What is the correct general form of Hamilton's equation?

Question

Usually, Hamilton's equations of motion are given by: $$ (1)\;\; \frac{dp}{dt} = -\frac{\partial H}{\partial q} \;\;\; \text{ and } \;\;\;(2)\;\; \frac{dq}{dt} =\frac{\partial H}{\partial p}.$$ Using Poisson's bracket one can generalize Hamilton's equation i.e. the time derivative of anything is given by the Poisson bracket of that thing with the Hamiltonian: $$\frac{dF}{dt} = \{F,H\} $$ where $F$ is a function of $p$ and $q$, $H$ is the Hamiltonian, and Poisson bracket is defined as $$ \{A,B\} = \frac{\partial A}{\partial q}\frac{\partial B}{\partial p} - \frac{\partial A}{\partial p}\frac{\partial B}{\partial q}. $$ See page 108 of 'No-nonsense Classical Mechanics' by Jakob Schwichtenberg and page 172 of 'The Theoretical Minimum' by Susskind & Hrabovsky for details. However, in Wikipedia, they are saying that the most general form of the Hamilton's equation of motion is $$\frac{dF}{dt} = \{F,H\} + \frac{\partial F}{\partial t}.$$ From where is the extra $\frac{\partial F}{\partial t}$ coming from? Which one is correct?

Answer 1

Most generally the observable $F$ depends not only on $q(t)$ and $p(t)$, but also depends on $t$ explicitly, i.e. $$F=F(q,p,t)$$ Differentiating this with respect to time $t$ you get $$\begin{align} \frac{dF}{dt} &=\frac{\partial F}{\partial q}\frac{dq}{dt} +\frac{\partial F}{\partial p}\frac{dp}{dt} +\frac{\partial F}{\partial t} \\ &=\frac{\partial F}{\partial q}\frac{\partial H}{dp} -\frac{\partial F}{\partial p}\frac{\partial H}{dq} +\frac{\partial F}{\partial t} \\ &=\{F,H\} +\frac{\partial F}{\partial t} \end{align}$$