"An odd kind of sympathy"

One of the major technological problems still unsolved in 1665 was to develop a system for determining the longitude of a ship at sea accurately.

This problem is solved by building a clock able to measure time with sufficient precision despite the movements of the ship. One of the most precise clocks then available was the pendulum clock, invented by Christiaan Huygens a few years earlier, in 1657. Huygens was in fact studying, in collaboration with the Royal Society, the behaviour of pendulum clocks for use at sea, when he noticed a strange coincidence.

Confined to bed by an illness, he was able to observe a pair of pendulum clocks for a long time, and in a letter of 27 February^[2] he said he had

In essence, even starting from random initial conditions, after a certain time the clocks always ended up oscillating in antiphase. Huygens also noticed that the phenomenon did not occur if the clocks were placed far apart, or if they were made to swing in perpendicular planes. He soon concluded that the phenomenon was due to the coupling between the pendulums caused by the imperceptible movements of the common support on which they were mounted. He also tried to formulate a more quantitative theory of the phenomenon, but the time was not ripe for this (Newton published his Principia only twenty years later).

Partly because of these observations, it was soon understood that the pendulum clock could not be used to solve the problem of determining longitude, since it was too sensitive to the movements of the ship, and to other even subtler ones, such as those that cause the "odd sympathy".

However, the phenomenon of synchronization, observed for the first time by Huygens in a mechanical context, has turned out to be an important phenomenon, common to many different classes of systems.

The feature common to all these systems, responsible for the emergence of self-organization and synchronization phenomena, is that their dynamics is non-linear.

A musical example concerns organ pipes. Each pipe is a self-sustained oscillator: the air jet striking the lip of the pipe keeps the air column oscillating, just as the escapement keeps the pendulum swinging. If two pipes with slightly different natural frequencies are placed side by side, the sound emitted by each reaches the mouth of the other and gives its air jet a slight push at every period: these small periodic pushes are enough to make the two pipes sound at the same frequency. Lord Rayleigh already noted that in extreme cases two neighbouring pipes can almost reduce each other to silence: they synchronize in antiphase and the sound waves they emit largely cancel^[3].

Synchronization of metronomes

An amusing version of Huygens' observation can be made using a few metronomes, as shown in the film. In this case, unlike what happened with Huygens' pendulums, synchronization occurs in phase rather than in antiphase (and in a much shorter time), but the phenomenon keeps its magic. ^[4]

Before going into the details, let us look at the general features of the phenomenon, using Lissajous figures to analyse the simplest case: that in which there are only two pendulums.

• The left-hand column shows the relative phase differences of the two metronomes. We note that, while in the linear case the phase differences keep oscillating, indicating the presence of beats, in the non-linear case the phase difference decreases progressively, down to 0. When the phase difference is zero the metronomes are "synchronized", and oscillate "in unison".
• In the next two columns the angle of oscillation of one metronome is plotted as a function of the angle of oscillation of the other metronome. It is the same diagram, but the animated version shows the time evolution of the oscillation (the number indicates the number of oscillation periods of each metronome), while the last column shows the evolution over the whole simulated time, about 50 periods.
• While in the linear case the motion keeps oscillating within the limiting quadrilateral, in the non-linear case we see that, after an initial transient, the oscillations concentrate along the diagonal of the first and third quadrants, again indicating an in-phase oscillation (for the interpretation of these diagrams see the page on Lissajous figures) .

	Phase difference	Lissajous figure (animation)	Lissajous figure
Linear
Non-linear

Further reading

• The simple pendulum is not in itself a linear oscillator. Indeed, the sine function appears in its equation of motion, which means that the pendulum approximates a linear oscillator only for small oscillations (see the page on non-linear effects).

• Metronomes (like pendulum clocks) are not even simple pendulums. Indeed, these devices contain an ingenious mechanism, called the escapement, which allows the spring that powers them to release its energy little by little, through a series of small impulses. That is, the spring delivers "little kicks" to the pendulum only when, in its swing, it passes a certain angle. In this way the energy supplied by the spring compensates for the energy lost to friction, and the oscillation can be kept going until the spring has run down completely. The oscillations are free between one "kick" and the next, and remain isochronous. A rudimentary model of this device consists in adding a friction term and a non-linear term, called the "Van der Pol" term, which represents the action of the escapement

• When several metronomes are placed on a common base that is itself free to move, like the board rolling on cans in the film, each pendulum, as it swings, pushes the base back and forth, and the motion of the base in turn acts on all the pendulums: energy thus passes from one metronome to another through the base. Equivalently, we can say that the metronomes now feel an apparent force due to the motion of the base, since they are oscillating in a non-inertial frame. Another term must be added to the equation, and so, in the case of two metronomes, we have the system

where we have called

${\displaystyle {\omega }_0^2=\frac{g}{L_{eff}}}$

the natural frequency of each pendulum, if it were a harmonic oscillator,

${\displaystyle \beta =\frac{x_0{\omega }_0^2}{g}}$

the coefficient that determines the degree of coupling between the pendulums, and

${\displaystyle x_0=\frac{m{L}_{cm}}{2m+M}}$

the displacement of the base resulting from the motion of the pendulums: the base moves by ${\displaystyle x=-x_0[\sin ({\theta }_1)+\sin ({\theta }_2)]}$, in the direction opposite to the pendulums, so that the total horizontal momentum stays zero. Here m is the mass of each pendulum, M that of the base (including the bodies of the metronomes), ${\displaystyle L_{cm}}$ the distance between the pivot and the centre of mass of the pendulum, and ${\displaystyle L_{eff}}$ the length of the simple pendulum with the same period (effective length).

• Note now that, besides the non-linearities of each oscillator, the coupling term between the metronomes is now non-linear too. The two terms are governed by the constants ${\displaystyle \mu }$ and ${\displaystyle \beta }$.

Check your understanding