Classical Mechanics: Newtonian, Lagrangian, and Hamiltonian

Photo: Rhett Allain. Newtonian, Lagrangian and Hamiltonian solutions of the same problem.

In classical mechanics, there are three common approaches to solving problems. I’m going to solve the same situation three different ways. It’s going to be fun. Trust me.

Here is the problem. A ball is at ground level and tossed straight up with an initial velocity. The only force on the ball while it is in the air is the gravitational force. Oh, the ball has a mass of “m”.

Newtonian Mechanics

First, let’s get this out of the way. It wasn’t just Newton that did this stuff. Also, please don’t lecture me on “Newton’s Three Laws of Eternal Motion and the Rules of the Universe”. Yes, I think everyone spends too much time on “the three laws”.

In short, Newtonian mechanics works for cases in which we know the forces and we have a reasonable coordinate system. Yes, it’s true that we can use unreasonable coordinate systems and still have this stuff work. Also, it’s possible to deal with unknown forces (like the tension in a string with a swinging pendulum). But Newtonian mechanics works best if we know the forces.

If you know the forces (or just one force), then the following is true (in one dimension):

Honestly, I prefer to write the momentum principle -but let’s just go with this for now. Oh, in case you didn’t notice, I…