Page - 92 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 386 ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ κ= [ κ1 κ2 0 0 ] ∈G η= [ η1 η2 0 0 ] ∈ g∗ n= [ n1 n2 0 0 ] ∈ g ⇒ ⎧⎪⎪⎨⎪⎪⎩ 〈ad∗nη,κ〉= 〈η,adnκ〉= 〈η,nκ−κn〉 〈ad∗nη,κ〉= 〈[ −η2nT2 n1η2 0 0 ] ,κ 〉 ⇒ ⎧⎪⎪⎨⎪⎪⎩ ad∗nη= [ −η2nT2 n1η2 0 0 ] ad∗nη={n,η} (215) This co-adjoint operator will give the Euler-Poincaré equation. While the Euler-Lagrange equations isdefinedonthe tangentbundle (unionof the tangentspacesateachpoint)of themanifold andgive thegeodesics, theEuler-Poincaréequationgivesadifferential systemonthedualLiealgebra of thegroupassociatedwith themanifold. Wecanalsocomplete thesemapsbyusingadditionalones. First,p∈T∗MG themomentassociated with . M∈TMG in tangentspaceofGatMandalso twoothermomentsmaptheelementof thedual algebra indual tangentspace, respectivelyonthe leftandontheright:⎧⎪⎨⎪⎩ 〈ΠL,nL〉= 〈 dL∗M−1ΠL, . M 〉 〈 ΠL,dLM−1 . M 〉 = 〈 ΠL,M−1 . M 〉 ⇒ p=(M−1)TΠL (216) where dL∗M−1 : g ∗ L→T∗MG ΠL → p= ( M−1 )TΠL and dR∗M−1 : g ∗ R→T∗MG ΠR → p=ΠR ( M−1 )T (217) Fromtheserelations,wecanalsoobserve that: ΠL=nL=M−1 . M ⇒ ⎧⎨⎩ p= ( M−1 )TM−1 .M p=ΞM · . MwithΞM= ( M−1 )TM−1 (218) All thesemapscouldbesummarized in the followingFigure12: Figure12.Mapsbetweenalgebras. 92