0
votes
1answer
74 views

What are the properties of this Poisson algebra?

I have the following (real) quantities (which are from a Classical Mechanics problem): $$A_1=\frac 1 4(x^2 +p_x^2-y^2-p_y^2 ) \quad A_2=\frac 1 2(x y +p_x p_y)$$ $$A_3=\frac 1 2(x p_y - y p_x )$$ ...
0
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0answers
165 views

Short proof of Jacobi identity for the Poisson bracket — is this valid?

I've been trying to make a short proof for the Jacobi identity for the Poisson bracket on phase space. My idea goes like this, we know the following: $$ \frac{\mathrm{d}}{\mathrm{dt}} \{f,g\} = ...
4
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0answers
94 views

The Nambu bracket

Does anybody know how to show the Jacobi identity for the Nambu bracket in $\mathbb{R}^3$? The Nambu bracket with respect to $c \in \mathcal{F}(\mathbb{R}^3)$ is defined as $$\{F,G\}_c = \langle\nabla ...