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What's the importance of Poisson brackets?

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$$F=F(q,p,t)$$ $$\frac{dF}{dt}=\frac{\partial F}{\partial q}\frac{\partial q}{\partial t}+\frac{\partial F}{\partial p}\frac{\partial p}{\partial t}+\frac{\partial F}{\partial t}$$ $$=\frac{\partial F}{\partial q}\dot{q}+\frac{\partial F}{\partial p}\dot{p}+\frac{\partial F}{\partial t}$$ $$=\frac{\partial F}{\partial q}\frac{\partial H}{\partial p}-\frac{\partial F}{\partial p}\frac{\partial H}{\partial x}+\frac{\partial F}{\partial t}$$ $$=\{F,H\}+\frac{\partial F}{\partial t}$$

Here $H$ is Hamilton. And, {F,H} they are inside Poisson braces. My question is what's the importance of Poisson brace? Does everyone use Poisson braces to make equation shorter? It is possible to derive earlier equation from the Poisson braces. Or, is there something else which can be expressed using only Poisson braces? Recently figured out a thing which is if a function (not sure if that's called function rather than variable) is constant than poisson brackets will be 0.

I meant if $F$ is constant in $\{F,H\}$ then $\{F,H\}=0$

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Wikipedia provides a reasonable survey (1 comment)

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