Page - 9 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 370 Proof. Since for each X ∈ G the function (t,v) → 〈KL(t,v),X〉 keeps a constant value along the parametrized curve t → ( t, dγ(t) dt ) , the map KL itself keeps a constant value along that parametrizedcurve. Example3. Letusassumethat theLagrangianLdoesnotdependexplicitlyon the time t and is invariantby thecanonical lift to the tangentbundleof theactiononNof thesix-dimensionalgroupofEuclideandiplacements (rotationsandtranslations) of thephysical space. Thecorresponding infinitesimalactionof theLie algebraof infinitesimalEuclideandisplacements (considered as anactiononR×TN, the actionon the factorRbeing trivial) is anactionby infinitesimal symmetries of ̂L. The six componentsof theLagrangianmomentummap are the three componentsof the total linearmomentumandthe three componentsof the total angularmomentum. Remark3. These results arevalidwithoutanyassumptionofhyper-regularityof theLagrangian. 3.4. TheEuler–PoincaréEquation InashortNote [29]published in1901, thegreat frenchmathematicianHenriPoincaré (1854–1912) proposedanewformulationof theequationsofmechanics. Let N be the configuration manifold of a conservative Lagrangian system, with a smooth Lagrangian L : TN →Rwhichdoes not depend explicitly on time. Poincaré assumes that there existsanhomomorphismψofafinite-dimensional realLiealgebraG into theLiealgebraA1(N)of smoothvectorfieldsonN, suchthat foreachx∈N, thevaluesatxof thevectorfieldsψ(X),whenX varies inG, completelyfill the tangentspaceTxN. Theactionψ is thensaid tobe locally transitive. Ofcourse theseassumptions implydimG≥dimN. Under theseassumptions,HenriPoincaréprovedthat theequationsofmotionof theLagrangian systemcouldbewrittenonN×GoronN×G∗,whereG∗ is thedualof theLiealgebraG, insteadof on the tangentbundleTN. WhendimG=dimN (whichcanoccuronlywhen the tangentbundle TN is trivial) the obtained equation, called theEuler–Poincaré equation, is perfectly equivalent to the Euler–Lagrange equations and may, in certain cases, be easier to use. But when dimG > dimN, thesystemmadebytheEuler–Poincaréequation isunderdetermined. Letγ : [t0,t1]→N bea smoothparametrizedcurve inN. Poincaréproves that there exists a smoothcurveV : [t0,t1]→G in theLiealgebraG suchthat, foreach t∈ [t0,t1], ψ ( V(t) )( γ(t) ) = dγ(t) dt . (2) WhendimG>dimN thesmoothcurveV inG isnotuniquelydeterminedbythesmoothcurve γ inN.However, insteadofwritingthesecond-orderEuler–LagrangedifferentialequationsonTN satisfiedbyγwhenthiscurve isapossiblemotionof theLagrangiansystem,Poincaréderivesafirst orderdifferential equation for the curveV andproves that it is satisfied, togetherwithEquation(2), if and only ifγ is apossiblemotionof theLagrangiansystem. Letϕ :N×G→TNandL :N×G→Rbethemaps ϕ(x,X)=ψ(X)(x) , L(x,X)=L◦ϕ(x,X) . We denote by d1L : N×G → T∗N and by d2L : N×G → G∗ the partial differentials of L :N×G→Rwithrespect to itsfirstvariablex∈Nandwithrespect to its secondvariableX∈G. Themapϕ :N×G→TN isa surjectivevectorbundlesmorphismof the trivialvectorbundleN×G into the tangentbundleTN. Its transpose ϕT :T∗N→N×G∗ is thereforean injectivevector bundles morphism,whichcanbewritten ϕT(ξ)= ( πN(ξ), J(ξ) ) , 9