PC Vectors, Fields and Matrices Semester 1

Transcription

1 P 7 - ectos, Felds d Mtces Semeste. ectos. Deto, ddto d Sutcto s phscsts we e coceed wth oects whch epeset phscl quttes. Scl ucto / eld: (). hs could tell us the tempetue t posto. It s speced ume. No decto s metoed. E.g. Mss. ectos: Oects chctesed oth mgtude d decto. e.g. the eltve dsplcemet o two pots, P d Q. emple s veloct, whch mght e epeseted s. Othe emples clude Mometum d gul eloct. Q o 7m PQ s the dsplcemet vecto om P to Q. he vecto tells us tht Q les 7m (mgtude) om P t eg o o (Decto) P ecto ddto: deotes the dsplcemet oth m d the dsplcemet est m, the we c s. lel, lthough ddto s commuttve. It s lso cle tht ( ) ( ) o,, d. We c dee the egtvt o vecto s - So we c toduce sutcto o vectos, ovous (-), o - s ull vecto. Fd the popet tht s ( ) ( s scl)..e. ddto s dstutve. It ollows om esclg the vectos. We c solve umeous polems usg ths geometc eect: e.g. mss o g s t the mdpot o the ope suspeded s the setch. Wht s the teso the ope? 6 o g w g N I equlum the sum o ll the oces s eo. w

2 P 7 - ectos, Felds d Mtces Semeste Hece N N. ompoets So we eeded to dw dgm ode to dd o sutct vectos. See lgec epesetto. e.. D vecto c e epessed s sum o two othe vectos. oveet to choose sc vectos to e pllel to the co-odte s. s the: d e se vectos d hve ut legth. e.g. t Wht s compoets? (,) hs c e geelsed to D eed d ss sc vecto pllel to the -s. (,, ) c c c.. we ust pced up ght-hded coodte sstem (.e. -s pots up ) he mgtude o the vecto s oted Pthgos.

3 P 7 - ectos, Felds d Mtces Semeste I D: It s lso cle tht: ( ) ( ) ( ). Ut ectos ectos o mgtude e clled ut vectos (.e.,, ) e.g. to d ut vecto whch s pllel to (,,) (,, ). Posto ectos osde pot P. We c epeset the locto o P eltve to og O specg ts posto vecto. P(,,) lel OP OP lso Reltol postos e es to compute. e.g. cosde pots P d P t d. Wht s the eltve postos? P P P P P P P P hs lst esult c e used to deve the vecto equto o le.

4 P 7 - ectos, Felds d Mtces Semeste - o o ( ) o s some ume. It ves om - to to geete ll pots. d o e two pots o the le. ete o Mss: Gve sstem o ptcles o mss m, m,..., m t postos,,..., the locto o the cete o mss s deed to e t: m m... m R m m... m.6 Scl Poduct hs s the deto o the scl poduct. s scl. It s ote lso clled the dot poduct. o ( ) ( ) e.g. oce F cts o ptcle. lculte the wo doe F whe the ptcle s dsplced mout. F ssume F s costt. lel Not so cle: ( ) (Pove t!) Fo pllel vectos Fo pepedcul e.g. deve the e ule:

5 P 7 - ectos, Felds d Mtces Semeste ( ) ( ) c c c c c etc... ut ( ) π heeoe c. e.g. clculte (,,) d (-,,) ( ) ( ) etc etc So 6 hs geelses esl:.e. d the hs s scl qutt.e. t does t deped o the choce o coodtes. e.g. s ot scl!.7 Equto o Ple spec the ple uquel specg pot the ple ( o ) d vecto oml to the ple ( ). o R R We wt equto o. ( ) o.e. o hs s the desed equto. π- R o

6 P 7 - ectos, Felds d Mtces Semeste c I,, ( ) he c o o α α dstce o ple om og d. o o So we c wte: c d Equvlet equto o ple. e.g. d the equto o ple wth oml N (,, ) pssg though the pot (,, ) How s the ple om the og? Equto o ple d cd I c (: c ). d Equto s (,, ) whee (,, ). o o d o.8 ecto Poduct Deto: s Use ght-hd ule to wo out whch decto goes. (wstg t-clocwse, up s ) u to get coect vlue. 6

7 P 7 - ectos, Felds d Mtces Semeste N: s vecto. s pepedcul to d. lso hs geometc tepetto: h e o pllelogm h ut h s So e s. s s ths s ot commuttve!. Note: d ( ) Note tht o pllel vectos. ( s ) e.g. smpl ( ) ( µ ) ( ) µ ( ) ( µ )( ) ( d µ e scls) µ µ Let us t to gue out omul o We wll get thgs le: ( ) ( ) compoets. ollect compoets: ( ) ( ) ( ) othe w to wte ths s detemt: ) Select d cove up colum d ow t s ) oss-multpl the vlues whch em ucoveed,.e. ) Repet o - d. e.g.: 7

