First what is it? it’s also called Euler-Lagrangian Equation.

The Euler-Lagrange equation arises from the principle of stationary action (or the principle of least action), or, more precisely called the principle of stationary action. Let’s derive it:

While the principle of stationary action might seem mysterious or even arbitrary, its success in describing physical phenomena and its deep connections to both classical and quantum mechanics underline its fundamental nature in the workings of the universe.

Naturally, you would ask why Euler-Langarangian, or why the objects always follow the least action? in relativity we found the always take the shortest path, and massless photon trace out zero path. Further digging, it can be explained by Richard Feynman’s integral of paths, where all paths are possible but the least action path will be the weighted and left, let’s dig more into QM to get this clear and lucid.