1.19 Curvilinear Coordinates: Curved Geometries

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Ανατόλιος Βούλγαρης
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1 Scton.9.9 Curlnr Coordnts: Curd Gomtrs In ths scton s mnd th spcl cs of two-dmnsonl curd surfc..9. Monoclnc Coordnt Systms Bs Vctors curd surfc cn b dfnd usn two cornt bs ctors wth th thrd bs ctor rywhr of unt sz nd norml to th othr two F..9. Ths bs ctors form monoclnc rfrnc frm tht s only on of th nls btwn th bs ctors s not ncssrly rht nl. Fur.9.: Gomtry of th Curd Surfc In wht follows n th nd notton Grk lttrs such s tk lus nd ; s bfor Ltn lttrs tk lus from... Snc nd 0 0 (.9.) th dtrmnnt of mtrc coffcnts s 0 J J 0 0 (.9.) 0 0 Th Cross Product Prtculrsn th rsults of.6.5 dfn th surfc prmutton symbol to b th trpl sclr product Sold Mchncs Prt III 75 lly

2 Scton.9 ε ε (.9.) whr ε ε s th Crtsn prmutton symbol ε ε nd zro othrws wth + ε ε δ δ δ δ (.9.4) From.9. (.9.5) nd so (.9.6) Th cross product of surfc ctors tht s ctors wth componnt n th norml ( ) drcton zro cn b wrttn s u u u u u u u (.9.7) Th Mtrc nd Surfc lmnts Consdrn ln lmnt lyn wthn th surfc so tht 0 surfc s th mtrc for th ( Δs) ( ) ( ) (.9.8) whch s n ths contt known s th frst fundmntl form of th surfc. Smlrly from.6.5 surfc lmnt s n by Δ S Δ Δ (.5.9) Chrstoffl Symbols Th Chrstoffl symbols cn b smplfd s follows. dffrntton of l to Sold Mchncs Prt III 76 lly

3 Scton.9 (.9.0) so tht from Eqn (.9.) Furthr snc / 0 0 (.9.) 0 Ths lst two qutons mply tht th jk nsh whnr two or mor of th subscrpts r. Nt dffrntt.9. to t (.9.) nd Eqns..8.6 now ld to (.9.4) From.8.8 usn.9. 0 (.9.5) nd smlrly { Problm } 0 (.9.6).9. Th Curtur Tnsor In ths scton s ntroducd tnsor whch wth th mtrc coffcnts compltly dscrbs th surfc. Frst lthouh th bs ctor mntns unt lnth ts drcton chns s functon of th coordnts nd ts drt s from.8. or.8.5 (nd usn.9.5) k k k k (.9.7) Dfn now th curtur tnsor to h th cornt componnts throuh Sold Mchncs Prt III 77 lly

4 Scton.9 (.9.8) nd t follows from nd.9.4 (.9.9) nd snc ths Chrstoffl symbols r symmtrc n th th curtur tnsor s symmtrc. Th md nd contrrnt componnts of th curtur tnsor follows from.6.8-9: (.9.0) nd th dot s not ncssry n th md notton bcus of th symmtry proprty. From ths nd.8.8 t follows tht (.9.) lso d ( ) ( ) ( ) ( ) (.9.) whch s known s th scond fundmntl form of th surfc. From.9.9 nd th dfntons of th Chrstoffl symbols th curtur cn b prssd s (.9.) shown tht th curtur s msur of th chn of th bs ctor lon th cur n th drcton of th norml ctor; ltrntly th rt of chn of th norml ctor lon n th drcton. Lookn t ths n mor dtl consdr now th chn n th norml ctor n th drcton F..9.. Thn d d d (.9.4) Sold Mchncs Prt III 78 lly

5 Scton.9 d + d dφ Fur.9.: Curtur of th Surfc Tkn th cs of 0 0 on hs d. From F..9. nd snc th norml ctor s of unt lnth th mntud d quls d φ th smll nl throuh whch th norml ctor rotts s on trls lon th coordnt cur. Th curtur of th surfc s dfnd to b th rt of chn of th nl φ : dφ (.9.5) nd so th md componnt th curtur n th drcton. s th curtur n th drcton. Smlrly ssum now tht 0 0. Eqn..9.4 now r d nd rfrrn F..9. th twst of th surfc wth rspct to th coordnts s s dϕ (.5.6) d + d dϕ Whn twst. Fur.9.: Twstn or th Surfc s th twst; whn thy r not qul s closly rltd to th ths s ssntlly th sm dfnton s for th spc cur of.5.; thr th nl φ κδs Sold Mchncs Prt III 79 lly

