Part 5/6:
The principle of least action is remarkably powerful. It unifies diverse areas of physics - from classical mechanics to electromagnetism and quantum theory - under a single mathematical framework. Rather than working with forces and vectors, one can simply write down the Lagrangian and apply the Euler-Lagrange equation to obtain the equations of motion.
This approach is particularly useful for complex systems, like the double pendulum, where the vector-based approach becomes unwieldy. The Lagrangian method provides a systematic way to derive the equations of motion, even in higher-dimensional or non-Cartesian coordinate systems.