1 Vectors & Tensors tensor

Transcription

1 Vctos & nsos h mthmtcl modln of th physcl wold qs nowld of qt fw dffnt mthmtcs sbcts sch s Clcls Dffntl Eqtons nd Ln lb. hs topcs slly ncontd n fndmntl mthmtcs coss. How n mo thooh nd n-dpth ttmnt of mchncs t s ssntl to dscb th physcl wold sn th concpt of th tnso nd so w bn ths boo wth comphns chpt on th tnso. h chpt s ddd nto th pts. h fst pt cos ctos (.-.7). h scond pt s concnd wth scond nd hh-od tnsos (.8-.5). h scond pt cos mch of th sm ond s don n th fst pt mnly nlzn th cto concpts nd pssons to tnsos. h fnl pt (.6-.9) (not qd n th st moty of pplctons) s concnd wth nlzn th l wo to cln coodnt systms. h fst pt compss bsc cto lb sch s th dot podct nd th coss podct; th mthmtcs of how th componnts of cto tnsfom btwn dffnt coodnt systms; th symbolc nd nd mt nottons fo ctos; th dffntton of ctos ncldn th dnt th dnc nd th cl; th ntton of ctos ncldn ln dobl sfc nd olm ntls nd th ntl thoms. h scond pt compss th dfnton of th tnso (nd -dfnton of th cto); dyds nd dydcs; th mnplton of tnsos; popts of tnsos sch s th tc tnspos nom dtmnnt nd pncpl ls; spcl tnsos sch s th sphcl dntty nd othoonl tnsos; th tnsfomton of tnso componnts btwn dffnt coodnt systms; th clcls of tnsos ncldn th dnt of ctos nd hh od tnsos nd th dnc of hh od tnsos nd spcl foth od tnsos. In th fst two pts ttnton s stctd to ctnl Ctsn coodnts (cpt fo bf foys nto cylndcl nd sphcl coodnts). In th thd pt cln coodnts ntodcd ncldn cont nd contnt ctos nd tnsos th mtc coffcnts th physcl componnts of ctos nd tnsos th mtc coodnt tnsfomton ls tnso clcls ncldn th Chstoffl symbols nd cont dffntton nd cln coodnts fo cd sfcs.

2

3 Scton.. Vcto lb.. Scls physcl qntty whch s compltly dscbd by snl l nmb s clld scl. Physclly t s somthn whch hs mntd nd s compltly dscbd by ths mntd. Empls tmpt dnsty nd mss. In th follown lowcs (slly G) ltts.. wll b sd to psnt scls... Vctos h concpt of th cto s sd to dscb physcl qntts whch h both mntd nd dcton ssoctd wth thm. Empls foc locty dsplcmnt nd cclton. Gomtclly cto s psntd by n ow; th ow dfns th dcton of th cto nd th mntd of th cto s psntd by th lnth of th ow F.... nlytclly ctos wll b psntd by lowcs bold-fc Ltn ltts.. q. h mntd (o lnth) of cto s dnotd by o. It s scl nd mst b non-nt. ny cto whos lnth s s clld nt cto; nt ctos wll slly b dnotd by. b c () (b) F..: () cto; (b) ddton of ctos.. Vcto lb h optons of ddton sbtcton nd mltplcton fml n th lb of nmbs (o scls) cn b tndd to n lb of ctos. Sold Mchncs Pt III Klly

4 Scton. h follown dfntons nd popts fndmntlly dfn th cto:. Sm of Vctos: h ddton of ctos nd b s cto c fomd by plcn th ntl pont of b on th tmnl pont of nd thn onn th ntl pont of to th tmnl pont of b. h sm s wttn c b. hs dfnton s clld th plllom lw fo cto ddton bcs n omtcl ntptton of cto ddton c s th donl of plllom fomd by th two ctos nd b F...b. h follown popts hold fo cto ddton: b b commtt lw bc b c ssoct lw. h Nt Vcto: Fo ch cto th sts nt cto. hs cto hs dcton oppost to tht of cto bt hs th sm mntd; t s dnotd by. omtcl ntptton of th nt cto s shown n F..... Sbtcton of Vctos nd th Zo Vcto: h sbtcton of two ctos nd b s dfnd by b ( b) F...b. If b thn b s dfnd s th zo cto (o nll cto) nd s psntd by th symbol o. It hs zo mntd nd nspcfd dcton. pop cto s ny cto oth thn th nll cto. hs th follown popts hold: o o 4. Scl Mltplcton: h podct of cto by scl s cto wth mntd tms th mntd of nd wth dcton th sm s o oppost to tht of ccodn s s post o nt. If 0 s th nll cto. h follown popts hold fo scl mltplcton: dstbt lw o ddton of scls b b dstbt lw o ddton of ctos ( ) ( ) ssoct lw fo scl mltplcton b b b () (b) F..: () nt of cto; (b) sbtcton of ctos Sold Mchncs Pt III 4 Klly

5 Scton. Not tht whn two ctos nd b ql thy h th sm dcton nd mntd dlss of th poston of th ntl ponts. hs bn F.... ptcl poston n spc s not ssnd h to cto t st hs mntd nd dcton. Sch ctos clld f to dstnsh thm fom ctn spcl ctos to whch ptcl poston n spc s ctlly ssnd. b F..: ql ctos h cto s somthn wth mntd nd dcton nd dfnd by th bo ls s n lmnt of on cs of th mthmtcl stct th cto spc. h cto spc s dscssd n th nt scton....4 h Dot Podct h dot podct of two ctos nd b (lso clld th scl podct) s dnotd by b. It s scl dfnd by b b cos. (..) h s th nl btwn th ctos whn th ntl ponts concd nd s stctd to th n 0 F...4. b b F..4: th dot podct n mpotnt popty of th dot podct s tht f fo two (pop) ctos nd b th lton b 0 thn nd b ppndcl. h two ctos sd to b othoonl. lso cos(0) so tht th lnth of cto s. noth mpotnt popty s tht th pocton of cto lon th dcton of nt cto s n by. hs cn b ntptd omtclly s n F...5. Sold Mchncs Pt III 5 Klly

6 Scton. cos F..5: th pocton of cto lon th dcton of nt cto It follows tht ny cto cn b dcomposd nto componnt plll to (nt) cto nd noth componnt ppndcl to ccodn to (..) h dot podct posssss th follown popts (whch cn b pod sn th bo dfnton) { Poblm 6}: () b b (commtt) () b c b c (dstbt) () b b (4) 0; nd 0 f nd only f o..5 h Coss Podct h coss podct of two ctos nd b (lso clld th cto podct) s dnotd by b. It s cto wth mntd b b sn (..) wth dfnd s fo th dot podct. It cn b sn fom th f tht th mntd of b s qlnt to th of th plllom dtmnd by th two ctos nd b. b b F..6: th mntd of th coss podct h dcton of ths nw cto s ppndcl to both nd b. Whth b ponts p o down s dtmnd fom th fct tht th th ctos b nd b fom ht hndd systm. hs mns tht f th thmb of th ht hnd s pontd n th Sold Mchncs Pt III 6 Klly

7 Scton. dcton of b nd th opn hnd s dctd n th dcton of thn th cln of th fns of th ht hnd so tht t closs shold mo th fns thoh th nl 0 bnn thm to b. Som mpls shown n F...7. b b b b F..7: mpls of th coss podct h coss podct posssss th follown popts (whch cn b pod sn th bo dfnton): () b b (not commtt) () b c b c (dstbt) () b b (4) b o o plll ( lnly dpndnt ) f nd only f nd b h pl Scl Podct h tpl scl podct o bo podct of th ctos w s dfnd by w w w pl Scl Podct (..4) Its mpotnc ls n th fct tht f th th ctos fom ht-hndd td thn th olm V of plllppd spnnd by th th ctos s ql to th bo podct. o s ths lt b nt cto n th dcton of of w on s h w nd w w F...8. hn th pocton h V (..5) Sold Mchncs Pt III 7 Klly

8 Scton. w h F..8: th tpl scl podct Not: f th th ctos do not fom ht hndd td thn th tpl scl podct ylds th nt of th olm. Fo mpl sn th ctos bo w. V..6 Vctos nd Ponts Vctos obcts whch h mntd nd dcton bt thy do not h ny spcfc locton n spc. On th oth hnd pont hs ctn poston n spc nd th only chctstc tht dstnshs on pont fom noth s ts poston. Ponts cnnot b ddd toth l ctos. On th oth hnd cto cn b ddd to pont p to nw pont q q p F...9. Smlly th dffnc btwn two ponts s cto q p. Not tht th noton of pont s dfnd h s slhtly dffnt to th fml pont n spc wth s nd on th concpt of on s not ncssy fo ths ponts nd th smpl optons wth ctos. q p F..9: ddn ctos to ponts..7 Poblms. Whch of th follown scls nd whch ctos? () wht () spcfc ht () momntm () ny () olm. Fnd th mntd of th sm of th nt ctos dwn fom common t of cb lon th of ts sds. Sold Mchncs Pt III 8 Klly

