Chaos in the Double Pendulum

Hi everyone!

Welcome to my blog :)

Today, I want to share a quick demonstration of chaotic behavior in the double pendulum system (along with some derivations and code!).

This video is a visualization of the evolution of 25 double pendulum systems with slightly different initial conditions.

Each pendulum starts with the same initial configuration, except that the angular velocities of the outer disks each differ from one another by 0.001. These small differences quickly get magnified until any information about the initial conditions is effectively lost.

Why does this happen?

I actually don’t understand the full story, but I hope that I can provide at least a bit more insight here than one can glean from the video alone. The double pendulum system is governed by the following Lagrangian:

The angles phi1 and phi2 should be understood as in the image below.

It is not particularly clear from looking at this Lagrangian what behavior the double pendulum system will exhibit, so we follow the recipe that one always follows when confronted with a Lagrangian: we solve the Euler-Lagrange equations. (If you are unfamiliar with this procedure, you can find a worked example here.)

We thus arrive at the equations of motion for this system:

I solved these equations numerically to make the video at the top of the blog post.

So why is the system chaotic? At this point in my education, I genuinely do not know, and so I cannot provide you all an answer. But I can at least state the problem more precisely: The double pendulum system is chaotic because the two differential equations above–-the double pendulum equations of motion––exhibit sensitive dependence on their initial conditions.

If we can understand why these differential equations have this property, then we understand chaos in the double pendulum.

Perhaps exploring this point will make for another blog post somewhere down the line, but I will leave things here for today.

As promised, before I go, here are some technical goodies:

If you are interested in the derivation of the Lagrangian and equations of motion, you can find that work here.
If you want the Mathematica code that I used to solve the equations and generate the visualization, you can find that here.

Thank you for reading my blog! I hope to see you soon :)

Sincerely,

-Matt