Hamiltonian dynamics Properties of hamiltonian Dynamics Volume Preservation(Liouville' s theorem ): the volume V of a region R in the phase space((g, p) space)is preserved under the transformation Ts Proof: letz=(q,p),V=J R(tq么 z·dS= dt ar(t) Rt(. i)dz, 卩·z=卩.(q,p) .+妞-∑两/∑Dnm1 a ah a 0H 0 i=1 dgi opi api dgiHamiltonian Dynamics • Properties of Hamiltonian Dynamics – Volume Preservation (Liouville’s theorem): the volume 𝑉 of a region 𝑅 in the phase space ((𝑞, 𝑝) space) is preserved under the transformation 𝑇𝑠 • Proof: let 𝑧 = (𝑞, 𝑝), 𝑉 = ׬�� �� 𝑑𝑧, 𝑑𝑉 𝑑𝑡 �� �𝜕�ׯ = 𝑧ሶ ⋅ 𝑑𝑆 = �� ��׬ 𝛻 ⋅ 𝑧ሶ 𝑑𝑧, 𝛻 ⋅ 𝑧ሶ = 𝛻 ⋅ 𝑞ሶ, 𝑝ሶ =