Page - 42 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 370 Letus consider amechanical systemmadebyapointparticle ofmassmwhosepositionand velocityat time t, in thereference frameallowingthe identificationofspace-timewithR3×R, are the vectors−→r and−→v ∈ R3. The action of an element of theGalilean group on−→r ,−→v and t can be writtenas⎛⎜⎝ −→r −→v t 1 1 0 ⎞⎟⎠ → ⎛⎜⎝A −→ b −→ d 0 1 e 0 0 1 ⎞⎟⎠ ⎛⎜⎝ −→r −→v t 1 1 0 ⎞⎟⎠= ⎛⎜⎝A−→r + t −→ b + −→ d A−→v +−→b t+e 1 1 0 ⎞⎟⎠ . Souriauhasshowninhisbook[14] that thisaction isHamiltonian,with themap J,definedonthe evolutionspaceof theparticle,withvalue in thedualG∗of theLiealgebraGof theGalileangroup, asmomentummap J(−→r ,t,−→v ,m)=m ( −→r ×−→v ,−→r − t−→v ,−→v , 1 2 ‖−→v ‖2 ) . Letb= ⎛⎜⎝j(−→ω) −→ β −→ δ 0 0 ε 0 0 0 ⎞⎟⎠beanelement inG. Its couplingwith J(−→r ,t,−→v ,m)∈G∗ isgivenby the formula〈 J(−→r ,t,−→v ,m),b〉=m(−→ω ·(−→r ×−→v )−(−→r − t−→v ) ·−→β +−→v ·−→δ − 1 2 ‖−→v ‖2ε ) . 7.3.3.One-ParameterSubgroupsof theGalileanGroup Inhisbook[14],SouriauhasshownthatwhentheconsideredLiegroupaction is theactionof the fullGalileangrouponthespaceofmotionsofan isolatedmechanical system, theopensubsetΩof theLiealgebraGof theGalileangrouponwhichtheconditionsspecifiedinSection7.2aresatisfied isempty. Inotherwords,generalizedGibbsstatesof the fullGalileangroupdonotexist.However, generalizedGibbsstates forone-parametersubgroupsof theGalileangroupdoexistwhichhavean interestingphysicalmeaning. Letus consideranelement bofG such that in itsmatrix expression (expression (9) above)we have ε =0. Theone-parametersubgroupG1 of theGalileangroupgeneratedbyb is thesetofmatrices exp(τb),withτ∈R.Wehave exp(τb)= ⎛⎜⎝A(τ) −→ b (τ) −→ d (τ) 0 1 τε 0 0 1 ⎞⎟⎠ , with A(τ)= exp ( τj(−→ω)) , −→ b (τ)= ( ∞ ∑ n=1 τn n! ( j(−→ω))n−1)−→β , −→ d (τ)= ( ∞ ∑ n=1 τn n! ( j(−→ω))n−1)−→δ +ε( ∞∑ n=2 τn n! ( j(−→ω))n−2)−→β , with theusualconventionthat ( j(−→ω))0 is theunitmatrix. Thephysicalmeaningof thisone-parametersubgroupof theGalileangroupcanbeunderstoodas follows. Letuscallfixed theaffineEuclideanreference frameofspace (O,−→ex ,−→ey ,−→ez)usedtorepresent, at time t=0,apoint inspacebyavector−→r orby its threecomponentsx,yand z. Letussetτ= t ε . 42