9 Scton.. Consd two non-colln (not plll) ctos nd b. Show tht cto lyn n th sm pln s ths ctos cn b wttn n th fom p qb wh p nd q scls. [Not: on sys tht ll th ctos n th pln spcfd by th bs ctos nd b.] 4. Show tht th dot podct of two ctos nd cn b ntptd s th mntd of tms th componnt of n th dcton of. 5. h wo don by foc psntd by cto F n mon n obct n dstnc s th podct of th componnt of foc n th n dcton tms th dstnc mod. If th cto s psnts th dcton nd mntd (dstnc) th obct s mod show tht th wo don s qlnt to F s. 6. Po tht th dot podct s commtt b b. [Not: ths s qlnt to syn fo mpl tht th wo don n poblm 5 s lso ql to th componnt of s n th dcton of th foc tms th mntd of th foc.] 7. Stch b f nd b s shown blow. b 8. Show tht b b b. 9. Sppos tht d body otts bot n s O wth nl spd s shown blow. Consd pont p n th body wth poston cto. Show tht th locty of p s n by ω wh ω s th cto wth mntd nd whos dcton s tht n whch ht-hndd scw wold dnc nd th otton. [Not: lt s b th c-lnth tcd ot by th ptcl s t otts thoh n nl on ccl of ds thn (snc s ds / dt ( d / dt) ).] ω O p 0. Show omtclly tht th dot nd coss n th tpl scl podct cn b ntchnd: b c b c.. Show tht th tpl cto podct b c ls n th pln spnnd by th ctos nd b. Sold Mchncs Pt III 9 Klly

10 Scton.. Vcto Spcs h noton of th cto psntd n th pos scton s h -cst n mo foml nd bstct wy sn som bsc concpts of Ln lb nd opoloy. hs mht sm t fst to b nncssly complctn mtts bt ths ppoch tns ot to b hlpfl n nfyn nd bnn clty to mch of th thoy whch follows. Som bcond thoy whch complmnts ths mtl s n n ppnd to ths Chpt.... h Vcto Spc h ctos ntodcd n th pos scton oby ctn ls thos lstd n... It tns ot tht mny oth mthmtcl obcts oby th sm lst of ls. Fo tht son th mthmtcl stct dfnd by ths ls s n spcl nm th ln spc o cto spc. Fst st s ny wll-dfnd lst collcton o clss of obcts whch cold b fnt o nfnt. n mpl of st mht b B (..) whch ds B s th st of obcts sch tht stsfs th popty. Mmbs of st fd to s lmnts. Consd now th fld of l nmbs R. h lmnts of R fd to s scls. Lt V b non-mpty st of lmnts b c wth ls of ddton nd scl mltplcton tht s th s sm b V fo ny b V nd podct V fo ny V R. hn V s clld (l) cto spc o R f th follown ht oms hold:. ssoct lw fo ddton: fo ny b c V on hs ( b) c ( b c). zo lmnt: th sts n lmnt o V clld th zo lmnt sch tht o o fo y V. nt (o ns): fo ch V th sts n lmnt V clld th nt of sch tht ( ) ( ) 0 4. commtt lw fo ddton: fo ny b V on hs b b 5. dstbt lw o ddton of lmnts of V: fo ny b V nd scl R ( b) b 6. dstbt lw o ddton of scls: fo ny V nd scls R ( ) fld s noth mthmtcl stct (s ppnd to ths Chpt.). Fo mpl th st of compl nmbs s fld. In wht follows th only fld whch wll b sd s th fml st of l nmbs wth th sl optons of ddton nd mltplcton. l snc th ssoctd fld s th ls. h wod l wll slly b omttd n wht follows fo bty. Sold Mchncs Pt III 0 Klly

11 Scton. 7. ssoct lw fo mltplcton: fo ny V nd scls R ( ) ( ) 8. nt mltplcton: fo th nt scl R fo ny V. h st of ctos s obcts wth mntd nd dcton dscssd n th pos scton stsfy ths ls nd thfo fom cto spc o R. How dspt th nm cto spc oth obcts whch not th fml omtc ctos cn lso fom cto spc o R s wll b sn n lt scton... Inn Podct Spc Jst s th cto of th pos scton s n lmnt of cto spc nt s ntodcd th noton tht th cto dot podct s on mpl of th mo nl nn podct. Fst fncton (o mppn) s n ssnmnt whch ssns to ch lmnt of st nq lmnt of st B nd s dnotd by f : B (..) n odd p b conssts of two lmnts nd b n whch on of thm s dsntd th fst lmnt nd th oth s dsntd th scond lmnt h podct st (o Ctsn podct) B conssts of ll odd ps b wh nd b B : B b b B (..) Now lt V b l cto spc. n nn podct (o scl podct) on V s mppn tht ssocts to ch odd p of lmnts y scl dnotd by y : V V R (..4) tht stsfs th follown popts fo y z V R :. ddtty: y z z y z. homonty: y y. symmty: y y 4. post dfntnss: 0 whn o Fom ths popts t follows tht f y 0 fo ll y V thn 0 cto spc wth n ssoctd nn podct s clld n nn podct spc. wo lmnts of n nn podct spc sd to b othoonl f Sold Mchncs Pt III Klly

12 Scton. y 0 (..5) nd st of lmnts of V y z sd to fom n othoonl st f y lmnt n th st s othoonl to y oth lmnt: y 0 z 0 y z 0 tc. (..6) h bo popts thos lstd n..4 nd so th st of ctos wth th ssoctd dot podct foms n nn podct spc. Ecldn Vcto Spc h st of l tplts nd th sl ls of ddton nd mltplcton foms cto spc R. Wth th nn podct dfnd by y y y y on hs th nn podct spc nown s (th dmnsonl) Ecldn cto spc nd dnotd by E. hs nn podct llows on to t dstncs (nd nls) btwn lmnts of E thoh th nom (lnth) nd mtc (dstnc) concpts dscssd nt... Nomd Spc Lt V b l cto spc. nom on V s l-ld fncton : V R (..7) tht stsfs th follown popts fo y V R :. postty: 0. tnl nqlty: y y. homonty: 4. post dfntnss: 0 f nd only f o cto spc wth n ssoctd nom s clld nomd cto spc. Mny dffnt noms cn b dfnd on n cto spc ch on n dffnt nomd ln spc. ntl nom fo th nn podct spc s (..8) It cn b sn tht ths nom ndd stsfs th dfnn popts. Whn th nn podct s th cto dot podct th nom dfnd by..8 s th fml cto lnth. Sold Mchncs Pt III Klly

13 Scton. On mpotnt consqnc of th dfntons of nn podct nd nom s th Schwz nqlty whch stts tht On cn now dfn th nl btwn two lmnts of V to b y y (..9) y : V V R (..0) y cos y h qntty nsd th cd bcts h s ncssly btwn nd by th Schwz nqlty nd hnc th nl s ndd l nmb...4 Mtc Spcs Mtc spcs blt on th concpt of dstnc btwn obcts. hs s nlzton of th fml dstnc btwn two ponts on th l ln. Consd st X. mtc s l ld fncton d : X X R (..) tht stsfs th follown popts fo y X :. post: d ( y) 0 nd d ( ) 0 fo ll y X. stctly post: f d ( y) 0 thn y fo ll y X. symmty: d( y) d( y ) fo ll y X 4. tnl nqlty: d( y) d( z) d( z y) fo ll y z X st X wth n ssoctd mtc s clld mtc spc. h st X cn h mo thn on mtc dfnd on t wth dffnt mtcs podcn dffnt mtc spcs. Consd now nomd cto spc. hs spc ntlly hs mtc dfnd on t: y y d (..) nd ths th nomd cto spc s mtc spc. Fo th st of ctos wth th dot podct ths s th dstnc btwn two ctos y. Sold Mchncs Pt III Klly

