December 10, 2007

Fig. 1: A schematic for Huygens' pendulum clock. This
figure is currently under a free license as stated
here. |

Christiaan Huygens, a Dutch scientist, invented the pendulum clock in 1657[1]. Until his invention of the accurate clock, the matter of timekeeping hindered advances in many fields of science and expeditions. An accurate timekeeping method can enable the precise measurement in Physics, as well as the applications in daily life. In addition, the Europe was entering "the age of exploration," which required explicit time measurement for the maritime voyage[2]. Although Galileo Galilei studied the mechanism of the pendulum and conceived the pendulum clock concept, no clock using the periodicity of pendulum was demonstrated. The problem to realize a clock from the physical concept was the period of pendulum varies when the pendulum makes wide swing. To regulate the swing of the pendulum, Huygens introduced a kind of escapement and cycloidal confinement of a flexible pendulum suspension, which is isochronous curve. In consequence, Huygens made a clock of which error was under 10 seconds a day. It was a great breakthrough in time measurement, compared with the previous limit - 15 minutes a day.

However, accurate regulation of the pendulum period was not the whole work from the pendulum clock. After combining two nearly identical pendulum clock in same support, Huygens discovered the "odd sympathy." [3]

"Of an odd kind of sympathy perceived by him in these watches (two maritime clocks) suspended by the side of each other."[4] |

Huygens saw that the two suspended pendulum's motion became the state of same frequency and exactly opposite direction, regardless of the initial condition. Even after he disturbed one pendulum, the system was back to the "antiphase state." At first, he thought that the reason was interference between two pendulums via air flow. But after several experiment, he concluded that the insensible movements and mechanical interaction on the frame caused the coincidence in motion.

The Huygens' experiment inspired the following studies of interacting nonlinear oscillators. The topic is basically of two mechanical oscillator with weak coupling, but it's notion of synchronization widely affected various problems from nonlinear dynamics to complex biological system. [5] In physics, the coupling of two oscillators are rediscoverd by novel magnetic and mechanical systems of nanometer scale. [6,7]

Fig. 2: Basic model for two coupled oscillator
system. |

Even though the synchronization in coupled pendulum have been known for over 300 years, the attempts to solve the phenomena were not quite satisfactory. [8,9] Recently, researchers from the Georgia Institute of Technology (Bennett et al.) tried to reexamined the old experiment both in experimental and theoretical ways. [10]

Fig. 2 shows the simplified schematic of the coupled
pendulums. Each pendulum consists of an identical point mass m and a
massless rigid rod with equal length l. φ_{k} is the angular
displacement of the kth pendulum about its own pivot point. And X is the
linear displacement of the system's center of mass. To simplify the
discussion, we can assume that the mass of platform M is much larger than
the point mass m. It also means that the coupling between two equal mass
is a weak coupling, since the effect of one pendulum's motion is small
compared to the inertial of the frame - which transmits the coupling
between two identical masses.

To solve such complex system analytically first, we start from the system without damping term and driving terms. With the basic assumption, the Lagrangian will be

Here we use the term of K, since the original experiment conducted by Huygens was settled on rest frame. However, further experiments reveals that the term only makes minor effect in the result. [10] And now we introduce viscous damping term for both angular motion and translational motion, as well as the driving term from the escapements. Then the equation of the motion become

where b is a damping coefficient for swinging and B
is a damping coefficient of translational motion. With the small angle
approximation and impulse-free condition, we can express the
characteristic equation of the system. It is convenient that the
variables are scaled as dimensionless forms. When we write the equation
with two variables σ = φ_{1} + φ_{2} and
δ = φ_{1} - φ_{2}, the equations are

where Y = X/l, γ = b (l/4g)^{1/2},
Γ = B (l/4g)^{1/2}/(M+2m), Ω^{2} = K/(M+2m),
μ = m/(M+2m), and the prime means differentiation with respect to
τ = t(g/l)^{1/2}. The system mass ration μ indicates the
magnitude of the coupling. And with the later discussion, we can figure
out that the value of μ does important role to determine the overall
behavior.

