## 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?