Page - 413 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 425 onthesymplectic spaceT∗FM. Letting πˆdenote theprojectiononthebundleT∗FM→FM, (8)gives H(z)= 1 2 〈gFM(z)|gFM(z)〉= 12‖z◦hπˆ(z)◦ πˆ(z)‖ 2 (Rd)∗= 1 2 d ∑ i=1 ξ(Hi(u))2. Normalgeodesics insub-Riemannianmanifoldssatisfy theHamilton–Jacobiequations [23]with Hamiltonianflow z˙t=XH=Ω#dH(z) (13) whereΩhere is thecanonical symplectic formonT∗FM (e.g., [25]).Wedenote (13) theMPPequations, andweletprojectionsxt=πT∗FM(zt)ofminimizingcurvessatisfying(13)bedenotednormalMPPs. Thesystem(13)has2(d+d2)degreesof freedom, incontrast to theusual2ddegreesof freedomforthe classicalgeodesicequation.Of these,d2 describes thecurrent frameat time t,while theremainingd2 allowsthecurve to“twist”whilestillbeinghorizontal.Wewill see thiseffectvisualized inSection6. Ina local canonical trivializationz=(u,ξ), (13)gives theHamilton–Jacobiequations u˙= ∂ξH(u,ξ)= gFM(u,ξ)= hu ( u(ξ(H1(u)), . . . ,ξ(Hd(u)))T ) ξ˙=−∂uH(u,ξ)=−∂u12‖ξ◦hu◦u‖ 2 (Rd)∗=−∂u 1 2 d ∑ i=1 ξ(Hi(u))2. (14) Using(3),wehavefor thesecondequation ξ˙=− d ∑ i=1 ξ(Hi(u))ξ(∂uhu(uei)) =− d ∑ i=1 ξ(Hi(u))ξ ( ψ(u,Ru(uei,π∗(∂u)))+∂hu(uei)ψ ( u,ψ−1(C(∂u)) ) +∂hu(uei)hu(π∗(∂u)) ) =−ξ(ψ(u,Ru(π∗(u˙),π∗(∂u)))+∂u˙ψ(u,ψ−1(C(∂u)))+∂u˙hu(π∗(∂u))). (15) Here∂u˙denotesu-derivative in thedirection u˙, equivalently∂u˙hu(v)= ∂t(hu)(v).While thefirst equationof (14) involvesonly thehorizontalpartofξ, thesecondequationcouples theverticalpartof ξ throughtheevaluationofξonthe termψ(u,Ru(π∗(u˙),π∗(∂u)). If theconnection iscurvature-free, whichinnon-flatcasesimpliesthat itcarriestorsion, thisvertical termvanishes.Conversely,whenM is Riemannian,C thegRLevi–Civitaconnection,andu0 isgRorthonormal,gFM(hu(v),hu(w))= gR(v,w) forallv,w∈Tπ(ut)M. In thiscase,anormalMPPπ(ut)willbeaRiemanniangRgeodesic. 4.2. Evolution inCoordinates Incoordinatesu=(xi,uiα,ξi,ξiα) forT ∗FM,wecanequivalentlywrite x˙i= gijξj+g ijβξjβ =W ijξj−WihΓjβh ξjβ X˙iα= g iαjξj+g iαjβξjβ =−ΓiαhWhjξj+ΓiαkWkhΓ jβ h ξjβ ξ˙i=−12 ( ∂yig hk y ξhξk+∂yig hkδ y ξhξkδ+∂yig hγk y ξhγξk+∂yig hγkδ y ξhγξkδ ) ξ˙iα =− 1 2 ( ∂yiαg hk y ξhξk+∂yiαg hkδ y ξhξkδ+∂yiαg hγk y ξhγξk+∂yiαg hγkδ y ξhγξkδ ) withΓhγk,i for∂yiΓ hγ k , andwhere ∂ylg ij=0 , ∂ylg ijβ =−WihΓjβh,l , ∂ylgiαj=−Γiαh,lWhj , ∂ylgiαjβ =Γiαk,lWkhΓ jβ h +Γ iα kW khΓ jβ h,l , 413