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edit approved Jun 25, 2013 at 11:36
agt
edit approved Jun 25, 2013 at 11:36
agt
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I'm trying to understand how this solution works. The question was to deduce momentum maps for right and left actions of $SO(3)$ on $GL(3,R)$, and got them as $J_R = \frac{1}{2}(Q^TP-P^TQ)$ and $J_L=\frac{1}{2}(PQ^T-QP^T)$ respectively, for $Q\in GL(3)$, $P\in GL(3)^*$. However, then I need to find if they Poisson commute. The solution is:

$ \{ J_L,J_R \} = \frac{1}{4}\{(PQ^T-QP^T),(Q^TP-P^TQ)\} = \frac{1}{4}((P-P^T)(Q-Q^T)-(Q^T-Q)(P-P^T)) $$$ \{ J_L,J_R \} = \frac{1}{4}\{(PQ^T-QP^T),(Q^TP-P^TQ)\} = \frac{1}{4}((P-P^T)(Q-Q^T)-(Q^T-Q)(P-P^T)) $$

But I don't understand how this works (specifically, how to compute a poisson bracket with matrices). There is no explicit definition of the poisson bracket in the solution. Elsewhere in the text there was a Poisson bracket defined as

$\{F,H\}=tr(\frac{\partial F}{\partial Q}\frac{\partial H}{\partial P}-\frac{\partial H}{\partial Q}\frac{\partial F}{\partial P})$$$\{F,H\}=\operatorname{ tr}\left(\frac{\partial F}{\partial Q}\frac{\partial H}{\partial P}-\frac{\partial H}{\partial Q}\frac{\partial F}{\partial P}\right)$$

but that is for scalar functions $F,H$. I guess you could apply that formula to every entry of the $P,Q$'s but there must be a shortcut. Sorry if i've not given enough context, please ask. Thanks for any help.

I'm trying to understand how this solution works. The question was to deduce momentum maps for right and left actions of $SO(3)$ on $GL(3,R)$, and got them as $J_R = \frac{1}{2}(Q^TP-P^TQ)$ and $J_L=\frac{1}{2}(PQ^T-QP^T)$ respectively, for $Q\in GL(3)$, $P\in GL(3)^*$. However, then I need to find if they Poisson commute. The solution is:

$ \{ J_L,J_R \} = \frac{1}{4}\{(PQ^T-QP^T),(Q^TP-P^TQ)\} = \frac{1}{4}((P-P^T)(Q-Q^T)-(Q^T-Q)(P-P^T)) $

But I don't understand how this works (specifically, how to compute a poisson bracket with matrices). There is no explicit definition of the poisson bracket in the solution. Elsewhere in the text there was a Poisson bracket defined as

$\{F,H\}=tr(\frac{\partial F}{\partial Q}\frac{\partial H}{\partial P}-\frac{\partial H}{\partial Q}\frac{\partial F}{\partial P})$

but that is for scalar functions $F,H$. I guess you could apply that formula to every entry of the $P,Q$'s but there must be a shortcut. Sorry if i've not given enough context, please ask. Thanks for any help.

I'm trying to understand how this solution works. The question was to deduce momentum maps for right and left actions of $SO(3)$ on $GL(3,R)$, and got them as $J_R = \frac{1}{2}(Q^TP-P^TQ)$ and $J_L=\frac{1}{2}(PQ^T-QP^T)$ respectively, for $Q\in GL(3)$, $P\in GL(3)^*$. However, then I need to find if they Poisson commute. The solution is:

$$ \{ J_L,J_R \} = \frac{1}{4}\{(PQ^T-QP^T),(Q^TP-P^TQ)\} = \frac{1}{4}((P-P^T)(Q-Q^T)-(Q^T-Q)(P-P^T)) $$

But I don't understand how this works (specifically, how to compute a poisson bracket with matrices). There is no explicit definition of the poisson bracket in the solution. Elsewhere in the text there was a Poisson bracket defined as

$$\{F,H\}=\operatorname{ tr}\left(\frac{\partial F}{\partial Q}\frac{\partial H}{\partial P}-\frac{\partial H}{\partial Q}\frac{\partial F}{\partial P}\right)$$

but that is for scalar functions $F,H$. I guess you could apply that formula to every entry of the $P,Q$'s but there must be a shortcut. Sorry if i've not given enough context, please ask. Thanks for any help.

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em12
em12
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Momentum maps and matrix poisson brackets.

I'm trying to understand how this solution works. The question was to deduce momentum maps for right and left actions of $SO(3)$ on $GL(3,R)$, and got them as $J_R = \frac{1}{2}(Q^TP-P^TQ)$ and $J_L=\frac{1}{2}(PQ^T-QP^T)$ respectively, for $Q\in GL(3)$, $P\in GL(3)^*$. However, then I need to find if they Poisson commute. The solution is:

$ \{ J_L,J_R \} = \frac{1}{4}\{(PQ^T-QP^T),(Q^TP-P^TQ)\} = \frac{1}{4}((P-P^T)(Q-Q^T)-(Q^T-Q)(P-P^T)) $

But I don't understand how this works (specifically, how to compute a poisson bracket with matrices). There is no explicit definition of the poisson bracket in the solution. Elsewhere in the text there was a Poisson bracket defined as

$\{F,H\}=tr(\frac{\partial F}{\partial Q}\frac{\partial H}{\partial P}-\frac{\partial H}{\partial Q}\frac{\partial F}{\partial P})$

but that is for scalar functions $F,H$. I guess you could apply that formula to every entry of the $P,Q$'s but there must be a shortcut. Sorry if i've not given enough context, please ask. Thanks for any help.