Double Pendulum Simulator
Simulate a double pendulum with proper Lagrangian equations, run two nearly identical copies side by side, and watch a microscopic difference grow into complete divergence.
Last reviewed by the Radiatus Cloud team
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Deterministic and unpredictable are not opposites
The double pendulum has no randomness in it at all: the equations are exact and the same starting conditions always produce the same motion. What it has is sensitive dependence, meaning that two starts differing by one part in a billion end up doing completely different things within a few seconds. Prediction fails not because the system is random but because you can never specify the starting state precisely enough, and the error you started with grows exponentially rather than staying small.
The divergence rate is measurable and it is the definition of chaos
Running two copies that differ by a tiny amount and plotting the separation on a logarithmic scale gives a straight line whose slope is the Lyapunov exponent. A positive exponent is what chaos means formally, and it tells you the horizon: at a separation growth of one order of magnitude per second, ten digits of initial precision buys about ten seconds of prediction. That is the entire practical content of the concept, and it is why weather forecasts stop rather than getting gradually vaguer.
Energy drift shows whether the integrator is trustworthy
Total energy is conserved exactly in the real system, so any drift in a simulation is numerical error. Watching it is the cheapest available check: if energy climbs, the integrator is adding energy and the trajectory is wrong in ways that look plausible. Reducing the step size reduces the drift, and reporting the drift alongside the result is the difference between a simulation and an animation.
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Frequently Asked Questions
Is the double pendulum random?
No. The equations are exact and identical starts give identical motion. It is unpredictable because tiny differences in the start grow exponentially, not because anything is random.
What is the Lyapunov exponent?
The rate at which nearby trajectories separate. A positive value is the formal definition of chaos, and it sets how long prediction remains possible for a given precision.
Why does energy drift matter?
Because energy is conserved exactly in the real system, so any drift is numerical error. It is the cheapest check that the simulation is trustworthy rather than merely smooth.
Does a smaller time step fix it?
It reduces the error and does not remove the sensitivity. Halving the step buys a fixed amount of extra accurate time, and the exponential divergence eats it quickly.
Is a single pendulum chaotic?
No. A single pendulum is regular and predictable for all time. Adding one more link is what introduces chaos, which is why the double pendulum is the standard example.
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How to Use
Set the starting angles and run the simulation.
Disclaimer: This tool is provided "as is" without warranty of any kind. Results are for educational and utility purposes.