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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)

# Comments on What's the importance of Poisson brackets?

Wikipedia provides a reasonable survey
Derek Elkins‭ wrote about 1 year ago:

You have a definition of the Poisson bracket, so there's nothing that can be expressed with this notion of Poisson bracket that can't be expressed without by simply using its definition. Besides that, what answer are you expecting that the Wikipedia page for Poisson bracket and the various pages it links to doesn't already provide?

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