Solution
= Solution
The supplied Jacobian identity and zero divergence give $J^{-1}\dot J=0$, so $J(t)=1$. Change of variables then shows $\operatorname{vol}\Phi_t(U)=\operatorname{vol}U$. This is <Liouville theorem in Hamiltonian mechanics>.
= Solution
The supplied Jacobian identity and zero divergence give $J^{-1}\dot J=0$, so $J(t)=1$. Change of variables then shows $\operatorname{vol}\Phi_t(U)=\operatorname{vol}U$. This is <Liouville theorem in Hamiltonian mechanics>.