I was reading Poisson structure on $\mathbb{R}^{n}$ , here is the

Proposition - "Let {.,.} be a Poisson Structure on $\mathbb{R}^{n}$ .then for any $f,g \in C^{\infty} (\mathbb{R}^{n},\mathbb{R})$" the following relation holds - $\{f,g\} = \sum_{i,j = 1}^{n} \{x_{i},x_{j}\} \frac{\partial{f}}{\partial{x_{i}}}\frac{\partial{g}}{\partial{x_{j}}}$ .

Actually after this i would construct a matrix with entries $\{x_{i},x_{j}\}$ i got that from the properties of Poisson bracket that the diagonal entries of this matrix will be $0$ and that it will be a skew symmetric matrix also for the remaining entries I would compute $\{x_{i},x_{j}\}$ , how do i do that, any approach?

I have read the properties of Poisson brackets like Skew symmetric , Jacobi identity , Leibniz rule.I think here $x_{i},x_{j}$ are generalized coordinates.

I am new to this and perhaps this can be a silly question but still i am unable to get a reference and even tried in chats too.

If anyone could cite a reference that would be extremely helpful! Thank you!

  • $\begingroup$ See the answers and links given in the comments here. $\endgroup$ – Dietrich Burde Mar 7 '17 at 12:55
  • $\begingroup$ I'm not sure this is relevant to the control tag, so I'm editing the post to remove it. $\endgroup$ – SZN Mar 17 '17 at 5:08

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