14 Scton...5 h ffn Spc Consd st P th lmnts of whch clld ponts. Consd lso n ssoctd cto spc V. n ffn spc conssts of th st P th st V nd two optons whch connct P nd V: () n two ponts p q P on cn dfn dffnc q p whch s nq lmnt of V.. q p V () n pont p P nd V on cn dfn th sm p whch s nq pont q of P.. q p P nd fo whch th follown popty holds fo : q p q p p q P. Fom th bo on hs fo th ffn spc tht p q P. p p o nd q p p q fo ll Not tht on cn t th sm of ctos ccodn to th stct of th cto spc bt on cnnot t th sm of ponts only th dffnc btwn two ponts. Fth th s no noton of on n th ffn spc. On cn choos som fd o P to b n on. In tht cs p o s clld th poston cto of p lt to o. Sppos now tht th ssoctd cto spc s Ecldn cto spc.. n nn podct spc. Dfn th dstnc btwn two ponts thoh th nn podct ssoctd wth V p q q p q p q p d (..) It cn b shown tht ths mppn d : P P R s mtc.. t stsfs th mtc popts nd ths P s mtc spc (lthoh t s not cto spc). In ths cs P s fd to s Ecldn pont spc Ecldn ffn spc o smply Ecldn spc. Whs n Ecldn cto spc th s zo lmnt th on ( 000) n Ecldn pont spc th s non pt fom tht th two spcs th sm nd pt fom ctn spcl css on dos not nd to dstnsh btwn thm. Sold Mchncs Pt III 4 Klly

15 Scton.. Ctsn Vctos So f th dscsson hs bn n symbolc notton tht s no fnc to s o componnts o coodnts s md mpld o qd. h ctos st ndpndntly of ny coodnt systm. It tns ot tht mch of cto (tnso) mthmtcs s mo concs nd s to mnplt n sch notton thn n tms of cospondn componnt nottons. How th mny ccmstncs n whch s of th componnt foms of ctos (nd tnsos) s mo hlpfl o ssntl. In ths scton ctos dscssd n tms of componnts componnt fom... h Ctsn Bss Consd th dmnsonl (Ecldn) spc. In ths spc consd th th nt ctos hn th popts 0 (..) so tht thy mtlly ppndcl (mtlly othoonl) nd (..) so tht thy nt ctos. Sch st of othoonl nt ctos s clld n othonoml st F.... Not fth tht ths othonoml systm s ht-hndd by whch s mnt (o o ). hs st of ctos foms bss by whch s mnt tht ny oth cto cn b wttn s ln combnton of ths ctos.. n th fom (..) F..: n othonoml st of bs ctos nd Ctsn componnts o bsolt o nnt o dct o cto notton Sold Mchncs Pt III 5 Klly

16 Scton. Sold Mchncs Pt III Klly 6 By ptd pplcton of Eqn... to cto nd sn.. th scls n.. cn b pssd s (s F...) (..4) h scls nd clld th Ctsn componnts of n th n bss. h nt ctos clld bs ctos whn sd fo ths ppos. Not tht t s not ncssy to h th mtlly othoonl ctos o ctos of nt sz o ht-hndd systm to fom bss only tht th th ctos not copln. h ht-hndd othonoml st s oftn th sst bss to s n pctc bt ths s not lwys th cs fo mpl whn on wnts to dscb body wth cd bonds (.. s.6.0). h dot podct of two ctos nd fd to th bo bss cn b wttn s (..5) Smlly th coss podct s (..6) hs s oftn wttn n th fom (..7) tht s th coss podct s ql to th dtmnnt of th mt

17 Scton... h Ind Notton h psson fo th coss podct n tms of componnts Eqn...6 s qt lnthy fo mo complctd qntts thns t nmnbly lon. hs shot-hnd notton s sd fo ths componnt qtons nd ths nd notton s dscbd h. In th nd notton th psson fo th cto n tms of th componnts nd th cospondn bss ctos s wttn s (..8) hs cn b smplfd fth by sn Enstn s smmton connton whby th smmton sn s doppd nd t s ndstood tht fo ptd nd ( n ths cs) smmton o th n of th nd ( n ths cs ) s mpld. hs on wts. hs cn b fth shotnd to smply. h dot podct of two ctos wttn n th nd notton ds Dot Podct (..9) h ptd nd s clld dmmy nd bcs t cn b plcd wth ny oth ltt nd th sm s th sm; fo mpl ths cold qlly wll b wttn s o. Fo th ppos of wtn th cto coss podct n nd notton th pmtton symbol (o ltntn symbol) cn b ntodcd: Fo mpl (s F...) f ( ) s n n pmtton of () f ( ) s n odd pmtton of () (..0) 0 f two o mo ndcs ql 0 o ndcl o sbscpt o sff notton n th cs of two-dmnsonl spc/nlyss Sold Mchncs Pt III 7 Klly

18 Scton. Not tht F..: schmtc fo th pmtton symbol (clocws s +) (..) nd tht n tms of th bs ctos { Poblm 7} (..) nd { Poblm 7}. (..) h coss podct cn now b wttn concsly s { Poblm 8} Coss Podct (..4) Intodc nt th Konc dlt symbol dfnd by 0 (..5) Not tht bt sn th nd notton. h Konc dlt llows on to wt th pssons dfnn th othonoml bss ctos (....) n th compct fom Othonoml Bss Rl (..6) h tpl scl podct (..4) cn now b wttn s w w w w w m w w m m m (..7) Sold Mchncs Pt III 8 Klly

19 Scton. Not tht snc th dtmnnt of mt s ql to th dtmnnt of th tnspos of mt ths s qlnt to w w w w (..8) H follow som sfl foml noln th pmtton nd Konc dlt symbol { Poblm }: pq p p p q q p (..9) Fnlly h som oth mpotnt dntts noln ctos; th thd of ths s clld Ln s dntty 4 { Poblm 5}: b b b b b c cb b b c d c d b c b d b c d b dc b cd b cd d b c d cb b d c c (..0).. Mt Notton fo Vctos h symbolc notton nd nd notton (o smply ) cn b sd to dnot cto. noth notton s th mt notton: th cto cn b psntd by mt ( colmn cto): Mtcs wll b dnotd by sq bcts so shothnd notton fo ths mt/cto wold b. h lmnts of th mt cn b wttn n th lmnt fom. h lmnt fom fo mt s ssntlly th sm s th nd notton fo th cto t psnts. 4 to b pcs th spcl cs of..0c..0 s Ln s dntty Sold Mchncs Pt III 9 Klly

20 Scton. Fomlly cto cn b psntd by th odd tplt of l nmbs. h st of ll ctos cn b psntd by nmbs: R th st of ll odd tplts of l R R (..) It s mpotnt to not th dstncton btwn cto nd mt: th fom s mthmtcl obct ndpndnt of ny bss th ltt s psntton of th cto wth spct to ptcl bss s dffnt st of bss ctos nd th lmnts of th mt wll chn bt th mt s stll dscbn th sm cto. Sd noth wy th s dffnc btwn n lmnt (cto) of Ecldn cto spc nd n odd tplt R. hs noton wll b dscssd mo flly n th nt scton. s n mpl th dot podct cn b wttn n th mt notton s shot mt notton fll mt notton H th notton dnots th mt (th ow cto). h slt s mt.. scl n lmnt fom...4 Ctsn Coodnts hs f th noton of n on hs not bn sd. Choos pont o n Ecldn (pont) spc to b clld th on. n on toth wth ht-hndd othonoml bss consttts (ctnl) Ctsn coodnt systm F.... (pont) o (pont) o (cto) F..: Ctsn coodnt systm Sold Mchncs Pt III 0 Klly

21 Scton. scond pont thn dfns poston cto o F.... h componnts of th cto o clld th (ctnl) Ctsn coodnts of th pont 5. Fo bty th cto o s smply lblld tht s on ss th sm symbol fo both th poston cto nd ssoctd pont...5 Poblms. Elt wh Po tht fo ny cto ( ) ( ) ( ). [Hnt: wt n componnt fom.]. Fnd th pocton of th cto on th cto Fnd th nl btwn 6 nd Wt down n psson fo nt cto plll to th sltnt of two ctos nd (n symbolc notton). Fnd ths cto whn 4 5 (n componnt fom). Chc tht yo fnl cto s ndd nt cto. 6. Elt wh. 7. Vfy tht m m. Hnc by dottn ch sd wth show tht. 8. Show tht. 9. h tpl scl podct s n by w w. Epnd ths qton nd smplfy so s to pss th tpl scl podct n fll (non-nd) componnt fom. 0. Wt th follown n nd notton:.. Show tht b s qlnt to b.. Vfy tht 6.. Vfy tht pq p q q p nd hnc show tht p p. 4. Elt o smplfy th follown pssons: () (b) (c) (d) 5. Po Ln s dntty..0c. 6. If s nt cto nd n bty cto show tht whch s noth psntton of Eqn... wh cn b sold nto componnts plll nd ppndcl to. 5 ht s componnts sd fo ctos nd coodnts sd fo ponts Sold Mchncs Pt III Klly

