What’s a Lagrangian? and something you can do with it: Harmonic Oscillator #2 – Lagrangian Formulation, at Built on Facts.

So why all that extra trouble? In this case, just to see how things work. But in other cases, this is our only option. There are plenty of times (for instance, a roller coaster on tracks) where the forces involved (such as the tracks on the car) are so blisteringly complicated as to be practically impossible to solve. However, it’s easy to write the relation between the track height and the potential energy, and the Lagrangian formulation can automatically give us a much simpler differential equation to solve.