newton revolving classical mechanics newton revolving identifies type central force needed angular speed particle factor affecting radial motion figures newton applied understanding overall rotation figure observed moon planets term radial motion signifies motion away center force angular motion perpendicular radial motion derived book published proposition showed added force central force magnitude depends distance particle point fixed space center proposition derived formula force showing cube force varies cube proposition newton extended arbitrary central forces assuming particle moved nearly circular orbit noted commentary newton remained largely unknown undeveloped centuries studied collaborators exact extension came work historical context motion astronomical bodies studied systematically thousands years stars observed rotate uniformly maintaining relative positions bodies observed background fixed stars bodies called planets greek word generally direction path sky individual planets reverse direction briefly exhibiting motion describe forward backward motion developed concept according planets carried rotating circles carried rotating circles orbit described sufficient number chosen approach corresponds modern transform roughly years later published developed system match best astronomical observations era explain adopted aristotle according planets confined concentric rotating spheres model universe authoritative nearly years modern understanding motion arose combined efforts physicist 16th century credited extremely accurate measurements motions able derive laws motion according laws planets sun earth second laws specific predictions planets sweep equal areas equal time square periods equals fixed constant times cube semi major axis subsequent observations showed long axis called line gradually time rotation known orbit points body closest away attracting center planets sun correspond closest publication roughly eighty years later provided physical theory accounted laws theory based newton laws motion law universal particular newton proposed force bodies central force varied square distance arguing laws motion newton showed orbit particle acted force section specifically infinity conclusion holds bodies present body problem motion bodies acting mutual body problem remained centuries newton solutions special cases discovered newton proposed planets sun largely elliptical sun dominant presence planets ignored analogy elliptical orbit moon earth dominated earth gravity sun gravity bodies solar system neglected newton stated gradual lunar effects neglected interactions particular stated moon orbit effects interactions sun newton revolving attempt understand according addition particular type central force cube force produce rotating orbit angular speed multiplied factor radial motion left unchanged restricted specific type force relevant square interactions planets unlikely sum exactly cube force applicable types forces newton found best arbitrary central force cube potential limit nearly circular elliptical low true solar system find newton developed infinite series viewed forerunner taylor expansion allowed newton estimate rate arbitrary central forces newton applied test models force causing moon orbit problem moon motion complex newton published accurate model moon accurate model models moon motion developed late 19th century newton general merely explaining describes effects adding cube force central force square forces newton law universal law newton problems classical mechanics eliminating cube forces consideration radial angular motions calculated cube force effect calculated angular speed particle formula mathematical statement consider particle moving arbitrary central force magnitude depends distance particle fixed center motion particle central force lies plane position particle described polar coordinates radius angle particle relative center force figure coordinates change time particle moves imagine second particle mass radial motion angular speed times faster particle words angles particles related equation newton showed motion second particle produced adding cube central force force acts particle formula magnitude particle angular momentum constant motion central forces greater negative number added cube force attractive observed green planet figures contrast positive number added cube force observed green planet figures red planet figures alteration particle path addition cube force changes path followed particle path particle ignores time radial angular motions relates radius angle purpose angle variable increase indefinitely particle revolves central point multiple times example particle revolves twice central point returns starting position final angle initial angle increased formally angle variable defined integral angular speed formula similar definition holds angle second particle path particle described form path second particle given function example let path particle formula path second particle given formula close equal second orbit resembles revolves gradually center force known figure greater orbit direction orbit figure orbit opposite direction orbit figure rotate uniformly constant angular speed true circular orbit angular speed angular speed second particle faster slower particle words angular speeds satisfy equation newton revolving states angular speeds related constant combining shows angular speed equals constant constant according conservation angular momentum changes radius formula particle mass angular momentum respectively constant constant radius constant orbit circle case orbit change example simplest illustration newton occurs initial force case particle stationary travels straight line travels straight line pass origin blue line figure equation line written polar coordinates formula angle distance figure distance begins infinity decreases gradually distance reaches minimum gradually increases infinity minimum distance impact defined length perpendicular fixed center line motion radial motion possible cube central force added cube central force form formula positive negative attractive cube force introduced newton says corresponding solutions shape called curves defined equation formula constant equals formula right hand equation positive real number solution corresponds argument equals goes zero radius goes infinity range allowed angles small force red curve right figure hand greater range allowed angles increases corresponding attractive force green blue curves left figure orbit particle wrap center times possible values range zero infinity corresponds values ranging negative infinity positive upper limit attractive cube forces negative corresponding orbit ones illustrated figure stronger forces correspond faster linear motion solution types given terms formula constant formula form corresponds figure possible values range zero infinity corresponds values greater positive number spiral motion occurs cube central forces applies case large given taking limit going zero yields form spiral called spiral spiral solution formula arbitrary curves result strength force exactly angular momentum mass term formula closed cube central forces types central forces increase distance law square forces newton law universal law unusual property particle moving type force returns starting place initial velocity provided lacks sufficient energy infinity words path bound particle closed motion repeats indefinitely matter initial position velocity shown property true types forces general particle return starting point velocity newton shows cubic force applied particle moving linear square