The Poissonian formulation of mechanics tells us that for a generating function $g(q,p,t)$, the Poisson bracket of some function/variable $f(q,p,t)$ with the generating function corresponds with an infinitesimal change in $f$ along the transformation or "motion" generated by $g$.
$$\delta f = \epsilon \left\{f,g \right\}$$
An example of this is momentum conservation due to invariance under infinitesimal translations. To show this, take $f$ to be the Hamiltonian and $g$ to be $\mathbf{p}\cdot\hat{n}$, where $\mathbf{p}$ is the momentum $p_x \hat{x}+p_y\hat{y}+p_z\hat{z}$ and $\hat{n}$ is an arbitrary unit vector. The canonical transformation generated by $\mathbf{p}\cdot \hat{n}$ is an infinitesimal translation along the $\hat{n}$ direction of the system variables with which the Hamiltonian is evaluated.
$$\begin{align*}
\epsilon\left\{H,\mathbf{p}\cdot\hat{n}\right\}&=\epsilon\left(\sum_i \frac{\partial H}{\partial q_i}\frac{\partial\,(\mathbf{p}\cdot\hat{n})}{\partial p_i}-\frac{\partial H}{\partial p_i}\frac{\partial\,(\mathbf{p}\cdot\hat{n})}{\partial q_i}\right)\\
&=\epsilon\left(\sum_i \frac{\partial H}{\partial q_i}(\hat{n})_i\right)\\
&=\epsilon (\nabla_q H)\cdot \hat{n}\\
&\\
&\implies \left\{H,\mathbf{p}\cdot\hat{n}\right\}=(\nabla_q H)\cdot \hat{n}
\end{align*}$$
Now, if we were to take an polar angle $\theta$ about some axis $\hat{n}$ to be a coordinate, the above procedure with $\mathbf{l}$, the angular momentum, in place of $\mathbf{p}$ would then translate as an infinitesimal "translation" of the $\theta$ variable – i.e. a rotation about the $\hat{n}$ axis. An example of this is given in Landau & Lifshitz, Goldstein, and many other mechanics textbooks – the rotation of a constant vector $\mathbf{c}$ about a specified axis.
$$\left\{\mathbf{c},\mathbf{l}\cdot\hat{n}\right\}=\hat{n}\times\mathbf{c}$$
In terms of the interpretation of the Poisson brackets through generating functions (which I just gave), I can see why this would be true. The vector $\mathbf{c}$ changes by an amount $d\theta(\hat{n}\times\mathbf{c})$ when rotated by an infinitesimal angle $d\theta$ about an axis $\hat{n}$, and that result can be reached by simple analytical geometry. However, by direct evaluation of the Poisson bracket, I can't see why this isn't zero (as $\mathbf{c}$ is a constant). The angular momentum operator (vector-valued function in terms of phase space variables) is given by
$$\begin{align*}
\mathbf{l}&=\mathbf{r}\times\mathbf{p}\\
&=(yp_z-zp_y)\hat{x}+(zp_x-xp_z)\hat{y}+(xp_y-yp_x)\hat{z}
\end{align*}$$
Note that this, assuming a typical classical Hamiltonian, entirely in terms of phase space variables. Now, the Poisson bracket of this with a constant vector is
$$\begin{align*}
\left\{\mathbf{c},\mathbf{l}\cdot\hat{n}\right\}&=\sum_i\left(\frac{\partial \mathbf{c}}{\partial q_i}\frac{\partial (\mathbf{l}\cdot\hat{n})}{\partial p_i}-\frac{\partial \mathbf{c}}{\partial p_i}\frac{\partial (\mathbf{l}\cdot\hat{n})}{\partial q_i}\right)\\
&=0\,\,\,\,(\mathbf{c}\textrm{ doesn't depend on phase space variables)}
\end{align*}$$
Please, could you tell me how to resolve this paradox?
P.s: I originally wrote this question extremely briefly because I thought somebody would certainly know what I'm talking about.
Best Answer
The starting point is that the $3$-vector $\vec{\bf c}$ transforms in the $3$-dimensional irreducible vector representation of the rotation group $SO(3)$,
$$\tag{1} \{ \vec{\bf c}, \vec{\bf L}\cdot \hat{\bf n} \}_{PB}~=~ \hat{\bf n}\times \vec{\bf c},$$
where $\hat{\bf n}$ is an arbitrary unit vector, whose Poisson bracket (PB) with anything vanishes
$$\tag{2} \{ \hat{\bf n}, \cdot \}_{PB}~=~0.$$
We assume that $\vec{\bf c}$ is not identically zero. Since the PB with $\vec{\bf c}$ does not vanish, the $3$-vector $\vec{\bf c}$ cannot be a constant. It must be a function of the phase space variables $\vec{\bf r}$ and $\vec{\bf p}$. It can be thought of as being of the form
$$\tag{3} \vec{\bf c}~=~\vec{\bf r}f+ \vec{\bf p}g+ \vec{\bf L}h,$$
where
$$\tag{4} f~=~f(r^2,p^2,\vec{\bf r}\cdot\vec{\bf p},L^2), \quad g~=~g(r^2,p^2,\vec{\bf r}\cdot\vec{\bf p},L^2), \quad\text{and}\quad h~=~h(r^2,p^2,\vec{\bf r}\cdot\vec{\bf p},L^2), $$ are three suitable functions of the phase space $SO(3)$ scalars
$$\tag{5} r^2,\quad p^2,\quad \vec{\bf r}\cdot\vec{\bf p},\quad\text{and}\quad L^2.$$
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