Total energy without damping
Kinetic and potential energy trade back and forth. If the total drifts noticeably, the cause is usually damping or numerical integration error.
Chaos · nonlinear dynamics
Double Pendulum
How long do two nearly identical releases of a double pendulum follow the same path? Only for a short while. The motion is fully deterministic, but the nonlinear dynamics amplifies a tiny difference in the start exponentially. Two releases a thousandth of a degree apart soon swing along completely different paths.
A 3D metal double pendulum swings in real time, integrated with the RK4 method. A velocity arrow and a glowing trail follow the tip and leave its path in space. A violet twin, drawn in a parallel plane, starts one-thousandth of a degree away; watch how long it keeps pace with the original.
Time0.0 s
Total energy0.00 J
Tip speed0.00 m/s
Twin gap0.00°
Physics tutorial
Double pendulum: from deterministic motion to chaos
BackgroundA simple pendulum is close to an oscillator with a steady frequency. Hang a second rod from it, and gravity and inertia couple the two angles nonlinearly.
Why it mattersWeather, orbits and many complex systems obey exact equations yet resist long-term prediction. The double pendulum is the smallest model that shows why.
Start with the essentials
- Focus question
- Why do two nearly identical double pendulums eventually follow completely different paths?
- One-sentence intuition
- Chaos is not randomness: the nonlinear motion amplifies tiny initial errors exponentially.
Core mathematical model
How nearby paths separate
When the Lyapunov exponent is positive, small initial differences grow exponentially. Its size sets how far ahead the motion can be predicted.
Common difficulties
Chaos is not randomness
Typical misconceptionThe path looks irregular, so no deterministic equation can be behind it.
Better mental modelThe same equations govern every instant. The uncertainty comes from the starting state, which can only be measured to finite precision.
Conserved energy does not mean periodic motion
Typical misconceptionIf total energy is fixed, the trajectory must repeat.
Better mental modelEnergy conservation limits which states the system can reach. It does not force the motion back along a closed path.
Run the experiment
- 01
Start with predictable motion
Choose Gentle, reduce damping to zero, and reset.
What to observe: The total energy holds steady and the motion is nearly periodic. - 02
Make the motion strongly nonlinear
Switch to Chaotic and keep the tip trail visible.
What to observe: The arms start to flip over, and the path stops repeating. - 03
Test sensitivity to the start
Turn on the chaotic twin, which starts only 0.001 degrees away.
What to observe: The two pendulums overlap at first, then quickly separate.