Variational equation ($\dot{Y}=Db(t,X(t))Y$)

Differentiating a smooth characteristic flow map in its initial point gives this linear equation, with $Y(s)=I$. The Liouville formula for a fundamental matrix then gives $\det Y(t)=\exp ({\int }_s^t\mathrm{div}b(r,X(r))dr)$. Incompressible Hamiltonian flows preserve Lebesgue measure because this divergence vanishes.