B E (c) Figure 2.1.1: (a) a material particle in a body, (b) a place in space, (c) a configuration of the body

Transcription

1 Kcs Kcs s cocr wh chrcrs o. h ol s o prss hcl for h foro oo of rls. I wh follows r of por qs ly cors sco-orr sors r roc. Ech of hs qs for pl h locy foro r or r of foro sor llows o o scr prclr spc of for rl. o cosro s o wh s cs h foro o h cs s h co of forcs o h rl hs wll scss h chpr. h frs sco rocs h rl spl coors scrpos. h sco hr scos scss h sr sors. h forh ffh sh scos l wh rs of foro rs of ch of c qs. h hory s spcls o sll sr foros sco 7. h oo of ocy s scss sco 8 h fl scos 9- l wh cs s h coc coor sys cl h por oos of psh-forwr pll-c h L r. 99

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3 Sco.. Moo.. h Mrl Boy Moo Physcl rls h rl worl r ol s src hcl y cll oy. hs oy cosss of f r of rl prcls. Show... s oy B wh rl prcl. O sshs w hs oy h spc whch rss hroh whch rls. Show... s cr plc flr Ecl spc E. B E c r..: rl prcl oy plc spc c cofro of h oy By f h rl prcls of h oy o plcs spc o hs cofro of h oy...c. cofro c prss s pp of h prcls o h plcs.. oo of h oy s fly of cofros prrs y.. y Eq... s h loco spc of h rl prcl.... hr r r of ffr wys whch h oo c scr. Eq... s h rl scrpo whch h oo s scr rs of h rl prcls whch p h oy. hs prcls r o h scr ss prcls of wo chcs rhr hy r ry sll poros of coos r; h of prcl s prcs h fos whch follow Sol Mchcs Pr III

4 Sco. r..: oo of rl h frc Crr Cofros Choos ow so rfrc cofro.... h oo c h sr rl o hs cofro. h rfrc cofro h h cofro occp y h rl whch cs s of cll h l cofro. or sol h rl o choos cofro for whch h rl s srss-fr whch cs s of cll h for cofro. Howr h choc of rfrc cofro s coplly rrry. Iroc Crs coor sys wh s cors E for h rfrc cofro. rl prcl po h rfrc cofro c h ss q poso cor E rl o h or of h s. h coors of h prcl r cll rl coors or Lr coors or rfrl coors. So lr sy h rl occps ffr cofro whch wll cll h crr cofro or for cofro. Iroc sco Crs coor sys wh s cors for h crr cofro.... I h crr cofro h s prcl ow occps h loco po whch c ow lso ss poso cor. h coors r cll spl coors or Elr coors. Ech prcl hs hs wo ss of coors ssoc wh. h prcl s rl coors sy wh hroho s oo. h prcl s spl coors ch s os. Sol Mchcs Pr III

5 Sco. rfrc cofro crr cofro E E r..: rfrc crr cofros I prcc h rl spl s r slly o coc so h h s cors E r h s s...4. rhlss h s of ffr s cors E for h rfrc crr cofros s sfl wh h rl spl s r coc sc hlps ssh w qs ssoc wh h rfrc cofro hos ssoc wh h spl cofro s lr. E E r..4: rfrc crr cofros wh coc s I rs of h poso cors h oo.. c prss s rloshp w h rl spl coors Mrl scrpo.. or h rs rlo Spl scrpo..4 If o ows h rl coors of prcl h s poso h crr cofro c r fro... lrly f o focss o so loco spc h crr cofro h h rl prcl occpy h poso c r fro..4. hs s llsr h follow pl. Sol Mchcs Pr III

6 Sco. Epl Eso of Br Cosr h oo..5 hs qos r of h for.. sy h h prcl h ws orlly poso s ow poso. hy rprs spl rslo l so of rl s show...5. o h. cofro cofros > r..5: rslo so of rl los of h for..4 c o y r..5: hs qos sy h h prcl h s ow poso ws orlly poso. Coc Coors h rl spl coor syss s hr r f Crs syss. lr ho of scr oo s o ch h rl coor sys o h rl l for wh h rl. h oo s h scr y f how hs coor sys chs. hs s h coc coor sys. I rl h s of coc sys wll o r lly orhool crlr sys s rqr. Coc coors wll.... h Mrl Spl Dscrpos y physcl propry sch s sy prr c. or c propry sch s splc or locy of oy c scr rs of hr h rl coors or h spl coors sc hy c rsfor o ch ohr s..-4. rl or Lr scrpo of s s o whr h Sol Mchcs Pr III 4

7 Sco. rl coors r h p rls. spl or Elr scrpo of s s o whr h spl coors r s. Epl prr of Boy Sppos h prr θ of oy s rl coors θ..6 h spl scrpo θ...7 ccor o h rl scrpo..6 h prr s ffr for ffr prcls h prr of ch prcl rs cos or. h spl scrpo..7 scrs h -p prr spcfc loco spc...6. Dffr rl prcls r flow hroh hs loco or. oo of l rl prcls r..6: prcls flow hroh spc I h rl scrpo h o s focs o spcfc rl. h pc of r r cosro y ch shp sy locy so o s lwys h s pc of rl. O h ohr h h spl scrpo o s focs o f loco spc. Mrl y pss hroh hs loco r h oo so ffr rl s r cosro ffr s. h spl scrpo s h o os of s l Mchcs sc hr s o rl rfrc cofro of h rl s s coosly o. Howr oh h rl spl scrpos r s Sol Mchcs whr h rfrc cofro s slly h srss-fr cofro... Sll Prros lr r of por prols ol rls whch for oly y rlly sll o. pl wol h sl srcrl cols l r os lo. I hs yp of prol hr s rlly o sco o Sol Mchcs Pr III 5

8 Sco. w h wo wpos o h lyss s splf rly s lr o Sll Sr hory Prols. h sy of rl s y ρ h oo s y h qos. wh of scrpo s hs for h sy wh of scrpo s hs for h oo? r-wr h sy rs of wh s h o hs scrpo of h sy? c s h sy of y rl prcl ch wh? r h oo qos so h s h p rl wh s h o hs scrpo of h oo? rw h l l o h or o sch h poso of hs l of rl s. Sol Mchcs Pr III 6

9 Sco.. Dforo Sr r of sfl wys of scr h foro of rl r scss hs sco. o s rsrc o h rfrc crr cofros. o cosro s o h prclr sqc y whch h crr cofro s rch fro h rfrc cofro so h foro c cosr o p of. I wh follows prcls h rfrc cofro wll of r for hos h crr cofro for. I ch fro Chpr lowr cs lrs wll ow rsr for oh cor- sor- fcos of h spl coors whrs ppr-cs lrs wll rsr for fcos of rl coors. hr wll cpos o hs shol clr fro h co wh s pl... h Dforo r h foro r s h fl sr of foro co chcs. I s h sco orr sor whch ps l ls h rfrc cofro o l ls coss of h s rl prcls h crr cofro. Cosr l l fro poso h rfrc cofro whch cos h crr cofro.... h s.. r cpl s s o r o phss h hs s r wh rspc o h rl coors h rl r /... r..: h Dforo r c o l l o c h rl rs spl rs of rl or spl fls s lr Sol Mchcs Pr III 7

10 Sco. I s csory o o h oo cor-fco.. sply y.. so h r J Dforo r.. J wh co of.. J J Lowr cs cs r s h oo o o qs ssoc wh h spl ss { } whrs ppr cs cs r s for qs ssoc wh h E. rl ss { } I o h s ffrl qy hs prsso hs so rror ssoc wh ; h rror o rs of orr hhr lc fro ylor srs s o zro s h ffrl. h foro r whos copos r f hs chrcrss h foro h horhoo of po pp fsl l ls fro h rfrc cofro o h fsl l ls fro h crr cofro.... for fr r..: foro of rl prcl Epl Cosr h c of rl wh ss of lh llsr y o ls.... I s for o h rclr prs llsr hs col ch for pl y coos roo srch oo. h rl spl coor s r coc. h rl scrpo of h foro s h spl scrpo s f 6 Sol Mchcs Pr III 8

11 Sco. f E E E 6 B E D E C C B D r..: for c h J / 6 / or pl l l [] [ / ]. Hooos Dforos hooos foro s o whr h foro r s for.. p of h coors h ssoc oo s r ff. Ery pr of h rl fors s h whol os srh prlll ls h rfrc cofro p o srh prlll ls h crr cofro s h o pl. Mos pls o cosr wh follows wll of hooos foros; hs ps h lr o hooos foro lyss s sfl slf sc os of h sc prl s of rls.. h l sl s ol hooos foros. Boy oos rslos O c cos cor c o h oo c who ch h foro r c r. hs h foro r os o o cco r-oy rslos of os spc. If oy oly rsls s r oy spc h I c. If hr s o oo h o oly s I. Sol Mchcs Pr III 9

12 Sco. If h oy ros s r oy wh o rslo h..8. roo sor h Irs of h Dforo r h rs foro r l l. I s f s crrs h spl l l o h rl I r I Irs Dforo r..4 so h co of I I..5 wh s Eq..5. I M M δ..6 Crs Bs Vcors Eplcly rs of h rl spl s cors s.4. J E J I J E I E J..7 / /. J J M M J J so h E E Bcs c o cors o cofro o proc cors h ohr cofro hy r r wo-po sors. hy r f oh cofros. hs s hhlh y hr h oh rfrc crr s cors E hr Crs rprso..7. Hr follow so por rlos whch rl sclr- cor- sco-orr sor-l fcos h rl spl scrpos { Prol }. rφ rφ r rv r :..8 Hr φ s sclr; V r h s cor h forr fco of h rl coors h rl scrpo h lr fco of h spl Sol Mchcs Pr III

13 Sco. Sol Mchcs Pr III coors h spl scrpo. Slrly s sco orr sor h rl for s h ql spl for. Epl Cosr h foro 5 E E E so h 5 Cosr h cor whch h rl scrpo s 5 5 E E E V h rl spl rs r 6 r r V c s h V r 6 6 r.. h Cchy-r Sr sors h foro r scrs how l l h rfrc cofro ps o l l h crr cofro. Ohr sfl srs of foro r h Lf Cchy-r Sr h Cchy-r Sr sors. hy sr of how h lhs of l ls ls w l ls hroh h cor o proc ch w cofros.

