# What is the analogue for symplectic structure in case of spin variables?

The phase space for spin is the two-sphere $S^2$ with the symplectic form being the area 2-form $$ \omega= J \sin\theta d\theta\wedge d\phi. $$ Here $\theta$ and $\phi$ are the polar angles. Then, with $$ S_x= J \sin\theta \cos\phi,\\ S_y= J \sin\theta \sin\phi,\\ S_z= J \cos\theta, $$ we have $\{S_x,S_y\}= S_z$ etc.

Angular momentum operators $\hat{J}_a$ satisfy an $so(3)$ Lie algebra $$ [\hat{J}_a,\hat{J}_b]~=~i\hbar \epsilon_{abc} \hat{J}_c,\qquad a,b,c~\in~\{1,2,3\},\tag{C}$$ which at the classical level is a Poisson algebra $$ \{J_a,J_b\}~=~ \epsilon_{abc} J_c,\qquad a,b,c~\in~\{1,2,3\}.\tag{P}$$ However, the Poisson structure (P) on $\mathbb{R}^3$ is not invertible/non-degenerate, so it is technically not a symplectic structure. But $\mathbb{R}^3$ equipped with (P) is a discrete union of symplectic leaves (namely concentric 2-spheres and the origin $\{0\}$).