8 P 7 - ectos, Felds d Mtces Semeste (,, ) (,, ) Show tht (,, ) lculte ( ) ( ) ( ) ( ) ( ) ( ) ( 6) ( ) ( ) whe (,, ) (,, ) ( 6) ( ) ( ) (,, ).9 pplctos o the ecto Poduct Some emples om dmcs: oque τ F F τ F s whch s sesle. τ s to the ppe tells us sese whch toque teds to duce otto. lso meet gul momet L P (gul mometum eltve to some og) dl Ke equto ottol dmcs s τ (See lte) dt dp F dt ecto poduct lso ppes electct d mgetsm. F q (Fq o Fe) q electc chge veloct mgetc eld e.g.: gd od ottes out s though. wth gul veloct ω. Show tht the le veloct v o pot P the od wth posto s v ω. 8

9 P 7 - ectos, Felds d Mtces Semeste ω R s to ppe od ottes out s though pllel to ω ( gul veloct). v R ω whee R dus o ccle s So v ω Fom setch we see tht v ω e.g. ptcle o mss g s ottg out the -s t d.s - (.e. ω(,,)d.s - ) t ed dstce om d t ed gle to the -s., ω mg ) Wht s ts veloct whe t s t (,,) m? (,, ) v ω ms ) Wht s the gul mometum out whe t s t ths pot? L P P mv (,, ) L ( 8 ).. L s ot pllel to ω (o LIω). L I ω s tue...) Note: we ust evluted two vecto poducts.e.:- L m( ω) hee s quce w to evlute such tple vecto poduct.. ple ecto Poduct Idett: ( ) ( ) ( ). Poo? Wht s ( )? ( ) ( ) ( ) e.g. we ust woed out 9

10 P 7 - ectos, Felds d Mtces Semeste L m ω (,, ) m (,, ) m g ( ω ) m[ ( ω) ( ) ω] d. s ( ) ( )(,, ) (,,8 )uts L,, Whch s s ove.. Scl ple Poduct We lso compute ( ) Scl ple Poduct. sde ( ) s thd w to multpl thee vectos. It s vecto, d s es to compute. Questo: s ( ) teestg? It s megless, s dot s scl, d ou c t do s poduct wth scl. φ he ove s o omed, d. Pllelepped (ll oces pllel ps) ( ) ϕ olume o o (e o se)(pepedcul heght) ϕ ( ) hs tepetto llows us to see tht: ( ) ( ) ( ) ( ) ( ) etc... e.g show tht: ( ) ( ) ( ) ( ) ( ) [ ( )...](...). ( ) ( )

11 P 7 - ectos, Felds d Mtces Semeste. Itesectg Ples e.g. d the pot o tesecto o the ples: () ( ) ( ) solve ute oce : Usg (): Ito (): Put these to ():! hee s o soluto! he ples do ot meet pot. Must e ths oleoe coguto ecuse, d e ot pllel. Whe do ples ot meet pot? Loog ed o he othe cses whch hve o soluto lso hve the popet tht the oml vectos le ple. I. pllel ples No soluto t ll. II. ples the sme: III. ples the sme. ll pots e solutos to the ogl equto. I. pllel, cled o soluto.

12 P 7 - ectos, Felds d Mtces Semeste. cocdetl, cled. Soluto s le. I. ll meet le. II. oleoe o soluto. ll hve the popet tht omls le the ple o the lcod / pge. ( ) ( ) to. So we could hve checed to see soluto ested clcultg: ( ) (,, ) theeoe o uque soluto.. Deetto o ectos It s stght owd to deette vectos. Geell (s) s vecto whch depeds o s we c dee: d lm ( s s) ( s) ds s s I compoets ths educes to: d d d d ds ds ds ds N:, d e ed. e.g.:- d d d d v dt dt dt dt s the veloct vecto o ptcle t posto (,, ) Newto s Secod Lw ecomes: dpdt F. md P mv dt d d d lel ( ) d ds ds ds d ( ) ds d d. ds ds

