Chapter 15 The Kepler Problem

Transcription

1 Chapte 5 The eple Poble eple, Newto, osieu Betad ad the Hea-Beoulli-aplace- Hailto-Gibbs-Ruge-ez vecto: the geeal solutio to the eple poble. If the solutios ae coic sectios, what exactly is the coe? Ad why does this oelativistic, classical poble lead us to elativity ad quatu echaics? ast Update: 3// Fotuitous, was t it? The aciet Geeks had a thoough kowledge of the geoety of coic sectios. Cosequetly, so did Newto. What olly good fotue that the otio of the celestial bodies tued out to be ust these vey coic sectios. Good luck o ot, it has to be oe of the uiciest bits of physics fo the cobiatio of utility ad atheatical elegace. Ad the eple poble has ot u out of supises yet. Ask youself, if the plaetay otios ae coic sectios, what exactly is the coe? But fist, to eple. By the sweat of his bow, by dit of aual ube-cuchig ove ay yeas, ad with o little ispiatio, eple coued his faous thee laws of plaetay otio out of a outai of obsevatioal data. They ae, [] The plaets ove i ellipses with the Su at oe focus; [] The lie fo the Su to a plaet sweeps out equal aeas i equal ties; [3] The atio of the squae of the peiod to the cube of the sei-ao axis is the sae fo all plaets. With these laws as the clue, Newto deduced the ivese-squae law of gavity. But how ay of these laws do you eed to deduce the ivese-squae law? We ca stat, as Newto did, fo a appeciatio that the geeal law of otio is F p, whee p is the oetu, the tie depedet positio vecto, ad the ass of the obitig body. Reakably oly about oe-ad-a-half of the laws ae equied to deduce the ivese squae law. oe pecisely we eed the secod law ad pat of the fist law, plus a little bit oe. Specifically what we eed is, (i) The plaets ove i closed, stable obits; (ii) The lie fo the Su to a plaet sweeps out equal aeas i equal ties; (iii) Gavity gets weake with iceasig distace. The fist of these is a educed vesio of eple s fist law. Ellipticity is ot etioed. The last, (iii), is the little bit oe, though ost people ight egad it as self evidet. Hece, eple s thid law is ot ecessay, the secod law is uchaged, ad the fist law has bee cosideably weakeed. The thee weake laws, (i), (ii) ad (iii) above, suffice to deduce eple s [] ad [3]. To be oe pecise, it tus out that what eple [3] tells us is that the ivese squae gavitatioal foce is popotioal to the ass of the plaet. eple s secod law is equivalet to the cosevatio of agula oetu, which i tu is equivalet to otio i a cetal foce. A cetal foce is defied as a foce

2 whose agitude depeds oly upo the adial distace fo soe oigi, ot upo the agula coodiates, ad whose diectio is adial, thus, F p f () This is equivalet to a potetial which is a fuctio of the adial distace, oly, V, so that, The agula oetu is defied as, F V V, i.e., V f () p (3) This is a costat of the otio i ay cetal foce, as ca be see fo, d dt p p The fist te i (4) is idetically zeo. The secod te is zeo by vitue of the foce beig cetal, i.e., of the fo (). If the body oves i tie t, the it sweeps out a aea /. Cosequetly the ate of sweepig out aea is ust the agitude of / p / /. So the ate of sweepig out aea is costat, which is eple s secod law. What has bee show is that the assuptio of a cetal foce is sufficiet to deive the secod law. Covesely if the potetial fuctio depeded upo the agula positio as well as the adial positio the the foce would have a o-zeo copoet i a diectio pepedicula to, popotioal to V, ad so would ot be a costat of otio, ad hece eple s secod law would ot hold. So the secod law is also sufficiet to deduce that the foce is cetal. The othe thig that the costacy of tells us is that the plaetay obits ae plaa. It is clea fo (3) that is pepedicula to the istataeous plae of the otio. So the costacy of eas that otio is i a fixed plae. So fa, so good, but kowig that gavity is a cetal foce does ot assist us with its adial depedece. The fuctio f could be aythig at all povided that it depeds oly upo the adial distace ad ot upo the agula coodiate. The secod law is the espected. So how ca we deduce the ivese squae law with o exta ifoatio tha ou substitute laws (i) ad (iii), above? That the ivese squae foce law is the oly possibility which obeys these coditios is kow as Betad s Theoe. It was fist poved oly i 873, ealy two cetuies afte Newto s Picipia (687). A Eglish taslatio of the Fech oigial has bee published by Satos et al (007). The key wod i ou alteative law (i) is stable. Of couse ay cetal foce will adit cicula obits if the iitial coditios ae suitably cotived. Howeve they will always be ustable obits, becoig o-closed ude the sallest of petubatios. Betad (873) poved that thee ae oly two cetal foces which poduce stable, closed obits. These ae the ivese squae law ad a foce which is popotioal to the adius (oughly speakig f 0 (4)

