Double Pendulum

Model. Two point masses move in a vertical plane on rigid, massless rods under gravity g = 9.81 m/s². There is no friction, collision or rod flexibility. Both initial angles are measured in degrees from the downward vertical; both angular velocities start at zero. The markers have illustrative sizes.

Steps and playback. RK4 uses a fixed 0.0005 s step. One step advances exactly this amount and pauses. Reset returns to the selected initial conditions and pauses. Playback speed changes model time per wall-clock second, not the numerical step. Slow frames are bounded and excess wall time is discarded under load. Hidden tabs and hidden tool views suspend playback. With reduced motion enabled, Start first prepares a paused model.

Comparison. The cyan pendulum adds ε only to its first initial angle. Its other parameters and initial velocities are identical. At ε = 0, both states remain identical. Tip separation measures their current physical distance; it is neither a Lyapunov exponent nor proof of chaos. Long chaotic trajectories are sensitive to numerical error even when energy drift is small.

Energy. E includes coupled kinetic energy and gravitational potential energy, with the potential zero at the pivot height. ΔE = E − E(0). The positive normalization scale is Eₛ = g[(m₁ + m₂)l₁ + m₂l₂], so the displayed ratio remains meaningful when E(0) is near zero. RK4 does not conserve energy exactly. The small fixed step improves short runs; it does not guarantee arbitrary long-term accuracy.

View and inputs. Initial-angle, length, mass and comparison edits require a fresh run. View, palette and trail changes preserve the model state. Each retained trail point represents a 0.01 s model-time sample; shortening a trail permanently discards older points. The 3D camera views the same planar motion with z = 0. PNG exports the current view; 3D export is disabled when WebGL is unavailable. Navigation retains parameter drafts, not live trajectories.

References. myPhysicsLab: Newtonian equations and energy; University of Maryland: small-angle normal modes, problem I.1.