6 Scton.9 Two mportnt quntts r oftn usd to dscrb th curtur of surfc. Ths r th frst nd th thrd prncpl sclr nrnts: I III dt j + ε (.9.7) Th frst nrnt s twc th mn curtur M whlst th thrd nrnt s clld th Gussn curtur (or Totl curtur) of th surfc. Empl (Curtur of Sphr) Th surfc of sphr of rdus cn b dscrbd by th coordnts ( ).9.4 whr G F. sn cos sn sn cos Fur.9.4: sphrcl surfc Thn from th dfntons { Problm } + cos sn cos sn sn 0 0 sn + cos + sn 4 sn sn cos sn (.9.8) From.9.6 cos sn cos + sn sn + (.9.9) Sold Mchncs Prt III 80 lly

7 Scton.9 nd ths s clrly n orthoonl coordnt systm wth scl fctors (s.6.5) h h sn h (.9.0) Th surfc Chrstoffl symbols r from cos 0 sn (.9.) sn cos Usn th dfntons.8.4 { Problm } sn (.9.) wth th rmnn symbols 0. Th componnts of th curtur tnsor r thn from (.9.) 0 0 sn 0 [ ] [ ] Th mn nd Gussn curtur of sphr r thn M G (.9.4) Th prncpl curturs r dntly rcprocl of th rdus of curtur. nd. s pctd thy r smply th.9. Cornt Drts Vctors Consdr ctor whch s not ncssrly surfc ctor tht s t mht h norml componnt. Th cornt drt s Sold Mchncs Prt III 8 lly

8 Scton.9 (.9.5) Dfn now two-dmnsonl nlou of th thr-dmnsonl cornt drt throuh (.9.6) so tht usn th cornt drt cn b prssd s (.9.7) In th spcl cs whn th ctor s pln ctor thn 0 nd thr s no dffrnc btwn th thr-dmnsonl nd two-dmnsonl cornt drts. In th nrl cs th cornt drts cn now b prssd s ( ) ( ) + + (.9.8) From.8.5 th rdnt of surfc ctor s (usn.9.) Tnsors ( ) + rd (.9.9) Th cornt drts of scond ordr tnsor componnts r n by.8.8. For mpl j j j m mj j j m m j j j (.9.40) Hr only surfc tnsors wll b mnd tht s ll componnts wth n nd r zro. Th two dmnsonl (pln) cornt drt s Sold Mchncs Prt III 8 lly

9 Scton.9 (.9.4) lthouh 0 for pln tnsors on stll hs non-zro (.9.4) wth 0. From.8.8 th drnc of surfc tnsor s d + (.9.4).9.4 Th Guss-Codzz Equtons Som usful qutons cn b drd by consdrn th scond drts of th bs ctors. Frst from.8. + (.9.44) scond drt s + + (.9.45) Elmntn th bs ctors drts usn.9.44 nd.9.0b l to { Problm 4} ( ) + ( + ) (.9.46) Ths quls th prtl drt. Comprson of th coffcnt of for ths ltrnt prssons for th scond prtl drt l to (.9.47) From Eqn..8.8 Sold Mchncs Prt III 8 lly

10 Scton.9 (.9.48) nd so (.9.49) Ths r th Codzz qutons n whch thr r only two ndpndnt non-trl rltons: (.9.50) Rsn ndcs usn th mtrc coffcnts l to th smlr qutons (.9.5) Th Rmnn-Chrstoffl Curtur Tnsor Comprn th coffcnts of n.9.46 nd th smlr prsson for th scond prtl drt shows tht (.9.5) Th trms on th lft r th two-dmnsonl Rmnn-Chrstoffl Eqn..8. nd so R (.9.5) Furthr R R (.9.54) Ths r th Guss qutons. From.8. t sq. only 4 of th Rmnn-Chrstoffl symbols r non-zro nd thy r rltd throuh R (.9.55) R R R so tht thr s n fct only on ndpndnt non-trl Guss rlton. Furthr R ( ) ν ρ ρ ν ( δ δ δ δ ) ρ ν (.9.56) Usn.9.4b.9. Sold Mchncs Prt III 84 lly

