Welcome back to the Demonstration Highlight of the Week! This week, we’re taking a look at demonstration G1-60: Chaos with Two Bifilar Pendula. You can see the demonstration in action in this video featuring doctoral student Subhayan Sahu.
We have two essentially identical sets of physical pendulums suspended from a single rod. The physical laws governing their behaviour are quite simple, merely the conservation of linear and angular momentum and the force of gravity. The two pendulums are started into apparently identical oscillations, but starting the pendula with identical initial conditions is nearly impossible. So no matter what, their motion soon diverges. No matter how closely the motions of the two pendulums are started, they eventually must undergo virtually total divergence. This extreme sensitivity to initial conditions is a form of chaos, the mathematical study of irregularity in dynamical systems.
Wikipedia has a surprisingly good article on the mathematics of the double pendulum. (https://en.wikipedia.org/wiki/Double_pendulum) Also, Eric Neumann has created an online simulation that can be used to model one of the legs of the pendulum. Try experimenting with the simulation as well, and see how sensitive it can be to its initial conditions. (https://www.myphysicslab.com/pendulum/double-pendulum-en.html)