Gudermann And The Simple Pendulum

Gudermann and the Simple Pendulum
Author: Robertson, John S.
College Mathematics Journal, Vol. 28, no. 4, pp. 271-276, 1997.

The author demonstrates a practical and useful application of a little-known function in mathematics, the Gudermannian [gd x=tan-1(sinh x)], to solve the equation of motion for a simple pendulum. Specifically, this function emerges in the solution for the case when the pendulum bob comes infinitely close to reaching the unstable equilibrium at the top of its circular path. Properties of hyperbolic functions are then used to solve for the pendulum's parametric equation of motion. The steps taken to attain this solution are easily accessible to students in a basic ordinary differential equations class. (Summary by Ben Murphy, Pomona College '13)

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