Staying in Step

Late in the winter of 1665, an ailing Christiaan Huygens (1629-1695) was confined to his room for a few days. The Dutch physicist whiled away the hours of his confinement by closely observing and pondering the odd behavior of two pendulum clocks he had recently constructed. Huygens had obtained a patent on the first pendulum clock in 1656.

Huygens noticed that the pendulums of the two suspended clocks, hanging side by side from a common support, were swinging together. When one pendulum swung to the left, the other went to the right. The pendulums remained precisely in opposite phase for as long as he cared to watch.

His curiosity piqued, Huygens began to experiment. He deliberately disturbed the motion of one pendulum so that it no longer mirrored the other’s movements. Within half an hour, the two pendulums were back in opposite-phase motion.

Huygens suspected that the clocks were somehow influencing each other, perhaps through air currents or vibrations of their common support. To test this notion, he moved the clocks to opposite sides of the room. They gradually fell out of step, with one clock losing 5 seconds a day in relation to the other. The two pendulums no longer swung at exactly the same frequency and in opposite phase. Conversely, the clocks kept precisely the same time when placed side by side.

Huygens described his experiments and observations in a letter to his father, providing the first recorded example of synchronized oscillators.

Now, physicists Kurt Wiesenfeld and Michael Schatz of the Georgia Institute of Technology in Atlanta have revisited Huygens’s experiments. The researchers constructed two spring-powered pendulum clocks and attached them to a wooden, weighted-down platform. The platform was placed on wheels, which were free to move along a level metal track. They also included a laser system to record the pendulum swings.

PENDULUM

Wiesenfeld and Schatz reflected in the pendulum bob of a clock used to recreate Huygens's experiment. Photos by Gary Meek.

Though much smaller than the models Huygens had made, the replicas matched the originals in certain key characteristics. The relationship between the masses of the replica pendulum bobs and the mass of the overall platform was roughly the same, and the clocks’ periods were also comparable.

Like the clocks that Huygens observed, the Georgia Tech replicas always ended up swinging in opposite phase, even when they started out moving in the same direction. Unlike Huygens, the researchers observed an additional effect. Sometimes, instead of synchronizing, either one pendulum or both of them would eventually stop moving. The effect was more likely to occur when the platform bearing the clocks was lighter. Wiesenfeld and Schatz termed this behavior "amplitude death."

The physicists suggest that in-phase and opposite-phase movements of the pendulums interact with the supporting platform in different ways. In-phase motion can drive platform vibrations, an effect that drains energy out the system through friction between the platform and the surface on which it rests. In contrast, when the pendulums swing in opposite directions at the same frequency, the platform doesn't move. As a result, the system conserves energy, Schatz says.

"The heavier the platform, the smaller [is] the coupling between the two clocks," Schatz notes. "If it's really heavy, the platform doesn't move at all, and there is no coupling and no synchronization." On the other hand, if the platform is too light and there's too much motion, one pendulum or both of them come to a stop.

Undergraduate student Matthew Bennett adjusts the pendulum of one clock.

Interestingly, despite small differences between the two replica clocks and between the replicas and the originals fabricated by Huygens, the systems all displayed stable, opposite-phase synchronization. Such synchronization is a robust feature of oscillating systems, Wiesenfeld says.

Indeed, the mechanical clocks, with their gears, springs, weights, and levers, may offer a handy Newtonian perspective on the various types of oscillatory behavior displayed by laser systems and modern electronic devices such as superconducting Josephson junctions.

"Classical physics still has things to teach us," Wiesenfeld remarks.

References:

Huygens, C., and R.J. Blackwell, trans. 1986. The Pendulum Clock: Geometrical Demonstrations Concerning the Motion of Pendula as Applied to Clocks. Ames, Iowa: Iowa State University Press.

Peterson, I. 1998. The Jungles of Randomness: A Mathematical Safari. New York: Wiley.

______. 1996. Keeping the beat. Science News 149(April 13):236.

Toon, J. 2000. Out of time. Research Horizons. 18(No. 1):30. Available at http://www.gtri.gatech.edu/rh-f00/time.html.

A biography of Huygens can be found at http://www-history.mcs.st-and.ac.uk/history/Mathematicians/Huygens.html.

Comments are welcome. Please send messages to Ivars Peterson at ip@sciserv.org.

Ivars Peterson is the mathematics/computer writer and online editor at Science News (http://www.sciencenews.org). He is the author of The Mathematical Tourist, Islands of Truth, Newton's Clock, Fatal Defect, and The Jungles of Randomness. He also writes for the children's magazine Muse (http://www.musemag.com) and is working on a book about math and art.

Ivars Peterson will present "A Kaleidoscope of Mathematics and Art" at the Joint Mathematics Meetings in New Orleans on Jan. 13, 2001, at 10:05 a.m.