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1、CelestialMechanicsNoteSet3:GeneralThreeBodyProblemandtheOrbitalCon¯gurationsofEulerandLagrangeJ.D.MirelesJamesJanuary1,2007Contents1Introduction12NumericalStudyofLagrange'sEquilateralTriangularSolu-tions23NumericalStudyofEuler'sCollinearSolution84JacobiCoordina
2、tes125AccelerationinRotatingReferenceFrames19AMatLabCode23A.1Theprogramfortheequilateralsolutions.............23A.2EulerStraightLinePrograms....................291IntroductionThissetofnotescontainsmaterialaboutthegeneralthreebodyproblem.Thisproblemhaswellknowne
3、quilibriasolutions,whencastinrotatingcoordinates.Wepresentseveralnumericalexperimentsdevotedtothepropertiesofthesespecialsolutions.(ThederivationsoftheEulerandLagrangesolutionsaretakenforgrantedinthispresentversionofthenotes,althoughIwouldliketo¯xthatatsomelate
4、rdate).Asidefromthematerialonthesespecialsolutions,thissetofnotesalsocontainssomeanalyticresultsthatareusefulinthestudyofthethreebodyproblem.Speci¯cally,theequationsofmotionaredevelopedinJacobiCoordi-nates,andtheexpressionsforvelocityandaccelerationinarotatingr
5、eferenceframearederived.1Thislastitemcanbeusedtoderivetheequationsofmotionforthecircularrestrictedthreebodyproblem,whichisthesubjectofthenextsetofnotes.Wewillchoosenottouseritthere,infavorofacompletelybottomupapproach.Neverthelessitisausefulresulttohaveeasyrefe
6、renceto,andthederivationgivenheredi®ersfromtheonesusuallygiveninphysicstextsinthatnoappealismadetointuitionnorisanyusemadeofin¯nitesimalrotations".Infactititnotevennecessarytomakeanylimitingarguments;allthatisneededisamessycomputation.2NumericalStudyofLagrange
7、'sEquilateralTri-angularSolutionsThissectionmakesastudyoftheequilateraltrianglesolutionstothegeneralthreebodyproblem,followingthemethodputforwardintheaxillarynotes[Ocampo2006].GiventheinitialdataG=1,thattheinitialpositionofthecenterofmassforthesystemisrcm(0)=(0
8、;0;0),thattheinitialvelocityofthecenterofmassisvcm(0)=(¡1;1;0),andthatthemassesarem1=1:0,m2=2,m3=3theprogramhw3prob2computesasetofinitialconditions,givenininertialcoordinate
