crossings can have peculiar e
ffects on the dynamics and, in the regime between the adiabatic and di
abatic limits in both classical and quantum approaches, the full dynam
ics has a series of non-intuitive features.