= Solution
The <Hamiltonian flow> of $H$ solves
$$
\frac d{dt}\phi_H^t(p)=X_H(\phi_H^t(p)),\qquad\phi_H^0(p)=p,
\qquad\iota_{X_H}\omega=dH.
$$
Compactness and the absence of a boundary make this <smooth> <vector field> complete, so the flow exists for all real $t$. The defining equation and <Cartan's magic formula> give
$$
\mathcal L_{X_H}\omega=d\iota_{X_H}\omega+\iota_{X_H}d\omega=d^2H=0.
$$
Therefore the <pullback of a differential form> differentiation rule yields
$$
\frac d{dt}(\phi_H^t)^*\omega=(\phi_H^t)^*(\mathcal L_{X_H}\omega)=0,
$$
so $(\phi_H^t)^*\omega=\omega$. Pullback commutes with the <wedge product of differential forms>, and consequently
$$
\boxed{(\phi_H^t)^*\left(\frac{\omega^n}{n!}\right)=\frac{\omega^n}{n!}.}
$$
Thus the flow preserves the <symplectic volume>, in fact the entire <symplectic form>.
Back to article page
