We consider, in turn, three systems being acted upon by a regularly pulsed harmonic potential (PHP). These are (i) a classical particle, (ii) a quantum particle, and (iii) a directed line We contrast the mechanics of the first two systems by parametrizing their bands of stability and periodicity. Interesting differences due to quantum fluctuations are examined in detail. The fluctuations of the directed line are calculated in the two cases of a binding PHP, and an unbinding PHP. In the latter case there is a finite maximum line length for a given potential strength.