Let's Learn Physics: Coordinates? What Coordinates?

Dealing with a fixed set of coordinates can be clunky and inconvenient, but in Newtonian mechanics, this is often what we have to do. Then, using various tricks and logical gymnastics, we can work out sensible equations of motion. However, this becomes very difficult when systems start to get complicated. So, wouldn't it be great if we could figure out a system where we can choose whatever coordinates will be most convenient to the problem? This is exactly what we will find in Hamiltonian mechanics: a completely coordinate-independent framework of classical mechanics!