Double Pendulum Simulator
Classroom worksheet · Interactive simulation: https://lkforge.com/tools/physics/double-pendulum/
Name: Date:
Learning objectives
- Observe sensitive dependence on initial conditions (chaos).
- Contrast predictable and chaotic motion.
- Recognize that a deterministic system can be practically unpredictable.
Variables to change
- Initial angles
- Arm lengths
- Masses
Procedure
- Release the pendulum from a small starting angle and watch the motion.
- Release it again from a nearly identical angle and compare the two paths over time.
- Release it from a large angle and observe the motion.
Observations
Describe how quickly two nearly identical starts diverge from each other.
Questions
- What happened when you started from two almost-identical angles?
- Is the double pendulum random?
- How does a single pendulum differ from a double pendulum?
- Why can’t we predict its exact position far into the future?
- Name another real system that shows sensitive dependence on initial conditions.
Answer key (instructors)
- 1. The motions matched briefly, then diverged completely — the hallmark of chaos.
- 2. No. It is fully deterministic (governed by fixed equations) but extremely sensitive to initial conditions, so long-term prediction is impractical.
- 3. A single pendulum is periodic and predictable; adding a second arm couples the motion and produces chaos.
- 4. Tiny uncertainties in the starting state grow exponentially, so any measurement error eventually dominates.
- 5. Weather, turbulent fluids, or a Lorenz-type system — the “butterfly effect.”
The double pendulum is deterministic yet chaotic: nearby starting states diverge exponentially, so its long-term motion is unpredictable even though the equations are exact. It is a classic, visual demonstration of the butterfly effect.