Hamiltonian Dynamics Properties of hamiltonian Dynamics Volume Preservation(Liouville's theorem) Alternative proof: compute the Jacobian Bs of the infinitesimal transformation to (a,p)= +6 aH 02H 02H 1+6 +6 aH SO Bs= asH 8-H 02H det(Bs) 1+d 1, so it keeps the infinitesimal volume unchangedHamiltonian Dynamics • Properties of Hamiltonian Dynamics – Volume Preservation (Liouville’s theorem) • Alternative proof: compute the Jacobian 𝐵𝛿 of the infinitesimal transformation 𝑇𝛿 𝑞, 𝑝 = 𝑞 𝑝 + 𝛿 𝑞ሶ 𝑝ሶ = 𝑞 𝑝 + 𝛿 𝜕𝐻 𝜕𝑝 − 𝜕𝐻 𝜕𝑞 , so =1, so it keeps the infinitesimal volume unchanged