Double Trouble

by Dirk Brockmann

This explorable illustrates the beautiful dynamical features of the double pendulum, a famous idealized nonlinear mechanical system that exhibits deterministic chaos. The double pendulum is essentially two simple pendula joined by a bearing. It's a classic complex system in which a simple setup generates rich and seemingly unpredictable behavior. The only force that is acting on it is gravity. There's no friction.

Initially, the pendulum is raised to some position and has therefore some potential energy with respect to the central pivot. When you press the play button, gravity acts and sets the pendulum in motion. It start moving in a chaotic manner, while energy is conserved.

This is how it works

When you press play, you release the pendulum. With the trace toggle you can follow the most recent part of the tip's trajectory. Switch it off, if it bothers you. You can also hide the pendulum if you want to focus on the funky shape of the trajectory.

The double pendulum's configuration is set by the two angles each of the levers makes with the vertical axis. You can use the sliders to initialize the pendulum at a new position (Every time you do this the angular speed of each lever is set to zero).

When you press stop and have "show path on stop" switched on, the entire trajectory is displayed since you last set initial conditions, until you press play again.

The ghost

An important feature of chaotic dynamical systems is a sensitivity to initial conditions. That means, if you set up two systems with slightly different initial conditions, very quickly their trajectories will diverge. To investigate this, every time you reset the system to a specific position a ghost pendulum is also prepared with almost the exact initial condition (the difference in one of the initial angles is 0.001 degrees, that's about the angle a bacterium makes if you hold it at arm's length).

Do this: Set the angles to a new position and turn the ghost switch on. Press play. After a short while you will (almost always) observe how original and ghost systems diverge. This is not an error in the numerical integration the computer does, it's an intrinsic feature of the dynamical system.

Switching off the chaos

Here's another little experiment. Prepare the pendulum such that it has lots of potential energy, i.e. when the entire thing points somewhat upwards. Press play and let the pendulum gain some momentum. Then: switch off gravity. What happens is that now the whole system is say in a state of free fall or somewhere in outer space. It only moves, because it was set in motion inititally.

Observe that the motion still looks a bit complex but a lot more regular. In fact, the system is no longer chaotic. There's a deeper reason behind this that has to do with the symmetries of the system. In general the pendulum is governed by Newtonian mechanics which means the state of the system is defined by 4 numbers, the angle variables $\theta _1$,$\theta _2$ and the angular velocities $\dot{\theta} _1$ and $\dot{\theta} _2$. Because energy is conserved the system is moving on a three-dimensional manifold in the four dimensional phase space. Without gravity at work, the system conserves not only energy, but also angular momentum. This reduces the dimensionality of the dynamical system to two. And in this case the system cannot exhibit deterministic chaos.