14 Sco. h h Cchy-r Sr Cosr wo l ls h rfrc cofro o h l ls whch r pp h crr cofro. h s.. C co of C..9 whr y fo C s h rh Cchy-r Sr C C IJ I J h Cchy-r Sr.. I J I s syrc pos f cf... sor s Eq...7 low whch pls h hs rl pos ls hs hs por cosqcs s lr. Eplcly rs of h s cors C E I J I J E E E... J I J Js s h l l s cor f ssoc wh h rfrc cofro C s f ssoc wh h rfrc cofro c o cors h rfrc cofro so s cll rl sor. h rs of C C - s cll h Pol foro sor. h Lf Cchy-r Sr Cosr ow h follow s Eq...8c: co of.. whr y fo s h lf Cchy-r Sr lso ow s h r sor: K K Lf Cchy-r Sr.. K K hs s syrc pos f sor oly hr s f h crr cofro so s cll spl sor. h rs of - s cll h Cchy foro sor. rh cs s o h rh of h forl Sol Mchcs Pr III

15 Sco. I c s h h rh lf Cchy-r sors r rl hroh C - - C..4 o h sors c rl.. C wo-po.. or spl... Whr yp hy r hy c lwys scr s rl or spl coors hroh h oo pp.. h s s h rl or spl scrpos. hs o sshs w for pl spl sor whch s rsc propry of sor h spl scrpo of sor. h Prcpl Sclr Irs of h Cchy-r sors s.. r r rc r..5 hs hols lso for rrry powrs of hs sors Eq...7 h rs of C r ql. C r r hrfor fro.. h Srch h srch or h srch ro s f s h ro of h lh of for l l o h lh of h corrspo for l l: h Srch..6 ro h rlos ol h Cchy-r Srs l cross y h sqr of h lh of or Hr h qs C..7 / / r cors h rcos of. hs hroh hs rlos C r how ch l l srchs fro..7 C c s o pos f. O sys h l l s srch or coprss ccor o > or <. Sol Mchcs Pr III

16 Sco. Srch lo h Coor s Cosr ow hr l ls ly lo h hr coor s. Sppos h h rl fors spcl wy sch h hs l ls ro pr srch h s hy ch lh wh o ch h rh ls w h. If h srchs hs rcos r h..8 h foro r hs oly ol ls s r for: J δ J o s..9 h s r hs cs cll h prcpl s of h rl h srchs r cll h prcpl srchs. L ls o o hs hr s wll rl srch/corc ro rl o ch ohr. sphrcl l of rl wll for o llpso wh h s of h llpso coc wh h s cors. or pl l l [ α α] / wh [ α α ros f. ] h Cs of l Syrc srchs y Cosr ow ohr spcl foro whr s rl syrc sor whch cs h ls r rl h cors for orhoorl ss cf.... I y coor sys wll rl rsl h srch of l ls h ch of h ls w l ls. Howr f o chooss coor s o h cors of h fro Eq...- o c wr [ ].. whr r h ls of h cors 4 r h prcpl s h ls r h prcpl srchs. hs cs h s lo s s rl syrc o c lwys f coor sys lo whos s h rl ros pr srch wh o roo. sphrcl l of rl wll for o llpso ow h s of h llpso coc wh h prcpl s of. wh h rl spl ss cors coc 4 r h cors of of Ê wh I coc; wh h ss r o coc h I oo of ro l ls cos os hs opc wll lr h co of ocy Sol Mchcs Pr III 4

17 Sco...4 h r-lr Elr-ls Sr sors Whrs h lf rh Cchy-r sors foro o h ch l w l ls h srch of l ls h r-lr sr h Elr-ls sr sors rcly foro o h ch h sqr lh of ls. Spcfclly wh h r-lr sr E oprs o l l s hlf h ch h sqrs of h for for lhs: E { C } { C I } co of E.. whr C I I E C δ E IJ IJ IJ r-lr Sr.. I s syrc pos f rl sor. Slrly h syrc spl Elr-ls sr sor s f hroh co of.. I I Elr-ls Sr..4 Physcl M of h Copos of E l l h -rco [ ] so h [ ] sqr of h srch of hs l s C C E C h so s / E o hs. Do h so of y. h E E E..5 Sol Mchcs Pr III 5

18 Sco. slrly for h ohr ol ls E E. Wh h foro s sll E s sll coprso o E so h E E. or sll foros h h ol rs r ql o h sos. L θ o h l w h for ls whch wr lly prlll o h s. h cosθ E E E C C..6 slrly for h ohr off-ol ls. o h f θ π / so h hr s o l ch h E. f h foro s sll h E E r sll π θ π θ θ s cos E..7 I wors: for sll foros h copo E s hlf h ch h orl rh l...5 Srch oo sors h foro r c lwys copos o h proc of wo sors srch sor roo sor o of wo ffr wys rl or spl rsos. hs s ow s h polr coposo s scss..7. O hs Polr Dcoposo Mrl..8 Hr s propr orhool sor.. I sor. I s sr of h locl roo. wh cll h roo s q syrc sor cll h rh srch sor. I s sr of h locl srch or corco of rl. Cosr l l. h so { Prol }..9.. Sol Mchcs Pr III 6

19 Sco. hs hs s fo of C C h h Srch sor.. ro.. h rh Cchy-r sr C y cosqc h Elr-Lr sr E oly foro o h srch of l ls; os o foro o h roo h s prc y prcl r oo. h foro r howr cos foro o oh h srch roo. I c lso s fro..- h s rl sor. o h sc h for l l s frs srch y s h ro y o h for l h l y lso ro r oy rslo c...4. s wo-po sor. prcpl rl s fl cofro srch for r..4: h polr coposo Elo of I orr o l s cssry o l C. o l h sqr-roo C s frs o rlo o s prcpl s so h s ol h h sqr roo c of h ol ls s..6. h h sor s o rsfor c o h orl coor sys. Epl Cosr h oo h hooos foro of sqr h pl s s show...5. Sol Mchcs Pr III 7

20 Sco. Sol Mchcs Pr III 8 r..5: foro of sqr O hs [ ] [ ] [ ] E E C E ss : 5 5 ss : o h s o syrc so h h h oly o rl l fc hr os h copl ls h cors y o orhoorl. C o h ohr h y s ry fo s syrc; s fc pos f so hs pos rl ls for orhoorl s. o r h prcpl s of C s cssry o l h ls/cors of h sor. h ls r h roos of h chrcrsc qo.9.4 III II I C C C α α α h frs sco hr rs of h sor r y.9.7 so h 6 6 α α α wh roos 8 α. h hr corrspo cors r fo fro C C C C C C C C C α α α α α α hs orlz h cors so h hy r cors for rh-h s...6: for 8 α 7 E E for α E E for α 4 4 E

21 Sco. prcpl rl rcos r..6: foro of sqr hs h rh Cchy-r sr sor C wh rspc o coors wh s cors E E E h s rs of prcpl coors s 8 [ C] ss : hs rsl c chc s h sor rsforo forl.. [ C ] [ Q] [ C][ Q] whr Q s h rsforo r of rco coss s lso h pl h of.4. Q M M M M M / / M / /. h srch sor wh rspc o h prcpl rcos s [ ] [ C] ss : hs ls of whch r h sqr roo of hos of C r h prcpl srchs s for hy r ll. I h orl coor sys s h rs sor rsforo rl.. [ ] [ Q][ ][ Q] so h / [ ] / / ss : E E / Sol Mchcs Pr III 9

