Page - 20 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 370 Proposition6. UndertheassumptionsandwiththenotationsofTheorem6, themapθ :G→G∗ isacocycleof theLiegroupGwithvalues inG∗, for the coadjoint representation. Itmeans that it satisfies, forall g andh∈G, θ(gh)= θ(g)+Ad∗g−1 ( θ(h) ) . Morepreciselyθ is a symplectic cocycle. Itmeans that itsdifferentialTeθ :TeG≡G→G∗ at theneutral element e∈Gcanbeconsideredasa skew-symmetricbilinear formonG: Θ(X,Y)= 〈 Teθ(X),Y 〉 =−〈Teθ(Y),X〉 . The skew-symmetric bilinear formΘ is a symplectic cocycle of the Lie algebraG. It means that it is skew-symmetric andsatisfies, for allX,YandZ∈G, Θ ( [X,Y],Z ) +Θ ( [Y,Z],X ) +Θ ( [Z,X],Y ) =0. Proof. Thesepropertieseasilyfollowfromthefact thatwhenΦ isanactionontheleft, forgandh∈G, Φg◦Φh=Φgh (andasimilarequalitywhenΦ is anactionontheright). The interestedreaderwill findmoredetails in [9,12,14]. Proposition7. Stillunder theassumptionsandwith thenotationsofTheorem6, the composition lawwhich associates to eachpair (f,g)of smoothreal-valued functionsonG∗ the function{f,g}Θ givenby {f,g}Θ(x)= 〈 x, [df(x),dg(x)] 〉−Θ(df(x),dg(x)) , x∈G∗ , (G being identified with its bidual G∗∗), determines a Poisson structure on G∗, and the momentum map J :M→G∗ isaPoissonmap,MbeingendowedwiththePoissonstructureassociatedto itssymplecticstructure. Proof. The fact that the bracket (f,g) → {f,g}Θ onC∞(G∗,R) is a Poisson bracketwas already indicated inExample4. It canbeverifiedbyeasycalculations. The fact that J isaPoissonmapcanbe provenbyfirst lookingat linear functionsonG∗, i.e., elements inG. Thereaderwillfindadetailed proof in [12]. Remark11. Whenthemomentummap J is replacedbyanothermomentummap J1= J+μ,whereμ∈G∗ is a constant, the symplecticLiegroupcocycleθ andthe symplecticLiealgebracocycleΘare replacedbyθ1 and Θ1, respectively,givenby θ1(g)= θ(g)+μ−Ad∗g−1(μ) , g∈G , Θ1(X,Y)=Θ(X,Y)+ 〈 μ, [X,Y] 〉 , XandY∈G . These formulae showthatθ1−θ andΘ1−Θare symplectic coboundaries of theLiegroupGandtheLie algebraG. Inotherwords, thecohomologyclassesof thecocyclesθ andΘonlydependontheHamiltonianaction ΦofGonthe symplecticmanifold (M,ω). 5.6. TheUseofSymmetries inHamiltonianMechanics 5.6.1. Symmetriesof thePhaseSpace HamiltonianSymmetriesareoftenusedfor thesearchofsolutionsof theequationsofmotionof mechanical systems. Thesymmetriesconsideredare thoseof thephase spaceof themechanical system. Thisspace isveryoftena symplecticmanifold, either thecotangentbundle to theconfigurationspace with its canonical symplectic structure,oramoregeneral symplecticmanifold. Sometimes,after some simplifications, thephasespace isaPoissonmanifold. 20