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

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?

• In general $F=F(q,p,t)$. Then you can define a function $\mathcal F(t) := F(q(t),p(t),t)$. Calculating its derivative (and employing the equations of motion) yields your last equation. Long story short: Observables can, in principle, depend on time explicitly and not only through $q$ and $p$. May 6, 2022 at 21:09
• Okay. Just for clarification, does that mean in Susskind/Jakob's text, they assumed that F doesn't depend explicitly on time, right? May 6, 2022 at 21:18
• I don't know the context (i.e. I haven't read the specific parts of the books), but it seems very likely to me. May 6, 2022 at 21:20

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}