  Adiabatic evolution is an emergent design principle for time modulated
metamaterials, often inspired by insights from topological quantum computing
such as braiding operations. However, the pursuit of classical adiabatic
metamaterials is rooted in the assumption that classical and quantum adiabatic
evolution are equivalent. We show that this is only true in the limit where the
frequencies of all the bands are at infinite distance from $0$; and some
instances of quantum adiabatic evolution, such as those containing zero modes,
cannot be reproduced in classical systems. This is because mode coupling is
fundamentally different in classical mechanics. We derive classical conditions
to ensure adiabaticity and demonstrate that only under these conditions - which
are different from quantum adiabatic conditions -, the single band Berry phase
and Wilczek-Zee matrix for everywhere degenerate bands emerge as meaningful
quantities encoding the geometry of classical adiabatic evolution. Finally, for
general multiband systems we uncover a correction term in the non-Abelian gauge
potential for classical systems.
