Double Pendulum Chaos Simulation

A single pendulum repeats: give it the same start and you get the same swing, every time. Join a second rod to the bottom of the first and the same law of gravity produces motion that never repeats and cannot be predicted for long. Two copies released a tenth of a degree apart track each other perfectly at first, then the difference between them doubles, and doubles again, at a steady rate — so it grows exponentially rather than linearly. Within about ten seconds the two pendulums are doing completely different things. Nothing was random and nothing was approximated: the equations are exact and the energy of each copy holds steady while the two paths come apart. What fails is prediction, and it fails on a schedule you can read straight off the graph. Set both start angles small instead and the same pendulum behaves itself, the gap merely wobbles, and the graph goes flat — chaos here is a property of how hard you swing it, not of the apparatus. Students can change upper start angle and lower start angle and compare what happens. Use it for math inquiry with High School on a projector or student device.

استانداردها: Common Core CCSS.MATH.CONTENT.HSF.IF.B.4, CCSS.MATH.CONTENT.HSF.LE.A.1

• Run the Wild start and count the seconds before you can tell the two pendulums apart
• Switch to the Gentle start: the gap stops growing, so this pendulum is not always chaotic
• Divide the head start by ten and check how many extra seconds of prediction it actually bought you
• Read the log graph: the straight stretch is exponential growth, and its steepness is the doubling rate
• Watch the energy readout while the paths come apart — unpredictable is not the same as not conserved

1. 1Predict how changing upper start angle will affect the upper rod's kick at release, then test the prediction.
2. 2Use evidence from the live calculation and visual to explain the result.

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