  Synchronization induced by long-range hydrodynamic interactions is attracting
attention as a candidate mechanism behind coordinated beating of cilia and
flagella. Here we consider a minimal model of hydrodynamic synchronization in
the low Reynolds number limit. The model consists of rotors, each of which
assumed to be a rigid bead making a fixed trajectory under periodically varying
driving force. By a linear analysis, we derive the necessary and sufficient
conditions for a pair of rotors to synchronize in phase. We also derive a
non-linear evolution equation for their phase difference, which is reduced to
minimization of an effective potential. The effective potential is calculated
for a variety of trajectory shapes and geometries (either bulk or substrated),
for which the stable and metastable states of the system are identified. Finite
size of the trajectory induces asymmetry of the potential, which also depends
sensitively on the tilt of the trajectory. Our results show that flexibility of
cilia or flagella is {\it not} a requisite for their synchronized motion, in
contrast to previous expectations. We discuss the possibility to directly
implement the model and verify our results by optically driven colloids.
