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In my introductory mechanics class, we were given a home assignment to calculate the following Poisson bracket.

In $\mathbb{R}^3$, let $\vec{r}$ and $\vec{p}$ be the generalized coordinate and the conjugate momentum, respectively. Now, let $\vec{F}$ be a smooth function in the form $$\vec{F}=f_1\,\vec{r}+f_2\,\vec{p}+f_3\,\vec{r}\times\vec{p}$$ where $f_i$ are scalar functions that only depend on $\vec{r}^2$, $\vec{p}^2$ and $\vec{r}\cdot\vec{p}$.

Denote by curly brackets $\{\cdot,\cdot\}$ the canonical Poisson bracket, i.e., $\{r_i,p_j\}=\delta_{ij}$ (Kronecker delta).

Also, denote the angular momentum by $\vec{M}=\vec{r}\times\vec{p}$, and let $\vec{n}$ be some constant vector.

Now, calculate the following: $$\{\vec{F},\vec{M}\cdot\vec{n}\}$$ Calculating this explicitly (using the Levi-Civita symbol), we get $\vec{n}\times\vec{F}$.

My question is: What is the intuition behind the identity $$\{\vec{F},\vec{M}\cdot\vec{n}\}=\vec{n}\times\vec{F}\,\,?$$

Or, rather, a more general question: What is the (geometric) intuition behind Poisson brackets?

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The intuition behind OP's identity is that the Hamiltonian vector field $X_{F}=\{F ,\cdot\}$ generators (infinitesimal) transformations dual to the generator $F$.

Examples:

  1. The momentum $F=\vec{p}\cdot\vec{n}$ along the direction $\vec{n}$ generates translations along the direction $\vec{n}$.

  2. The angular momentum $F=\vec{M}\cdot\vec{n}$ along the direction $\vec{n}$ generates rotations around the direction $\vec{n}$.

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