22 Scton.4.4 Mtcs nd Elmnt Fom.4. Mt Mt Mltplcton In th nt scton.5 dn cto tnsfomton qtons t wll b ncssy to mltply os mtcs wth ch oth (of szs nd ). It wll b hlpfl to wt ths mt mltplctons n shot-hnd lmnt fom nd to dlop som shot ls whch wll b bnfcl ht thoh ths chpt. Fst t hs bn sn tht th dot podct of two ctos cn b psntd by o s mt wth lmnt fom. Smlly th mt mltplcton o n fll hs typ of mt psnts th tnso podct of two ctos wttn n symbolc notton s (o smply ). nso podcts wll b dscssd n dtl n.8 nd.9. Nt th mt mltplcton s Q Q Q Q Q Q Q (.4.) Q Q Q mt wth lmnts Q Q { Poblm }. h lmnts of Q th sm s thos of Q whch n lmnt fom ds Q Q h psson Q s mnnlss bt Q Q Q. lmnts hs lds to th follown l:. f cto p-mltpls mt. f mt Q p-mltpls th cto t s. { Poblm } s mt wth Q t s th tnspos. f smmd ndcs bsd ch oth s th n Q o Q th mt s Q 4. f smmd ndcs not bsd ch oth s th n Q th mt s th tnspos Q Sold Mchncs Pt III Klly

23 Scton.4 Fnlly consd th mltplcton of mtcs. n ths follows th bsd B s th mt ch oth l fo th smmd nd. Fo mpl { Poblm 6} B B nd th mltplcton B B B. h s lso th mpotnt dntty B B s wttn s (.4.) Not lso th follown (whch ppls to both th nd notton nd lmnt fom): () f th s no f nd s n th s on lmnt (psntn scl) () f th s on f nd s n Q t s (o ) mt (psntn cto) () f th two f ndcs s n B t s mt (psntn s wll b sn lt scond-od tnso).4. h c of Mt noth mpotnt notton noln mtcs s th tc of mt dfnd to b th sm of th donl tms nd dnotd by t h c (.4.).4. Poblms. Show tht Q Q. o do ths mltply th mt nd th cto n Eqn..4. nd wt ot th sltn cto n fll; Show tht th th lmnts of th cto Q Q nd Q.. Show tht Q s fll). mt wth lmnts Q (wt th mtcs ot n. Show tht Q Q. 4. th th lmnts of Q th sm s thos of Q? 5. Wht s th lmnt fom fo th mt psntton of bc? 6. Wt ot th mtcs nd B n fll.. n tms of tht B B fo. 7. Wht s th lmnt fom fo B () () (th s no mbty h snc () B B 8. Show tht t. 9. Show tht dt[ ]. tc. nd fy ) Sold Mchncs Pt III Klly

24 Scton.5.5 Coodnt nsfomton of Vcto Componnts Vy oftn n pctcl poblms th componnts of cto nown n on coodnt systm bt t s ncssy to fnd thm n som oth coodnt systm. Fo mpl on mht now tht th foc f ctn n th dcton hs ctn l F..5. ths s qlnt to nown th componnt of th foc n n coodnt systm. On mht thn wnt to now wht foc s ctn n som oth dcton fo mpl n th dcton shown ths s qlnt to sn wht th componnt of th foc s n nw coodnt systm. f componnt of foc componnt of foc F.5.: cto psntd sn two dffnt coodnt systms h ltonshp btwn th componnts n on coodnt systm nd th componnts n scond coodnt systm clld th tnsfomton qtons. hs tnsfomton qtons dd nd dscssd n wht follows..5. Rottons nd nsltons ny chn of Ctsn coodnt systm wll b d to tnslton of th bs ctos nd otton of th bs ctos. tnslton of th bs ctos dos not chn th componnts of cto. Mthmtclly ths cn b pssd by syn tht th componnts of cto nd ths th qntts do not chn nd tnslton of bs ctos. Rotton of th bs ctos s ths wht on s concnd wth n wht follows..5. Componnts of Vcto n Dffnt Systms Vctos mthmtcl obcts whch st ndpndntly of ny coodnt systm. Intodcn coodnt systm fo th ppos of nlyss on cold choos fo mpl ctn Ctsn coodnt systm wth bs ctos nd on o F. Sold Mchncs Pt III 4 Klly

25 Scton In tht cs th cto cn b wttn s nd ts componnts. Now scond coodnt systm cn b ntodcd (wth th sm on) ths tm wth bs ctos. In tht cs th cto cn b wttn s wh ts componnts n ths scond coodnt systm s shown n th f. hs th sm cto cn b wttn n mo thn on wy: h fst coodnt systm s oftn fd to s th o systm nd th scond s th o systm. o F.5.: cto psntd sn two dffnt coodnt systms Not tht th nw coodnt systm s obtnd fom th fst on by otton of th bs ctos. h f shows otton bot th s (th sn connton fo ottons s post contclocws). wo Dmnsons Concnttn fo th momnt on th two dmnsons fom tonomty (f to F..5.) nd so OB B BD CP cos sn sn cos Sold Mchncs Pt III 5 Klly

26 Scton.5 cos sn sn cos cto componnts n fst coodnt systm cto componnts n scond coodnt systm In mt fom ths tnsfomton qtons cn b wttn s cos sn sn cos P C D o B F.5.: omty of th D coodnt tnsfomton Q. By p- Q on obtns th : h mt s clld th tnsfomton o otton mt mltplyn both sds of ths qtons by th ns of Q tnsfomton qtons tnsfomn fom to cos sn sn cos n mpotnt popty of th tnsfomton mt s tht t s othoonl by whch s mnt tht Q Q h Dmnsons Othoonlty of nsfomton/rotton Mt (.5.) It s stht fowd to show tht n th fll th dmnsons F..5.4 th componnts n th two coodnt systms ltd thoh Sold Mchncs Pt III 6 Klly

27 Scton.5 cos( ) cos( ) cos( ) cos( ) cos( ) cos( ) cos( ) cos( ) cos( ) wh cos( ) s th cosn of th nl btwn th nd s. hs nn qntts clld th dcton cosns of th coodnt tnsfomton. n dnotn ths by th ltt Q Q cos( ) Q cos( ) tc. so tht Q cos( ) (.5.) on hs th mt qtons Q Q Q Q Q Q Q Q Q o n lmnt fom nd shot-hnd mt notton Q Q (.5.) F.5.4: two dffnt coodnt systms n D spc Not: som thos dfn th mt of dcton cosns to consst of th componnts Q cos( ) so tht th sbscpt fs to th nw coodnt systm nd th to th old coodnt systm th thn th oth wy ond s sd h. nsfomton of Ctsn Bs Vctos h dcton cosns ntodcd bo lso lt th bs ctos n ny two Ctsn coodnt systms. It cn b sn tht Q (.5.4) Sold Mchncs Pt III 7 Klly

28 Scton.5 hs ltonshp s llsttd n F..5.5 fo. cos( ) cos( ) cos( ) F.5.5: dcton cosns Foml Dton of th nsfomton Eqtons In th bo th tnsfomton qtons Q w dd omtclly. hy cn lso b dd lbclly sn th nd notton s follows: stt wth th ltons nd post-mltply both sds by to t (th cospondn mt psntton s to th ht (lso s Poblm n.4.)): Q Q Q Q Q h ns qtons { Poblm } Othoonlty of th nsfomton Mt Q Q Q (.5.5) s n th two dmnsonl cs th tnsfomton mt s othoonl Q hs follows fom Empl Consd Ctsn coodnt systm wth bs ctos. coodnt tnsfomton s cd ot wth th nw bss n by Q. Sold Mchncs Pt III 8 Klly

29 Scton.5 n n n () () () n n n () () () n n () n () () Wht s th tnsfomton mt? Solton h tnsfomton mt conssts of th dcton cosns Q cos( ) so n n n () () () n n n () () () n n n () () ().5. Poblms. h nls btwn th s n two coodnt systms n n th tbl blow. o 5 o 90 o 60 o 45 o 0 o o o Constct th cospondn tnsfomton mt Q nd fy tht t s othoonl.. h o coodnt systm s obtnd fom th o coodnt systm by post (contclocws) otton of bot th s. Fnd th (fll th dmnsonl) tnsfomton mt Q. fth post otton bot th s s thn md to th o coodnt systm. Fnd th cospondn tnsfomton mt P. hn constct th tnsfomton mt R fo th complt tnsfomton fom th o to th o coodnt systm.. Bnnn wth th psson fomlly d th lton Q Q ). ( o 45 Sold Mchncs Pt III 9 Klly

