[comment_ran]

I still remember in high school, we only need two or three equations to solve a free-fall problem. In another book, basically the same question, but someone uses Hamiltonian framework to solve really complex PDEs and couple pages with those crazy equations to basically solve the same thing, and eventually got the same results.

I still remember that was mind-blowing. High school physics is so simple, whereas the Hamilton is so complex. I later on notice that Hamilton is kind of a more standard way to solve the problem. Never mind, I'm not an expert on it, but I'm just kind of amazed by the Hamiltonian mechanics.

[jfengel]

The Hamiltonian is a lot more flexible with respect to frame. The Newtonian formulation works great for simple cases but as it gets more complex it's harder and harder to pick a reference frame that's easy to compute.

Effectively, by working with energy rather than force, you can avoid working with vectors. That ends up being simpler as the components add up.