Geometric and Hamiltonian Analysis of the Large-Amplitude Pendulum

This paper examines the mathematical structure of the pendulum, focusing on the limitations of the small-angle approximation and the resulting error growth in large-amplitude motion. It develops a phase space and Hamiltonian framework to provide a global geometric description that overcomes these local approximation limitations. The paper concludes with a numerical simulation in Python illustrating both the linearized and nonlinear pendulum dynamics and their phase space representations.