22 Sco. / / [ ] [ ] / / ss : E c rf h s roo sor.. s propr orhool. r o h foro of h sqr h srch roo r s llsr...7 h co of s c y h rrows for h sqr o h o o prlllor whrs ros h prlllor hroh 45 s r oy o s fl poso; o h h l l lo h ol c y h hy l ls lo prcpl rco of hrfor ros pr srch. r..7: srch roo of sqr Spl Dscrpo polr coposo c h spl scrpo. I h cs Polr Dcoposo Spl.. Hr s syrc pos f sco orr sor cll h lf srch sor whr s h lf Cchy-r sor. s h s roo sor s pprs h rl scrpo. hs ll sphr c rr s frs srch o llpso whos s r h prcpl rl s h prcpl s of h ro; or frs ro h srch o llpso whos s r h prcpl spl s h prcpl s of. h rsl s h s. h lop h spl scrpo s slr o h o for h rl scrpo o fs y loy wh.... I h o pl rs o h s h spl ol for Sol Mchcs Pr III

23 Sco. [ ] ss : so h sqr ros frs h ros pr srch lo h coor s whch r h prcpl spl s h sqc s ow s show r..8: srch roo of sqr spl scrpo loshp w h Mrl Spl Dcoposos Copr h wo coposos o ss h h rl spl sors ol r rl hroh C..4 rhr sppos h hs l cor. h so h. B so. hs lso hs l cor. ro hs s s h h roo sor ps h prcpl rl s o h prcpl spl s. I lso follows h c wr plcly rs of h rl spl prcpl s copr h frs of hs wh..5 5 :..5 h foro r cs o h prcpl s s cors ccor o { Prol 4}..6 5 hs s o spcrl coposo of lss hpps o syrc whch s orr o h spcrl coposo Sol Mchcs Pr III

24 Sco. Sol Mchcs Pr III h rprso of rs of oh rl spl prcpl s cors..5 hhlhs hr wo-po chrcr. Ohr Sr Msrs So ohr sfl srs of sr r h Hcy sr sr: H l rl or h l spl h Bo sr sr: I B rl or I spl h Hcy sr s l y frs l lo h prcpl s so h h lorh c of h ol ls. h rl sors H B C E r col sors wh h s cors. Slrly h spl sors h r col wh h s cors. ro h fos h spcrl coposos of hs sors r l l / B h H E C So Spl Dforos I hs sco so lry foros r cosr. Pr Srch hs foro hs lry s ow c w s spcl cs of h polr coposo. h oo s Pr Srch..8 h foro r s

25 Sco. Hr I hr s o roo. h prcpl rl s r coc wh h rl coor s. h ls of r h prcpl srchs. Srch wh roo Cosr h oo so h cosθ sθ sθ cosθ scθ scθ whr θ. hs coposo shows h h foro cosss of rl srch y scθ h prcpl srchs lo ch of h s follow y r oy roo hroh l θ o h s...9. h foro s rlly spl cs h prcpl rl s r l wh h rl coor s so h s ol. h foro of h sqr s s show...9. θ r..9: srch wh roo Pr Shr Cosr h oo Pr Shr..9 Sol Mchcs Pr III

26 Sco. so h whr sc s syrc hr s o roo. Sc h roo s zro o c wor rcly wh o h o cosr C. h ls of h prcpl srchs r wh corrspo prcpl rcos E E E E E. h foro of h sqr s s show.... h ol c y h hy l srchs y o whrs h ohr ol corcs y o. l of rl lo h ol wll ro pr srch s c y h srch of h o o. r..: pr shr Spl Shr Cosr h oo Spl Shr..4 so h C h rs of C r IC IIC IIIC h chrcrsc qo s so h prcpl ls of C r ± 4. h prcpl ls of r h pos sqr-roos of hs: ±. 4 Sol Mchcs Pr III 4

27 Sco. hs c wr s scθ ± θ y l θ. hr corrspo cors of C r E E E E E 4 4 or orlz so h hy r of sz wr rs of θ sθ sθ sθ sθ E E E E E h rsforo r of rco coss s h [ Q ] sθ / sθ sθ / sθ / / so h s h rs rsforo forl [ ] [ ][ ][ ] rs of h orl coors hc Q Q o os cosθ sθ sθ cosθ cosθ sθ s sθ θ / cosθ o h foro of h sqr s show... for. θ 5.7. h sqr frs ros pr srch/corco lo h prcpl s s h roh o s fl poso y clocws roo of θ. or hs foro s wll show low hs s h h spl shr foro s ol-prsr. θ r..: spl shr Sol Mchcs Pr III 5

28 Sco...7 Dsplc Dsplc rs h splc of rl prcl 6 s h o ros h rso fro h rfrc cofro o h crr cofro. hs... 7 Dsplc Mrl Dscrpo..4 Dsplc Spl Dscrpo..4 o h h h s ls hy s h ffr rs. r..: h splc Dsplc rs h splc r h rl spl scrpos / / r rl o h foro r h rs foro r hroh r r I I δ δ..4 s clr h h splc rs r rl hroh s Eq...8 r r..44 h foro c ow wr rs of hr h rl or spl splc rs: 6 I sol chcs h oo foro r of scr rs of h splc. I fl chcs howr h prry fl qy scr h c proprs s h locy h cclro s lr. 7 h rl splc hr s o o cofs wh h rh srch sor scss rlr. Sol Mchcs Pr III 6

29 Sco. r r..45 Srs rs of Dsplc rs h srs c wr rs of h splc rs. s.. E I r I r I I r r r r E IJ I J J I K K I J..46 I I I r I r r r r r..46 Sll Sr If h splc rs r sll h h qrc rs hr procs r sll rl o h rs hsls y lc. Wh hs sspo h r-lr sr E h Elr-ls sr rcs o h sll-sr sor I J r r ε IJ J I ε..47 Sc hs cs h splc rs r sll os o r whhr o rfrs h srs o h rfrc or crr cofros h rror s of h s orr s h qrc rs lry lc 8 so h sll sr sor c qlly wll wr s r r ε ε Sll Sr sor lhoh lr r oy roos s o llow s.7 Sol Mchcs Pr III 7

30 Sco...8 h Dforo of r Vol Els L ls rsfor w h rfrc crr cofros hroh h foro r. Hr h rsforo of r ol ls s. h Jco Dr h Jco r of h foro s f s h r of h foro r J h Jco Dr..49 Eqlly c cosr o h Jco of h rsforo fro rl o spl coors s pp.b.. ro Eq...7 h Jco c lso wr h for of h rpl sclr proc J..5 Cosr ow ol l h rfrc cofro prlllpp o y h hr l-ls. h ol of h prlllpp 9 s y h rpl sclr proc Eqs...4: V..5 fr foro h ol l s o y h hr cors ol of h for l s s.9.6f V so h h..5 hs h sclr J s sr of how h ol of rl l hs ch wh h foro for hs rso s of cll h ol ro. J V Vol o..5 9 h cors shol for rh-h s so h h ol s pos. Sol Mchcs Pr III 8

31 Sco. Sc ols co o s ss o physcl ros h J >. lso sc hs rs J. hs o hs h rsrco J >..54 o h r oy roo os o lr h ol so h ol ch s coplly chrcrs y h srch sor. hr l ls ly lo h prcpl rcos of for l wh ol V h ro pr srch o w l ls f l of ol V whr r h prcpl srchs.... h ch ol s hrfor lso V V..55 rfrc cofro crr cofro V V prcpl rl s r..: ch ol or pl h ol ch for pr shr s ol crs for spl shr s zro cf. Eq...9 sq. sc θ θ scθ θ. coprssl rl s o for whch h ol ch s zro.. h foro s sochorc. or sch rl J h hr prcpl srchs r o p r cosr y Icoprssly Cosr..56 so s orl Cosr r l h rfrc cofro wh r S orl o y h cors...4. h S..57 h ol of h l o y h cors so rrry l l s V S. h r l s ow for o l of Sol Mchcs Pr III 9

32 Sco. r s wh orl o y h l ls w l o y h r l s h. h ol of h s s J S..58 r..4: ch of srfc r hs sc s rrry s.. s J S so s orl..59 so s forl shows how h cor l of r s h crr cofro s rl o h cor l of r S h rfrc cofro...9 Isly Oro Cosrs cosr o h prcpl srchs ws roc for coprssl rl..56. Ohr cosrs rs prcc. or pl cosr rl whch s sl cr rco f y cor  h rfrc cofro. I follows h h cosr c prss s..7 C Isly Cosr..6 If hr r wo sch rcos pl f y  B ls θ φ rspcly wh h prcpl rl s h [ cosθ sθ ] cosθ sθ cos θ φ. I follows h φ θ φ θ π θ φ or π.. o foro. cos π θ φ or Sol Mchcs Pr III

33 Sco. Slrly o c h oro cosrs. or pl sppos h h rco ssoc wh h cor  s h rco. h for so sclr μ >. μ Oro Cosr..6.. Prols. I qos..8 o hs fro h ch rl φ φ r φ φ E E rφ Dr h ohr wo rlos.. s I.. o show h. or h foro Dr h Dforo r h h Cchy-r sors Cosr h wo l ls fro. s h h Cchy r sor o r whhr hs ls h crr cofro r prpclr. c s h rh Cchy r sor o l h srch of h l l hc r whhr h l corcs srchs or sys h s lh fr foro. Dr h r-lr Elr sr sors Dcopos h foro o srch roo chc h s syrc s orhool. Wh r h prcpl srchs? 4. Dr Eqos or h foro Dr h splc cor oh h rl spl fors Dr h splc loco of h prcls h for s whch orlly coprs h pl crclr srfc / h fsl c wh s lo h coor s of lh ε Sch h splc cofros f / 6. or h foro Dr h splc cor oh h rl spl fors Clcl h fll rl sr sor h fll spl sr sor c Clcl h fsl sr sor s r fro h rl spl sors copr h for h cs of ry sll. Sol Mchcs Pr III

