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 unchanged
Hamiltonian 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
Hamiltonian Dynamics Properties of hamiltonian dynamics Volume Preservation (Liouville's theorem) More general conclusion: if z=(q, p) evolves under the Hamiltonian dynamics, and p(z)is a distribution over Z, then dp dt at +之·Vp=0 Proof dt j(pt(e)dz=0(definition of pdf 小9lt9R(02)·dS( Reynolds transport theore +四·(p2))dz( Gauss's theorem) R(t)at Thus, P+V. (pi)=0 for arbitrary dynamics For hamiltonian dynamics v. Z=0,so 0 +卩·(p2) p ap t at 之·Vp+pV at 么V≡a
Hamiltonian Dynamics • Properties of Hamiltonian Dynamics – Volume Preservation (Liouville’stheorem) • More general conclusion: if z = (𝑞, 𝑝) evolves under the Hamiltonian dynamics, and 𝜌𝑡 𝑧 is a distribution over 𝑧, then 𝑑𝜌 𝑑𝑡 ≡ 𝜕𝜌 𝜕𝑡 + 𝑧ሶ ⋅ 𝛻𝜌 = 0 • Proof: 𝑑 �� �� �𝑑� 𝜌𝑡 𝑧 𝑑𝑧 = 0 (definition of pdf.) �� �� = 𝜕𝜌 �� �𝜕�ׯ + �𝑑� �𝜕� 𝜌𝑧ሶ ⋅ 𝑑𝑆 (Reynolds transport theorem) �� �� = 𝜕𝜌 𝜕𝑡 + 𝛻 ⋅ 𝜌𝑧ሶ 𝑑𝑧 (Gauss’s theorem). Thus, 𝜕𝜌 𝜕𝑡 + 𝛻 ⋅ 𝜌𝑧ሶ = 0 for arbitrary dynamics. For Hamiltonian dynamics, 𝛻 ⋅ 𝑧ሶ = 0, so 0 = 𝜕𝜌 𝜕𝑡 + 𝛻 ⋅ 𝜌𝑧ሶ = 𝜕𝜌 𝜕𝑡 + zሶ ⋅ 𝛻𝜌 + 𝜌𝛻 ⋅ 𝑧ሶ = 𝜕𝜌 𝜕𝑡 + zሶ ⋅ 𝛻𝜌 ≡ 𝑑𝜌 𝑑𝑡
Hamiltonian Dynamics Properties of hamiltonian Dynamics Symplecticness(辛性) at H(z), =0axd laxd Bs is the Jacobian of dz d×d 0 d×d the transformation ts. Symplecticness implies BS-lB=J Symplecticness implies volume conservation det(bt)det(-)det(Bs ) det(J-) det(b 2=1
Hamiltonian Dynamics • Properties of Hamiltonian Dynamics – Symplecticness(辛性) • 𝑑𝑧 𝑑𝑡 = 𝐽𝛻𝐻 𝑧 , 𝐽 = 0𝑑×𝑑 𝐼𝑑×𝑑 −𝐼𝑑×𝑑 0𝑑×𝑑 , 𝐵𝑠 is the Jacobian of the transformation 𝑇𝑠 . Symplecticness implies 𝐵𝑠 𝑇 𝐽 −1𝐵𝑠 = 𝐽 −1 , 𝐽 −1 = −𝐽 • Symplecticness implies volume conservation: det 𝐵𝑠 2 = 1
Hamiltonian Dynamics Discretized methods for computing Hamiltonian evolutⅰon Euler's method d n(t+e)=p2()+s()=p() de gi(t+a) )+:()=a( dt Cannot preserve volume Modified euler 's method Pi(t+e)= pi(t) 9(t+e)=9()+2(+) m Volume conversation is guaranteed. ("Shear transformation: update of one variable does not depend on the other, so the Jacobian is triangular
Hamiltonian Dynamics • Discretized methods for computing Hamiltonian evolution – Euler’s method • Cannot preserve volume – Modified Euler’s method • Volume conversation is guaranteed. (“Shear” transformation: update of one variable does not depend on the other, so the Jacobian is triangular.)
Hamiltonian Dynamics Discretized methods for computing Hamiltonian evolution The leapfrog method P(t+/2)=p(t)-(=/2)(q(t) q(t+) q(t)+c22(t+/2) l P2(t+)=p1(t+/2)-(/2)mx(q(t+) Preserves volume(each is a"shear"transformation)
Hamiltonian Dynamics • Discretized methods for computing Hamiltonian evolution – The leapfrog method • Preserves volume (each is a “shear” transformation)