3 a elastic stig). The pupose of ou little bit exta, law (iii), is to exclude the latte. Hece the ivese squae law is all that eais. Newto would ot have bee awae of this eas of deducig the ivese squae law. He was, though, well awae of the solutio to the equatio of otio fo both the ivese squae law ad the elastic stig, discussig both i the Picipia. Both adit elliptic solutios, though oly the ivese squae foce has the oigi (the Su) at the focus. The elastic stig is tied to the cete of the ellipse. Cosequetly Newto would have bee able to eect the elastic stig case o the basis of eple s oigial law []. Havig aived at the coect fo of the gavitatioal foce law, we ca poceed to pove that the otio ust be a coic sectio specifically ellipses fo closed obits ad thece pove eple s fist ad thid laws. Fo auseet, ad because it will pove useful late, we shall show that the solutios ae the coic sectios without itegatig the equatios of otio. We kow that diffeetial equatios caot be ecessay, afte all Newto (687) did it usig oly Euclidea geoety. The tick is to ote that i additio to eegy ad agula oetu, thee is aothe quatity which is a costat of the otio, aely, whee is the uit vecto i the adial diectio, ivese squae gavitatioal foce, p (5) F /, ad is the costat i the The attactive atue of gavity eas that 0. The costacy of is easily established as follows, d dt p whee we have used the idetity p a b c a c b a b c ad also the fact that (which follows fo ad the fact that 0). The costat vecto defied by (5) is geeally kow as the Ruge-ez vecto: oe of its agic latte. Fo ow we shall use it to solve the eple poble without itegatig the equatios of otio. 0 (6) (7) That is, i the deivatio of the eegy levels of the hydoge ato. Just as i this Chapte the taectoies ae foud without diectly solvig the equatios of otio, so i Chapte 35 we shall show how the eegy levels of the hydoge ato ca be foud without solvig the Schodige equatio, i both cases the agic beig povided by the Ruge-ez vecto,. The Ruge-ez vecto appeas to bea this ae oly because it was discussed by these authos i the ealy 0 th cetuy ad thece picked up by Pauli who attached thei ae to it. Howeve it was fist discoveed i piitive fo by Jakob Hea i 70 ad foud its way via Beoulli to aplace late the sae cetuy. It was ediscoveed seveal ties i the ext two cetuies.

4 We ca wite p p whee is a uit vecto pepedicula to the diectio of otio (ad lyig i the plae of otio). Hece (5) ca be e-aaged as, This shows that p p / ad that the taectoy of the heavely body obeys, (8) (9) This is effectively a equatio fo the shape (but ot the size) of the taectoy, because ad ae costats. It is i a o-stadad fo which specifies the oal to the taectoy i tes of the agula positio. All that eais is to show that (9) is equivalet to a abitay coic sectio. To do so ecall that a abitay coic sectio ca be witte, i pola coodiates, Acos whee the oigi is at oe focus. Without loss of geeality we ca assue A 0. The diffeet types of coic sectio occu fo, B A 0: Ellipse B 0; A 0 : Cicle B A: Paabola B 0 B A : Hypebola (+ bach) ; A B 0 : Hypebola (- bach) To establish that (0) ca be witte as (9), take as defiig the x-axis ( gadiet of the taectoy is, dy dx (0) 0 ). The x cos () si y But i pola coodiates fo which x cos, y si the gadiet is, dy dx si cos d d cos si d d () Usig (0) we fid, d A si d (3) So () becoes, dy dx A A Asi si si cos si Asi cos Asi Asi cos A B cos B si cos cos si Acos Acos B cos B si (4)