34 Sco. 7. I h pl o o h polr coposo..5 chc h h rlos C r ssf wh rspc o h orl s. Chc lso h h rlos C r ssf hr h cors r h cors h sco coor sys h prcpl rcos of C C s wh rspc o hs s.. s ol. Sol Mchcs Pr III

35 Sco.. Dforo Sr: rhr opcs.. Volrc Isochorc Dforos Wh lys rls whch r oly slhly coprssl s sfl o copos h foro r lplcly ccor o ro hs fo { Prol } J / I J /.... so chrcrss ol prsr sorol or sochorc foro. h sor J / I chrcrss h ol-ch lol or olrc copo of h foro wh J / I J. hs cocp c crr o o ohr c sors. or pl wh C / / C J J C... C r cll h of foro r h of rh Cchy-r sor rspcly. h sqr of h srch s y { C } C / J..4 so h J / whr s h of srch o h co of C. Slrly h of prcpl srchs r wh J / h cs of spl shr scss rlr s pl of sochorc foro whch h foro r h of foro r coc / J I I... l Dforo I s sl o s h cofro s h rfrc cofro f qs sch s h foro r rl o hs rfrc cofro. s o y cofro c o h rfrc cofro w Sol Mchcs Pr III

36 Sco. foro r c cosrc wh rspc o hs w rfrc cofro. rhr h rfrc cofro os o h o f col o lso. I y css s sfl o choos h crr cofro o h rfrc cofro for pl wh l rs of ch of c qs s lr. o hs roc hr cofro: hs s h cofro so τ h poso of rl prcl hr s o y τ whr s h oo fco. h foro hs τ rl o h crr cofro s cll h rl foro s o y τ s llsr.... τ τ τ cofro τ τ rl foro l cofro crr cofro r..: h rl foro h rl foro r s f hroh τ..7 lso sc τ o hs h rlo τ τ..8 Slrly rl sr srs c f for pl h rl rh Cchy- r sr sor s Epl Cosr h wo-sol oo C τ τ τ..9 Sol Mchcs Pr III 4

37 Sco. Sol Mchcs Pr III 5 Ir hs s h spl scrpo / h rl foro s / τ τ τ τ τ τ h foro rs r / E E E τ τ τ.. Drs of h Srch I hs sco so sfl forl ol h rs of h srchs wh rspc o h Cchy-r sr sors r r. Drs wh rspc o rs h srchs o fcos of h lf Cchy-r sr. Wr s h spl prcpl rcos s ss..7 so h h ol ffrl c prss s [ ].. Sc δ h [ ] o s or.. hs ls follows sc h ch cor of cos lh s lwys orhool o h cor slf s h crr lyss of.6.. s h propry : o hs s or h o or h ; hr δ / : : :.. h sc / : / s lso ql o o hs

38 Sco. Sol Mchcs Pr III 6 : :.. h ch rl h s h sco r. h o lyss s for sc prcpl srchs. Wh h I I. lso / so I / or I : :..4 B / : / I I : so hs cs / I /. slr clclo c crr o for wo ql ls. I sry o s or 4 o s or I..5 Drs wh rspc o C h srch c lso cosr o fco of h rh Cchy-r sr C. h rs of h srchs wh rspc o C c fo cly h s wy s for h lf Cchy-r sr. h rsls r h s s..5 cp h rfrr o..7 s rplc y C s rplc y...4 h Drcol Dr of Kc Qs h rcol r of cors sors ws roc rcol rs of c qs s of ry sfl for pl lrs qos orr o pply rcl solo lorhs h Dforo r rs cosr h foro r s fco of h crr poso or oo s l :

39 Sco. h rcol r [] [ ] o..6 / c prss s ε ε ε ε r r [] ε ε..7 h ls l rsl fro..8. I follows h h rcol r of h foro r h rco of splc cor fro h crr cofro s [ ] r..8 O h ohr h cosr h foro r s fco of s l : ow [ ]..9 ε [ ] ε ε ε ε ε ε r r ε ε.. whr. Ohr Kc Qs h rcol r of h r-lr sr h rh lf Cchy-r sors h Jco h rco of splc fro h crr cofro r { Prol } Sol Mchcs Pr III 7

40 Sco. E[ ] ε C[ ] ε [ ] J [ ] J r r.. whr ε s h sll-sr sor..48. h rcol r s lso sfl for r ros rlos w h c rls. or pl for rrry cor s h ch rl h rc rlos rj J[ ] J[ [] ] J[ r ] J : r Jr r Jr r Jr r J.. so h fro.4.6 wh cos r J J....5 Prols. s..6c o show h.. s h rlo E I Eq...8 [ ] r h proc rl of ffro o r.. E[ ] ε whr ε s h sll sr sor. l C [ ] rs of ε h sll sr sor c l [ ] rs of r l J [ ] rs of J ; s h ch rl [ ] [ [ ] J J wh J [ ] r Sol Mchcs Pr III 8

41 Sco.4.4 Mrl Drs o s o rs oh h rl r h locl r. h oo s ow llow o fco of.4. Vlocy cclro h locy of o prcl s h r of ch of h poso of h prcl. ro.. y fo I h oo prsso V.4. r p rls so s p of o h prcl for whch h locy s clcl. h locy c hs wr s / or o h oo y s / or /. h spl scrpo of h locy fl y o fro h rl scrpo y sply rplc wh.. V.4. s wh splcs oh scrpos hr s oly o locy V hy r s rs of ffr coors. h locy s os of prss h spl scrpo s locy.4. o prcs h rh h s hr ols whch s fco of h rl coors s rsoo h h sso c o spl coors s.4. s. Slrly h cclro s f o V.4.4 h Locl of Ch o h h r V /.4.4 wh f s o h s s h r / wh f. h forr s h cclro of rl prcl. h lr s h r of ch of h locy of prcls f loco spc hs s Sol Mchcs Pr III 9

42 Sco.4 cll h locl r of ch of ; rl ffr rl prcls wll occpy poso ffr s..4. h Mrl Dr Sppos h h locy rs of spl coors s ow; for pl o col h sr sr whch rcors h locy spcfc loco h oo slf s ow. I h cs o l h cclro h ch rl of ffro s ppl: or r cclro spl scrpo.4.5 h cclro c ow r cs h rs c r sr who ow h oo. I h o h rl r or ol r of h prcl s locy ws o o h cclro. I rl o c h r of y physcl or c propry prss h spl scrpo: r Mrl Dr.4.6 or pl h r of ch of h sy ρ ρ of prcl sosly s ρ ρ ρ r ρ.4.7 h frs r ρ / s h locl r of ch of sy whrs h sco r r ρ s h ch o h prcl s oo s cll h coc r of ch. h rl r / c ppl o y sclr cor or sor: α α α rα r r.4.8 Sol Mchcs Pr III 4

43 Sco.4 Sol Mchcs Pr III 4 ohr oo of s for h rl r s D D / : f f f f D Df.4.9 Sy for lows I sy flow qs r p of so h locl r of ch s zro for pl ρ ρ r. I for flow qs r p of poso so h for pl / ρ ρ Epl Cosr h oo h locy cclro c l hroh V O c wr h oo h spl scrpo y r h rl scrpo: 4 4 Ss hs qos h s h spl scrpo of h locy cclro: f V f lrly h cclro c o rcly fro h spl locy fl:

44 Sco.4 Sol Mchcs Pr III r s for. h loshp w h Dsplc Vlocy h locy c r rcly fro h splc..4:.4. or r.4. Wh h splc fl s rl for o hs V Prols. h sy of rl s y ρ h locy fl s y Dr h r of h sy cr poso spc of rl prcl sosly occpy poso.

