 Moser's trick

Let $M$ be compact and let ${\omega }_t$ be a smooth family of symplectic forms with constant de Rham cohomology class. Choose ${\sigma }_t$ smoothly so that ${\dot{\omega }}_t=d{\sigma }_t$, solveand let ${\varphi }_t$ be the flow of $X_t$. Thenso ${\varphi }_t^{\ast }{\omega }_t={\omega }_0$.