Page - 19 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 370 Hamiltonians, respectively, the smooth functions f andgonthemanifoldM.The function f remainsconstant oneach integral curveofXg if andonly if g remainsconstantoneach integral curveofXf. Proof. Thefunction f is constantoneach integral curveofXg if andonly if i(Xg)df=0, sinceeach integral curveofXg is connected.Wecanuse thePoissonbracket, evenwhenM isapresymplectic manifold, since thePoissonbracket of twoHamiltoniansonapresymplecticmanifold still canbe defined. Sowecanwrite i(Xg)df={g, f}=−{f,g}=−i(Xf)dg . Corollary2 (ofNoether’sTheoreminHamiltonianFormalism). Letψ :G→A1(M)beaHamiltonian actionof afinite-dimensionalLie algebraG onapresymplectic or symplecticmanifold (M,ω), or onaPoisson manifold (M,Λ), and let J : M→G∗ be amomentummapof this action. LetXH be aHamiltonianvector field onMadmitting asHamiltonian a smooth functionH. If for eachX∈Gwehave i(ψ(X))(dH) = 0, themomentummap J remainsconstantoneach integral curveofXH. Proof. Thisresult isobtainedbyapplyingTheorem5tothepairsofHamiltonianvectorfieldsmade byXH andeachvectorfieldassociatedtoanelementofabasisofG. 5.5. SymplecticCocycles Theorem6 (J.M.Souriau [14]). LetΦbe aHamiltonianaction (either on the left or on the right) of aLie groupGonaconnected symplecticmanifold (M,ω)and let J :M→G∗ beamomentummapof this action. There exists anaffine actionA(either on the left or on the right) of theLie groupGon thedualG∗ of itsLie algebraG such that themomentummap J is equivariantwith respect to theactionsΦofGonMandAofGon G∗, i.e., such that J◦Φg(x)=Ag◦ J(x) for all g∈G , x∈M . TheactionAcanbewritten,withg∈Gandξ∈G∗,⎧⎨⎩A(g,ξ)=Ad ∗ g−1(ξ)+θ(g) ifΦ is anactiononthe left, A(ξ,g)=Ad∗g(ξ)−θ(g−1) ifΦ is anactiononthe right. Proof. LetusassumethatΦ isanactiononthe left. ThefundamentalvectorfieldXM associatedto eachX∈G isHamiltonian,with the function JX :M→R, givenby JX(x)= 〈 J(x),X 〉 , x∈M , asHamiltonian. Foreachg∈G thedirectimage(Φg−1)∗(XM)ofXMbythesymplecticdiffeomorphism Φg−1 isHamiltonian,with JX◦Φg asHamiltonian.Aneasycalculationshowsthat thisvectorfield is the fundamentalvectorfieldassociatedtoAdg−1(X)∈G. The function x → 〈J(x),Adg−1(X)〉= 〈Ad∗g−1◦J(x),X〉 is thereforeaHamiltonianfor thatvectorfield. Thesetwofunctionsdefinedontheconnectedmanifold M,whichbothareadmissibleHamiltonians for thesameHamiltonianvectorfield,differonlybya constant (whichmaydependong∈G).Wecanset, foranyg∈G, θ(g)= J◦Φg(x)−Ad∗g−1◦J(x) andcheck that themapA : G×G∗→G∗defined in the statement is indeedanaction forwhich J isequivariant. Asimilarproof,withsomechangesofsigns,holdswhenΦ isanactionontheright. 19