45 Sco.5.5 Dforo s I hs sco rs of ch of h foro sors roc rlr C E c. r l spcl sors s o sr foro rs r scss for pl h locy r l h r of foro h sp sor w..5. h Vlocy r h locy r s s s sr of h r whch rl s for. Cosr wo f hor pos..5.. h locs of h rl prcls hs pos y s r h rl locy w h pos s l.5. wh l f o h spl locy r r l l Spl Vlocy r.5. r.5.: locy r h spl locy r s cooly s oh sol fl chcs. Lss cooly s s h rl locy r whch s rl o h r of ch of h foro r: V r V.5. s hs of h fc h sc r p rls rl rs rl rs co. Sol Mchcs Pr III 4

46 Sco.5 Sol Mchcs Pr III Mrl Drs of h Dforo r h spl locy r y wr s or l so h h rl r of c prss s l Mrl Dr of h Dforo r.5.4 lso c show h { Prol }... l l h of Dforo Sp sors h locy r c copos o syrc sor sw-syrc sor s follows s..: w l.5.6 whr s h r of foro sor or r of srch sor w s h sp sor or r of roo or orcy sor f y w l l w l l of Dforo Sp sors.5.7 h physcl of hs sors s. h of Dforo Cosr frs h r of foro sor o h l.5.8

47 Sco.5 h r whch h sqr of h lh of s ch s h l.5.9 h ls qly follow fro D cross y o h ls of srch pr srch h rco.5. whr / s h srch / s orl h rco of. hs h r of foro s h r of srch of l ls. h ol copos of for pl rprs rs of so h coor rcos. o: Eq..5. c lso r s follows: l orl h rco of h corrspo orl h rco of. h or. Dffr s l or l. Corc oh ss wh ls o / l. B so y h ch rl cofr h cor of cos lh s orhool o ch h cor h rsl follows Cosr ow h r of ch of h l θ w wo cors l l l l. s.5. whch rcs o.5.9 wh s. lr prsso for hs o proc Sol Mchcs Pr III 45

48 Sco.5 cosθ cosθ Eq ls o cosθ cosθ sθ θ cosθ sθ θ.5. cosθ sθ θ.5. whr / s h srch rco of. / s orl h I follows fro.5. h h off-ol rs of h r of foro sor rprs shr rs: h r of ch of h rh l w l ls l wh h coor rcos. or pl h s cors.5. rcs o whr θ s h orl rh l w h s. h Sp θ.5.4 Cosr ow h sp sor w; sc s sw-syrc c wr rs of s l cor ω Eq...4 cll h lr locy cor: ω w crl w w.5.5 h cor ω s cll h orcy or sp cor.. hs wh s zro h oo cosss of roo o so s lr locy ω ω cf. h of.. wh ω r r sr fro po o h s wr ω r. O h ohr h wh rrool. l w o hs ω o h oo s cll Sol Mchcs Pr III 46

49 Sco.5 Epl Shr low Cosr spl shr flow whch h locy profl s rlr s show..5.. hs yp of flow c r ls pproly y fls y cof h fl w pls sc h pr y sl h ppr pl or h lowr o cos locy V. If h rl prcls c o h ppr pl h locy V h h locy fl s γ whr γ V / h. h l γ γ γ w γ γ fro.5.4 γ θ h r of ch of h l show..5.. V h γ r.4.: shr flow h ls of r ± γ / h prcpl rs Eq...7 r I II 4 γ III. or γ / h cor s [ ] for γ / s [ ] for s. h ls cors of w r copl. l o h ss of cors γ / γ / so o 45 hr s sos pr r of srch/corco of rl. Choos h for s o h rfrc cofro so γ γ γ γ E. h Sol Mchcs Pr III 47

50 Sco.5 γ C γ γ γ hs s spl shr wh γ Eq Ohr s of Sr sors ro..9.. C E.5.6 hs c lso wr rs of spl l ls: [ E ] E.5.7 B fro.5.9 hs lso ql whch ls o prssos for h rl rs of h rh Cchy-r r-lr sr sors lso hr r prssos for h rs of h lf Cchy-r Elr-ls sors { Prol } C E l l l l.5.8 o h E E so h h rl of h r of r-lr sr s ph p prclr h rl of E ro y clos loop so h h fl cofro s h s s h l cofro s zro. Howr rl h rl of h r of foro s o p of h ph hr s o rsl fco h sch h h / wh h. hs h rl or clos ph y o-zro hc h rl of h r of foro s o oo sr of h ol sr. Sol Mchcs Pr III 48

51 Sco.5 h Hcy Sr h Hcy sr s Eq...7 l h whr r h prcpl spl s. hs f h prcpl spl s o o ch wh h /. Wh h lf srch follows h slrly for h corrspo rl sors l h l. or pl cosr so h coor rcos so. h oo locy r H o s so / o s h. rhr h. o h s o o hs prsso os o hol rl os hs cs of for so..5.5 Mrl Drs of L r Vol Els h rl r of l l / hs r f hroh.4.8. or r ol ls s cssry frs o l h rl r of h Jco r J. ro h ch rl o hs s Eqs Hc { Prol 4} J J J : J :.5.9 J J r l J rr J.5. Sc l w r w lso follows h J Jr. s o rlr sochorc oo s o for whch h ol s cos hs y of h follow ss chrcrs h cssry sffc coos for sochorc oo: J J r :.5. pply so s forl..59 h rl r of r cor l s { Prol 6} Sol Mchcs Pr III 49

52 Sco.5 Sol Mchcs Pr III 5 s s l.5. lly fro..5 h rl r of ol l s JV JV.5. Epl Shr Srch Cosr spl of rl ro h follow oo..4.. r.4.: shr srch h foro r rl sr sors r E C h Jco J h spl sr sors r δ γ

53 Sco.5 Sol Mchcs Pr III 5 hs foro c lso prss s srch follow y spl shr: h locy s / / V h locy r s / / l h r of foro sp r [ ] [ ] [ ] [ ] / / / / / w lso C s pc fro.5. / Jr J J.5.6 Prols. Dffr h rlo I s.5.4 l o r.5.5 l..

54 Sco.5 Dffr h rlo. I s.5.4 l.. o r.5.5c l.. or h locy fl r h r of srch pr srch h rco of h cor 4 / 5 h rco of?. Dr h rlo.5.8 C rcly fro C s h fos I / o r h rlos.5.8c: l l l l 4. s h..6 o r or h oo rfy h l. Wh s h ro of h ol l crrly occpy o s ol h for cofro? wh s h r of ch of hs ol l pr crr ol? 6. s so s forl..59 h proc rl of ffro c o r h rl r of cor r l.5. o h orl h for cofro s cos. Sol Mchcs Pr III 5

55 Sco.6.6 Dforo s: rhr opcs.6. loshp w l w h r of ch of Cosr h polr coposo. Sc s orhool I ffro of hs qo ls o Ω.6. wh Ω sw-syrc s Eq..4.. s hs rlo h prsso h fos of w Eq..5.7 o fs h { Prol } l w sw sy Ω [ ] [ ] Ω Ω l.6. o h Ω sw-syrc s coss wh w sw-syrc h oh w ol h r of ch of. Wh h oo s r oy roo h w Ω Dforo sors h Prcpl Mrl Spl Bss h r of ch of h srch sor rs of h prcpl rl s cors s { }.6.4 Cosr h cs wh h prcpl rl s sy cos s c hpp so spl foros. I h cs r col s..5:.6.5 Sol Mchcs Pr III 5

56 Sco.6 Sol Mchcs Pr III 54 wh s pc fro.5.5 Ω w h s y sp s o r oy roo. Slrly fro..7 ffr I { } E..6.6 lso ffr δ ls o so h prsso W.6.7 s l pro W r h copos of sw-syrc sor W W. hs ls o lr prsso for h r-lr sor: W E.6.8 Slrly fro..7 h lf Cchy-r sor c prss rs of h prcpl spl s cors: { }.6.9 h fro spco of.5.8c l l h locy r c prss s { Prol } l s of Ch h l Dforo Js s h rl r of h foro r s f s o c f h rl r of h rl foro r cf... h r of ch rl o h crr cofro: τ τ τ.6.8

57 Sco.6 Sol Mchcs Pr III 55 ro..8 τ τ so h r wh rspc o τ s ow f s τ s h fro.5.4 l.6.9 s pc h locy r s h r of ch of foro rl o h crr cofro. rhr s h polr coposo τ τ τ Dffr wh rspc o τ s τ h s l o h crr cofro I so fro.4.4 l.6. Wh syrc sw-syrc r rspcly syrc sw-syrc follows h w.6. s pc h r of foro s h sos r of srch h sp s h sos r of roo. h Corool Dr h corool r of cor s w o. orlly s f hroh { } [ ] { } [ ] { } { } w w w I Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ L L l l l l o.6.

58 Sco.6 h fo shows h h corool r ols cor h crr cofro ro wh h r oy roo pr of h oo..6.. I s hs w ro cor whch s copr wh h cor Δ whch hs ro roo srch. Δ τ Δ τ Δ r.6.: roo srch of cor.6.4 l-ercs sors h -h l-ercs sor s f s τ L τ C.6. whr C τ s h rl rh Cchy-r sr. Sc C I τ τ I. o l h l-ercs sor o s h rs of h rl foro r; fro h wh.5.5 τ τ [ τ ] l τ τ l τ τ.6.4 τ τ / τ τ l τ [ τ l τ l τ τ ] l l τ hs h sor s sr of h r of srch of rl l ls s Eq..5.. Slrly hhr l-ercs sors sr of hhr orr srch rs so o. τ Sol Mchcs Pr III 56

59 Sco h Drcol Dr h Mrl Dr h rcol r of fco h rco of cr s y fo s for pl Eq..5.7 or S Δ s h ch rl.5.8 [ Δ] Δ.6.5 [ Δ] Δ.6.6 [] [ [] ] [ ].6.7 h rl r s hs ql o h rcol r h rco of h locy cor..6.6 Prols. Dr h rlos.6... s.6.9 o rfy.5.8 l l. Sol Mchcs Pr III 57

