Page - 7 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 370 3. LagrangianSymmetries 3.1.AssumptionsandNotations In this sectionN is theconfigurationspaceofaconservativeLagrangianmechanical systemwith asmooth,maybetimedependentLagrangianL :R×TN→R. Let ̂LbethePoincaré-Cartan1-form ontheevolutionspaceR×TN. Severalkindsofsymmetriescanbedefinedforsuchasystem.Veryoften, theyarespecial cases of infinitesimal symmetries of the Poincaré-Cartan form, whichplay an important part in the famous Noether theorem. Definition 2. An infinitesimal symmetry of the Poincaré-Cartan form ̂L is a vector field Z onR×TN such that L(Z)̂L=0, L(Z)denoting theLiederivativeofdifferential formswith respect toZ. Example1. 1. Letusassumethat theLagrangianLdoesnotdependonthe time t∈R, i.e., is a smooth functiononTN. ThevectorfieldonR×TNdenotedby ∂ ∂t ,whoseprojectiononR is equal to1andwhoseprojectionon TN is0, is an infinitesimal symmetryof ̂L. 2. LetXbea smoothvectorfieldonNandXbe its canonical lift to the tangentbundleTN.Westill assume that L doesnot depend on the time t. Moreoverwe assume thatX is an infinitesimal symmetry of the LagrangianL, i.e., thatL(X)L=0.ConsideredasavectorfieldonR×TNwhoseprojectiononthe factor R is0,X isan infinitesimal symmetryof ̂L. 3.2. TheNoetherTheoreminLagrangianFormalism Theorem2 (E.Noether’sTheoreminLagrangianFormalism). LetZbean infinitesimal symmetryof the Poincaré-Cartan form ̂L. For eachpossiblemotionγ : [t0,t1]→Nof theLagrangian system, the function i(Z)̂L,definedonR×TN,keepsaconstantvaluealong theparametrizedcurve t → ( t, dγ(t) dt ) . Proof. Letγ : [t0,t1]→Nbeamotionof theLagrangiansystem, i.e., a solutionof theEuler–Lagrange equations. TheEuler-CartanTheorem1proves that, forany t∈ [t0,t1], i ( d dt ( t, dγ(t) dt )) d̂L ( t, dγ(t) dt ) =0. SinceZ isan infinitesimalsymmetryof ̂L, L(Z)̂L=0. Using the well known formula relating the Lie derivative, the interior product and the exteriorderivative L(Z)= i(Z)◦d+d◦ i(Z) 7