13 P 7 - ectos, Felds d Mtces Semeste d d d ( ) ds ds ds d d d ds ds ds (Eecse: pove these) ( ) Note tht the ode mttes vecto poducts. e.g. clculte the veloct d cceleto o od whose posto s R ωt, R sωt, (R, ω e postve costts) ( ) v (t) ωt R We epect: v ωr v ω R ω R s the ut vecto potg dll outwds. d v & R dt d && R dt ( ω sωt, ω ωt,) ( ω ωt, ω sωt,) ω R( ωt,sωt,) ω R ω e.g. L P sc equto o ottol dmcs s: d L F dt Deve ths o ptcle t posto d mometum P. dl d( P) d dp P F dt dt dt dt S ss vectos hve ee ed sometmes t s useul to use movg ss (!) e.g. ple pol cods: osde ptcle movg ple. Its posto t tme t s (,). use d s ss vectos. (,)

14 P 7 - ectos, Felds d Mtces Semeste, e ut vectos d e othogol ( ) d dt d dt d dt. ut the deped upo tme, t. So: d lm d dt t dt dt & & Hece: & & & eloct ple pol coodtes. e.g. uom ccul moto: & & " v ω" && && & & & & ccel : && & & s st! Loo c t dgm:

15 P 7 - ectos, Felds d Mtces Semeste ( ) & & Susttute to & & gves: ( ) ( ) ( ) ( ) & & && & && && & & && & & & & && && Fo uom ccul moto, & & &.. Rottos We mght wt to covet om oe ss (,, ) to othe (,, ). How do we do t? e.g. cosde D cse: s s s s e.g. show tht s vt ude otto o coodte s (Stc to D) ( )( ) ( )( ) s s s s s s

16 P 7 - ectos, Felds d Mtces Semeste So s scl. e.g. So s ot scl. he geelsto to D s pcple stght owd. Need gles to spec geel otto. s o Rotto ψ φ d φ spec s o otto. Ψ speces gle o otto out tht s. Qute t moe volved th the D cse. hee s et w to epess D ottos: s s hs s ottol s. R Mtces e the suect o Secto o the couse. 6

17 P 7 - ectos, Felds d Mtces Semeste. Detemts. Le Equtos I phscs we ote ecoute quttes whch deped lel o sevel othes. e.g. ottos D: s s comto o d e.g. Hooe s Lw F- osde the cse o msses coected spgs. Let F c Foce o let hd mss (oveto, F c > oce cts to ght) Let F Foce o ght hd mss. Fc ( ) F l( ) hese e le equtos d. I geel (.e. o uequl msses d spgs) we would hve: Fc ( K K ) F ( K K ) I ou cse: K K K K We c wte these equtos s: F K Mt Equto Fc F F K K K K K Now let s s the ollowg questo: Stt om equlum d ppl etel oces F d F to the let d ght msses espectvel. How do the msses move eoe the g come to est? I sttc equlum, the etel oces must lce those om Hooe s lw,.e. F F, F c F Equvletl Fet F F Fet F F s colum vecto. F K F et 7

18 P 7 - ectos, Felds d Mtces Semeste t solve o ust dvdg K sce t s mt..e. we must solve: K K F () F K F () Solvg gves: K F F F F KK KK F K F F F KK KK Soluto ests povdg the deomto s ot eo..e. K K KK. K K ut K K KK DetK K K K hs s emscet o the tesectg ples polem..e.: hee s uque soluto.. Detemts Let s stt wth the geel ule o evlutg detemts: he cocto o s, etc. he sg o the cocto s detemed the ptte: etc... Wht s the cocto o? choose NY ow o colum to evlute detemt. e.g. suppose s the elemet o mt t ow d colum, the det (.e. the detemt o ) s: det.... lels ow.... NN s colum. e.g. lculte the detemt o the mt: M 8

19 P 7 - ectos, Felds d Mtces Semeste 9 hoose the st ow: det M Do t g o the secod ow: det M. Uses o Detemts We hve see emples (Spg d ples) o uses led. I geel we c test to see sstem o le equtos hs soluto evlutg the ppopte detemt. N,,..., hs s set o le equtos whch we mght wt to solve o the. e.g. pc hese equtos hve soluto : det M M M M I det the ethe the equtos e cosstet o thee s t o solutos. e.g. does hve uque soluto? Wll 8

20 P 7 - ectos, Felds d Mtces Semeste. Felds. Itoducto eld s used to desce phscl qutt whose vlue depeds upo posto. I the vlue s ust ume the t s scl eld. I t s vecto the t s vecto eld. e.g. clss the ollowg: empetue ths oom scl eld. () Gvttol eld ths oom vecto eld. g() Mgetc eld oud mget vecto eld. (). Fuctos o sevel vles Fuctos o ol oe vle e pett e phscs. e.g. (,,), U(p,) etc Fuctos o two vles c e epeseted s suces dmesos, o cotou mp (e.g. Odce Suve o wethe mp sos). Hllsde Le o costt Z. Ptl Devtves Rtes o chge e cucl phscs. Need to udestd how to do clculus wth uctos o > vle. Fo (,): Lm ( δ, ) (, ) Dee δ δ (,) (δ,) δ lel Dee tells us the te o chge o (,) the -decto. Lm (, δ ) (, ) δ δ δ (,δ) (,) lel tells us the te o chge o (,) the -decto.

