Page - 73 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 386 Therefore, thecohomologyclassof thesymplectic cocycle θonlydependsontheHamiltonian actionΦ, notonthechoiceof itsmomentmap J.Wehavealso: Θ˜′(X,Y)= Θ˜(X,Y)+〈μ, [X,Y]〉 (89) Thisproperty isusedbyJean-MarieSouriau[10] toofferaverynicecohomological interpretation of the totalmassofa classical (nonrelativistic) isolatedmechanical system.He [10]proves that the spaceofallpossiblemotionsof thesystemisasymplecticmanifoldonwhichtheGalileangroupacts byaHamiltonianaction. Thedimensionof thesymplecticcohomologyspaceof theGalileangroup (thequotientof thespaceofsymplecticone-cocyclesbythespaceofsymplecticone-coboundaries) is equal to1. Thecohomologyclassof thesymplecticcocycleassociatedtoamomentmapof theactionof theGalileangrouponthespaceofmotionsof thesystemis interpretedas the totalmassof thesystem. ForHamiltonianactionsofaLiegrouponaconnectedsymplecticmanifold, theequivariance of themomentmapwith respect to anaffineactionof thegroupon thedualof itsLie algebrahas beenprovedbyMarle [110]. Marle [110] has alsodeveloped thenotionof symplectic cocycle and hasprovedthatgivenaLiealgebrasymplecticcocycle, thereexistsontheassociatedconnectedand simplyconnectedLiegroupauniquecorrespondingLiegroupsymplectic cocycle.Marle [104]has alsoprovedthat thereexistsa two-parameter familyofdeformationsof theseactions (theHamiltonian actionsofaLiegrouponitscotangentbundleobtainedbylifting theactionsof thegrouponitselfby translations) intoapairofmutuallysymplecticallyorthogonalHamiltonianactionswhosemoment mapsareequivariantwithrespect toanaffineaction involvinganygivenLiegroupsymplecticcocycle. Marle [104]hasalsoexplainedwhyareductionoccurs forEuler-Poncaréequationmainlywhenthe Hamiltonian canbe expressed as themomentmap composedwith a smooth functiondefinedon thedualof theLiealgebra; theEuler-Poincaréequation is thenequivalent to theHamiltonequation writtenonthedualof theLiealgebra. 6.5.DualSpacesofFinite-DimensionalLieAlgebras Letgbeafinite-dimensionalLiealgebra,andg∗ itsdualspace. TheLiealgebragcanbeconsidered as thedualofg∗, thatmeansas thespaceof linear functionsong∗, andthebracketof theLiealgebrag isacomposition lawonthisspaceof linear functions. Thiscomposition lawcanbeextendedto the spaceC∞(g∗, )bysetting: {f,g}(x)= 〈x, [df(x),dg(x)]〉 , f andg∈C∞(g∗, ), x∈ g∗ (90) Ifweapply this formula forSouriauLiegroupthermodynamics,andforentropy s(Q)depending ongeometricheatQ: {s1,s2}(Q)= 〈Q, [ds1(Q),ds2(Q)]〉 , s1 and s2∈C∞(g∗, ), Q∈ g∗ (91) ThisbracketonC∞(g∗, )definesaPoissonstructureong∗, called its canonicalPoissonstructure. It implicitlyappears in theworksofSophusLie,andwasrediscoveredbyAlexanderKirillov [111], BertramKostantandJean-MarieSouriau. TheabovedefinedcanonicalPoissonstructureong∗ canbemodifiedbymeansofasymplectic cocycle Θ˜bydefiningthenewbracket: {f,g}Θ˜(x)= 〈x, [df(x),dg(x)]〉−Θ˜(df(x),dg(x)) (92) with Θ˜asymplecticcocycleof theLiealgebragbeingaskew-symmetricbilinearmap Θ˜ : g×g→ whichsatisfies: Θ˜([X,Y] ,Z)+Θ˜([Y,Z] ,X)+Θ˜([Z,X] ,Y)=0 (93) 73