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In my book, it is written that for any vector $\mathbf{v}$, we have

$$\{\mathbf{v},\mathbf{L}\cdot \mathbf{n}\}=\mathbf{n}\times\mathbf{v}.\tag{1}$$

For me it is absurd... For example, if we take $\mathbf{v}=(\cos(x),0,0)$

the Poisson bracket, say we consider $L_z=xp_y-yp_x$:

$$(\{\mathbf{v},\mathbf{L}\cdot \mathbf{n}\})_x=\frac{\partial v_1}{\partial x}\frac{\partial L_z}{\partial p_x}-\frac{\partial L_z}{\partial x}\frac{\partial v_1}{\partial p_x}+\frac{\partial v_1}{\partial y}...=-\sin(x)x.$$ I can see that it already fails sincer the left-hand side is $\mathbf{n}\times\mathbf{v}$ can not contain $\sin(x)$...

Or maybe I misunderstood the proposition?

Screenshot of the book's content:

enter image description here

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    $\begingroup$ could you cite the book? $\endgroup$ – Fabio Di Nocera Jul 16 '19 at 16:37
  • $\begingroup$ Poisson brackets are normally written with curly braces, not square ones. $\endgroup$ – G. Smith Jul 17 '19 at 0:43
  • $\begingroup$ @G.Smith isn't curly brackets anticommutator? $\{a,b\}=ab+ba$. $\endgroup$ – Upc Jul 18 '19 at 0:17
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    $\begingroup$ @Upc Can you still cite the book? A screenshot is not a replacement for a citation. It is more like a replacement for a quotation. $\endgroup$ – Dvij Mankad Jul 18 '19 at 0:31
  • $\begingroup$ @Upc en.wikipedia.org/wiki/Poisson_bracket $\endgroup$ – G. Smith Jul 18 '19 at 2:09
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  1. Well, there are different notions of vectors in mathematics and physics.

  2. The notion of vectors ${\bf v}$ that seems to be relevant here are maps $$\{\text{phase space}\}\times\{\text{time axis}\}\quad\stackrel{\bf v}{\longrightarrow}\quad \mathbb{R}^3$$ that transform in the vector representation of the 3D rotation group $SO(3)$. Here the angular momentum $L_x$, $L_y$, and $L_z$ belong to the corresponding $so(3)$ Lie algebra. OP's eq. (1) is the characteristic property of such vectors ${\bf v}$. [Here ${\bf n}$ is a so-called rotation vector, although it is not a vector in above sense].

  3. Similarly, scalars $s$ in this context are maps $$\{\text{phase space}\}\times\{\text{time axis}\}\quad\stackrel{s}{\longrightarrow}\quad \mathbb{R}$$ that transform in the trivial representation of $SO(3)$.

  4. OP's example does not fulfill eq. (1) and is hence not a vector in the sense of point 2.

  5. Similarly, a constant 3-tuple ${\bf v}=(1,-5,2)$ is not a vector in the sense of point 2 because it stays constant rather than rotate under 3D rotations.

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