Page - 90 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 386 Tostudythegeodesic trajectoriesof thegroup,weconsider theLagrangianfromthetotalkinetic energy(aquadratic formonspeeds). Itmaytherefore inparticularbewritten in the leftalgebra“left”, with thescalarproductassociatedwith themetric. EL= 1 2 〈nL,nL〉= 12Tr [ nTLnL ] (200) Ifweconsiderasscalarproduct: 〈., .〉 : g∗×g→R k,n → 〈k,n〉=Tr(kTn) (201) andleftalgebra: nL= ⎡⎣ R−1/2 .R1/2 R−1/2 .m 0 0 ⎤⎦ (202) weobtain for the totalkineticenergy EL= 1 2 ( Tr ( R−1 . R ) + . mTR−1 .m ) (203) Wewill then introduce the coadjoint operator that will enable us towork on the elements of the dual algebra of the Lie algebra defined above. Like algebra,which is physically the space of instantaneous speeds, the dual algebra is the space ofmoments. For the dual of left algebra, themoment isgivenby: ΠL= ∂EL ∂nL =nL (204) WhereEL is thekineticenergyof thesystemandiscurrentlyassociatedwithΠL isanelementof the leftalgebra. Themomentspace is thedualalgebra,denotedg∗, associatedwith theLiealgebrag. Thisvalue isdeducedfromthecomputation:〈 ∂EL ∂nL ,δU 〉 =Lim ε→0 EL(nL+ε ·δU)−EL(nL) ε withEL(nL+ε ·δU)= 12 〈nL+ε.δU,nL+ε ·δU〉= 1 2 (nL+ε ·δU)T (nL+ε ·δU)〈 ∂EL ∂nL ,δU 〉 =2 · 1 2 tr ( ηTLδU ) = 〈nL,δU〉⇒ ∂EL∂nL =nL (205) Thenthemomentmapisgivenby: αM : g→ g∗ nL →ΠL=ηL (206) We can observe that the application that turns left algebra into dual algebra is the identity applicationbut,physically, thefirstaremomentsandthesecondsare instantaneousspeeds. WecanalsodefinethemomentΠR associatedto therightalgebraηRby: 〈ΠL,nL〉= 〈 ΠL,M−1nRM 〉 = 〈ΠR,nR〉 (207) 90