5 This is exactly as () with A ad B. It eais to fid the absolute size of the taectoy by deteiig the costat. To do so we appeal to the oe costat of otio that we have ot yet used the eegy. Of couse we kow that the total eegy is ust the su of the potetial ad kietic eegies ad hece is, E (5) But eally we should pove that this is a costat. This is easily doe, de dt by vitue of, as oted aleady. Now coside the poit at which the taectoy is closest to the oigi. Call this sallest distace. At this poit we have 0 ad so the velocity is puely i the -diectio. The kietic eegy at this poit is theefoe ad so, 0 (6) E (7) All tes i (7) ae costats, ad hece it fos a equatio fo which to deteie. The solutio(s) is/ae, E E E (8) Oly solutios with positive adial coodiates ae physical. Cosequetly we ca distiguish two cases, (i) E 0 : I this case the fist te i (8) is egative ad so the positive optio ust be chose fo the sig, which yields a uique, positive outcoe fo ; E E E (9) (ii) E 0 : I this case the fist te i (8) is positive ad so eithe optio ca be chose fo the sig sice both esult i a positive outcoe fo. Witig these two possibilities explicitly, E E E ad E E E (0) Now we ealise that the coditio 0 which we have used to deive these esults is ot stictly the coditio fo a iiu distace, but athe fo ay tuig poit of. The two esults i (0) ae espectively the iiu,, ad axiu,, distace of the taectoy fo the oigi. The existece of a fiite axiu distace iplies that the taectoy is a closed obit. I cotast, case (i) yields o such axiu ad hece coespods to a ope taectoy i which the body escapes to ifiity. These coclusios ae cosistet with the eegy equieet fo the two cases. I case (ii)

6 escape to ifiity is clealy ipossible sice the total eegy is egative 3. Rathe less obviously, case (i) establishes that positive total eegy is sufficiet to esue escape. Sice we have aleady deduced that the shape of the taectoies ae the coic sectios, we ca coclude that case (i) gives paabolic o hypebolic taectoies, ad case (ii) elliptic o cicula obits. To deive the agitude of i tes of the othe two costats of the otio, ad E, ote that the oetu whe is p / ad so, x ad is kow i tes of ad E fo (9) o (0). Note that always poits fo the oigi to the peihelio (the poit of closest appoach, ). [The RHS of () ca be show to always be geate tha o equal to zeo by akig use of (9) ad (0)]. Fially, whilst we have deived the taectoy i the fo yet to fid the scale facto,. But this is ow tivial sice usig (), ad hece taectoy as, cos () we have. So fially we have obtaied the geeal solutio fo the whee is give by (9) o the fist of (0). Thus the total eegy ad the agula oetu deteie the solutio. Note that tie has played o pat i this deivatio. The geoety of the taectoy ca be obtaied without cosideig the equatio of otio. But this eas, of couse, that we have ot deived the positio o the velocity as a fuctio of tie. Nevetheless, the velocity is kow as a fuctio of positio. Give, the the adial coodiate follows fo (). The the copoet v of velocity follows fo cos v ad the adial copoet of velocity follows fo, () E v. Note also that, fo closed obits, (0) ad () give diffeet but equivalet elatios betwee the peihelio distace ( ) ad the aphelio distace ( ), E ad Oly eple s thid law eais. It ay see that we will have a difficulty with this sice it ivolves the peiod of the obit, ad we have ot solved fo the tepoal depedecies. But actually it is o poble. We kow that the ate of sweepig out aea is ust /, so the peiod is ust A / whee A is the aea of the obit. (3) 3 This is because, at ifiity, the potetial eegy is zeo so the kietic eegy equals the total eegy. But kietic eegy caot be egative.

7 Because we ow kow that we ae dealig with a ellipse, the aea is A ab whee the sei-ao axis is a / / E, fo (3), ad the sei-io axis ca easily be foud to be b / E. Hece, A 3 3 a 4 a E 3 E E (4) 3 This establishes eple s thid law, aely that / a is the sae fo all plaets, oly if / is idepedet of the ass of the plaet,. This equies that the costat i the gavitatioal foce equatio, (6), is popotioal to the plaet s ass. But by syety the sae ust be tue fo the ass of the Su. So eple s thid law is see to be equivalet to equiig the gavitatioal foce to be popotioal to the poduct of the two asses, G (5) whee G is a uivesal gavitatioal costat. It is salutay to ecall that ou udestadig of the atue of Newto s G is essetially the sae ow as it was i 687. Whee ou kowledge has advaced, thaks to Eistei, is i the appeciatio that (5), which expesses the equivalece of gavitatioal ad ietia ass, ca be udestood as a cosequece of geeal covaiace ad the geoetical itepetatio of gavity. So uch fo eple s laws, but what about the questio posed at the stat of this Chapte: give that the taectoies ae coic sectios, what is the coe? Of couse we could always cotive a coe which fits ay of the obits. But such coes see abitay ad without eaig, ad thee would be a plethoa of diffeet coes fo diffeet taectoies. It would see cazy to suggest that, i fact, thee is oe sigle coe whose sectios povide evey possible obit, ope ad closed. This caot be, you ague, sice a ellipse of a give ecceticity could aise i ay oietatio i space ad all these ellipses obviously caot be geeated as sectios of ust oe fixed coe. Quite tue so log as we cofie ouselves to coes i 3D. But let us see what happes whe we allow ouselves a fouth diesio. Now we ca coside a coe whose ight-sectios ae sphees athe tha cicles. Callig the exta diesio t the coe is, t x y z fo t 0 (6) Aazigly ay obit is a sectio of this coe which is, of couse, ust the fowad light coe. Coside the sectio with espect to the plae z 0, t costat. Clealy this is ust a cicle i the x, y plae. But cicula obits i ay plae ca be geeated i the sae way sice (6) ivolves sphees i the 3-space x, y, z ad sectios of sphees with espect to ay plae ae cicles. But how ae othe coic sectios poduced? To do so the plae which defies the sectio ust be tilted so that the t -diectio has a o-zeo poectio oto it. Suppose the plae is defied by z 0 ad a oal vecto lyig i the t, x plae. If is tie-like the the obits ae ellipses. If is space-like the taectoies ae hypebolae. If is ull the taectoy is a paabola. As with the cicula obits, the spatial oietatio of the taectoy ca be otated abitaily owig to the spheical

