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The science
Nonlinear pendulum
For a point mass on a rigid rod the equation of motion is θ'' = −(g/L) sinθ − qθ' + A cos(ωdt). For small angles sinθ ≈ θ gives the harmonic period T0 = 2π√(L/g), independent of amplitude. The exact undamped period is T = 4 K(k) √(L/g) with k = sin(θ0/2) and K the complete elliptic integral of the first kind (computed here by the arithmetic-geometric mean); for small amplitudes T/T0 ≈ 1 + θ0²/16 + 11θ0&sup4;/3072, and T diverges as θ0 → 180 degrees.
In the phase portrait the energy E/(mgL) = ω²/(2ω0²) + 1 − cosθ is conserved, so undamped motion follows closed contours; the separatrix E = 2 mgL divides oscillation from full rotation. Damping spirals the state into the origin; a periodic drive near ω0 gives resonance, and a strong drive can produce chaos.
Honest simplifications
A point bob on a massless rigid rod, no air drag other than the linear term q, no pivot friction. Integration is fourth-order Runge-Kutta with a 2 ms step. The 3D page shows several pendulum setups; this page concentrates on the maths of one pendulum.
What to try
- Compare 5 deg and 120 deg: the blue (exact) bob lags the orange (small-angle) one because its period is longer.
- Release at 175 deg: it hangs near the top for a long time (T / T0 is about 3).
- Driven chaotic preset: the phase trail never closes.