🔮 Double Pendulum

A double pendulum — two pendulums connected end-to-end — is the simplest mechanical system that exhibits chaotic motion. The equations of motion derived from the Lagrangian show that nearby initial conditions lead to exponentially diverging trajectories. Starting just 0.001° apart, two pendulums look identical for a few seconds, then separate completely. Enable Chaos Mode to launch a rainbow of pendulums with tiny perturbations and watch the butterfly effect unfold. The coloured trail records the tip position. 🇺🇦 Українська

Pendulum

Length L₁ (m) 1.0 Length L₂ (m) 1.0 Mass m₁ (kg) 1.0 Mass m₂ (kg) 1.0

Initial angles

θ₁ (°) 120 θ₂ (°) -30

Gravity g (m/s²) 9.81 Speed 2

θ₁—

θ₂—

Energy E—

Lagrangian Mechanics

The double pendulum has two degrees of freedom (θ₁, θ₂). The Lagrangian L = T − V gives equations of motion that are solved here with RK4 (dt = 0.02/speed). Lyapunov exponent λ ≈ 3–5 s⁻¹ for typical initial conditions — trajectories double their separation in about 0.1–0.3 s. Energy is conserved (no damping); small drift is due to finite step size.