# 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 $$\{0\}$$).