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0 answers  ·  posted 1y ago by deleted user  ·  edited 1y ago by deleted user

#2: Post edited by deleted user · 2021-09-08T16:30:03Z (about 1 year ago)
• $$F=F(q,p,t)$$
• $$\frac{dF}{dt}=\frac{\partial F}{\partial q}\frac{\partial q}{\partial t}+\frac{\partial F}{\partial p}\frac{\partial p}{\partial t}+\frac{\partial F}{\partial t}$$
• $$=\frac{\partial F}{\partial q}\dot{q}+\frac{\partial F}{\partial p}\dot{p}+\frac{\partial F}{\partial t}$$
• $$=\frac{\partial F}{\partial q}\frac{\partial H}{\partial p}-\frac{\partial F}{\partial p}\frac{\partial H}{\partial x}+\frac{\partial F}{\partial t}$$
• $$=\\{F,H\\}+\frac{\partial F}{\partial t}$$
• 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?
• $$F=F(q,p,t)$$
• $$\frac{dF}{dt}=\frac{\partial F}{\partial q}\frac{\partial q}{\partial t}+\frac{\partial F}{\partial p}\frac{\partial p}{\partial t}+\frac{\partial F}{\partial t}$$
• $$=\frac{\partial F}{\partial q}\dot{q}+\frac{\partial F}{\partial p}\dot{p}+\frac{\partial F}{\partial t}$$
• $$=\frac{\partial F}{\partial q}\frac{\partial H}{\partial p}-\frac{\partial F}{\partial p}\frac{\partial H}{\partial x}+\frac{\partial F}{\partial t}$$
• $$=\\{F,H\\}+\frac{\partial F}{\partial t}$$
• 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 $\\{F,H\\}=0$
#1: Initial revision by deleted user · 2021-09-08T15:42:27Z (about 1 year ago)
What's the importance of Poisson brackets?
$$F=F(q,p,t)$$
$$\frac{dF}{dt}=\frac{\partial F}{\partial q}\frac{\partial q}{\partial t}+\frac{\partial F}{\partial p}\frac{\partial p}{\partial t}+\frac{\partial F}{\partial t}$$
$$=\frac{\partial F}{\partial q}\dot{q}+\frac{\partial F}{\partial p}\dot{p}+\frac{\partial F}{\partial t}$$
$$=\frac{\partial F}{\partial q}\frac{\partial H}{\partial p}-\frac{\partial F}{\partial p}\frac{\partial H}{\partial x}+\frac{\partial F}{\partial t}$$
$$=\\{F,H\\}+\frac{\partial F}{\partial t}$$

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?