Page - 10 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 370 whereπN : T∗N → N is the canonical projectionof the cotangent bundle and J : T∗N →G∗ is a smoothmapwhoserestriction toeachfibreT∗xNof thecotangentbundle is linear, andis the transpose of themapX →ϕ(x,X)=ψ(X)(x). Remark4. Thehomomorphismψof theLie algebraG into theLie algebraA1(N)of smoothvectorfieldsonN isanactionof thatLiealgebra, in thesensedefinedbelowDefinition11. Thatactioncanbecanonically lifted into aHamiltonianactionofG onT∗N,endowedwith its canonical symplectic formdθN Definition13. Themap J is in fact aHamiltonianmomentummap for thatHamiltonianactionProposition5. LetLL=dvertL :TN→T∗NbetheLegendremapdefinedinDefinition1. Theorem3 (Euler–PoincaréEquation). With theabovedefinednotations, letγ : [t0,t1]→Nbeasmooth parametrizedcurve inNandV : [t0,t1]→G bea smoothparametrizedcurve such that, for each t∈ [t0,t1], ψ ( V(t) )( γ(t) ) = dγ(t) dt . (3) Thecurveγ is apossiblemotionof theLagrangiansystemif andonly ifV satisfies the equation( d dt −ad∗V(t) )( J◦LL◦ϕ ( γ(t),V(t) ))− J◦d1L(γ(t),V(t))=0. (4) The interestedreaderwillfindaproofof that theoremin local coordinates in theoriginalNoteby Poincaré [29].More intrinsicproofscanbefoundin[12,30].Anotherproof ispossible, inwhichthat theoremisdeducedfromtheEuler-CartanTheorem1. Remark5. Equation (3) is called the compatibility conditionandEquation (4) is theEuler–Poincaré equation. It canbewrittenunder the equivalent form( d dt −ad∗V(t) )( d2L ( γ(t),V(t) ))− J◦d1L(γ(t),V(t))=0. (5) Examplesofapplicationsof theEuler–Poincaréequationcanbefoundin[5,6,12,30]and, foran application in thermodynamics, [31]. 4.TheHamiltonianFormalism TheLagrangian formalism can be applied to any smooth Lagrangian. Its application yields secondorderdifferential equationsonR×N (in local coordinates, theEuler–Lagrange equations)which in generalarenot solvedwith respect to the secondorderderivatives of theunknown functionswith respect to time. Theclassicalexistenceandunicity theoremsfor thesolutionsofdifferentialequations (suchas theCauchy-Lipschitz theorem) therefore cannotbeapplied to these equations. Under theadditional assumption that theLagrangian ishyper-regular, averyclever changeof variablesdiscoveredbyWilliamRowanHamilton (LagrangeobtainedhoweverHamilton’sequations beforeHamilton, butonly inaspecial case, for theslow“variationsofconstants”suchas theorbital parametersofplanets in thesolarsystem[32,33]).Hamilton[21,22]allowsanewformulationof these equations in the frameworkof symplecticgeometry. TheHamiltonian formalismdiscussedbelowis the useof thesenewequations. Itwas latergeneralized independentlyof theLagrangianformalism. 10