When I first learned about the Poisson bracket in classical mechanics I was very excited. I had learned about the commutator in quantum first and this looked a lot alike. For any dynamical quantity that we’re interested in,
Where H is the Hamiltonian. But here’s where I was a bit confused. If we put in the coordinates (on ) in for then,
So I wondered, what happened to the (where can stand for either position or momentum)? I was flummoxed because my experience has been to find these letters as functions of time,
where they seem to be explicit functions of time. But they are not. We’re really working on a cotangent bundle, which the Hamiltonian is a function on. Hamilton’s equations define a vector field on and what we’re trying to find are integral curves. The vector field:
Generates a flow which acts on . So if our initial data is then there is a curve that satisfies Hamilton’s equations for our system and initial conditions. We can describe in local coordinates where stands for the way that component of evolves with time.
Its a subtle point that I’m still digesting but it does resolve the above issue. Comments?