8 syety of (6). So, we ca poduce ay coic sectio i the 3-space eas. x, y, z by this But what is goig o hee? Just what is the light coe, the epitoe of elativistic costucts, doig poppig up i a o-elativistic poble? You ay vey well ask! These attes u deep. The iteested eade should seek a oe copetet guide, such as Guowu eg (0) who discoveed this siple itepetatio of the eple coic sectios, o Guillei ad Stebeg (990). Howeve, we ca give soe clue as to what ight have bought the light coe ito the poble. It is dow to the agic of the Ruge-ez vecto. Recall that the Poisso backet is defied by, 3 f g f g f, g (7) x p p x i We aleady kow what this will yield fo the copoets of the agula oetu vecto, k k i i i i, (8) whee k is the alteatig teso. With a little tedious algeba the defiitio of the Ruge-ez vecto, (5), leads to, ad,, (9) k k E, k k (30) Istead of deivig (8-30) via the Poisso backet we could equivaletly have coveted the oetu to the quatu echaical opeato by the eplaceet p i, so that ad also becoe opeatos [otig that (5) has to be eplaced with the syetised fo p p / ]. Equs.(8-30) would the hold fo the coutatos betwee these opeatos, with the io chages that each RHS would pick up a facto of i ad the te E o the RHS of (30) would be i eplaced by the Hailtoia opeato, H. Eve i the quatu case, though, we ay cofie attetio to a sub-space of states with the sae eegy, E, to avoid this latte coplicatio. I the case of boud states (elliptic obits) with E 0, we ca adopt a oalised Ruge-ez vecto (o opeato) such that, E So that (8-30) becoe, adoptig the quatu echaical opeato itepetatio ow, k k (3), i (3), i (33) k k ad,, i (34) k k

9 I (3-34) the eade ay ecogise the coutatos of the six geeatos of the goup of otatios i 4-diesioal Euclidea space, the goup SO 4. If ot, the geeatos of this goup ca be epeseted by coutato x p x p with the usual x, p i ad whee the Geek subscipts hee spa the age, 4. The eade ca check fo hiself that these geeatos epoduce the coutatio stuctue of (3-34), specifically with i ik k / ad i i 4 (ati subscipts takig values,,3 oly). I the case of paabolic o hypebolic taectoies, with E 0, the oalisatio of the Ruge-ez vecto (o opeato) is, E This leads to a ius sig o the RHS of (34) ad the esultig coutato stuctue idicates that the goup has chaged fo SO 4, fo the boud states, to SO 3, fo the fee taectoies. The sigificace of this is that SO 3, is ust the coected pat of the hoogeeous oetz goup. This is the esult that we have bee wokig towads. The suggestio is that it is the algebaic stuctue of the ad vectos (o opeatos) which ceates the geoetical lik with elativity despite the eple poble itself beig o-elativistic. The athe stuig coclusio that the eple taectoies ae coic sectios because they ae sectios of the fowad light coe is atched by the equally elegat obsevatio that this fact was always lukig i the agic of the Ruge-ez vecto. (35) Refeeces J.Betad J (873) Théoèe elatif au ouveet d'u poit attié ves u cete fixe, C.R. Acad. Sci. Pais F C Satos, V Soaes ad A C Tot, A Eglish taslatio of Betad s theoe, at.a. J. Phys. Educ. 5 (0) 684 (also axiv: ). I.Newto (687) Philosophiae Natualis Picipia atheatica, (Royal Society of odo). Guowu eg (0): Geealized eple Pobles I: Without agetic Chages, axiv: Victo Guillei ad Shloo Stebeg (990): Vaiatios o a Thee by eple, Povidece, R.I., Aeica atheatical Society. F.C. Satos, V. Soaes, ad A.C. Tot

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