30 Scton.6.6 Vcto Clcls - Dffntton Clcls noln ctos s dscssd n ths scton th nttly t fst nd mo fomlly towd th nd of ths scton..6. h Odny Clcls Consd scl-ld fncton of scl fo mpl th tm-dpndnt dnsty of mtl (t). h clcls of scl ld fnctons of scls s st th odny clcls. Som of th mpotnt concpts of th odny clcls wd n ppnd B to ths Chpt.B...6. Vcto-ld Fnctons of scl Consd cto-ld fncton of scl fo mpl th tm-dpndnt dsplcmnt of ptcl (t). In ths cs th dt s dfnd n th sl wy d dt ( t t) ( t) lm t 0 t whch tns ot to b smply th dt of th coffcnts d dt d d d dt dt dt d dt Ptl dts cn lso b dfnd n th sl wy. Fo mpl f s fncton of th coodnts ) thn ( lm 0 ( ) ( ) Dffntls of ctos lso dfnd n th sl wy so tht whn ndo ncmnts d d d th dffntl of s d d d d nd th dffntl nd ctl ncmnt ppoch on noth s 0. ssmn tht th bs ctos do not dpnd on t Sold Mchncs Pt III 0 Klly

31 Scton.6 Spc Cs h dt of cto cn b ntptd omtclly s shown n F..6.: s th ncmnt n consqnt pon n ncmnt t n t. s t chns th nd-pont of th cto (t) tcs ot th dottd c shown t s cl tht s t 0 ppochs th tnnt to so tht d / dt s tnntl to. h nt cto tnnt to th c s dnotd by τ : d / dt τ (.6.) d / dt s τ (t) ( t t) s ds d d () (b) F.6.: spc c; () th tnnt cto (b) ncmnt n c lnth Lt s b ms of th lnth of th c msd fom som fd pont on. Lt s b th ncmnt n c-lnth cospondn to ncmnts n th coodnts F..6.b. hn fom th odny clcls (s ppnd.b) so tht ds d d d ds dt d dt d dt d dt Bt d dt d dt d dt d dt so tht d dt ds dt (.6.) Sold Mchncs Pt III Klly

32 Scton.6 hs th nt cto tnnt to th c cn b wttn s τ d / dt ds / dt d ds (.6.) If s ntptd s th poston cto of ptcl nd t s ntptd s tm thn d / dt s th locty cto of th ptcl s t mos wth spd ds / dt lon. Empl (of ptcl moton) ptcl mos lon c whos pmtc qtons t t 4t t 5 wh t s tm. Fnd th componnt of th locty t tm t n th dcton. Solton h locty s d d t dt dt 4 h componnt n th n dcton s n 8 4 / 7. Ct t 4t t 5 t t ˆ wh â s nt cto n th dcton of h scl ct (s) of spc c s dfnd to b th mntd of th t of chn of th nt tnnt cto: dτ d ( s) (.6.4) ds ds Not tht τ s n dcton ppndcl to τ F..6.. In fct ths cn b pod s follows: snc τ s nt cto τ τ s constnt ( ) nd so dτ τ/ ds 0 bt lso d ds dτ τ τ τ ds nd so τ nd d τ / ds ppndcl. h nt cto dfnd n ths wy s clld th pncpl noml cto: Sold Mchncs Pt III Klly

33 Scton.6 dτ ν (.6.5) ds R ( s) s ν(s) ν( s ds) s τ(s) τ τ( s ds) F.6.: th ct hs cn b sn omtclly n F..6.: fom Eqn..6.5 τ s cto of mntd s n th dcton of th cto noml to τ. h ds of ct R s dfnd s th cpocl of th ct; t s th ds of th ccl whch st tochs th c t s F..6.. Fnlly th nt cto ppndcl to both th tnnt cto nd th pncpl noml cto s clld th nt bnoml cto: b τ ν (.6.6) h plns dfnd by ths ctos shown n F..6.; thy clld th ctfyn pln th noml pln nd th oscltn pln. ν Noml pln Oscltn pln τ b Rctfyn pln F.6.: th nt tnnt pncpl noml nd bnoml ctos nd ssoctd plns Sold Mchncs Pt III Klly

34 Scton.6 Rls of Dffntton h dt of cto s lso cto nd th sl ls of dffntton pply d dt d ( t) dt d d dt dt d d dt dt (.6.7) lso t s stht fowd to show tht { Poblm } d dt d dt d dt d dt d dt d dt (.6.8) (h od of th tms n th coss-podct psson s mpotnt h.).6. Flds In mny pplctons of cto clcls scl o cto cn b ssoctd wth ch pont n spc. In ths cs thy clld scl o cto flds. Fo mpl () tmpt scl fld ( scl-ld fncton of poston) () locty cto fld ( cto ld fncton of poston) hs qntts wll n nl dpnd lso on tm so tht on wts ( t) o ( t). Ptl dffntton of scl nd cto flds wth spct to th bl t s symbolsd by / t. On th oth hnd ptl dffntton wth spct to th coodnts s symbolsd by /. h notton cn b md mo compct by ntodcn th sbscpt comm to dnot ptl dffntton wth spct to th coodnt bls n whch cs / / nd so on..6.4 h Gdnt of Scl Fld Lt () b scl fld. h dnt of s cto fld dfnd by (s F..6.4) Gdnt of Scl Fld (.6.9) h dnt s of consdbl mpotnc bcs f on ts th dot podct of wth d t s th ncmnt n : Sold Mchncs Pt III 4 Klly

35 Scton.6 d d d d ( d) ( d) (.6.0) d F.6.4: th dnt of cto If on wts d s d d wh s nt cto n th dcton of d thn d d n dcton d dn (.6.) hs qntty s clld th dctonl dt of n th dcton of nd wll b dscssd fth n.6.. h dnt of scl fld s lso clld th scl dnt to dstnsh t fom th cto dnt (s lt) nd s lso dnotd by d (.6.) Empl (of th Gdnt of Scl Fld) Consd two-dmnsonl tmpt fld. hn Fo mpl t ( 0) nd t ( ) F Not th follown: () ponts n th dcton noml to th c const. () th dcton of mmm t of chn of s n th dcton of n ths contt dnt s dt wth spct to poston cto bt th tm dnt s sd mo nlly thn ths.. s.4 Sold Mchncs Pt III 5 Klly

36 Scton.6 () th dcton of zo d s n th dcton ppndcl to () (0) h cs const. F.6.5: dnt of tmpt fld clld sothms (cs of constnt tmpt). In nl thy clld so-cs (o so-sfcs n th dmnsons). Mny physcl lws n n tms of th dnt of scl fld. Fo mpl Fo s lw of ht condcton lts th ht fl q (th t t whch ht flows thoh sfc of nt ) to th tmpt dnt thoh q (.6.) wh s th thml condctty of th mtl so tht ht flows lon th dcton noml to th sothms. h Noml to Sfc In th bo mpl t ws sn tht ponts n th dcton noml to th c const. H t wll b sn nlly how nd why th dnt cn b sd to obtn noml cto to sfc. Consd sfc psntd by th scl fncton f ( ) c c constnt 4 nd lso spc c C lyn on th sfc dfnd by th poston cto ( t) ( t) ( t). h componnts of mst stsfy th qton of th sfc so f ( ( t) ( t) ( t)) c. Dffntton s df dt f d dt f d dt f d dt 0 th fl s th t of flow of fld ptcls o ny thoh n sfc; th fl dnsty s th fl p nt bt s h ths s mo commonly fd to smply s th fl 4 sfc cn b psntd by th qton f ( ) c ; fo mpl th psson 4 s th qton fo sph of ds (wth cnt t th on). ltntly th sfc cn b wttn n th fom ) fo mpl ( 4 Sold Mchncs Pt III 6 Klly

37 Scton.6 whch s qlnt to th qton d f d / dt 0 nd s sn n.6. d / dt s cto tnntl to th sfc. hs d f s noml to th tnnt cto; d f mst b noml to ll th tnnts to ll th cs thoh p so t mst b noml to th pln tnnt to th sfc. ylo s Ss Wtn s fncton of th bls (omttn tm t) so tht ( ) thn cn b pndd n th-dmnsonl ylo s ss: ( d d d ) ( ) d Nlctn th hh od tms ths cn b wttn s whch s qlnt to ( d) ( ) d d d d.6.5 h Nbl Opto h symbolc cto opto s clld th Nbl opto 5. On cn wt ths n componnt fom s (.6.4) On cn nls th d of th dnt of scl fld by dfnn th dot podct nd th coss podct of th cto opto wth cto fld ccodn to th ls h follown tmnoloy s sd: (.6.5) d d cl (.6.6) 5 o dl o th Gdnt opto Sold Mchncs Pt III 7 Klly