force orbit remains closed provided equals rational number number called rational written fraction cases addition cubic force causes particle complete center force time original particle completes method producing closed violate added cubic force depends initial velocity particle special types closed closed trajectory called orbit formula example green planet figures green orbit figure resulting orbit original orbit conversely closed trajectory called orbit formula example green planet figure green orbit figure resulting orbit called original orbit unlikely occur nature helpful illustrating newton limit nearly circular proposition newton applies revolving develop method finding force laws govern motions planets noted planets moon long axis determined accurately astronomical measurements long axis defined line connecting positions minimum maximum distances central point line connecting illustration long axis planet mercury defined line successive positions time long axis bodies gradually generally degrees complete revolution bodies attracting body general effects effects newton method uses sensitive probe type force applied planets newton describes effects adding cube central force newton extends arbitrary central forces restricting attention nearly circular low true seven solar system newton applied planet mercury roughly suggested orbit roughly justification method suggested wilson harper according argument newton considered angle angle successive minimum maximum distance center smooth continuous function square force equals positions minimum maximum distances lie line initially low quasi circular general equal isolated values randomly chosen value unlikely observed slow rotation suggest force gravity square law formula newton writes terms new function formula average radius nearly circular orbit newton expands series known taylor expansion powers distance appearances series resulting cube force term cube force revolving newton derives equivalent angular scaling factor nearly circular formula words application arbitrary central force nearly circular elliptical orbit accelerate angular motion factor affecting radial motion significantly elliptical orbit stationary particle center force moves end long axis corresponding angle general central force equals general law examples newton illustrates formula examples central force power law proportional formula indicates angular motion multiplied factor angle equals angular scaling seen gradual rotation long axis figure noted orbit mean angular speed equals mean angular speed particle stationary particle requires time implies time long axis rotate angle square law newton law universal equals angular scaling angle elliptical orbit stationary final illustration newton considers sum power laws formula angular speed factor formula applies power law sum power laws examine moon orbit moon orbit motion moon measured accurately noticeably complex planets ancient greek noted periodic variations moon orbit small orbit plane generally occur monthly twice monthly time scale line gradually period roughly years line turns circle roughly double time years accounts roughly year called cycle lines experience small fluctuations motion monthly time scale published reasonably accurate model moon motion moon assumed follow elliptical orbit sufficiently accurate simple method predicting moon motion solved navigational problem determining ship newton time goal predict moon position arc minutes correspond error terrestrial model predicted lunar position errors arc minutes comparison diameter moon roughly arc minutes newton revolving ways account moon showed moon observed accounted changing force law gravity square law power law roughly formula adopted approach modifying square law slightly explain planet mercury observed ironically hall theory ruled careful astronomical observations moon currently accepted explanation involves theory general adds force varies fourth power distance second approach explaining moon newton suggested influence sun moon motion approximately equivalent additional linear force formula term corresponds attraction moon earth moon distance earth second term newton reasoned represent average force sun gravity system force law result earth surrounded spherical dust cloud uniform density formula nearly circular estimates newton showed force law account moon predicted angle observed revolution long axis rotate roughly half observed published book noted commentary newton remained largely unknown undeveloped centuries newton discovered newton assumed angular motion second particle times faster particle contrast newton require radial motion particles required related linear equation formula transformation changes path particle path particle written second particle path written formula motion particle produced central force showed motion second particle produced following force formula according equation second force obtained scaling force changing argument adding square cube central forces comparison newton revolving corresponds case case original force scaled argument unchanged cube force added square term path second particle consistent formula given newton newton found section specifically based largely geometry proposition problem required body curve revolves center force manner body curve rest newton proposition depends proposition derived earlier proposition provides test net force acting point mass particle central force newton showed force central particle sweeps equal areas equal times measured center newton begins particle moving arbitrary central force motion particle force described radius center function time angle time particle sweeps approximate right triangle area formula force acting particle assumed central force particle sweeps equal angles equal times newton proposition expressed way rate sweeping area constant formula constant velocity calculated follows positions closest distance attracting center velocity radius perpendicular angular momentum mass particle written related rate sweeping areas formula consider second particle orbit identical radius angular variation multiplied constant factor formula velocity second particle equals particle multiplied factor formula constant second particle sweeps equal areas equal times proposition second particle acted central force conclusion proposition proposition difference forces bodies equally fixed orbit revolving varies cube common altitudes find magnitude original central force newton calculated difference geometry definition acceleration proposition showed difference proportional cube radius specifically formula given writes terms constant formula proposition problem find motion approaching near circles proposition newton derives consequences revolving limit nearly circular generally valid orbit moon earth allows newton consider great variety central force laws merely square cube force laws modern modern newton published assumption second angular speed times faster formula behavior time angular related factor formula equation motion radius particle mass moving central potential given formula applying general formula yields equation formula arranged form formula equation relating radial forces understood follows difference angular speeds angular causes difference force requirement offset radial force altered cube force newton expressed terms potential energy defined central forces formula radial force equation written terms potential energies formula integrating respect distance states fold change angular speed results adding square potential energy given potential energy formula
Newton's theorem of revolving orbits