60 Sco.7.7 Sll Sr hory Wh h foro s sll fro..4-4 I r I r I r.7. lc h proc of r wh r sc hs r sll qs. hs o c r r hr s o sco o w h for for cofros. h foro r s of h for I α whr α s sll..7. Dcoposo of Sr y sco orr sor c copos o s syrc syrc pr ccor o..8 so h ε Ω ε Ω.7. whr ε s h sll sr sor..48 Ω h -syrc pr of h splc r s h sll roo sor so h c wr s I ε Ω Sll Sr Dcoposo of h Dforo r.7. I follows h for h clclo of o c s h rlo I δ I δ δ for sll C I ε E ε.7.4 oo Sc Ω s syrc c wr rs of l corω cf... so h for y cor Ω ω ω Ω Ω Ω.7.5 h rl splc c ow wr s Sol Mchcs Pr III 58

61 Sco.7 r ε ω.7.6 h copo of rl splc y ω s prpclr o so rprss pr roo of h rl l l..7.. r.7.: pr roo Prcpl Srs Sc ε s syrc s h hr lly orhool cors h prcpl s of sr hr corrspo rl ls h prcpl srs whch c pos or cf... h ffc of ε s hrfor o for ll sphr o ll llpso whos s r h prcpl s whos lhs r. Mrl frs hs prcpl rcos r srch oly whch cs h foro s cll pr foro; frs ohr rcos wll srch ro. h r ε.7.6 hrfor corrspos o pr srch lo h prcpl s. h ol foro s h s of pr foro rprs y ε r oy roo rprs y Ω. hs rsl s slr o h o for h c f sr hory hr h coposo s rhr h lplc. I hr h corrspo sll sr srch roo sors r I ε I Ω so h Epl Cosr h spl shr c.f. Eq...4 I ε Ω.7.7 whr s sll. h splc cor s so h r Sol Mchcs Pr III 59

62 Sco.7 h foro c wr s h coposo ε Ω or ε ω wh ε / / Ω / / ω /. or h roo copo o c wr I Ω / / whch sc for sll θ cos θ sθ θ c s o roo hroh l θ / clocws roo. h prcpl ls of ε r ± / wh corrspo prcpl rcos / / / /. hs h spl shr wh sll splcs cosss of roo hroh l / sprpos po pr shr wh l /..6.. θ / θ r.6.: spl shr.7. oos Sll Sr Cosr ow pr roo o h s wh h c f sr hory wh Sol Mchcs Pr III 6

63 Sco.7 cosθ sθ sθ cosθ.7.8 hs roo os o ch h lh of l ls. ccor o h sll sr hory howr cosθ ε cosθ Ω sθ sθ whch os prc l l lh chs whch c lc f θ s sll. or pl f h roo s of h orr 4 r h ε ε. Howr f h roo s lr h rrors wll pprcl; h cs r oy roo rocs orcl o-lrs whch s l wh s h f foro hory. hs h sll sr hory s rsrc o o oly h cs of sll splc rs lso sll r oy roos..7. Vol Ch ll c wh s of lh h rcos of h prcpl s fors o c wh s of lhs so h ch ol of h c s V V O.7.9 Sc sco orr qs h lry lc roc h sll sr sor hy s lc hr. Hc h crs ol pr ol cll h lo or lo s δv V rε Dlo.7. Sc y ll ol c cosrc o of f r of sch ll cs hs rsl hols for y ll ol rrspc of shp..7.4 of Dforo Sr Sp sors ow h prssos.4.7 for h r of foro sp sors. plc hs prssos y o hs Sol Mchcs Pr III 6

64 Sco.7 w l l l l w.7. or sll srs o c h r os y cosr h o rl coors p of : w.7. h r of foro hs co s s o h r of sr s s o h r of roo w Ω. ε h sp h sos oo of rl prcl c hc rr s h s of hr ffcs: rslo y so h rl Δ h prcl hs splc y Δ pr foro y ε r oy roo y Ω.7.5 Coply Coos Sppos h h srs ε oy r ow. If h splcs r o r h h sr-splc prl ffrl qos ε.7. o r. Howr hr r s p sr copos oly hr splc copos. hs pls h h srs r o p r rl so wy. h rlos w h srs r cll coply coos c show h hy r y ε ε ε ε.7.4 hs r 8 qos oly s of h r sc hs s qos r cssry sffc o l h splc fl. Sol Mchcs Pr III 6

65 Sco.8.8 Ocy Oc sors.8. Dpc o Osrr Cosr rclr loc of rl rs o crclr l. prso ss osrs h rl for..8.. h sh ls c h for rl whrs h sol l cs h crr s. sco osrr s s s h h frs o sp lr hs osrr ss h rl s.8.. o hr osrr s s ro h l 45 fro h frs ss h rl s..8.c. h foro c scr y ch osrr s cocps l splc locy sr so o.. Howr s clr h h hr osrrs wll rl rcor ffr ls for hs srs sc hr prspcs ffr. h ol wh follows s o r whch of h cl sors r fc p of osrr. Sc h lws of physcs scr h rspos of for rl s p of y osrr s hs prclr sors whch wll or rly s prssos o scr rl rspos. c r.8.: for rl s s y ffr osrrs o h..8. c rpr ohr ql wy. O c o sc osrr hs wh h rl o o hr ffr posos. hs wpo wll rr o h sco..8. Ch of frc r Cosr wo frs of rfrc h frs coss of h or o h ss { } h sco coss of h or h f s h poso cor h sco fr. Wh h ors o rl hroh Eq..5. o coc o h ss { }..8.. po spc s h frs fr poso cor h cor copos Q or Q whr [ ] Q s h r Sol Mchcs Pr III 6

66 Sco.8 rsforo r.5.4 or Q. Q. lrly o hs Eq..5.5 Q o o r.8.: wo frs of rfrc Wh h shf or o o o hs Q.8. whr. lrly Q.8. whr wh Q. orl.8.- rl h coors of h poso cor o po spc s osr fro o fr of rfrc o h coors of h poso cor o h s po s osr fro ffr fr of rfrc. lly cosr h poso cor whch s f rl o h fr o. o osrr h fr o h s poso cor wol ppr s.8.. o hs cor hroh h ss { } h cor h procs. Q h sor whch ros h ss { } : o Q.8. hs rlo wll scss frhr low. Sol Mchcs Pr III 64

67 Sco.8 Q Q o o r.8.: lo w cors Eq Ch of Osrr h ch of fr copss y Eqs..8.- s or prcsly cll pss ch of fr rly ols rsforo w cor copos. O wol sy h hr s o osrr h hs osrr s s wo frs of rfrc. Hr follows ffr cocp c ch of fr lso cll ch osrr whch hr r wo osrrs ch wh hr ow fr of rfrc. osrr s soo who c sr rl posos spc wh rlr ss of wh cloc. h physcl worl for pl rl prcl s prc y osrr s occrr prclr po spc prclr. O c rr osrr O o p of E h physcl worl o po po spc cf...5 rl r. sl E s rcor s h pr y osrr O rl y ffr pr y sco osrr O E O O r.8.4: rcors y wo osrrs of h s L h wo osrrs rcor hr pos corrspo o hr s hs pos f cors spc s h ffrc w h pos cf...5. I s ss h oh osrrs s h s Ecl ory h s f o osrr ss llps h h ohr osrr wll s h s llps prhps poso ffrly spc. o sr h hs s so osr cors s rl hroh so orhool sor Q for pl Q.8.4 Sol Mchcs Pr III 65

68 Sco.8 sc hs rsforo wll oclly prsr scs w pos ls w cors s..7 for pl Q Q.8.5 O O r.8.5: rcors of wo osrrs of hr spr s lhoh ll orhool sors Q o prsr lh ls s h h Q s propr orhool.. roo sor cf...8 so h oro s lso prsr. rhr s ss h Q Q whch prsss h fc h h osrrs c o rl o ch ohr or. Osrrs s lso r o rls w s. L osrr O rcor cr sco osrr O rcor h s s occrr. h h s s rl hroh α Osrr rsforo.8.6 whr α s cos. If ow h osrrs rcor sco s occrr sy o hs s rqr. h osrr rsforo.8.4 ols h cors s sch os o rqr h oo of or or coor sys; s src syolc oo for osrr rsforo. Howr or o for O o for O c roc h h pos c rr s poso cors spc h rsforo.8.4 c ow prss h of-s for whr c Q Osrr Spl rsforo.8.7 c Q.8.8 h rsforo.8.7 s cll Ecl rsforo sc prsrs h Ecl ory. Sol Mchcs Pr III 66

69 Sco.8 o O O o r.8.6: poso cors for wo osrrs of h s s Coor Syss Ech osrr c roc y Crs coor sys wh ss cors { } { } sy. hy c h rsol h poso cors o cor copos. hs ss cors c or wh rspc o ch ohr y wy h s hy wll rl hroh whr s y roo sor. I ch osrr c ch hr ss ffc coor rsforo. o p o roc spcfc coor syss wll hr sc hy r coplly cssry o h oo of osrr rsforo wol oly rly cofs h ss. loshp o Pss Ch of r cll h pss ch of fr copss Eqs If o sss h cl for Eq..8. o hs: Q.8.9 hs s clrly osrr rsforo rl h poso cor s s y o osrr o h poso cor s s y sco osrr hroh orhool sor cor s Eq I h pss ch of fr Q r h copos of h orhool sor Q Eq...5 whch ps h ss oo ch ohr: Q. hs h rsforo.8.- c f qly y h pr Q. I h ss h pss ch of fr os f c ch of fr.. ch of osrr hroh Eq Howr h cocp of osrr scss o s h prfrr wy of f osrr rsforo..8.4 Oc Vcors sors h osrr rsforo.8.7 cpsls h ffr wpos osrrs h of h physcl worl. hy wll s h s ocs rl hy wll s hs ocs or ffrly loc ffr posos. h ol ow s o s Sol Mchcs Pr III 67

