The effect of a stationary electric field on a water droplet with a diameter of several tens micrometers in oil was examined. Such a droplet exhibits repetitive translational motion between the electrodes in a spontaneous manner. The state diagram of this oscillatory motion was deduced; at 0-20 V the droplet is fixed at the surface of the electrode, at 20-70 V the droplet exhibits small-amplitude oscillatory motion between the electrodes, and at 70-100 V the droplet shows large-amplitude periodic motion between the electrodes. The observed rhythmic motion is explained in a semiquantitative manner by using differential equations, which includes the effect of charging the droplet under an electric field. We also found that twin droplets exhibit synchronized rhythmic motion between the electrodes.