38 Scton.6 hs ltt two dscssd n th follown sctons..6.6 h Dnc of Vcto Fld Fom th dfnton (.6.5) th dnc of cto fld () s th scl fld d Dnc of Vcto Fld (.6.7) Dffntl Elmnts & Physcl ntpttons of th Dnc Consd flown compssbl 6 mtl wth locty fld ( ). Consd now dffntl lmnt of ths mtl wth dmnsons wth bottom lft-hnd con t ( ) fd n spc nd thoh whch th mtl flows 7 F h componnt of th locty n th dcton wll y o fc of th lmnt bt f th lmnt s smll th locts wll y lnly s shown; only th componnts t th fo cons of th fc shown fo clty. Snc [dstnc = tm locty] th olm of mtl flown thoh th ht-hnd fc n tm t s t tms th olm bondd by th fo con locts (btwn th ht-hnd fc nd th pln sfc dnotd by th dottd lns); t s sthtfowd to show tht ths olm s ql to th olm shown to th ht F..6.6b wth constnt locty ql to th locty whch occs t th cnt of th fc. hs th olm of mtl flown ot s 8 t nd th olm fl.. th t of olm flow s. Now ( ) Usn ylo s ss pnson nd nlctn hh od tms ( ) 6 tht s t cn b compssd o pndd 7 ths typ of fd olm n spc sd n nlyss s clld contol olm 8 th locty wll chn by smll mont dn th tm ntl t. On cold s th t) ( t t) bt n th lmt s t 0 ths wll dc to locty n th clclton.. ( t) ( Sold Mchncs Pt III 8 Klly

39 Scton.6 wth th ptl dts ltd t ) so th olm fl ot s ( ( ) ( ) ( ) ( ) ( ) ( ) () (b) F.6.6: dffntl lmnt; () flow thoh fc (b) olm of mtl flown thoh th fc h nt olm fl ot (t of olm flow ot thoh th ht-hnd fc mns th t of olm flow n thoh th lft-hnd fc) s thn / nd th nt olm fl p nt olm s /. Cyn ot sml clclton fo th oth two coodnt dctons lds to nt nt olm fl ot of n lmntl olm: d (.6.8) whch s th physcl mnn of th dnc of th locty fld. If d 0 th s nt flow ot nd th dnsty of mtl s dcsn. On th oth hnd f d 0 th nflow qls th otflow nd th dnsty mns constnt sch mtl s clld ncompssbl 9. flow whch s dnc f s sd to b sochoc. cto fo whch d 0 s sd to b solnodl. Nots: h bo slt holds only n th lmt whn th lmnt shns to zo sz so tht th t tms n th ylo ss tnd to zo nd th locty fld s n ln fshon o fc consd th locty t fd pont n spc ( t). h locty t lt tm ( tt) ctlly s th locty of dffnt mtl ptcl. hs s shown n F..6.7 blow: th mtl ptcls mon thoh spc nd whs ( t) psnts th locty of ptcl ( t t) now psnts th locty of ptcl whch hs mod nto poston. hs pont s mpotnt n th consdton of th nmtcs of mtls to b dscssd n Chpt 9 lqd sch s wt s mtl whch s ncompssbl Sold Mchncs Pt III 9 Klly

40 Scton.6 Sold Mchncs Pt III Klly 40 F.6.7: mon mtl ptcls noth mpl wold b th dnc of th ht fl cto q. hs tm sppos lso tht th s som nto of ht nsd th lmnt ( soc) ntn t t of p nt olm bn scl fld. n ssmn th lmnt to b smll on ts to b ctn t th md-pont of th lmnt nd on consds ) (. ssm stdy-stt ht flow so tht th (ht) ny wthn th lmntl olm mns constnt wth tm th lw of blnc of (ht) ny thn qs tht th nt flow of ht ot mst ql th ht ntd wthn so ) ( q q q Ddn thoh by nd tn th lmt s 0 on obtns q d (.6.9) H th dnc of th ht fl cto fld cn b ntptd s th ht ntd (o bsobd) p nt olm p nt tm n tmpt fld. If th dnc s zo th s no ht bn ntd (o bsobd) nd th ht ln th lmnt s ql to th ht ntn t..6.7 h Lplcn Combnn Fo s lw of ht condcton (.6.) q wth th ny blnc qton (.6.9) q d nd ssmn th condctty s constnt lds to. Now (.6.0) tm t t t tm ) ( t ) ( t t

41 Scton.6 hs psson s clld th Lplcn of. By ntodcn th Lplcn opto on hs (.6.) hs qton ons th stdy stt ht flow fo constnt condctty. In nl th qton s clld Posson s qton. Whn th no ht socs (o sns) on hs Lplc s qton 0. Lplc s nd Posson s qton s n mny oth mthmtcl modls n mchncs lctomntsm tc..6.8 h Cl of Vcto Fld Fom th dfnton.6.5 nd.6.4 th cl of cto fld () s th cto fld cl Cl of Vcto Fld (.6.) It cn lso b pssd n th fom cl (.6.) Not: th dnc nd cl of cto fld ndpndnt of ny coodnt systm (fo mpl th dnc of cto nd th lnth nd dcton of cl ndpndnt of th coodnt systm n s) ths wll b -dfnd wthot fnc to ny ptcl coodnt systm whn dscssn tnsos (s.4). Physcl ntptton of th Cl Consd ptcl wth poston cto nd mon wth locty ω tht s wth n nl locty bot n s n th dcton of ω. hn { Poblm 7} ω ω cl (.6.4) hs th cl of cto fld s ssoctd wth ottonl popts. In fct f s th locty of mon fld thn smll pddl whl plcd n th fld wold tnd to ott n ons wh cl 0 n whch cs th locty fld s clld ot Sold Mchncs Pt III 4 Klly

42 Scton.6 fld. h pddl whl wold mn sttony n ons wh cl 0 n whch cs th locty fld s clld ottonl..6.9 Idntts H som mpotnt dntts of cto clcls { Poblm 8}: d d d cl d d d d cl cl d( ) d d d d d cl cl d cl cl d o cl 0 d d d d cl d (.6.5) (.6.6).6.0 Cylndcl nd Sphcl Coodnts Ctsn coodnts h bn sd clsly p to ths pont. In mny pctcl poblms t s s to cy ot n nlyss n tms of cylndcl o sphcl coodnts. Dffntton n ths coodnt systms s dscssd n wht follows 0. Cylndcl Coodnts Ctsn nd cylndcl coodnts ltd thoh (s F..6.8) cos y sn z z hn th Ctsn ptl dts bcom y tn y / (.6.7) z z y sn cos cos sn y y (.6.8) 0 ths scton lso ss s n ntodcton to th mo nl topc of Cln Coodnts cod n.6-.9 Sold Mchncs Pt III 4 Klly

43 Scton.6 z y z z z y z y h bs ctos ltd thoh F.6.8: cylndcl coodnts y z z cos sn sn cos z z cos y sn sn y cos (.6.9) so tht fom Eqn..6.4 ft som lb th Nbl opto n cylndcl coodnts ds s { Poblm 9} z z (.6.0) whch llows on to t th dnt of scl fld n cylndcl coodnts: z z (.6.) Ctsn bs ctos ndpndnt of poston. How th cylndcl bs ctos lthoh thy lwys of nt mntd chn dcton wth poston. In ptcl th dctons of th bs ctos dpnd on nd so ths bs ctos h dts wth spct to : fom Eqn..6.9 (.6.) wth ll oth dts of th bs ctos wth spct to h dnc cn now b ltd: z ql to zo. Sold Mchncs Pt III 4 Klly

44 Scton.6 Sold Mchncs Pt III Klly 44 z z z z z z (.6.) Smlly th cl of cto nd th Lplcn of scl { Poblm 0} z z z z z z (.6.4) Sphcl Coodnts Ctsn nd sphcl coodnts ltd thoh (s F..6.9) cos sn sn cos sn z y y z y z y / tn / tn (.6.5) nd th bs ctos ltd thoh cos sn sn sn cos cos cos cos sn sn cos sn sn cos cos sn cos sn sn sn cos cos cos sn y z y z y z y (.6.6) F.6.9: sphcl coodnts In ths cs th non-zo dts of th bs ctos z y y z z y

45 Scton.6 Sold Mchncs Pt III Klly 45 cos sn cos sn (.6.7) nd t cn thn b shown tht { Poblm } sn cot sn sn sn sn (.6.8).6. h Dctonl Dt Consd fncton. h dctonl dt of n th dcton of som cto w s th chn n n tht dcton. Now th dffnc btwn ts ls t poston nd w s F..6.0 w d (.6.9) F.6.0: th dctonl dt w 0 () ( w) ) ( D w

