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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
zurück zum  Buch Differential Geometrical Theory of Statistics"
Differential Geometrical Theory of Statistics
Titel
Differential Geometrical Theory of Statistics
Autoren
Frédéric Barbaresco
Frank Nielsen
Herausgeber
MDPI
Ort
Basel
Datum
2017
Sprache
englisch
Lizenz
CC BY-NC-ND 4.0
ISBN
978-3-03842-425-3
Abmessungen
17.0 x 24.4 cm
Seiten
476
Schlagwörter
Entropy, Coding Theory, Maximum entropy, Information geometry, Computational Information Geometry, Hessian Geometry, Divergence Geometry, Information topology, Cohomology, Shape Space, Statistical physics, Thermodynamics
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Differential Geometrical Theory of Statistics