70 Sco.8 whch of h cl sors r p of hs ffr wpos. s frs sp s roc h cocp of oc sor. Sppos h ffr osrrs r for rl. I orr o scr h rl h osrrs srs. hs wll ol srs of spl ocs ssoc wh h crr cofro for pl h locy or sp. I wll lso ol rl ocs ssoc wh h rfrc cofro for pl l ls h cofro. I wll lso ol wo-po sors sch s h roo or foro r whch r ssoc wh oh h crr rfrc cofros. I s ss h ll osrrs osr h rfrc cofro o h s h s hy rcor h s s of pos for h rl prcls h rfrc cofro. h osrrs h o rl o ch ohr hr srs of ocs ssoc wh h crr cofro wll rl ffr. O wol pc w ffr osrrs o h s sr of rl ocs sp hs rl o; hs o sys h rl cors sors r oc rl cors oc rl sors f hy r ch r h osrr rsforo spl cor o h ohr h s s o oc spl cor f ssfs h osrr rsforo s.8.4: Q Ocy qr for Spl Vcor.8. for ll roo sors Q. oc spl sor s f o o whch rsfors oc cor o oc cor. Cosr sor osr s y wo ffr osrrs. oc cor whch s osr s l. h for o oc Q Q QQ.8. so h sor s oc pro QQ Ocy qr for Spl sor.8. Vros s c r; for pl for oc cors oc sors B { Prol } hs os o ffc h rly of wh follows; h oo of oc sor s p of h chos rfrc cofro h rsforo.8.6 s rl os o ffc h rlos o r Sol Mchcs Pr III 68

71 Sco.8 B B B : B : B B.8. or sclr φ φ Ocy qr for Sclr.8.4 I ohr wors oc sclr s o whch hs h s l o ll osrrs. lly cosr wo-po sor. Sch sor s s o oc f ps oc rl cor o oc spl cor. Cosr h wo-po sor osr s. oc rl cor whch s osr s l. rl cor s oc f s ffc y osrr rsforo so Q Q Q.8.5 so h sor s oc pro Q Ocy qr for wo-po sor.8.6 hs h ocy rqr for wo-po sor s h s s h for spl cor..8.5 Oc Kcs r h ros c cors sors roc h rlr scos hr ocy ss s r. h oo s osr y o osrr s y sco osrr s. h osrr rsforo s Q c α.8.7 so h oo s o oc cor.. Q. Sol Mchcs Pr III 69

72 Sco.8 h Vlocy cclro Dffr.8.7 s h oo s of locy r h osrr rsforo s for ry h Q Q c.8.8 whch os o coply wh h ocy rqr for spl cors.8.. I ohr wors ffr osrrs wll sr ffr s for h locy. h locy prsso c p for slr o h of lry chcs h o-oc rs r o h rh c c Q Ω Q.8.9 whr Ω Q QQ.8. s sw-syrc s Eq..4.; hs sor rprss h r oy lr locy w h osrrs s Eq..6.. o h h locy s oc pro Q c o for whch Q c whch s cll -p r rsforo. Slrly for h cclro c show h c Ω c Ω c c Q Ω Q Q Q.8. h frs hr rs o h rh-h s r cll h Elr cclro h crfl cclro h Corols cclro rspcly. h cclro s oc pro c Q r cos for whch Q c wh c o whch s cll ll rsforo whr h wo cofros r rl y r roo rslol oo wh cos locy. h Dforo r Cosr h oo. s o osrrs osr h rfrc cofro o h s:. h foro s h osr s so h Q Q Q.8. Q.8. so ccor o.8.6 h foro r s oc. Sol Mchcs Pr III 7

73 Sco.8 h Cchy-r Sr sors or h rh lf Cchy-r sors C Q Q C Q Q QQ.8.4 hs h rl sor C h spl sor r oc. h Jco Dr or h Jco r s..6 Q Q J J so s oc ccor o.8.4. h oo Srch sors h polr coposo s whr s h orhool roo sor s h rh srch sor. h Q Q. Sc Q s orhool h prsso Q s l pro Q.8.6 hs h wo-po sor h rl sor r oc. h Vlocy r llow Q o fco of for h locy r s c l Q Q Q QlQ.8.7 Ω Q whr Ω Q s h lr locy sor.8.. O h ohr h wh l w spr o h syrc sw-syrc prs QQ w QwQ.8.8 Ω Q hs h locy r s o oc. hs s o srprs h h locy s o oc. Howr sfcly h r of foro sr of h r of srch of rl s oc. So hors f sco orr sor o oc oly f.8. s ssf rrlss of whhr s spl wo-po or rl; wh hs fo C wol f s o-oc 4 o h Q s roo sor o s orhool sor hr Sol Mchcs Pr III 7

74 Sco.8 h Spl r Cosr h spl r of oc cor : r r.8.9 Sc Q h ch rl s Q Q.8. I follows h.8. r Q Q hs h spl r s oc. I rl c show h h spl r of sor fl of orr s oc for pl h r of sclr φ { Prol } r φ. rhr for cor { Prol } s oc. Oc s Cosr oc cor fl. h rl r s o oc. Howr h co-rool r Eq..6. w s oc. o show hs corc.8.8 w QwQ QQ o h rh wh Q o prsso for Q : Q w Q Qw.8. h o Q Q Q w Q Q w w Q Q.8. o h w Q or oc cor. o o o Q so h h co-rool r of cor s s of spl sors c lso of orr o cosrc oc rs. or pl cosr oc spl sor so QQ. h QQ QQ QQ.8.4 whch s clrly o oc. Howr hs c r-rr s.8. o Sol Mchcs Pr III 7

75 Sco.8 w w Q w w Q.8.5 so h qy w w.8.6 s oc r cll h J r. Ohr oc rs of sors c cosrc slr fsho for pl h Cor-l r f y { Prol 4} Sry of Oc Kc Ocs l l.8.7 l.8. srss h ocy of so por c ocs: oc fo yp rsforo Jco r Sclr J J Dforo r -po Q oo -po Q h Cchy-r C Mrl C C sr r-lr E C I Mrl E E sr of r- Mrl Lr sr E E h Srch C Mrl Lf Cchy-r Spl QQ sr I Spl QQ Elr-ls sr Lf Srch Spl QQ Spl Vlocy l r Spl l QlQ QQ r of Dforo l l Spl QQ l w l Spl w QwQ QQ Sp l.8.: Oc c ocs.8.6 Oc cos I slr wy fcos r f o oc s follows: sclr-l fco φ of for pl sor s oc f rsfors h s wy s oc sclr Sol Mchcs Pr III 7

76 Sco.8 φ φ.8.8 spl cor-l fco of sor s oc f rsfors h s wy s oc cor Q.8.9 spl sor-l fco f of sor s oc f rsfors ccor o f Qf Q.8.4 Oc fcos of h Dforo r Cosr oc sclr-l fco φ of h foro r φ. h fco s oc f φ φ. B lso φ Q φ φ.8.4 s h polr coposo hor φ Q r-oy roo Q h ls o whch ls o h rc for φ. Choos h prclr φ φ.8.4 φ φ.8.4 hs for h sclr fco φ o oc s p of h rool pr of ps oly o h srch pr; co fco of h p copos of h foro r oly of h s p copos of h rh srch sor. Cosr oc spl sor-l fco f of h foro r f. ccor o h fo of ocy of sco orr sor.8.: B lso Q f Qf.8.44 f Q f f.8.45 s h polr coposo hor choos h prclr r-oy roo Q ls o f f.8.46 Sol Mchcs Pr III 74

77 Sco.8 whch ls o h rc for f f.8.47 hs for f o oc s pc o s hroh rrry fco of ohr wh or plc pc o h roo sor Epl Cosr h sor fco f α. h [ Q Q ] Q [ ] Q Qf f Q α α Q so h ocy rqr s ssf. ccor o h o h o c l f f α h rc for s f 4 α α lso sc C E C I lr rc fors r C f f E f f lly cosr spl sor fco f of rl sor. h I follows h hs s r oly h spcl cs fco f s o oc. f Qf Q f f f.8.48 f QfQ.8.49 Q I so s o r rl. I follows h h.8.7 Prols. Dr h rlos.8.. Show h h spl r of sclr φ s oc.. Show h h rc of spl cor s oc. [H: s h fo..9 y.9.] 4. Vrfy h h l-cor r of sor l l s oc. Sol Mchcs Pr III 75

