The Moser's argument is introduced in my lecture in the following way
Try to find $\varphi_t$, s.t. $\varphi_0=id, \varphi_t^{*} \alpha_0=\alpha_t$. \ If such a family $\varphi_t$ exists, then we denote by $X_t$ the generating vector field, i.e. $X_t\circ \varphi_t=\dfrac{d \varphi_t}{dt}$. Then \begin{align*} \dfrac{d \alpha_t}{dt}|_{t=t_0}=\dfrac{d}{dt}|_{t=t_0}(\varphi^{*}_t \alpha_0) =L_{X_{t_0}}(\varphi^{*}_{t_0} \alpha_0)=L_{X_{t_0}}(\alpha_{t_0})=d(\iota_{X_{t_0}} \alpha_{t_0})+ \underbrace{\iota_{X_{t_0}} d \alpha_{t_0}}_0 \end{align*} So we see that a necessary condition for the existence of the family $\varphi_t$ is that $\overset{.}{\alpha_t}$ should be exact at all $t \in [0,1]$. Sometimes this condition is also sufficient.
Mosers theorem
$M$ closed, connected, (oriented). $\Omega_0, \Omega_1$ volume forms on $M$ with $\int_M \Omega_0= \int_M \Omega_1$. Then there exists an isotopy $\varphi_t:M \rightarrow M$, s.t. $\varphi^{*}_1 \Omega_0=\Omega_1$.
Proof: We connect $ \Omega_0$ and $\Omega_1$ by the family $\Omega_t=(1-t) \Omega_0+ t \Omega$. Since $\Omega_0$ and $\Omega_1$ give the same orientation on $M$, these are non-degenerate volume forms. Note that $\overset{.}{\Omega}_t=\Omega_1-\Omega_0$ is independent of $t$. By assumption we know \begin{align*} \int_M \Omega_1-\Omega_0=0 \end{align*} so by de Rham's theorem \begin{align*} \exists \beta \in \Omega^{dim \ M -1} (M) \text{ s.t. } d \beta =\Omega_1-\Omega_0 \end{align*} According to Moser's ansatz, we need to solve \begin{align*} d \beta= d(\iota_{X_t} \alpha_t) \text{ for all } t \in [0,1] \end{align*} which we can do by solving \begin{align*} \beta= \iota_{X_t} \Omega_t \end{align*} Because all the $\Omega_t$ are nondegenerate, this equation determines a uniwue family of vector fields $X_t$. Since $M$ is closed, this family integrates to a family of diffeomorphisms with \begin{align*} \overset{.}{\varphi_t}=X_t \circ \varphi_t , \varphi_0=id \end{align*} Moser's calculation shows that \begin{align*} \varphi^{*}_t \Omega_0=\Omega_t \end{align*} In particular, the family $\varphi_t$ is our required isotopy. \
I don't see what the argument is here and why in the last part of the theorem the claim should follow. How can I conclude from
$\overset{.}{\Omega}= d(\iota_{X_t} \Omega_t)$ that $\varphi^{*}_t \Omega_0=\Omega_t$, it really doesn't make any sense for me.
Can somebody explain this?