Consider the standard two dimensional torus $M=\mathbb{R}^2 / \Gamma$ where $$ \Gamma:=\{(n,m)\in \mathbb{R}^2:n,m\in \mathbb{Z}\}. $$ I want to
$1)$ construct a symplectic form $\omega$ on $M$, and
$2)$ describe the image of $$C^{\infty}(M) \to Ham(M), \ H \mapsto X_H.$$
My feeling for $1)$ is that it should be along the lines of $$\omega=d\theta\wedge d\varphi$$ where $\theta, \varphi$ are the angle co-ordinates. But I cannot make this precise. Please help with this and $2)$!