Page - 129 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 254 thatallowsustosatisfyautomatically (35).Next,owingto (32),Equation(33)canbesimplified as follows: q=Rq−τ0p+mk , thatallowstodetermine thespatial translationkwithrespect toRandtheclockchangeτ0: k= 1 m (q−Rq+τ0p). (37) Finally,becauseof (32),Equation(34) is simplifiedas follows: l=Rl−u×(Rq)+k×p . Substituting(37) into the last relationgives: l=Rl−u×(Rq)+ 1 m q×p− 1 m (Rq)×p . Owingto (32)andthedefinitionof thespinangularmomentum l0 l0= l−q×p/m , leads to: l0=Rl0 . (38) Thesequantitybeinggiven,wehave todetermine therotationssatisfying thepreviousrelation. It turnsout that twocasesmustbeconsidered. – Genericorbits :massiveparticlewithspinorrigidbody. If l0doesnotvanish, thesolutionsof (38) are therotationsofanarbitraryangleϑabout theaxis l0.Weknowby(36)and(37) thatuand karedeterminedinauniquemannerwithrespect toRandτ0. The isotropygroupofμcan beparameterisedbyϑandτ0. It isaLiegroupofdimension2. Thedimensionof theorbitof μ is10−2=8. Themaximumnumberof independent invariant functions is10−8=2.A possible functionalbasis is composedof: s0=‖ l0 ‖ , (39) e0= e− 12m ‖ p‖ 2 , (40) ofwhichthevaluesareconstantontheorbitwhichrepresentsamassiveparticlewithspinor arigidbody(seenfromalongwayoff). – Singularorbits : spinlessmassiveparticle. Intheparticularcase l0=0,all therotationsofSO(3) satisfy (38), then the isotropygroup isofdimension4. Bysimilar reasoning to thecaseof nonvanishing l0,weconcludethatdimensionof theorbit is6andthenumberof invariant functions is4.Apossible functionalbasis iscomposedof e0 andthe threenull components of l0. For theorbitswithm=0, thereader is referredto [6] (pp. 440,441). Tophysically interpret thecomponentsof themomentum, let consideracoordinatesystemX′ inwhichaparticle is at rest andcharacterizedby the components p′= 0, q′= 0, l′= l0 and e′= e0 of themomentumtensor. LetusconsideranothercoordinatesystemX=PX′+Cwitha Galileanboostvandatranslationof theoriginatk= x0 (henceτ0=0andR=1R3),providing the trajectoryequation: x= x0+vt , (41) 129