From Simple to Double Pendulum: Theoretical Foundations and Numerical Solutions

This document provides a complete, self-contained, step-by-step examination of the theoretical background, foundations and numerical computation of the Simple and Double Pendulum systems, covering all essential mathematics. It starts with harmonic motion and its connection to general oscillatory dynamics. The simple pendulum is analyzed under Newtonian, Lagrangian, and Hamiltonian mechanics, showing how different formalisms converge to a single equation of motion. The double pendulum is then explored in both Lagrangian and Hamiltonian frameworks, highlighting its non-linear and chaotic behavior. The equations of motion are expressed in a compact matrix form, suitable for theoretical analysis and numerical computation. Finally, several numerical integration methods are presented, including Euler, Runge–Kutta 4, and symplectic algorithms such as Symplectic Euler and Verlet. A comparative survey evaluates their accuracy, stability, and energy conservation, making the document a complete guide for both theory and computation.