Chaos Lab

Continuous flow · dim 4

Double pendulum

The simplest mechanical system that exhibits chaotic motion. Conservative (no friction) chaos for moderate-to-large initial energies.

Bob traces of two double pendulums starting ε ≈ 10⁻³ apart — SDIC visualised.

Equations

Coupled Euler-Lagrange equations for two pendula joined end to end (4-D phase space: θ_1, θ_2, dθ_1/dt, dθ_2/dt).

At a glance

Parametersm_1, m_2, l_1, l_2, g, initial conditions
Chaotic formoderate-to-large initial angles
HistoryStudied since Daniel Bernoulli (1733); chaos understood in the 1960s-70s.

Try it

Open the interactive playground at /tools/double-pendulum.

See also

Quick quiz

Test yourself on double-pendulum

8 multiple-choice questions. Pick an answer for each, then submit to see explanations.

  1. Q1.Why is the double pendulum chaotic?

  2. Q2.How many degrees of freedom does a double pendulum have?

  3. Q3.For small angle amplitudes the double pendulum is:

  4. Q4.Energy of the unforced double pendulum is:

  5. Q5.A useful way to visualise double-pendulum chaos:

  6. Q6.First analysed historically by:

  7. Q7.A pedagogical advantage of the double pendulum:

  8. Q8.Triple pendulums extend this to:

0 of 8 answered