- #1
Emspak
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Proving Some Poisson Bracket identities -- a notational question
I need some help just understanding notation, and while this might count as elementary it has to do with Hamiltonians and Lagrangians, so I posted this here.
Prove the following properties of Poisson's bracket:
[A,A] = 0 and [A,B] = [-B,A]
Now honestly this looks a lot like matrix math to me, and I know it's related. But I a jut trying to make sure I get what it is that [A,A] wants me to do – add the two quantities? Multiply them? And understand what it is I am seeing when I see that notation.
So I know that Poisson's bracket is:
[tex]
\sum_i \left(\frac{\partial f}{\partial q_i} \frac{\partial \mathcal H}{\partial p_i} - \frac{\partial f}{\partial p_i} \frac{\partial \mathcal H}{\partial q_i} \right)
[/tex]
and the poisson bracket would be [itex][f, \mathcal H][/itex]. So does that mean the above question would be, prove that
[tex]
\sum_i \left(\frac{\partial A}{\partial q_i} \frac{\partial A}{\partial p_i} - \frac{\partial A}{\partial p_i} \frac{\partial A}{\partial q_i} \right) = 0
[/tex]
and
[tex]
\sum_i \left(\frac{\partial A}{\partial q_i} \frac{\partial B}{\partial p_i} - \frac{\partial B}{\partial p_i} \frac{\partial A}{\partial q_i} \right) = \sum_i -\left(\frac{\partial B}{\partial p_i} \frac{\partial A}{\partial q_i} + \frac{\partial B}{\partial p_i} \frac{\partial A}{\partial q_i} \right)
[/tex]
Am I reading this correctly? Maybe this sounds terribly elementary but notation I often find confusing rather than illuminating.
Anyhow, if someone could let me know if I have read this right that would go a long way.
I need some help just understanding notation, and while this might count as elementary it has to do with Hamiltonians and Lagrangians, so I posted this here.
Homework Statement
Prove the following properties of Poisson's bracket:
[A,A] = 0 and [A,B] = [-B,A]
Now honestly this looks a lot like matrix math to me, and I know it's related. But I a jut trying to make sure I get what it is that [A,A] wants me to do – add the two quantities? Multiply them? And understand what it is I am seeing when I see that notation.
So I know that Poisson's bracket is:
[tex]
\sum_i \left(\frac{\partial f}{\partial q_i} \frac{\partial \mathcal H}{\partial p_i} - \frac{\partial f}{\partial p_i} \frac{\partial \mathcal H}{\partial q_i} \right)
[/tex]
and the poisson bracket would be [itex][f, \mathcal H][/itex]. So does that mean the above question would be, prove that
[tex]
\sum_i \left(\frac{\partial A}{\partial q_i} \frac{\partial A}{\partial p_i} - \frac{\partial A}{\partial p_i} \frac{\partial A}{\partial q_i} \right) = 0
[/tex]
and
[tex]
\sum_i \left(\frac{\partial A}{\partial q_i} \frac{\partial B}{\partial p_i} - \frac{\partial B}{\partial p_i} \frac{\partial A}{\partial q_i} \right) = \sum_i -\left(\frac{\partial B}{\partial p_i} \frac{\partial A}{\partial q_i} + \frac{\partial B}{\partial p_i} \frac{\partial A}{\partial q_i} \right)
[/tex]
Am I reading this correctly? Maybe this sounds terribly elementary but notation I often find confusing rather than illuminating.
Anyhow, if someone could let me know if I have read this right that would go a long way.