The Hamiltonian in Covariant Theory of Gravitation

55Copyright©CanadianResearch&DevelopmentCenterofSciencesandCulturesISSN1715-7862[PRINT]ISSN1715-7870[ONLINE]:10.3968/j.ans.1715787020120504.2023TheHamiltonianinCovariantTheoryofGravitationSergeyG.Fedosin[a],*[a]Perm,PermRegion,Russia.*Correspondingauthor.Received20October2012;accepted25December2012AbstractIntheframeworkofcovarianttheoryofgravitationtheEuler-LagrangeequationsarewrittenandequationsofmotionaredeterminedbyusingtheLagrangefunction,inthecaseofsmalltestparticleandinthecaseofcontinuouslydistributedmatter.FromtheLagrangiantransitiontotheHamiltonianwasdone,whichisexpressedthroughthree-dimensionalgeneralizedmomentuminexplicitform,andalsoisdefinedbythe4-velocity,scalarpotentialsandstrengthsofgravitationalandelectromagneticfields,takingintoaccountthemetric.Thedefinitionofgeneralized4-velocity,andthedescriptionofitsapplicationtotheprincipleofleastactionandtoHamiltonianisdone.Theexistenceofa4-vectoroftheHamiltonianisassumedandtheproblemofmassisinvestigated.Tocharacterizethepropertiesofmassweintroducethreedifferentmasses,oneofwhichisconnectedwiththerestenergy,anotheristheobservedmass,andthethirdmassisdeterminedwithouttakingintoaccounttheenergyofmacroscopicfields.Itisshownthattheactionfunctionhasthephysicalmeaningofthefunctiondescribingthechangeofsuchintrinsicpropertiesastherateofpropertimeandrateofriseofphaseangleinperiodicprocesses.Keywords:Euler-Lagrangeequations;Lagrangian;Hamiltonian;Generalizedmomentum;Generalized4-velocity;EquationsofmotionPACS:03.30.+p,04.20.Fy,04.40.-b,11.10.EfSergeyG.Fedosin(2012).TheHamiltonianinCovariantTheoryofGravitation.AdvancesinNaturalScience,5(4),55-75.Availablefrom::(CTG)appearedin2009[1],asaconsequenceoftherelativisticgeneralizationoftheLorentz-invarianttheoryofgravitation(LITG).LITGequationsaresimilarbytheirformtoMaxwell’sequationsandcanbederivedonthebasisofaxioms[2].RecentlyderivationofCTGequationswasmadebasedontheprincipleofleastaction[3].BasedontheresultingformoftheLagrangiannowitispossibletomakethenextstepandgototheHamiltoniancorrespondingtotheCTGtheory.AfterabriefpresentationoftheEuler-Lagrangeequationsweusethemtodescribethemotionofasmalltestparticle,aswellasinthecaseofcontinuouslydistributedmatter.ThenwefindtheHamiltonianinitstwoforms,withthehelpof4-velocityandthegeneralizedmomentum,andsubstitutetheHamiltonianintoHamiltonequationstoverifythemotionequations.Attheendofthispaperweintroduceforconsiderationthefour-dimensionalgeneralizedvelocitytosimplifytheexpressionsfortheLagrangianandHamiltonian.Thetransitionwasdonefromthe4-vectorofthegeneralizedvelocitytoanew4-vectoroftheHamiltonian,specifyingtheenergyandthemomentumofsubstanceinfundamentalfields.ThecomparisonwiththeLagrangianapproachismade,inwhichtheenergyandthemomentumarecalculated56Copyright©CanadianResearch&DevelopmentCenterofSciencesandCulturesTheHamiltonianinCovariantTheoryofGravitationthroughenergy-momentumtensors.Theproblemofmassisanalyzedwiththehelpofformulasfortheenergy.Inthelastpart,wedescribetheactionfunctionasafunctionhavinganindependentmeaninginphysics–itcanhelptodeterminetheeffectsoftimedilation,arisingfromthechangeofvelocityofbodies’motionorundertheinfluenceoffields.THEPRINCIPLEOFLEASTACTIONInthissectionweshallwritedownknownrelationsfortheLagrangefunctionandtheprincipleofleastactionforthecovarianttheoryofgravitation(CTG).Accordingtothelatter,theequationsofmotionofsubstanceandfieldscanbefoundbyvaryingtheactionfunctionSLdt=∫.Inthecoordinatesxμ=(ct,x,y,z)theLagrangiandependsonthecoordinatesxμ,onthe4-velocityofsubstancemotioncdxudsµµ=(wherec–thespeedoflight,dsindicatestheintervalforthemovingsubstanceunit),on4-potentialDμofgravitationalfieldand4-potentialAμofelectromagneticfieldandonmetrictensorgμvofthereferenceframe.Iftomoveonfromxμanduμtothethree-dimensionalcoordinates,timeandvelocity,thentheLagrangianfunctionwiththesevariablescanbewrittenintheform:(,,,,,,,,,)LLtxyzxyzDAgµµµν=.Herethequantitiesdxxdt=,dyydt=,dzzdt=arethecomponentsof3-vectorofcoordinatevelocity(,,)xyz=v.Whenmovingalongacertaintrajectorythecurrentcoordinatesx,y,zofasubstanceunit,anditsvelocities,,xyzarefunctionsofcoordinatetimet.Ingeneral4-potentialsDμandAμ,whichactonthesubstance,andthemetrictensorgμvdependonthecoordinatesandtime.Ifwetakethecoord