Equation of Motion: The exact nonlinear differential equation for a simple pendulum is $\ddot{\theta} + \frac{g}{L}\sin\theta = 0$.
Small-Angle Approximation: For small angular displacements ($\sin\theta \approx \theta$), motion is simple harmonic with natural period $T_0 = 2\pi\sqrt{\frac{L}{g}}$.
Conservation of Mechanical Energy: Total energy is $E = \frac{1}{2}m(L\dot{\theta})^2 + mgL(1 - \cos\theta) = \text{const}$, oscillating continuously between kinetic and potential forms.
Operating Instructions: Set length $L$, gravity $g$, mass $m$, and initial release angle $\theta_0$. Observe real-time trajectory, angular velocity, and energy partition charts.