46 Scton.6 n ppomton to d cn b obtnd by ntodcn pmt nd by consdn th fncton w; on hs w 0 nd w w. If on tts s fncton of ylo s ss bot 0 s d ( ) (0) d d 0 d 0 o wtn t s fncton of w d ( w) ( ) d 0 w By sttn th dt h cn b sn to b ln ppomton to th ncmnt d Eqn hs s dfnd s th dctonl dt of th fncton () t th pont n th dcton of w nd s dnotd by [ w] d d 0 w h Dctonl Dt (.6.40) h dctonl dt s lso wttn s D w. h pow of th dctonl dt s dfnd by Eqn s ts nlty s sn n th follown mpl. Empl (th Dctonl Dt of th Dtmnnt) Consd th dctonl dt of th dtmnnt of th mt n th dcton of scond mt (th wod dcton s obosly sd loosly n ths contt). On hs d d dt [ ] dt d d 0 0 h Dctonl Dt nd h Gdnt Consd scl-ld fncton of cto z. Lt z b fncton of pmt z z z. hn Sold Mchncs Pt III 46 Klly

47 Scton.6 d dz dz d z d z d hs wth z w [ w] d d dz z d z w 0 0 (.6.4) whch cn b compd wth Eqn..6.. Not tht fo Eqns..6. nd.6.4 to b consstnt dfntons of th dctonl dt w h shold b nt cto..6. Foml tmnt of Vcto Clcls h clcls of ctos s now ttd mo fomlly n wht follows follown on fom th ntodctoy scton n.. Consd cto h n lmnt of th Ecldn cto spc E h E. In od to b bl to sp of lmts s lmnts bcom smll o clos to ch oth n ths spc on qs nom. H t th stndd Ecldn nom on E Eqn...8 h h h h h (.6.4) Consd nt scl fncton h f M h s h o thn on wts f : E R. If th s constnt M 0 sch tht h O h f s h o (.6.4) hs s clld th B Oh (o Lnd) notton. Eqn..6.4 stts tht zo t lst s fst s h. n psson sch s f h os to thn mns tht h h Smlly f f h h O h (.6.44) f s smll thn h fo h sffcntly clos to o. h f h 0 s h o (.6.45) thn on wts h oh h. f s o h. hs mpls tht h f os to zo fst thn Sold Mchncs Pt III 47 Klly

48 Scton.6 fld s fncton whch s dfnd n Ecldn (pont) spc E. scl fld s thn fncton f E : R. scl fld s dffntbl t pont E f th Df sch tht sts cto E In tht cs th cto th symbol f ). f h f Df h o h fo ll h E (.6.46) Df s clld th dt (o dnt) of f t (nd s n Now sttn h w n.6.46 wh w E s nt cto ddn thoh by nd tn th lmt s 0 on hs th qlnt sttmnt f d d w f w 0 fo ll w E (.6.47) whch s.6.4. In oth wods fo th dt to st th scl fld mst h dctonl dt n ll dctons t. Usn th chn l s n.6. Eqn cn b pssd n tms of th Ctsn bss hs mst b t fo ll w nd so n Ctsn bss whch s Eqn f f f w w w (.6.48) f f (.6.49).6. Poblms. ptcl mos lon c n spc dfnd by t 4t t 4t 8t t H t s tm. Fnd () nt tnnt cto t t () th mntds of th tnntl nd noml componnts of cclton t t d d d. Us th nd notton (..) to show tht. Vfy ths dt dt dt slt fo t t t t. [Not: th pmtton symbol nd th nt ctos ndpndnt of t; th componnts of th ctos scl fnctons of t whch cn b dffnttd n th sl wy fo mpl by sn th podct l of dffntton.] Sold Mchncs Pt III 48 Klly

49 Scton.6. h dnsty dstbton thohot mtl s n by. () wht sot of fncton s ths? () th dnsty s n n symbolc notton - wt t n nd notton () lt th dnt of () nt cto n th dcton n whch th dnsty s ncsn th most () nt cto n ny dcton n whch th dnsty s not ncsn () t ny nt cto oth thn th bs ctos nd th oth ctos yo sd bo nd clclt d / d n th dcton of ths nt cto () lt nd stch ll ths qntts fo th pont (). In pts (-) yo nsw n () symbolc (b) nd nd (c) fll notton. 4. Consd th scl fld dfnd by y z. () fnd th nt noml to th sfc of constnt t th on (000) () wht s th mmm l of th dctonl dt of t th on? () lt d / d t th on f d ds( ). 5. If dtmn d nd cl. 6. Dtmn th constnt so tht th cto s solnodl. 7. Show tht cl ω (s lso Poblm 9 n.). 8. Vfy th dntts (.6.5-6). 9. Us (.6.4) to d th Nbl opto n cylndcl coodnts (.6.0). 0. D Eqn. (.6.4) th cl of cto nd th Lplcn of scl n th cylndcl coodnts.. D (.6.8) th dnt dnc nd Lplcn n sphcl coodnts.. Show tht th dctonl dt D ( ) of th scl-ld fncton of cto ( ) n th dcton s.. Show tht th dctonl dt of th fnctonl U n th dcton of () s n by l l d d ( ) EI d d ( ) d ( ) d d EI d 0 0 l 0 l 0 p( ) ( ) d p( ) ( ) d. Sold Mchncs Pt III 49 Klly

50 Scton.7.7 Vcto Clcls - Intton.7. Odny Intls of Vcto cto cn b nttd n th odny wy to podc noth cto fo mpl 5 5 t t t dt 6.7. Ln Intls Dscssd h s th noton of dfnt ntl noln cto fncton tht nts scl. Lt b poston cto tcn ot th c C btwn th ponts p nd p. Lt f b cto fld. hn p p s n mpl of ln ntl. Empl (of Ln Intl) f d f d f d f d f d C C ptcl mos lon pth C fom th pont ( 000) to ( ) wh C s th stht ln onn th ponts F..7.. h ptcl mos n foc fld n by f Wht s th wo don on th ptcl? f d C F.7.: ptcl mon n foc fld Sold Mchncs Pt III 50 Klly

51 Scton.7 Solton h wo don s 6 d 4 d 0 d W f d C C h stht ln cn b wttn n th pmtc fom t t t so tht d W 0t t 6t dt o W f dt f dt dt 0 C C If C s closd c.. loop th ln ntl s oftn dnotd d. Not: n fld mchncs nd odynmcs whn s th locty fld ths ntl d s clld th cclton of bot C. C C.7. Const Flds If fo cto f on cn fnd scl sch tht thn p () f d s ndpndnt of th pth C onn p nd p p C () f d 0 ond ny closd c C f (.7.) In sch cs f s clld const cto fld nd s ts scl potntl. Fo mpl th wo don by const foc fld f s p p f d p p d p p d p p d ) ( p ) ( p whch clly dpnds only on th ls t th nd-ponts p nd p nd not on th pth tn btwn thm. It cn b shown tht cto f s const f nd only f cl f o { Poblm }. n nl of cos th dos not st scl fld sch tht f ; ths s not spsn snc cto fld hs th scl componnts whs s dtmnd fom st on Sold Mchncs Pt III 5 Klly

52 Scton.7 Sold Mchncs Pt III Klly 5 Empl (of Const Foc Fld) h ttonl foc fld f m s n mpl of const cto fld. Clly o f cl nd th ttonl scl potntl s m : ) ( ) ( p p p p m d m d m W p p p p Empl (of Const Foc Fld) Consd th foc fld ) ( f Show tht t s const foc fld fnd ts scl potntl nd fnd th wo don n mon ptcl n ths fld fom ) ( to ) 4 (. Solton On hs o f / / / cl so th fld s const. o dtmn th scl potntl lt f f f. Eqtn coffcnts nd nttn lds to ) ( ) ( ) ( q p whch f on chooss 0 q p so tht to whch my b ddd constnt. h wo don s

53 Scton.7 W ( 4) ( ) 0 Hlmholtz hoy s mntond const cto fld whch s ottonl.. f mpls f o nd c s. Smlly t cn b shown tht f on cn fnd cto sch tht f wh s clld th cto potntl thn f s solnodl.. f 0 { Poblm 4}. Hlmholtz showd tht cto cn lwys b psntd n tms of scl potntl nd cto potntl : yp of Vcto Condton Rpsntton Gnl f Iottonl (const) f o f Solnodl f 0 f.7.4 Dobl Intls h most lmnty typ of two-dmnsonl ntl s tht o pln on. Fo mpl consd th ntl o on R n th pln F..7.. h ntl R d d thn s th of R nd st s th on dmnsonl ntl of fncton s th nd th c th ntl R f ( ) dd s th olm nd th (n nl cd) sfc f ( ). hs ntls clld dobl ntls. ths dcomposton cn b md nq by qn tht f 0 s ; n nl f on s n f thn nd cn b obtnd by soln nmb of dffntl qtons Sold Mchncs Pt III 5 Klly

