Poisson Bracket

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 f\in F(T^*M) that we’re interested in,

\frac{df}{dt}=\{f,H\}+\partial_t f

Where H is the Hamiltonian. But here’s where I was a bit confused. If we put in the coordinates (on T^*M) in for f then,

\dot{q}=\{q,H\}
and
\dot{p}=\{p,H\}

So I wondered, what happened to the \partial_t \xi (where \xi can stand for either position or momentum)? I was flummoxed because my experience has been to find these letters as functions of time,

q=q(t) and p=p(t)

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 T^*M and what we’re trying to find are integral curves. The vector field:

\dot{q}^{\alpha}\partial_{\alpha}+\dot{p}_{\alpha}\partial_{\alpha}

Generates a flow \phi_t which acts on T^*M. So if our initial data is q_0, p_0 then there is a curve \gamma(t)=\phi_t (q_0,p_0) that satisfies Hamilton’s equations for our system and initial conditions. We can describe \gamma in local coordinates \gamma(t)=(q^1(t), q^2(t),...,p_1(t), p_2(t),...) where q^i(t) stands for the way that component of \gamma evolves with time.

Its a subtle point that I’m still digesting but it does resolve the above issue. Comments?

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About because0fbeauty

Fascinated by the way mathematics and physics interact, captivated by visual and tactile mathematics and hoping to become a better expositor of these things is why I blog...occasionally...when I remember.
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