Fig. 3: The "truncated" phase diagram of single
pendulum with periodic impulse. |

One important feature is derived from above equations. Let's compare the differential equations of σ and of δ. For the extreme case with no viscous damping(γ = 0),σ would damp out due to the Y'' term. That is, among two different modes of pendulums, antiphase mode(δ) survives at last whereas inphase mode(σ) vanishes. Although this analysis is induced with the condition between periodic kicking - no external impulse, the characteristic of the mode is valid after the simulation considering the energy input.

Until now, we only treat the isolated system from
the outside driving. In order to consider the energy input, it requires
the normal mode evolutions and the analysis of nonlinear Poincare
section. I will not discuss the details of the works done by Bennet and
the much since it includes numerical analysis with several strong
restrictions to converge the calculation. The brief strategy is
following; at first, we can derive the normal modes and neglect one mode
since we assume that "imperceptible movement - weak coupling." And among
the four dimensional phase space, we can focus just on three dimensional
map. In addition, for counting outside impact, we introduce the
"truncated" orbit in phase space due to the forced change of the
pendulum movement (Fig. 3). With the consideration of both damping and
energy input, the numerical simulation shows that the three dimensional
map reveals two stable states and one unstable state asymptotically. Two
stable states are antiphase state(φ_{1} = -
φ_{2}) and the trivial state, whereas the inphase
state(φ_{1} = φ_{2}) is unstable. [10] This is a
consistent with Huygens' observation.

Fig. 4: Phase diagram from the theoretical
simulation: (a)Δ - μ plane with Γ fixing
(b)Γ - μ plane with Δ fixing. |

One last thing we should consider is that the
non-identical condition of two pendulums in real experiment. Therefore,
we also need to verify in what condition such antiphase collapse in
terms of the difference between two pendulums. The difference is
represented by the "detuning" variable Δ, which is accounted for
the difference in pendulum frequencies. And with the previous
constriction for the simulation, the critical detuning
Δ_{c} is given as the below expression.

Fig. 4. describes the overall conditions determining the system's behavior. There are three parts in each phase diagram. Quasiperiodic domain means two forced pendulum do periodic motion with different frequency. And in the 'Beating Death' domain, one of the pendulum stops to move even the other pendulum swings periodically. The antiphase mode survives only with the proper condition - nearly identical conditions and intermediate relative magnitudes of the damping and the coupling terms. Fortunately, Huygens' clock satisfies all of the conditions above; so that the secret of 'sympathy' in nature comes out to us finally.

© 2007 Seung Sae Hong. The author grants permission to copy, distribute and display this work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.

[1] C. Huygens, *The Pendulum Clock or Geometrical
Demonstrations Concerning the Motion of Pendula as Applied to
Clocks*, ed. by R. Blackwell (Iowa State University Press,
1986).

[2] J. G. Yoder, *Unrolling Time: Christiaan
Huygens and the Mathematization of Nature* (Cambridge, 1990).

[3] C. Huygens, *Oeuvres Completes de Christiaan
Huygens*, ed.by M. Nijhoff (Societe Hollandaise des Sciences,
1893).

[4] T. Birch, *Philosophical Transactions* vol.
2. (Johnson Reprint, 1968), p. 19.

[5] J. Buck and E. Buck, Science **159**, 1319
(1968).

[6] S. Kaka *et al*., Nature **437**, 389
(2005).

[7] F. B. Mancoff *et al*., Nature **437**,
393 (2005).

[8] D. J. Korteweg, *Les Horloges Sympathiques de
Huygens* Serie II. Tome XI, ed. by M. Nijhoff (Archives
Neerlandaises, 1906), p. 273.

[9] I. I. Blekhman, *Synchronization in Science and
Technology* (ASME Press, 1988).

[10] M. Bennett *et al*., Proc. R. Soc. London A
**458**, 563 (2002).