Plane Pendulum and Beyond by Phase Space Geometry

The small angle approximation often fails to explain experimental data, does not even predict if a plane pendulum's period increases or decreases with increasing amplitude. We make a perturbation ansatz for the Conserved Energy Surfaces of a one-dimensional, parity-symmetric, anharmonic oscillator. A simple, novel algorithm produces the equations of motion and the period of oscillation to arbitrary precision. The Jacobian elliptic functions appear as a special case. Thrift experiment combined with recursive data analysis provides experimental verification of well-known predictions. Development of the quantum/classical analogy enables comparison of time-independent perturbation theories. Many of the useful notions herein generalize to integrable and non-integrable systems in higher dimensions.