  A Schr\"odinger equation may be transformed by unitary operators into
dynamical equations in different interaction pictures which share with it a
common physical frame, i.e., the same underlying interactions, processes and
dynamics. In contrast to this standard scenario, other relations are also
possible, such as a common interaction-picture dynamical equation corresponding
to several Schr\"odinger equations that represent different physics. This may
enable us to design alternative and feasible experimental routes for operations
that are a priori difficult or impossible to perform. The power of this concept
is exemplified by engineering Hamiltonians that improve the performance or make
realizable several shortcuts to adiabaticity.