78 Sco.9.9 Boy oos of Cofros I hs sco r scss r oy roos o h crr rfrc cofros..9. Boy oo of h Crr Cofro s o.8. h crcsc of wo osrrs o rl o ch ohr f cofro h crr cofro s ql o o osrr srs of wo ffr cofros o rl o ch ohr. h ocy rqrs of h ros c ocs scss h pros sco c hs lso y cosr r oy roos rslos of h crr cofro. y r oy roo rslo of h crr cofro c prss h for Q c.9. whr Q s roo sor. hs s llsr h crr cofro s o y S h ro cofro y S. Js s h foro r for h cofro S rl o h rfrc cofro S s f hroh. ro.9. s.8.5 s Eq..8. slrly for h rh lf Cchy-r sors C Q C QQ.9. hs h foros : S S : S S h rh Cchy r sors C C r h s h lf Cchy r sors r ffr rl hroh QQ. ll h ohr rsls o h ls sco h co of osrr rsforos for pl for h Jco srch sors c. hol lso for h cs of roos o h crr cofro. lhoh ql hr s ffrc: o hr r wo osrrs who rcor o rl prcl sy s wo ffr pos h ohr hr s o osrr who rcors wo ffr s h plc whr h o rl prcl s wo ffr cofros Sol Mchcs Pr III 76

79 Sco.9 S S Q rfrc cofro S c r.9.: r oy roo rslo of h crr cofro.9. Boy oo of h frc Cofro Cosr ow r-oy roo o h rfrc cofro. Sch roos ply por rol h oo of rl syry s Chpr 5. h rfrc cofro s o y S h ro/rsl cofro y S..9.. h foro r for h crr cofro S rl o S s f hroh Q. B so slrly for h rh lf Cchy-r sors C Q QCQ.9. hs h ch o h rh lf Cchy-r sr sor r roo o h rfrc cofro s h s s h ch o h lf rh Cchy-r sr sor r roo of h crr cofro. Sol Mchcs Pr III 77

80 Sco.9 Q S Q S S rfrc cofro r.9.: r oy roo of h rfrc cofro Sol Mchcs Pr III 78

81 Sco.. Coc Coors I hs sco h foro sr sors scr.- r ow scr s coc coors s.6. o h ll h sor rlos prss syolc oo lry scss sch s C l r p of coor sys hol lso for coc coors... Coc Coors Iroc h crlr coors Θ. h rl coors c h wr s Θ Θ Θ.. so E E Θ.. whr corr s cors r h cor s cors h rfrc cofro wh corrspo... wh δ.. rfrc cofro crr cofro E E r..: Crlr Coors h coor crs crs of cos Θ for h for cofro. O sys h h crlr coors r coc or h s h coor crs r ch o rl prcls for wh h oy so h ch rl Sol Mchcs Pr III 79

82 Sco. prcl hs h s ls of h coors cofros. Θ oh h rfrc crr I h crr cofro h spl coors c prss rs of w crr s of crlr coors Θ Θ Θ..4 wh corrspo cor s cors corr s cors wh Θ..5 Epl Cosr oo whry c of rl wh ss of lh L s rsfor o cylr of rs hh H.... L H L r..: c for o cylr pl w of o qrr of h c cylr r show.... L P p r..: c for o cylr Sol Mchcs Pr III 8

83 Sco. Sol Mchcs Pr III 8 h oo rs oo r y L H L L ss: H L L L ss: E Iroc s of coc coors...4 h rl spl coors r Θ Θ Θ Θ Θ Θ Θ H L L L hs r sply cylrcl coors Θ Θ Θ s cos Θ Θ Θ Θ Θ ypcl rl prcl o y p s show...4. o h h poso cors for p h h s Θ ls sc hy rprs h s rl prcl.

84 Sco. Θ 4 π Θ p p Θ Θ Θ Θ r..4: crlr coor crs.. h Dforo r Wh coc crlr coors h foro r s..6 whch s coss wh Θ Θ..7 h foro r h rspos h rss p h s cors o cofro oo h s cors h ohr cofro: Dforo r..8 hs h sors p h cor s cors o ch ohr whrs h sors p h corr s cors o ch ohr s llsr...5. Sol Mchcs Pr III 8

85 Sco. Sol Mchcs Pr III 8 r..5: h foro r s rspos h rss Copos of hs ffr copos wh rspc o h ffr ss: f f f f f f f f f f f f f f f f..9 h copos of wh rspc o h rfrc ss { } { } r cor ss corr ss

86 Sco. Sol Mchcs Pr III 84 Θ Θ Θ Θ.. slrly for h copos wh rspc o h crr ss. Copos of h Bs Vcors ffr Bss ow δ δ.. show h so of h copos of h foro r c w lso s copos of h s cors. Slrly f f.. or h corr s cors o hs δ δ.. f f..4.. co o Mrl Spl Coors Mrl Coors Sppos h h rl coors wh Crs ss r s rhr h h coc coors wh crlr ss...6. h

87 Sco. Sol Mchcs Pr III 85 E E E E E E Θ Θ Θ Θ Θ..5 E E E E r r..6 whch r Eqs hs r s h oo for o s wh h rl coors r s o scr h foro. r..6: Mrl coors for ss Spl Coors Slrly wh h spl coors r o s s p rls h E E E E Θ Θ Θ Θ Θ..7 E E crr cofro rfrc cofro

88 Sco. E E r r..8 h scrpos r llsr...7. o h h s cors s ch of hs css crlr rl spl. r o h E E E r E r r..7: foro scr s ffr p rls Sol Mchcs Pr III 86

89 Sco...4 Sr sors h Cchy-r sors h rh Cchy-r sor C h lf Cchy-r sor r f y Eqs..... C C C C..9 hs h cor copos of h rh Cchy-r sor r h rc coffcs h cor copos of h y sor wh rspc o h coc ss h crr cofro I. I s possl o l ohr copos of C.. C lso s copos wh rspc o h crr ss hroh..4 oly h copos C wh rspc o h rfrc ss wll s h lyss. Slrly for wh rspc o h crr cofro wll s. h copos h Srch ow loos o..9.. s S C.. so h h srchs r loos o..7 s S S s C C C.. h r-lr Elr-ls sors h r-lr sr sor E h Elr-ls sr sor r f hroh....4 Sol Mchcs Pr III 87

90 Sco. Sol Mchcs Pr III 88 I E I C S s S s.. h copos of E c l hroh wr I h y sor prss rs of h s cors h rfrc cofro I h y sor prss rs of h s cors h crr cofro E C E.. o h h copos of E wh rspc o hr ss r ql E lhoh hs s o r rr hr ohr copos.. E...5 Ir Cofros Srch oo sors h polr coposos h scr..5. h coposos r llsr...8. I h rl coposo h rl s frs srch y h ro y. L h s cors h ssoc r cofro { } ĝ. Slrly h spl coposo h rl s frs ro y h srch y. L h s cors h ssoc r cofro hs cs { }. h loos o Eq...8 { Prol }..4..5

91 Sco. Sol Mchcs Pr III 89 o h syrc so Slrly for h roo sor wh orhool h o rlos c chc s Eqs...8 c. r..8: h rl spl polr coposos Vros rlos w h s cors c r for pl L L L.. { } { } ĝ { } { } Ĝ

92 Sco. Sol Mchcs Pr III 9 Dforo r loshp w Bss h ros s cors r rl o hroh h srch roo sors. h r ss r rl rcly hroh h foro r. or pl fro I h s wy.. sor Copos h srch roo sors c copos lo y of h ss. or h os rl ss wol { } { } for pl.. wh. O lso hs..4 wh slr syry. lso

93 Sco. Sol Mchcs Pr III wh slr syry. o h copr s....7 ow o h roos prsr cors lhs prclr prsr h rc....8 hs s h corr copos of h o sors r lso ql. s o h sors c copos lo ohr ss for pl Ecors Els loos o..5 h ls of C r r fro h l prol

94 Sco. l o h chrcrsc qo..5 wh prcpl sclr rs..6-7 C I..4 C I II III..4 C C C C C C I II III C C C rc C [rc r C ] C C C C C ε C C C C C C C C C C C C C C..4 h cors r h prcpl rl rcos h spcrl coposo s h C I wh..4 C..44 whr C h r h srchs. h r spcrl coposos..7 hol lso. o lso h h roo sor rs of prcpl rcos s s whr r h spl prcpl rcos...7 Dsplc Dsplc rs Cosr h splc of rl prcl. hs c wr rs of cor copos :...46 h cor r of c prss s Θ..47 Sol Mchcs Pr III 9

95 Sco. Sol Mchcs Pr III 9 h sl l rfrs o cor ffro wh rspc o h for ss.. h Chrsoffl syols o s r fcos of h. h ol l rfrs o cor ffro wh rspc o h for ss.. h Chrsoffl syols o s r fcos of h. lrly h cor r c prss s Θ Θ Θ..48 so f δ δ..49 h ls qls follow fro..-. h copos of h r-lr Elr-ls sr sors.. c wr rs of splcs s rlos..49 { Prol }: E..5 I rs of spl coors E Θ / / h copos of h Elr-Lr sr sor r E δ δ..5 whch s h Dforo of r Vol Els Dffrl Vol El Cosr ffrl ol l for y h ls Θ h for cofro Eq..6.6: