Let $U,V\subset\mathbb{R}^2$ be two bounded opens of same area, wrt to the standard symplectic form $\omega_0$. If there's a diffeomorphism $\phi:U\to V$ preserving orientation, show that there exists a symplectomorphim form $(U,\omega_0)$ to $(V,\omega_0)$.
My thoughts: Try to apply Moser trick. Set $\omega_1=\phi^*w_0$ and $\omega_t=t\omega_1+(1-t)\omega_0$ on $U$. Then $\omega_t$ is symplectic. As $\omega_0$ is exact on $\mathbb{R}^2$, so is $\omega_t$ on $U$. Let $d\eta=\omega_0-\omega_1$. We solve $i_{X_t}\omega_t=\eta$. But $X_t$ may not be complete, so its flow may not exist up to time $1$.