# Hiding behind chaos and error in the double pendulum

If you want a visual intuition for just how unpredictable chaotic dynamics can be then the go-to toy model is the double pendulum. There are lots of great simulations (and some physical implementations) of the double pendulum online. Recently, /u/abraxasknister posted such a simulation on the /r/physics subreddit and quickly attracted a lot of attention.

In their simulation, /u/abraxasknister has a fixed center (block dot) that the first mass (red dot) is attached to (by an invisible rigid massless bar). The second mass (blue dot) is then attached to the first mass (also by an invisible rigid massless bar). They then release these two masses from rest at some initial height and watch what happens.

The resulting dynamics are at right.

It is certainly unpredictable and complicated. Chaotic? Most importantly, it is obviously wrong.

But because the double pendulum is a famous chaotic system, some people did not want to acknowledge that there is an obvious mistake. They wanted to hide behind chaos: they claimed that for a complex system, we cannot possibly have intuitions about how the system should behave.

In this post, I want to discuss the error of hiding behind chaos, and how the distinction between microdynamics and global properties lets us catch /u/abraxasknister’s mistake.

A number of people on Reddit noticed the error with /u/abraxasknister’s simulation right away. But the interesting part for me was how other people then jumped in to argue that the correctors could not possibly know what they were talking about.

For example, /u/Rickietee10 wrote:

It’s based on [chaos] theory. … Saying it doesn’t [look] right isn’t even something you can say, because it’s completely random.

Or /u/chiweweman’s dismissing a correct diagnosis of the mistake with:

That’s possible, but also double pendulums involve chaos theory. It’s likely this is just a frictionless simulation.

These detractors were trying to hide behind complexity. They thought that unpredictable microdynamics meant that nothing about the system is knowable. Of course, they were wrong.

But their error is an interesting one. This seems like an unfortunately common misuse of chaos in some corners of complexology. We say that a some system (say the economy) is complex. Thus it is unknowable. Thus, people offering liner theories (say economists) cannot possibly know what they are talking about. They cannot possibly be right.

Have you encountered variants of this argument, dear reader?

This kind of argument is wrong. And in the case of the double pendulum, /u/GreatBigBagOfNope responded best:

You can’t just slap the word chaos on something and expect the conservation of energy to no longer apply

So let us use the conservation of energy to explain why the simulation is wrong.

From the initial conditions, we can get an estimate of the system’s energy. This is particularly easy in this case since the masses start at rest at some height — thus all energy is potential energy. From this — due to the time-invariance of the Hamiltonian specifying the double pendulum — we know by Noether’s theorem that this initial energy will be conserved. In this particular case, this means that we cannot ever have both of the masses above their initial position at the same time. If that happened then (just) the potential energy of this configuration will be strictly higher than the total initial energy. Since we see both of the masses simultaneously above their initial position in the gif, we can conclude that there is an error in /u/abraxasknister simulation.

I enjoy this kind of use of global abstract argument to reason without knowing the details of microdynamics. For me, this is the heart of theoretical computer science.

Based on this violation of energy conservation, many theories were discussed for what the error in the simulation might have been. And the possibility of energy-pumping from finite step size was a particularly exciting candidate. A ‘bug’ (that can be a ‘feature’) that I’ll discuss another day in the context of replicator dynamics.

The actual main mistake turned out to be much less exciting: a typo in the code. A psy instead of phi in one equation.

I’m sure that all of us that have coded simulations can relate. If only we always had something as nice as the conservation of energy to help us debug.

Of course, this doesn’t mean that there aren’t interesting discussions to be had on chaos and prediction. I’ve written before on computer science on prediction and the edge of chaos and on how stochasticity, chaos and computations can limit prediction. But we shouldn’t use chaos or complexity to stop ourselves from asking questions or making predictions. Instead, we should use apparent complexity as motivation to find the limits of more linear theories. And whatever system we work with, we should look for overarching global principles like the conservation of energy that we can use to abstract over the chaotic microdynamics.