---
title: "Double Pendulum & Chaos Interactive Lab | Sandbox Physics"
description: "How long do two nearly identical releases of a double pendulum follow the same path? Only for a short while."
canonical_url: "https://sandboxphysics.com/labs/double-pendulum/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/double-pendulum/"
last_modified: "2026-10-27"
---

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.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

Rotate the view and you can see that both rods always swing in the same plane. With damping at zero, total energy stays nearly constant. The twin still drifts away, which shows sensitive dependence on initial conditions.

Physics tutorial

## Double pendulum: from deterministic motion to chaos

**Background** A 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 matters** Weather, 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

#### Total energy without damping

$$
E=T+U\approx\text{constant}
$$

Kinetic and potential energy trade back and forth. If the total drifts noticeably, the cause is usually damping or numerical integration error.

#### How nearby paths separate

$$
\delta(t)\approx\delta_0e^{\lambda_{\mathrm L}t}
$$

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 misconception** The path looks irregular, so no deterministic equation can be behind it.

**Better mental model** The 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 misconception** If total energy is fixed, the trajectory must repeat.

**Better mental model** Energy conservation limits which states the system can reach. It does not force the motion back along a closed path.

### Run the experiment

1. 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.

2. 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.

3. 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.

Physical conclusion

Deterministic equations do not guarantee long-term predictability.

Motion & dynamics · Starter path

## From a falling ball to an orbit

Start by measuring gravity and splitting motion into components. Then add contact forces and collisions, and follow the same laws all the way to an orbit.

[Start here → **Free Fall & Measuring Gravity** Capture a few drops and fit the camera positions to estimate gravitational acceleration. Then add air damping and watch the fit drift from the truth. Observe · 14 min](https://sandboxphysics.com/labs/free-fall/)

1. [1 Free Fall & Measuring Gravity](https://sandboxphysics.com/labs/free-fall/)
2. [2 Projectile Flight Designer](https://sandboxphysics.com/labs/projectile-flight/)
3. [3 Inclined Plane & Friction](https://sandboxphysics.com/labs/inclined-plane/)
4. [4 One-Dimensional Collision Bench](https://sandboxphysics.com/labs/collision-bench/)
5. [5 Mass–Spring Oscillator](https://sandboxphysics.com/labs/mass-spring/)
6. [6 How Orbits Take Shape](https://sandboxphysics.com/labs/orbit/)

[View the full path →](https://sandboxphysics.com/topics/mechanics/#starter-path)