# Intuition behind Poisson bracket of Arbitrary Function, & dot product of Angular Momentum & Constant Vector being the Cross Product of the Latter Two

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?

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