Page - 16 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 370 Thisbracket still satifies theproperties stated inDefinition6, thereforedefinesonG∗ aPoissonstructure called its canonicalLie-PoissonstructuremodifiedbyΘ. 4.4.2. PropertiesofPoissonManifolds The interestedreaderwillfindtheproofsof theproperties recalledhere in [8–11]. 1. On aPoissonmanifoldP, the Poisson bracket {f,g} of two smooth functions fand g can be expressedbymeansofasmoothfieldofbivectorsΛ: {f,g}=Λ(df,dg) , f andg∈C∞(P,R) , calledthePoissonbivectorfieldofP. TheconsideredPoissonmanifold isoftendenotedby (P,Λ). ThePoissonbivectorfieldΛ identicallysatisfies [Λ,Λ]=0, the bracket [ , ] in the left hand side being theSchouten-Nijenhuis bracket. That bivector field determinesavectorbundlemorphismΛ :T∗P→TP,definedby Λ(η,ζ)= 〈 ζ,Λ (η) 〉 , whereηandζ∈T∗Pare twocovectorsattachedto thesamepoint inP. Readers interested in the Schouten-Nijenhuis bracketwill find thoroughpresentations of its properties in [40,41]. 2. Let (P,Λ)beaPoissonmanifold.A(maybetime-dependent)vectorfieldonPcanbeassociated toeach (maybe time-dependent) smooth functionH :R×P→R. It is called theHamiltonian vectorfieldassociatedto theHamiltonianH, anddenotedbyXH. Itsexpression is XH(t,x)=Λ (x) ( dHt(x) ) , where dHt(x) = dH(t,x)− ∂H(t,x) ∂t dt is the differential of the function deduced from H by considering tasaparameterwithrespect towhichnodifferentiation ismade. TheHamiltonequationdeterminedbythe (maybetime-dependent)HamiltonianH is dϕ(t) dt =XH( ( t,ϕ(t) ) =Λ (dHt) ( ϕ(t) ) . 3. AnyPoissonmanifold is foliated,byageneralizedfoliationwhose leavesmaynotbeallof the samedimension, into immersedconnectedsymplecticmanifoldscalledthe symplectic leavesof thePoissonmanifold. The value, at anypoint of a Poissonmanifold, of the Poissonbracket of twosmooth functionsonlydependson the restrictionsof these functions to the symplectic leaf through the consideredpoint, and can be calculated as the Poisson bracket of functions definedonthat leaf,with thePoissonstructureassociatedto thesymplectic structureof that leaf. ThispropertywasdiscoveredbyAlanWeinstein, inhisvery thoroughstudyof the local structure ofPoissonmanifolds [42]. 5.HamiltonianSymmetries 5.1. Presymplectic,Symplectic andPoissonMapsandVectorFields LetMbeamanifoldendowedwithsomestructure,whichcanbeeither • a presymplectic structure, determinedbyapresymplectic form, i.e., a 2-formω which is closed (dω=0), 16