Recovering Hamilton's Equations from Poisson brackets

[brotherbobby]

TL;DR

The online lecture notes I am following (see screenshot below) (correctly) derives the time derivative of a dynamical variable $u(q,p,t) = \{u,H\}+\dfrac{\partial u}{\partial t}$.
It then takes in stages, $u=\q_i$ and then $u = p_i$ to find Hamilton's equations of motion, if only as a form of reassurance. Namely, $\dot{q}_i = \partial H/\partial p_i\quad \dot{p}_i=-\partial H/\partial q_i$. But there's a serious problem in the derivation.

The issue : Let me start by copying and pasting the relevant passage from the text, thanks to modern day methods of computing.

The trouble is, in equation (4.79), it completely ignores the partial derivative of $q_i$ with respect to time, i.e. it puts $\partial q_i/\partial t=0$.

But $q_i$ is a dynamical variable of $t$, or $q_i(t)$. In the derivation of Hamilton's equations from the Hamiltonian, viz. $H = p_i \dot q_i-L$, nowhere did we assume that $\partial q_i/\partial t=0$.

It goes on to the do the same (next page, 70) with the second Hamilton's equation, putting again $\partial p_i/\partial t=0$.

Question : Clearly, these partial derivatives are not zero. And yet, if we don't put them to zero, Hamilton's equations are not recovered.
What is going on?

Attempt :
Let me try to "fill in" the mathematics that the author has left out, with the first equation.

$\small{\dot{q}_i=\{q_i,H\}+\partial q_i/\partial t=\partial q_i/\partial q_j \partial H/\partial p_j- \partial q_i/\partial p_j \partial H/\partial q_j+\partial q_i/\partial t=\delta_{ij}\, \partial H/\partial p_j+\partial q_i/\partial t=\partial H/\partial p_i+\partial q_i/\partial t\ne \partial H/\partial p_i\;\text{unless}\; \partial q_i/\partial t=0}$.

Request : Is $\partial q_i/\partial t=0$? A help or a hint would be most welcome.

 

[wrobel]

Such a way of presentation can really confuse a reader.
In (4.76) the function u depends on p,q and t. But in formulas (4.77) and (4.78) it is assumed that u=u(p(t),q(t),t) where p(t),q(t) is a solution to the Hamilton equations.

The trouble is, in equation (4.79), it completely ignores the partial derivative of qi with respect to time, i.e. it puts ∂qi/∂t=0.

that is because of that $u(p,q,t)=q_i$ does not depend on t explicitly. In such cases I usually recommend changing a textbook.

UPD

We can look at this as follows.
For functions f(t,q,p), g(t,q,p) introduce a Poisson brackets:
$$\{f,g\}:=\frac{\partial f}{\partial q}\frac{\partial g}{\partial p}-
\frac{\partial g}{\partial q}\frac{\partial f}{\partial p}.$$
Particularly if $f=q_i$ then
$\{q_i,g\}=\frac{\partial g}{\partial p_i}$ . Thus the Hamilton equations can be presented as follows
$\dot q_i=\{q_i,H\}$
analogously
$\dot p_i=\{p_i,H\}$

 

[div_grad]

.Request : Is $\partial q_i/\partial t=0$? A help or a hint would be most welcome.

@brotherbobby Equation (4.78) in your notes is valid for functions defined on the phase space, like $u=u(q,p,t)$. For such functions, the independent variables are taken to be $q$, $p$ and $t$. Therefore, if you want to use Eq.(4.78) to calculate total time derivative of $q$, you need to formally treat $q$ as a function defined on phase space - in this case you see that the particular phase space function $u(q,p,t)=q$ does not depend on time explicitly (since the independent variables for functions $u(q,p,t)$ are $q$, $p$ and $t$), hence the partial time derivative of this phase space function is zero. Thus you recover the Hamilton's equations.