Symmetry of pulsating ratchets

Using an exact expression for the average velocity of inertialess motion of pulsating ratchets, a simple proof is given for the recently discovered hidden space-time symmetry of Cubero–Renzoni (D. Cubero, F. Renzoni, 2016). The conditions are revealed for the absence of the ratchet effect in systems with potential energies described by products of periodic functions of coordinate and time possessing the symmetry of the main types. In particular, it is shown that the ratchet effect is absent for the time dependence of the universal symmetry type (which combines three standard symmetries), and this restriction is removed when inertia is taken into account, unless the coordinate dependence of the potential energy is related to symmetric or antisymmetric functions.