I've recently noticed that the Poisson brackets of the three dimensional angular momentum $$\{L_i,L_j\}$$ in classical mechanics follow the same commutator relations as the standard basis of the Lie algebra $\mathfrak{so}(3)$. This means that these to Lie algebras are isomorphic.

Also $\mathfrak{so}(3)$ is the Lie algebra of the Lie group SO$(3)$, which is the group of three dimensional rotations.

This seems very geometric to me. My question is therefore: Is there a geometric way to interpret Poisson brackets (of angular momenta or in general)?

I've recently noticed that the Poisson brackets of the three dimensional angular momentum $$\{L_i,L_j\}$$ in classical mechanics follow the same commutator relations as the standard basis of the Lie algebra $\mathfrak{so}(3)$. This means that these to Lie algebras are isomorphic.

Also $\mathfrak{so}(3)$ is the Lie algebra of the Lie group ${\rm SO}(3)$, which is the group of three dimensional rotations.

This seems very geometric to me. My question is therefore: Is there a geometric way to interpret Poisson brackets (of angular momenta or in general)?