Warmup to Statistical MechanicsWhat Exactly is a Hamiltonian ASystem?In ordinary Mechanics, you might begin with position and velocity. Hamiltonian Mechanics rewrites the same motion in a different language. Instead of position and velocity, it uses position and momentum. We write the position variables as q and the momentum variables as p. Then the full state of the system at one instant is(q, p)That pair is one point in phase space.Why do we do this?Because in these variables, the equations of

Phase Space Is the Real Stage of Dynamics. So, Stop Watching the Object and Watch the State.Ordinary Space tells you where something is. Phase Space tells you what state it is in. Position alone is not enough because you also need the variable that tells you where the system is trying to go.In the animation, the left panel shows the particle moving in Ordinary Space, while the right panel shows the same system as a moving point in Phase Space.For one degree of freedom, the state is(x, p)and as t