54 Scton.7 f ( ) R F.7.: ntton o on Chn of bls n Dobl Intls o lt ntls of th typ f ( ) dd t s oftn connnt to m R chn of bl. o do ths on mst fnd n lmntl sfc n tms of th nw bls t t sy qlnt to tht n th coodnt systm ds d d. h on R o whch th ntton ts plc s th pln sfc ( ) 0. Jst s c cn b psntd by poston cto of on snl pmt t (cf..6.) ths sfc cn b psntd by poston cto wth two pmts t nd t : ( t t ) ( t t ) Pmtsn th pln sfc n ths wy on cn clclt th lmnt of sfc ds n tms of t t by consdn cs of constnt t t s shown n F..7.. h ctos bondn th lmnt d () () d dt const d d dt const (.7.) t t t t so th of th lmnt s n by wh J s th Jcobn of th tnsfomton () () ds d d dtdt J dtdt (.7.) t t fo mpl th nt ccl 0 cn b psntd by t cos t t sn t 0t 0 t ( t t bn n ths cs th pol coodnts spctly) Sold Mchncs Pt III 54 Klly

55 Scton.7 J t o J (.7.4) t t t t t t t h Jcobn s lso oftn wttn sn th notton d d Jdt dt J t t h ntl cn now b wttn s R f ( t t ) Jdtdt () d ds () d t t t t t t F.7.: sfc lmnt Empl Consd on R th qt nt-ccl n th fst qdnt 0. h momnt of nt bot th s s dfnd by I d d R 0 nsfom th ntl nto th nw coodnt systm t t by mn th sbstttons4 t cost t sn t. hn J t t t t cost sn t t sn t t cost t 4 ths th pol coodnts t t ql to spctly Sold Mchncs Pt III 55 Klly

56 Scton.7 so I / 0 0 t sn tdtdt Sfc Intls Up to now dobl ntls o pln on h bn consdd. In wht follows consdton s n to ntls o mo compl cd sfcs n spc sch s th sfc of sph. Sfcs n cd sfc cn b pmtzd by t t now by th poston cto ( t t ) ( t t ) ( t t ) On cn nt c C on th sfc S by tn t ( t s) t ( t s) so tht C hs poston cto F..7.4 s t s) t ( ) ( s cto tnnt to C t pont p on S s fom Eqn..6. d dt dt ds t ds t ds (s) C s ( t t) S F.7.4: cd sfc Mny dffnt cs C pss thoh p nd hnc th mny dffnt tnnts wth dffnt cospondn ls of dt / ds dt / ds. hs th ptl dts / t / t mst lso both b tnntl to C nd so noml to th sfc t p s n by th coss-podct nd nt noml s Sold Mchncs Pt III 56 Klly

57 Scton.7 n / t t (.7.5) t t In som css t s possbl to s non-pmtc fom fo th sfc fo mpl ( ) c n whch cs th noml cn b obtnd smply fom n d / d. Empl (Pmtc Rpsntton nd th Noml to Sph) h sfc of sph of ds cn b pmtsd s 5 sn t t t 0 t cost sn t sn t cos 0 H lns of t const plll to th pln ( pllls ) whs lns of t const mdn lns F If on ts th smpl pssons t s t / s o 0 s / on obtns c C onn ( 00 ) nd ( 00) nd pssn thoh ( / / / ) s shown. t / n C F.7.5: sph h ptl dts wth spct to th pmts t t cost sn t cost sn t cost sn t sn t cost sn t so tht t t sn t cost sn t sn t sn t cost 5 ths th sphcl coodnts (s.6.0); t t Sold Mchncs Pt III 57 Klly

58 Scton.7 Sold Mchncs Pt III Klly 58 nd nt noml to th sphcl sfc s cos sn sn cos sn n t t t t t Fo mpl t 4 / t t (ths s on th c C ) on hs 4 / 4 / n nd s pctd t s n th sm dcton s. Sfc Intls Consd now th ntl ds S f wh f s cto fncton nd S s som cd sfc. s fo th ntl o th pln on const const dt dt t t d d ds t t only now ds s not flt nd s th dmnsonl. h ntl cn b ltd f on pmtss th sfc wth t t nd thn wts dt dt t t S f On wy to lt ths coss podct s to s th lton (Ln s dntty Poblm 5.) c b d d b c d c b (.7.6) so tht t t t t t t t t t t t t (.7.7) Empl (Sfc of Sph) Usn th pmtc fom fo sph n bo on obtns 4 sn t t t so tht

59 Scton.7 S ds 0 0 sn t dt dt 4 Fl Intls Sfc ntls oftn nol th noml to th sfc s n th follown mpl. Empl If f 4 lt f n ds wh S s th sfc of th cb S bondd by ; 0; 0 nd n s th nt otwd noml F n n n n F.7.6: th nt cb Solton h ntl nds to b ltd o th s fcs. Fo th fc wth n nd S f nds 4 dd 4dd Smlly fo th oth f sds whnc f nds. S Intls of th fom f nds nown s fl ntls nd s qt oftn n S pplctons. Fo mpl consd mtl flown wth locty n ptcl th flow thoh smll sfc lmnt ds wth otwd nt noml n F h olm of mtl flown thoh th sfc n tm dt s ql to th olm of th slntd cylnd shown whch s th bs ds tms th hht. h slntd hht s (= Sold Mchncs Pt III 59 Klly

60 Scton.7 locty tm) s dt nd th tcl hht s thn ndt. hs th t of flow s th olm fl (olm p nt tm) thoh th sfc lmnt: nds. ndt n dt F.7.7: flow thoh sfc lmnt h totl (olm) fl ot of sfc S s thn 6 olm fl: nds (.7.8) S Smlly th mss fl s n by mss fl: S nds (.7.9) Fo mo compl sfcs on cn wt sn Eqn S f nds S dtdt t t f Empl (of Fl Intl) Compt th fl ntl f n S ds wh S s th pbolc cylnd psntd by nd f F Solton 0 0 Mn th sbstttons t t so tht by th poston cto t th sfc cn b psntd 6 f cts n th sm dcton s n.. pontn otwd th dot podct s post nd ths ntl s post; f on th oth hnd mtl s flown n thoh th sfc nd n n oppost dctons nd th dot podct s nt so th ntl s nt Sold Mchncs Pt III 60 Klly

61 Scton.7 t t 0 t t t 0 hn / t t / t nd t t t so th ntl bcoms t tt t dtdt 0 0 n f F.7.8: fl thoh pbolc cylnd Not: n ths mpl th l of th ntl dpnds on th choc of n. If on chooss n nstd of n on wold obtn. h noml n th oppost dcton (on th oth sd of th sfc) cn b obtnd by smply swtchn t nd t snc / t / t / t / t. Sfc fl ntls cn lso b ltd by fst contn thm nto dobl ntls o pln on. Fo mpl f sfc S hs pocton R on th pln thn n lmnt of sfc ds s ltd to th poctd lmnt d d thoh (s F..7.9) cos ds n ds dd nd so S f nds f n dd R n Sold Mchncs Pt III 6 Klly

62 Scton.7 n F.7.9: pocton of sfc lmnt onto pln on h Noml nd Sfc Elmnts It s somtms connnt to ssoct spcl cto d S wth dffntl lmnt of sfc ds wh ds n ds so tht d S s th cto wth mntd ds nd dcton of th nt noml to th sfc. Fl ntls cn thn b wttn s S f nds S f ds.7.6 Volm Intls h olm ntl o tpl ntl s nlston of th dobl ntl. Chn of Vbl n Volm Intls Fo olm ntl t s oftn connnt to m th chn of bls ( ) ( t t t ). h olm of n lmnt dv s n by th tpl scl podct (Eqns ) dv dtdt t t t dt Jdt dt dt (.7.0) wh th Jcobn s now Sold Mchncs Pt III 6 Klly

63 Scton.7 Sold Mchncs Pt III Klly 6 o t t t t t t t t t J t t t t t t t t t J (.7.) so tht V V dt dt J dt t t t t t t t t t ddydz f f.7.7 Intl homs nmb of ntl thoms nd ltons psntd h (wthot poof) th most mpotnt of whch s th dnc thom. hs thoms cn b sd to smplfy th lton of ln dobl sfc nd tpl ntls. hy cn lso b sd n os poofs of oth mpotnt slts. h Dnc hom Consd n bty dffntbl cto fld ) ( t dfnd n som fnt on of physcl spc. Lt V b olm n ths spc wth closd sfc S bondn th olm nd lt th otwd noml to ths bondn sfc b n. h dnc thom of Gss stts tht (n symbolc nd nd notton) V S V S dv ds n nds dv d Dnc hom (.7.) nd on hs th follown sfl dntts { Poblm 0} V S V S V S dv ds dv ds dv ds n n n cl d ) d( (.7.) By pplyn th dnc thom to y smll olm on fnds tht V ds S V n 0 lm d tht s th dnc s ql to th otwd fl p nt olm th slt.6.8.