Page - 6 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 370 whereγ : [t0,t1]→N is a smoothcurve inNparametrizedby the time t. Theseequationsexpress the fact that theaction integral IL(γ) is stationarywithrespect toanysmooth infinitesimalvariation ofγwithfixedend-points ( t0,γ(t0) ) and ( t1,γ(t1) ) . This fact is todaycalledHamilton’s principle of stationaryaction. Thereader interested inCalculusofVariationsanditsapplications inmechanicsand physics is referredto thebooks [23–25]. 2.4. TheEuler-CartanTheorem TheLagrangian formalism is theuseofHamilton’sprincipleof stationaryactionfor thederivation of the equations of motion of a system. It is widely used in mathematical physics, often with moregeneralLagrangians involvingmore thanone independentvariableandhigherorderpartial derivativesofdependentvariables. Forsimplicity Iwill considerhereonly theLagrangiansof (maybe time-dependent) conservativemechanical systems. An intrinsic geometric expression of the Euler–Lagrange equations, wich does not use local coordinates,wasobtainedbythegreatFrenchmathematicianÉlieCartan (1869–1951). Letus introduce theconceptsusedbythestatementof this theorem. Definition1. LetNbe the configurationspaceof amechanical systemand let its tangentbundleTNbe the space of kinematic states of that system. Weassume that the evolutionwith timeof the state of the system is governedby theEuler–Lagrange equations fora smooth,maybe time-dependentLagrangianL :R×TN→R. 1. ThecotangentbundleT∗Niscalled thephase spaceof the system. 2. ThemapLL :R×TN→T∗N LL(t,v)=dvertL(t,v) , t∈R , v∈TN , where dvertL(t,v) is the vertical differential of L at (t,v), i.e., the differential at v of the the map w →L(t,w),withw∈τ−1N ( τN(v) ) , is called theLegendremapassociated to L. 3. ThemapEL :R×TN→Rgivenby EL(t,v)= 〈LL(t,v),v 〉−L(t,v) , t∈R , v∈TN , is called the the energy functionassociated to L. 4. The1-formonR×TN ̂L=L∗LθN−EL(t,v)dt , whereθN is theLiouville1-formonT∗N, is called theEuler–Poincaré1-form. Theorem1 (Euler-CartanTheorem). Asmoothcurveγ : [t0,t1]→Nparametrizedby the time t∈ [t0,t1] is a solutionof theEuler–Lagrange equations if andonly if, for each t∈ [t0,t1] thederivativewith respect to t of themap t → ( t, dγ(t) dt ) belongs to thekernel of the2-formd̂L, inotherwords if andonly if i ( d dt ( t, dγ(t) dt )) d̂L ( t, dγ(t) dt ) =0. The interestedreaderwillfindtheproofof that theoremin[26], (Theorem2.2,Chapter IV,p.262) or, forhyper-regularLagrangians (anadditionalassumptionwhich in fact, isnotnecessary) in [27], Chapter IV,Theorem2.1,p. 167. Remark 1. In his book [14], Jean-Marie Souriau uses a slightly different terminology: for him the odd-dimensional spaceR×TN is the evolutionspaceof the system,and the exact2-formd̂L on that space is theLagrange form.Hedefines that2-forminasettingmoregeneral than thatof theLagrangian formalism. 6