Page - 93 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 386 HeniPoincaréprovedthatwhenaLiealgebraacts locallyandtransitivelyontheconfiguration spaceofaLagrangianmechanicalsystem,theEuler-Lagrangeequationsareequivalenttoanewsystem ofdifferentialequationsdefinedontheproductof theconfigurationspacewith theLiealgebra. Ifweconsider that the followingfunction is stationary foraLagragian l(.) invariantwithrespect to theactionofagrouponthe left: S(ηL)= b a l(ηL)dtwithδS(ηL)=0and l : g→R (219) Thesolution isgivenbytheEuler-Poincaréequation: d dt δl δηL = ad∗ηL δl δηL δηL= . Γ+adηLΓwhereΓ(t)∈ g (220) Ifwetakeforthefunction l(.), thetotalkineticenergyEL,usingΠL=M−1 . M= ∂EL∂nL ∈ gL, thenthe Euler-Poincaréequation isgivenby: dΠL dt = ad∗nLΠLwith δl δηL = ∂EL ∂nL =ΠL∈ gL (221) Thefollowingquantitiesareconserved: dΠR dt =0 (222) With thissecondtheorem, it ispossible towrite thegeodesicnot fromitscoordinatesystembut fromthequantityofmotion,and inaddition todetermineexplicitlywhat theconservedquantities along the geodesic are (conservations are related to the symmetries of the variety andhence the invarianceof theLagrangianunder theactionof thegroup). Forouruse-case, theEuler-Poincaréequation isgivenby: { . ηL1=−ηL2ηTL2 . ηL2=ηL2ηL1 with ⎧⎨⎩ ηL1=R−1/2 . R 1/2 ηL2=R−1/2 . m ⇒ ⎧⎪⎪⎨⎪⎪⎩ ( R−1/2 . R 1/2 )• =−R−1/2 .m .mTR−1/2( R−1/2 .m )• = . R −1/2 . R 1/2 R−1/2 .m (223) IfweremarkthatwehaveR−1/2 . R 1/2 =R−1/2 ( R−1/2 . R ) =R−1 . R, thentheconservedSouriau momentcouldbegivenby: ΠR= [ R−1/2 . R 1/2 +R−1 .mmT R−1 .m 0 0 ] = [ R−1 . R+R−1 .mmT R−1 .m 0 0 ] (224) Componentsof theSouriaumomentgive theconservedquantities thatare theclassicalelements givenbyEmmyNoetherTheorem(Souriaumoment isageometrizationofEmmyNoetherTheorem): dΠR dt = ⎡⎢⎣ d ( R−1 . R+R−1 .mmT ) dt d(R−1 . m) dt 0 0 ⎤⎥⎦=0⇒ ⎧⎨⎩ R −1 .R+R−1 .mmT=B= cste R−1 .m= b= cste (225) Fromthisconstant,wecanobtainareducedequationofgeodesic:⎧⎨⎩ . m=Rb . R=R ( B−bmT) (226) 93