About the Double Pendulum | Chaos & Light Trails

What is a double pendulum?

Hang one pendulum from a fixed pivot, then hang a second pendulum from the end of the first. You now have two angles—usually written θ₁ and θ₂—and four degrees of freedom once angular velocities are included. Gravity pulls both bobs; the joints couple their motion so energy sloshes between the upper and lower arms.

Why the motion becomes chaotic

For small swings a double pendulum can look almost regular. Push the energy higher and tiny differences in the starting angles or speeds grow until two nearly identical starts paint completely different paths. That sensitive dependence on initial conditions is the hallmark of deterministic chaos: the rules are exact, yet long-term prediction is practically impossible.

Equations in brief

The standard model uses Lagrangian mechanics with lengths L₁, L₂ and masses m₁, m₂. The resulting ordinary differential equations for θ₁ and θ₂ are nonlinear and coupled—you integrate them numerically (for example with RK4) rather than solving them in closed form. Change a length or mass and the whole family of trails shifts.

Light trails and long exposures

Photographers hang LEDs on physical double pendulums and open the shutter to capture glowing ribbons. Digitally, you sample the tip position over thousands of steps and stroke those points onto a canvas. Fade the trail for neon streaks, keep every sample for a dense long-exposure lace, or warp a background along the path for distortion art.

Chaotic double-pendulum tip trails rendered in PictorX

Double pendulum vs painting pendulum

A painting (Blackburn) pendulum combines two harmonic oscillators and paints Lissajous figures—ordered loops when the frequency ratio is rational. A double pendulum is a single mechanical chain whose nonlinear coupling produces chaos. Both make beautiful trails; the mathematics and the look of the curves are different.

Try Double Pendulum in PictorX

Open /double-pendulum to simulate the tip in your browser. Tune lengths, masses, gravity, and damping; choose trails, long-exposure, or distort modes; set paint color and style; then save a PNG or open the result in the editor. Nothing is uploaded. Pair it with Painting Pendulum, Lorenz Attractor, or Spirograph for more generative play.

Frequently asked questions

Why is a double pendulum chaotic?

Its equations are nonlinear and coupled. At sufficient energy, tiny changes in the starting state grow rapidly, so long-term motion cannot be predicted in practice even though every step follows fixed physics.

What do L₁, L₂, m₁, and m₂ mean?

L₁ and L₂ are the lengths of the upper and lower rods; m₁ and m₂ are the masses at the joints (or bob masses in the usual idealized model). Together with gravity they set how fast and how wildly the tip moves.

How is this different from a painting pendulum?

A painting pendulum traces Lissajous curves from two independent harmonic oscillators. A double pendulum is one linked mechanism whose tip path becomes chaotic rather than a closed harmonic figure.

What are light-trail and long-exposure modes?

Light trails emphasize recent motion with glow and fade. Long exposure keeps a dense history of tip positions, like leaving a camera shutter open while an LED swings.

Does PictorX upload my image?

No. The double-pendulum simulation runs on a canvas in your browser. Save downloads a PNG you choose; optional distort modes can sample a local background without uploading it.