Page - 8 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 370 wecanwrite d dt ( i(Z)˜L ( t, dγ(t) dt )) = 〈 di(Z)̂L, d dt ( t, dγ(t) dt )〉 =− 〈 i(Z)d̂L, d dt ( t, dγ(t) dt )〉 =0. Example 2. When the Lagrangian L does not depend on time, application of EmmyNoether’s theorem to the vector field ∂ ∂t shows that the energy EL remains constant during any possible motion of the system, since i ( ∂ ∂t ) ̂L=−EL. Remark2. 1. Theorem2 isdue to theGermanmathematicianEmmyNoether (1882–1935),whoproved itundermuch moregeneral assumptions than thoseusedhere. ForaverynicepresentationofEmmyNoether’s theorems in amuchmore general setting and their applications inmathematical physics, interested readers are referred to theverynicebookbyYvetteKosmann-Schwarzbach [28]. 2. Several generalizations of theNoether theorem exist. For example, if instead of being an infinitesimal symmetryof ̂L, i.e., insteadof satisfyingL(Z)̂L=0 thevectorfieldZsatisfies L(Z)̂L=df , where f :R×TM→R is a smooth function,which implies of courseL(Z)(d̂L)=0, the function i(Z)̂L− f keepsaconstantvaluealong t → ( t, dγ(t) dt ) . 3.3. TheLagrangianMomentumMap TheLiebracketof two infinitesimal symmetriesof ̂L is tooan infinitesimal symmetryof ̂L. Letus thereforeassumethat thereexistsafinite-dimensionalLiealgebraofvectorfieldsonR×TN whoseelementsare infinitesimalsymmetriesof ̂L. Definition3. Letψ : G→ A1(R×TN) be aLie algebras homomorphismof a finite-dimensional real Lie algebraG into the Lie algebra of smooth vector fields onR×TN such that, for each X ∈ G, ψ(X) is an infinitesimal symmetryof ̂L. TheLiealgebrashomomorphismψ is said tobeaLiealgebraactiononR×TN by infinitesimal symmetries of ̂L. ThemapKL :R×TN→G∗,which takes itsvalues in thedualG∗ of the Lie algebraG, definedby〈 KL(t,v),X 〉 = i ( ψ(X) ) ̂L(t,v) , X∈G , (t,v)∈R×TN , is called theLagrangianmomentumof theLie algebraactionψ. Corollary1 (ofE.Noether’sTheorem). Letψ :G→A1(R×TM)beanactionofafinite-dimensional real Lie algebraG on the evolutionspaceR×TNofa conservativeLagrangiansystem,by infinitesimal symmetries of thePoincaré-Cartan form ̂L. For each possiblemotionγ : [t0,t1]→Nof that system, the Lagrangian momentummapKL keepsaconstantvaluealong theparametrizedcurve t → ( t, dγ(t) dt ) . 8