This is a theorem of Moser:
Assume $\omega_0, \omega_1$ are two volume forms (with the same total
mass) on
a compact manifold. Then there is a diffeomorphism $\phi$ on $M$
so that $\phi^*\omega_1=\omega_0$.
Proof:
Let
$$
\omega_s=\omega_0+s(\omega_1-\omega_0).
$$
Since $\omega_0$ and $\omega_1$ has the same total mass, they are
in the same cohomology class.
So there is an $n-1$ form $\eta$ so that
$$
\omega_1-\omega_0=d\eta.
$$
Observe, (this is most easily seen when writing all the forms
in a local coordinate system $(x_1,...,x_n)$,
there is a unique vector field $X_s$ so that
$$
\iota_{X_s}\omega_s=-\eta.
$$
Let $\phi_s$ is the one parameter group of diffeomorphism that is
generated by $X_s$.
Compute, at time $s=t$,
$$
\begin{aligned}
\frac{d}{ds}(\phi_s^*\omega_s)\Big|_{s=t}=&L_{X_t}(\phi_t^*\omega_t)+
\phi_t^*(\omega_1-\omega_0)\\
=&d\iota_{X_t}(\phi_t^*\omega_t)+\iota_{X_t}d(\phi_t^*\omega_t)+
\phi_t^*(\omega_1-\omega_0)\\
=&d\phi_t^*(-\eta)+\phi_t^*(d\eta)\\
=&0.
\end{aligned}
$$
Here notice, $(\phi_t)_*X_t=X_t$ as a vector field.
Thus
$$
\phi_1^*\omega_1=\phi_0^*\omega_0=\omega_0.
$$