Odd Sympathy

The phrase odd sympathy appears in the record of a letter by Dutch mathematician and physicist Christiaan Huygens (1629–95) to Sir Robert Moray as presented to the Royal Society of London, relating to the tendency of two pendulum clocks to synchronize with opposite phases when suspended side by side. Huygens' actual phrase was une espece de sympathie. Huygens, credited with being the inventor of the pendulum clock, first noticed the effect while lying in bed. His two pendulum clocks, mounted together, would end up swinging in exactly opposite directions, regardless of their respective individual initial motions. This was one of the first observations of the phenomenon of mode locking in coupled driven oscillators, which has many applications in physics.

Huygens originally believed the synchronization was due to air currents shared between the two pendulums, but he dismissed the hypothesis himself after several tests. Huygens would later attribute sympathetic motion of pendulums to imperceptible movement in the beam from which both pendulums are suspended. This idea was later validated by researchers from the Georgia Institute of Technology who tested Huygens' idea.

Using instruments capable of registering movement too small to have been measured in Huygens' time, the Georgia Tech researchers chronicled the nature of the forces at work on the supporting beam. They found that if the pendulums are moving in the same direction, together they tend to move the beam the opposite direction, giving rise to forces that resist motion in the same direction. If however, the pendulums are moving in opposite directions, these forces cancel each other out, causing the beam to remain nearly motionless. Thus, motion, in this example, tends to be perfectly asynchronous.

Huygens only observed anti-phase synchronization of his pendulum clocks. Bennett and co-workers reported both anti-phase and in-phase synchronization of their clocks as well as "death" states wherein one or both of the clocks stops. Synchronization of driven oscillators is also discussed in books written by I.I. Blekhman, and by Pikovsky and co-workers. A detailed analysis was provided by Fradkov and Andrievsky in 2007 regarding the conditions for in-phase or anti-phase synchronization of a 2-pendulum system.