Page - 41 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 370 Weobserve that ρb reaches itsmaximal value at the pointm ∈ M such that −→ Om = R −→ b ‖−→b ‖ and its minimalvalueat thediametrallyopposedpoint. 7.3.2. TheGalileanGroup, ItsLieAlgebraandItsActions Inviewof thepresentation,madebelow,of somephysicallymeaningfulgeneralizedGibbsstates forHamiltonianactionsof subgroupsof theGalileangroup,werecall in this sectionsomenotions about thespace-timeofclassical (non-relativistic)mechanics, theGalileangroup, itsLiealgebraandits Hamiltonianactions. The interestedreaderwillfindamuchmoredetailed treatmentonthesesubjects in thebookbySouriau[14]or in therecentbookbydeSaxcéandVallée [45]. Thepaper [62]presentsa niceapplicationofGalilean invariance in thermodynamics. The space-time of classicalmechanics is a four-dimensional real affine spacewhich, once an inertial reference frame,unitsof lengthandtime,orthonormalbasesof spaceandtimearechosen, can be identifiedwithR4≡R3×R (coordinates x,y, z, t). Thefirst threecoordinates x,yand z canbe consideredas the threecomponentsofavector−→r ∈R3, thereforeanelementof space-timecanbe denotedby (−→r ,t).However,as theactionof theGalileangroupwill show, thesplittingofspace-time intospaceandtime isnotuniquelydetermined, itdependsonthechoiceofan inertial reference frame. Inclassicalmechanics, thereexistsanabsolute time,butnoabsolutespace. Thereexists insteadaspace (which isanEuclideanaffinethree-dimensional space) foreachvalueof the time. Thespaces for two distinctvaluesof the timeshouldbeconsideredasdisjoint. Thespace-timebeing identifiedwithR3×Rasexplainedabove, theGalileangroupG canbe identifiedwith thesetofmatricesof the form⎛⎜⎝A −→ b −→ d 0 1 e 0 0 1 ⎞⎟⎠ , withA∈SO(3) ,−→b and−→d ∈R3 , e∈R , (8) the vector spaceR3 being oriented and endowed with its usual Euclidean structure, the matrix A∈SO(3)actingonit. The actionof theGalileangroupGon space-time, identifiedas indicatedabovewithR3×R, is theaffineaction ⎛⎜⎝ −→r t 1 ⎞⎟⎠ → ⎛⎜⎝A −→ b −→ d 0 1 e 0 0 1 ⎞⎟⎠ ⎛⎜⎝ −→r t 1 ⎞⎟⎠= ⎛⎜⎝A−→r + t −→ b + −→ d t+e 1 ⎞⎟⎠ . TheLiealgebraG of theGalileangroupG canbe identifiedwith thespaceofmatricesof the form⎛⎜⎝j(−→ω) −→ β −→ δ 0 0 ε 0 0 0 ⎞⎟⎠ , with−→ω ,−→β and−→δ ∈R3 , ε∈R . (9) Wehavedenotedby j(−→ω) the3×3skew-symmetricmatrix j(−→ω)= ⎛⎜⎝ 0 −ωz ωyωz 0 −ωx −ωy ωx 0 ⎞⎟⎠ . Thematrix j(−→ω) isanelement in theLiealgebraso(3), anditsactiononavector−→r ∈R3 isgiven bythevectorproduct j(−→ω)−→r =−→ω×−→r . 41