Complete Elliptic Integral and Pendulum Period

Worked Example

Consider a pendulum of length L = 1 m. The small angle period is T_0 = 2π√(1 / 9.81) ≈ 2.006 s. If we release the pendulum from an initial amplitude such that the modulus is k = 0.5 (which means \theta_0 = 2arcsin(0.5) = π/3 = 60^\circ), we must compute the complete elliptic integral K(0.5).

The arithmetic-geometric mean (AGM) provides an extraordinarily efficient way to compute K(k). Starting with a_0 = 1 and g_0 = √(1 - k²) = √(1 - 0.25) ≈ 0.866025:

1. a_1 = (a_0 + g_0)/2 ≈ 0.9330127
2. g_1 = √(a_0 g_0) ≈ 0.9306048
3. a_2 = (a_1 + g_1)/2 ≈ 0.93180875
4. g_2 = √(a_1 g_1) ≈ 0.93180838
5. a_3 ≈ 0.93180839
6. g_3 ≈ 0.93180839

Since a_n and g_n converge rapidly, we have AGM(1, √(0.75)) ≈ 0.93180839. The complete elliptic integral is K(0.5) = (π)/(2 · 0.93180839) ≈ 1.68575. The period is T = 2/(π) T_0 K(0.5) ≈ 2.1528 s. The period ratio T/T_0 is approximately 1.073, meaning the pendulum is about 7.3\% slower than the small-angle approximation predicts. This matches the Period of 2.1528 seconds shown in the experiment.

Experiment

Explore the exact period by varying the modulus k and the length L. Observe how the period diverges as k → 1, which represents a pendulum released from the straight up position (a separatrix orbit). You can interact with the experiment below.

1. Set Modulus k to 0.5 and Length L to 1.
2. Observe that the calculated Period is approximately 2.1528 s, and the Period ratio is 1.073.

Limitations

This model assumes an ideal, frictionless simple pendulum with a point mass on a massless, rigid rod. When k = 1, the theoretical period diverges to infinity, corresponding to the separatrix limit where the pendulum takes infinite time to reach the upright unstable equilibrium. For k > 1, the pendulum undergoes continuous rotation rather than oscillation, which requires incomplete elliptic integrals and is outside the scope of this module.

References

• NIST Digital Library of Mathematical Functions, Section 19.2 (Elliptic Integrals)
• NIST Digital Library of Mathematical Functions, Section 19.8 (Arithmetic-Geometric Mean)

The NIST DLMF sections 19.2 and 19.8 provide rigorous definitions and computational methods for complete elliptic integrals and the arithmetic-geometric mean iteration.