The Double Pendulum

When does a pendulum hanging from a pendulum look regular, and when does it become chaotic?

Interactive · live simulation

Two double pendulums released 0.0001 rad apart, in blue and gold. Identical equations, identical timestep — and within seconds their lower bobs are nowhere near each other. This is deterministic chaos.

The Double Pendulum — simulation render — fig 1. The lower bob's path. At small amplitude it weaves a quasi-regular two-mode pattern; at large amplitude two runs starting 1e-4 rad apart diverge into completely different trajectories.

Coupled and nonlinear

The Lagrangian equations of motion couple the two angular accelerations through the velocities. That velocity dependence is exactly why velocity-Verlet is no longer strictly symplectic for this system, so RK4 becomes the integrator of choice and explicit Euler is kept only as a cautionary tale.

fig 2. Total energy under RK4. The centripetal coupling makes Verlet non-symplectic here, so RK4 is the workhorse — and its drift is the diagnostic.

Regular at small angles

Start both arms at 15° and the motion is close to two coupled harmonic oscillators — the lower bob traces a tidy interference pattern of two normal modes. It looks almost periodic, and over short times it nearly is.

Chaos you can measure

Start both arms at 120° and the system is chaotic in the technical sense: the chaotic_perturbed run shifts the second angle by just 1e-4 rad, and within a few seconds the two paths — same integrator, same timestep — are unrecognizably different. Sensitive dependence on initial conditions, rendered as two diverging curves.

This chapter is drawn from the physics-lab study notes and renders. The longer write-ups and project essays live on the blog.