In 1657, Christiaan Huygens revolutionized the measurement of time by creating the first working pendulum clock. In early 1665, Huygens discovered ``..an odd kind of sympathy perceived by him in these watches [two pendulum clocks] suspended by the side of each other.'' The pendulum clocks swung with exactly the same frequency and 180 degrees out of phase; when the pendulums were disturbed, the antiphase state was restored within a half-hour and persisted indefinitely. Huygens deduced that the crucial interaction for this effect came from ``imperceptible movements'' of the common frame supporting the two clocks. He thought this synchronization could be used to provide the exquisite timing necessary to solve the famous ``longitude problem'' in maritime navigation and spent many years developing and testing designs for pendulum clocks that function at sea.

Ultimately, the innovation of the pendulum clock did not solve the longitude problem. Nevertheless, Huygens' synchronization observations have inspired modern studies of sympathetic oscillations in many areas of nonlinear science. Previous attempts to understand Huygens' findings are few and ultimately unsatisfying.

Our experiments and our theoretical model reproduce Huygens' observations. The pendulums are attracted to the antiphase state, never the inphase state. As Huygens surmised, the platform motion is the culprit: if we prevent the platform from moving, there is no synchronization at all. Our analysis shows that damping effects favor antiphase motion, since inphase oscillations induce additional counter-motion of the frame. And while the clock escapements can transfer energy into the inphase mode, it is never enough to overcome this frictional disadvantage.

We also find an outcome that Huygens never mentions, in which one or both pendulum motions die out--a state we call ``beating death''. Huygens may not have observed beating death because his clock systems were heavily weighted (a by-product of the maritime design--the weighting was necessary to keep the clocks upright when at sea). Beating death happens less often as the system mass is increased. For a system mass comparable to those of Huygens' studies, we observe only antiphase synchronization.

In sum, we conjecture that Huygens' observations of ``sympathy" depended on both talent and luck. His clocks contained heavy lead weights in order to keep them upright in stormy seas. If they had not been, the coupling would have been too large, and, eventually, beating death would have occurred. Neither would the sympathy have been possible if the coupling was too weak, since the small but inevitable difference in the clock frequencies would prevent frequency locking. Only clocks with sufficiently close frequencies could fall into antiphase lock-step. As it happened, Huygens' own inventions made such exquisite matching possible.

M. Bennett, M. F. Schatz, H. Rockwood, and K. Wiesenfeld, "Huygens' clocks ", to appear in Proceedings (A) of the Royal Society (2001). ( PDF format.)

APPEARING SOON: A movie demonstrating the anti-phase attractor (the state first observed by Huygens in 1665). ( format.)

APPEARING SOON: A movie showing the 'beating death' attractor ( format.)

This work was covered in the popular press by Scientific American, Science News and Wired .