CHAOTIC PENDULUMS

A SIMPLE PENDULUM’s measured behavior is a marvelous thing. A clock with one doesn’t need batteries. It doesn’t need a plug. There’s no solar panel or other wizardry. All pendulums, though, aren’t so well behaved; some are downright chaotic.

Displaced from its equilibrium position, the pendulum is acted upon by gravity which accelerates it back toward vertical. This restoring force combined with the pendulum’s mass causes it to overshoot. And then the process continues, at least in theory where there’s no friction nor air resistance.

Once displaced from vertical, a pendulum is encouraged by gravity to return.

The time of a pendulum’s swing, its period, depends on gravity, the pendulum’s length and also, just a tad, on the width of its swing, its amplitude.

For a typical pendulum swing of a clock, T ≈ 2π √(L/g), where T is the time for a complete oscillation, L is the length of the pendulum and g is the local acceleration of gravity. (A clock pendulum has a set-screw on its base to fine-tune its length for enhanced accuracy).

On average, g is around 9.8 m/sec²; about 32 ft/sec² in English units honoring Sir Isaac Newton. More precisely, g depends on one’s location. The official value of 9.80665 m/sec² is considered spot-on if you happen to reside at latitude 45 degrees 32 minutes 33 seconds.

I used my Microsoft Flight Simulator to identify where this precise latitude lay from my home airfield of Orange County, Santa Ana, California. Directly north of SNA at N 45 degrees 32 minutes 33 seconds is a location in northeast Oregon, not far from Smith Mountain Ranch. Elsewhere in the world, this latitude is south of London’s N 51 degrees, north of Monaco’s N 43 and very close to Milan’s N 45 degrees 28 minutes.

Enough already on latitude; what of pendulums? In particular, we’re only one short variation away from utter chaos.

A double pendulum.

As its name suggests, a double pendulum is a pendulum with another pendulum attached to its end. It’s fascinating because of its extreme sensitivity to initial conditions. Technically, as a dynamical system, a double pendulum is structurally unstable to the point of being chaotic.

Let’s pause here to give chaos something of a proper definition: A system is chaotic if small differences in initial conditions, in its startup, can yield arbitrarily large changes later on. One example of this is the Butterfly Effect, where an Amazonian wing flutter can lead to a Duluth blizzard.

Note, there’s nothing random about such behavior; it’s completely deterministic. However, there’s a difference between determinism and predictability. Mathematician/meteorologist Edward Lorenz summed it up concisely: “Chaos: When the present determines the future, but the approximate present does not approximately determine the future.”

A trace of a double pendulum’s end point. Image by George Ioannidis.

It’s possible, but far from straightforward, to describe a double pendulum’s behavior with the mathematical tool of differential equations. However, like many such expressions, there’s no means of solving its equations explicitly; the best that can be done is an approximation.

There’s a fascinating video of double pendulum motion with comments by its author, Paul Nathan, on how the video was constructed. Nathan observes, “In this universe, chaos truly is the rule rather than the exception.” ds

© Dennis Simanaitis, SimanaitisSays.com, 2016