i was looking for a symplectic structure on the $S^2 $. Originally i considered the Poisson-Structure of a rigid body, which was given by $\{F,G\}=\langle \Pi, \nabla F \times \nabla G \rangle$, for $F,G\in C^\infty(\mathbb{R}^3,\mathbb{R})$ . It turns out, that the sphere $S^2$ is invariant w.r.t. this structure. At the moment i´m looking at functions on the sphere which i´ve multiplied with a bump-function $\varphi(r)$ to extend the functions on the sphere in to the $\mathbb{R}^3$. In order to use the Poisson-structure i have to calculate $\nabla F$ and $\nabla G$ w.r.t. the Cartesian coordinates. Therefore i used the spherical coordinates. $\Pi_1=r sin(\theta) cos(\eta), \Pi_2=r sin\theta) sin(\eta), \Pi_3=r cos(\eta)$ and with the Jacobian J i´m able to calculate the gradient in Cartesian coordinates by
$ \nabla F=\left(\begin{smallmatrix} \frac{\partial F}{\partial \Pi_1}\\ \frac{\partial F}{\partial \Pi_2}\\ \frac{\partial F}{\partial \Pi_3}\end{smallmatrix} \right)=(J^{-1}(r,\theta,\eta))^T\left(\begin{smallmatrix} \frac{\partial F}{\partial r}\\ \frac{\partial F}{\partial \theta}\\ \frac{\partial F}{\partial \eta}\end{smallmatrix} \right)=\left(\begin{smallmatrix} \frac{\partial F}{\partial r}\sin(theta)\cos(\eta) + \frac{\partial F}{\partial\theta}\frac{1}{r}\cos(\theta) \cos(\eta)- \frac{\partial F}{\partial \eta}\frac{\sin(\eta)}{r\sin(theta)}\\ \frac{\partial F}{\partial r}\sin(\theta)\sin(\eta) + \frac{\partial F}{\partial \theta}\frac{1}{r}\cos(\theta) \sin(\eta) + \frac{\partial F}{\partial\eta}\frac{\cos(\eta)}{r\sin(theta)}\\ \frac{\partial F}{\partial r}\cos(\theta) -\frac{1}{r}\frac{\partial F}{\partial \theta}\sin(theta) \end{smallmatrix} \right)$
Now we put $\nabla F $ and $\nabla G$ into the Poisson-structure. $$\{F,G\}=-\langle \Pi, \nabla F \times \nabla G \rangle =$$ $$ -\biggr\langle \left( \begin{smallmatrix} r sin(\theta) cos(\eta)\\ r sin(\theta)sin(\eta) \\ r cos(\theta) \end{smallmatrix} \right), \left(\begin{smallmatrix} \frac{\partial F}{\partial r} sin(\theta) cos(\eta) + \frac{\partial F}{\partial \theta}\frac{1}{r}cos(\theta)cos(\eta)- \frac{\partial F}{\partial \eta}\frac{sin(\eta)}{r sin(\theta)}\\ \frac{\partial F}{\partial r} sin(\theta) sin(\eta) + \frac{\partial F}{\partial \theta}\frac{1}{r}cos(\theta)sin(\eta)+ \frac{\partial F}{\partial \eta}\frac{cos(\eta)}{r sin(\theta)}\\ \frac{\partial F}{\partial r}cos(\theta) -\frac{1}{r}\frac{\partial F}{\partial \theta} sin(\theta) \end{smallmatrix} \right) \times \left(\begin{smallmatrix} \frac{\partial G}{\partial r} sin(\theta) cos(\eta) + \frac{\partial G}{\partial \theta}\frac{1}{r}cos(\theta)cos(\eta)- \frac{\partial G}{\partial \eta}\frac{sin(\eta)}{r sin(\theta)}\\ \frac{\partial G}{\partial r} sin(\theta) sin(\eta) + \frac{\partial G}{\partial \theta}\frac{1}{r}cos(\theta) sin(\eta) + \frac{\partial G}{\partial \eta}\frac{cos(\eta)}{r sin(\theta)}\\ \frac{\partial G}{\partial r}cos(\theta) -\frac{1}{r}\frac{\partial G}{\partial \theta} sin(\theta) \end{smallmatrix} \right) \biggr\rangle $$
My solution is the following: $$\frac{1}{r\sin(\theta)}[\frac{\partial G}{\partial \theta} \frac{\partial F}{\partial \eta}-\frac{\partial F}{\partial \theta} \frac{\partial G}{\partial \eta}]$$ I checked it with Matlab but i expected a symplectic structure and my structure is not defined at $\theta=0$ or $\theta = \pi$. Are there any mistakes in my ansatz? Thanks for help