What it Shows. A double pendulum executes simple harmonic motion (two normal modes) when displacements from equilibrium are small. However, when large displacements are imposed, the non-linear system becomes dramatically chaotic in its motion and demonstrates that deterministic systems are not necessarily predictable.

The double pendulum is one of the simplest and neatest examples of chaos. ​“These systems are extremely sensitive to initial conditions — so much that even if things change by a bit, you're going to get a completely different outcome,” says Ramkrishna.

They are used to determine the angle θ, the angular velocity ω = dθ/dt, and the angular acceleration α = d2θ/dt2. As shown in Fig. 1, the pendulum angle θ is defined from the vertical with θ = 0 for the straight-up, or inverted, position. Clockwise angles are taken positive.

The three-body problem represents the idea that an analytical solution to the motion of three bodies in a system where all bodies' forces affect each other is impossible to solve, since the forces of each body interact with the others simultaneously. In short, the double pendulum is impossible to solve analytically.