Chaos & the Double Pendulum pt. 1

Why is it that physics is able to predict exactly where Mars will be in ten years, but is unable to predict whether or not it will rain in two weeks? Some systems are inherently less predictable than others over long time-scales, and we call these systems chaotic. This doesn’t mean that they’re random in nature, but rather that they’re highly sensitive to uncertainties in our measurements. Take the double pendulum as an example.

You’d be forgiven for thinking that the pendulum below is moving around randomly, but in fact its motion is governed by deterministic laws. It’s just that the equations that describe the motion of the pendulum are highly sensitive to initial conditions. Small changes to the pendulum’s length, mass, or initial position lead to large changes in its trajectory over time, like the butterfly effect. In theory, if we were to measure the mass, position, and velocity of the pendulum at a point in time, we should be able to use the laws of physics to perfectly predict how it will behave arbitrarily far in the future. The problem is that in practice, measurements are never perfectly precise. Even if the uncertainty in our measurement lies in the sixth or seventh decimal place, this can lead to a large uncertainty in the pendulum’s position after a relatively short amount of time. It’s the very same phenomenon that prevents us from accurately predicting the weather (and many other complex systems) over the long-term. Play around with the double pendulum simulation below to get a feel for the chaotic nature of the double pendulum.

Check out part 2 for some pretty pictures that can emerge from chaos.