Page - 21 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 370 TheMarsden-Weinsteinreductionprocedure [43,44]oroneof itsgeneralizations [10] is themethod mostoftenusedtofacilitate thedeterminationofsolutionsof theequationsofmotion. Inafirst step, apossiblevalueof themomentummap is chosenand the subset of thephase spaceonwhich the momentummaptakes thisvalue isdetermined. Ina secondstep, that subset (when it is a smooth manifold) isquotientedbyits isotropic foliation. Thequotientmanifold isasymplecticmanifoldofa dimensionsmaller thanthatof theoriginalphasespace,andonehasaneasier tosolveHamiltonian systemonthat reducedphasespace. WhenHamiltoniansymmetriesareusedfor thereductionof thedimensionof thephasespace ofamechanical system, thesymplectic cocycleof theLiegroupof symmetriesaction,orof theLie algebraofsymmetriesaction, isalmostalways thezero cocycle. Forexample, if thegroupofsymmetries is thecanonical lift to thecotangentbundleofagroupof symmetriesof theconfigurationspace,notonly the canonical symplectic form, but theLiouville1-formof thecotangentbundle itself remains invariantunder theactionof thesymmetrygroup,andthis fact implies that thesymplecticcohomologyclassof theaction iszero. 5.6.2. Symmetriesof theSpaceofMotions A completely different way of using symmetries was initiated by Jean-Marie Souriau, who proposed to consider the symmetries of the manifold of motions of the mechanical system. Heobservedthat theLagrangianandHamiltonianformalisms, in theirusual formulations, involve the choiceof aparticular reference frame, inwhichthemotion isdescribed. Thischoicedestroysapartof the natural symmetries of the system. Forexample, inclassical (non-relativistic)mechanics, thenatural symmetrygroupofan isolated mechanical systemmust contain the symmetrygroupof theGalilean space-time, called theGalilean group. Thisgroupisofdimension10. It containsnotonly thegroupofEuclideandisplacements of space which isofdimension6andthegroupof time translationswhich isofdimension1, but thegroupof linear changesofGalileanreference frameswhich isofdimension3. IfweusetheLagrangian formalismortheHamiltonian formalism, theLagrangianortheHamiltonian of the systemdepends on the reference frame: it is not invariantwith respect to linear changes ofGalilean reference frames. Itmayseemstrange toconsider thesetofallpossiblemotionsofasystem,which isunknownas longaswehavenotdeterminedall thesepossiblemotions.Onemayask if it is reallyusefulwhenwe want todeterminenotallpossiblemotions,butonlyonemotionwithprescribed initialdata, since that motion is justonepointof the (unknown)manifoldofmotion! Souriau’sanswers to thisobjectionare the following. 1. Weknowthat themanifoldofmotionshasa symplectic structure, andveryoftenmanythingsare knownabout its symmetryproperties. 2. Inclassical (non-relativistic)mechanics, thereexistsanaturalmathematicalobjectwhichdoesnot dependon the choice of a particular reference frame (even if thedecriptionsgiven to thatobjectby differentobserversdependon the reference frameusedby theseobservers): it is the evolution spaceof thesystem. Theknowledgeoftheequationswhichgovernthesystem’sevolutionallowsthefullmathematical descriptionof the evolutionspace, evenwhentheseequationsarenotyet solved. Moreover, thesymmetrypropertiesof the evolutionspaceare thesameas thoseof themanifold ofmotions. Forexample, the evolutionspaceofaclassicalmechanical systemwithconfigurationmanifoldN is 1. in the Lagrangian formalism, the spaceR×TN endowedwith the presymplectic formd̂L, whosekernel isofdimension1whentheLagrangianL ishyper-regular, 2. intheHamiltonianformalism,thespaceR×T∗Nwiththepresymplecticformd̂H,whosekernel too isofdimension1. 21