11 Scton.9 R ε ε ρν ρ ν (.9.57) nd so th Guss rlton cn b prssd succnctly s R (.9.58) G whr G s th Gussn curtur.9.7b. Thus th Rmnn-Chrstoffl tnsor s zro f nd only f th Gussn curtur s zro nd n ths cs only cn th ordr of th two cornt dffrnttons b ntrchnd. Th Guss-Codzz qutons.9.50 nd.9.58 r qulnt to st of two frst ordr nd on scond ordr dffrntl qutons tht must b stsfd by th thr ndpndnt mtrc coffcnts nd th thr ndpndnt curtur tnsor coffcnts. Intrnsc Surfc Proprts n ntrnsc proprty of surfc s ny quntty tht rmns unchnd whn th surfc s bnt nto nothr shp wthout strtchn or shrnkn. Som mpls of ntrnsc proprts r th lnth of cur on th surfc surfc r th componnts of th surfc mtrc tnsor (nd hnc th componnts of th Rmnn-Chrstoffl tnsor) nd th Gussn curtur (whch follows from th Guss quton.9.58). dlopbl surfc s on whch cn b obtnd by bndn pln for mpl pc of ppr. Empls of dlopbl surfcs r th cylndrcl surfc nd th surfc of con. Snc th Rmnn-Chrstoffl tnsor nd hnc th Gussn curtur nsh for th pln thy nsh for ll dlopbl surfcs..9.5 Godscs Th Godsc Curtur nd Norml Curtur Consdr cur C lyn on th surfc wth rc lnth s msurd from som fd pont. s for th spc cur.6. on cn dfn th unt tnnt ctor τ prncpl norml ν nd bnorml ctor b (Eqn..5. t sq.): d τ dτ ν b τ ν (.9.59) κ so tht th cur psss lon th ntrscton of th oscultn pln contnn τ nd ν (s F..6.) nd th surfc Ths ctors form n orthonorml st but lthouh ν s norml to th tnnt t s not ncssrly norml to th surfc s llustrtd n F. Sold Mchncs Prt III 85 lly

12 Scton For ths rson form th nw orthonorml trd ( τ τ ) τ ls n th pln tnnt to th surfc. From so tht th unt ctor τ τ (.9.60) ν τ C Fur.9.5: cur lyn on surfc Nt th ctor d τ / wll b dcomposd nto componnts lon τ nd th norml. Frst dffrntt.9.59 nd us.9.44b to t { Problm 5} dτ d + (.9.6) Thn dτ κ τ + κ n (.9.6) whr κ n d κ (.9.6) Ths r formul for th odsc curtur κ nd th norml curtur κ n. Mny dffrnt curs wth rprsnttons (s) cn pss throuh crtn pont wth n tnnt ctor τ. Form.9.59 ths wll ll h th sm lu of d / nd so from.9.6 ths curs wll h th sm norml curtur but n nrl dffrnt odsc curturs. Sold Mchncs Prt III 86 lly

13 Scton.9 cur pssn throuh norml scton tht s lon th ntrscton of pln contnn τ nd nd th surfc wll h zro odsc curtur. Th norml curtur cn b prssd s κ τ τ (.9.64) n If th tnnt s lon n nctor of thn κ n s n nlu nd hnc mmum or mnmum norml curtur. Surfc curs wth th proprty tht n nctor of th curtur tnsor s tnnt to t t ry pont s clld ln of curtur. connnt coordnt systm for surfc s on n whch th coordnt curs r lns of curtur. Such systm wth contnn th mmum lus of κ hs t ry pont curtur tnsor of th form n j [ ] 0 0 ( κ ) n 0 m 0 ( κ ) n mn (.9.65) Ths ws th cs wth th sphrcl surfc mpl dscussd n.9.. Th Godsc odsc s dfnd to b cur whch hs zro odsc curtur t ry pont lon th cur. Form.9.6 prmtrc qutons for th odscs or surfc r d 0 (.9.64) It cn b prod tht th odsc s th cur of shortst dstnc jonn two ponts on th surfc. Thus th odsc curtur s msur of th dnc of th cur from th shortst-pth cur. Th Godsc Coordnt Systm If th Gussn curtur of surfc s not zro thn t s not possbl to fnd surfc coordnt systm for whch th mtrc tnsor componnts qul th ronckr dlt δ rywhr. Such omtry s clld Rmnnn. Howr t s lwys possbl to construct coordnt systm n whch δ nd th drts of th mtrc coffcnts r zro t prtculr pont on th surfc. Ths s th odsc coordnt systm..9.6 Problms Dr Eqns Dr th Crtsn componnts of th curlnr bs ctors for th sphrcl surfc Eqn Sold Mchncs Prt III 87 lly

14 Scton.9 Dr th Chrstoffl symbols for th sphrcl surfc Eqn Us Eqns nd.9.0b to dr Us Eqns nd.9.44b to dr.9.6. Sold Mchncs Prt III 88 lly

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