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The science
The double pendulum
Two point masses m1, m2 on rigid massless rods l1, l2 have the Lagrangian L = ½(m1 + m2) l1² θ1'² + ½ m2 l2² θ2'² + m2 l1 l2 θ1' θ2' cos(θ1 − θ2) + (m1 + m2) g l1 cosθ1 + m2 g l2 cosθ2. Its Euler-Lagrange equations are coupled and nonlinear, with no closed form for large angles; here they are solved with a fourth-order Runge-Kutta scheme.
Sensitive dependence
For two trajectories that start a distance d0 apart, chaos means d(t) ≈ d0 eλt with a positive Lyapunov exponent λ, until the gap saturates at the size of the accessible space. The slope of ln d against time in the growing region is estimated live (a typical value for a vigorous double pendulum is a few per second). Reducing the initial difference by a factor of a million only delays the divergence by ln(106)/λ seconds: this is the butterfly effect, and it limits long-range prediction.
Order and chaos
Small angles give nearly harmonic motion with two normal modes (for equal masses and lengths ω² = (g/l)(2 ± √2)); then λ is about zero and the twins stay together. Energy is conserved by the physics, so the energy-error graph is a test of the numerical method, not of the system.
Honest simplifications: frictionless pivots unless damping is switched on (damping drains energy and tames the chaos), rigid massless rods, no air, fixed time step, and floating-point rounding that itself acts as a tiny perturbation: after a long time even the 'identical' runs of a computer would eventually part.
What to try
- Start from the Small swing preset: the twins stay together, because small oscillations are regular.
- Use Horizontal release and watch the separation graph climb as a straight line on the log scale.
- Change the initial difference from 10-6 to 10-10 rad and note that divergence is only delayed.
- Add damping and see the chaos die out.