Welcome to the Double Pendulum!

This page shows a library of double pendulum motions and a phase space map of initial angles (θ₁, θ₂). Instructions on how to explore:

Playback Controls

Play / Pause: Toggles the animation. When paused, additional buttons appear to step the simulation forward/backward.
Speed + / Speed -: Change playback speed. Also changes the size of the simulation time step buttons.
Next / Prev: Step to the next or previous orbit along the same orbit family (continuation sequence).
Editable values: The Time, Period, Speed, and Energy values in the top corners are editable — click any underlined value, type a number, and press Enter. Time (only when paused) seeks within the current orbit; Period and Energy navigate to the nearest orbit in the current family; Speed sets the playback rate directly (0.02–32×). Press Esc to cancel.

Phase Space Mini Map

Top-right mini plot shows the entire phase space.
The black dot marks the initial condition of the current orbit.
Click the mini map to open the full Phase Space overlay.

Phase Space Overlay

This is a Poincaré section at θ̇₁ = 0: each orbit is plotted at the unique initial condition where the upper arm is momentarily at rest. This ensures every periodic orbit appears as exactly one point, even though the same orbit could be started at many other moments in time.
Hover the large plot to see initial conditions and properties of orbits (initial θ₁, θ₂, |θ̇₂|, Energy, Period).
Click (or tap) a point to load a new orbit.
Zoom with the mouse wheel or trackpad, the − / + controls, or a two-finger pinch. Drag the plot with a mouse or one finger to pan.
Hold the arrow keys to pan smoothly (hold Shift to move faster), and press 0 or Reset to restore the full view.
Use the buttons (Initial θ̇₂, Energy, Period) to recolor all points based on that parameter. Press 1 / 2 / 3 as shortcuts.
Hover a point and press Enter (or click "Navigate to Point") to load that orbit.

Tips

Press Esc or Enter to close this overlay.
Press Space to play / pause.
On the main animation, press ↑ / ↓ to step through the orbit family.
On the main animation, press ← / → to step one frame backward / forward when paused.
On the main animation, press Shift+← / Shift+→ to step 10 frames backward / forward when paused.
On the main animation, press + / - to increase / decrease playback speed.
On the main animation, press 0 to reset speed to default.
Press ? to open this help page.
Press P to open the Phase Space overlay.
The white pendulum trace shows the path of the lower mass.
You can always find these instructions again by pressing the "Help" button.

Conventions

Angles: 0° means the pendulum arm hanging straight down; positive angles are counterclockwise.
Energy: Zero corresponds to the stable fixed point — both masses hanging at rest. All other parameters (masses, arm lengths, g) are set to 1.

About

Research conducted by Leo Yao, Ziming Liu, and Max Tegmark. Presented at Machine Learning for Physical Sciences Workshop, NeurIPS 2024 (Poster, Workshop Paper). Journal paper under review (arXiv).

Visualization created by Leo Yao (leoy@mit.edu).