Series Approximation for the Nonlinear Pendulum
The plot shows the difference between the closed form of the solution for the nonlinear pendulum (in blue), and the series approximation for the corresponding elliptic integral of the first kind , plotting the initial amplitude versus the oscillation period , for a given frequency . The nonlinear oscillation arises when no assumption of small oscillations is made; that is, the approximation is not assumed here.
The elliptic integral of the first kind is represented in Mathematica by EllipticK.
The complete analytical treatment can be found in G. Baumann, Mathematica for Theoretical Physics, New York: Springer, 2nd. ed., 2005.