21 P 7 - ectos, Felds d Mtces Semeste e.g. Gve (,), clculte the slope o (,) t geel pot ) the -decto; ) the -decto. ) I the -decto the slope s: ( ) Smll emds us to eep ed. Ptl d d s ltetve otto. ) I the -decto the slope s: ( ) 8 Hghe devtves c e computed e.g. I ou cse Smll 8 lso ( ) 8 ( ).e. hs s geell tue,.e. ode o deetto s umpott. e.g. o mole o del gs pr. Wht e p d v P? R p R p p p p p ), ( R R p p p. otl Deetls How does (,) chge s we go om (,) to ( ) δ δ,? Recll, o ucto o vle g() ( ) δ d dg g δ g g δ ) ( (lo s theoem) δ δ δ

22 P 7 - ectos, Felds d Mtces Semeste I the lmt δ, δ we c wte d d d d s clled the totl deetl o [N geeles to dmesos d d d d ] mples,.e. ll othe vlues em costt. e.g. (,) -. Estmte (.,.) (,) wted moe ccutel. δ δ 8 δ δ ( ) ( ) δ δ. (.,.). e.g. Suppose we wl log pth ((t), (t), (t)) whee (t) s heght ove se level. Wht s the te t whch we g heght? d ( t) dt Wht ou wee gve (,) d (t) d (t)? d ( t) susttute o (t) d (t) to (,) the do dt d d d dt dt dt We c wo ths out wthout susttutg. d d d Geelto o the ch ule dt d dt Suppose we ow s o the chge lttude o tesml dsplcemet the - decto log the pth. δ (te o chge o lttude wth ) δ t s ot ths sce ves s we move log the pth! d d d d d d d d totl devtve o wth espect to d d ew om ove:

23 P 7 - ectos, Felds d Mtces Semeste (,) (δ,δ) (, ) ( δ, ) (, ) δ d d δ δ e.g. Suppose (,) s such tht (u,v) d (u,v). Wht s [g, t s possle to susttute c.] ut much ese (usull) s to do: d d d v u v u v u lm d ( δ ) ll. chges. tesml v u?. Stto Pots e pots whee ll slopes vsh. g g.e. o g(,) stto pots e t See hdout o test o tue. Fo ucto o vles thee s ew tpe o tug pot. It s clled sddle pot. Stto pot.6 ecto Felds How would ou setch the eth s gvttol eld? Gm g( ) E

24 P 7 - ectos, Felds d Mtces Semeste Use legth o the ow to spec stegth o eld. e.g. Setch the eld ( ) e.g. Setch ( ) e.g. setch D( )

25 P 7 - ectos, Felds d Mtces Semeste e.g. Setch E( ) ( ).7 Gdet ecto pcll we wll wt the te o chge o ucto some ptcul decto. (e. ot ust etc) Stt o dmesos. (, ). Ptch o the suce û (,) wht s the te o chge o (,) the û decto? We wt: d lm ( uδs ) ( ) ds δs δs u d ds d ds u d d d d d ds ds u d ds u û d ds dsu d d d ds

26 P 7 - ectos, Felds d Mtces Semeste d u ds So d ds u u d ds u. u s the gdet vecto. (, ) (,, ) Dectol devtve c e wtte s decto û.. u d ds u. u slope o the ucto the û Hece pots the steepest uphll decto d s the slope tht decto. It s lso es to show tht evluted t some pot s pepedcul to the cotou les ()costt t tht pot. Poo: ()cost.. u û e.g. () Setch the eld les o d the cotous o costt. - 6

27 P 7 - ectos, Felds d Mtces Semeste 7 e.g. Fd the gdet vecto o the scl eld ( ) ) ( g Setch g d the cotou les. g g Summe popetes o : Slope decto û s u ds d u. pots steepest uphll slope steepest uphll decto. s oml to suces o costt. N: ll these sttemets ee to ptcul pot spce. Potetl eeg s scl ucto U() e.g. uom gvttol eld U()mg Feel the eect o the gvttol eld s oce F. F-mg I geel: ) ( ) ( U F I ths cse mg U U U F e.g. he PE o ptcle t some pot s: α U ) ( Wht s the oce ctg o ths ptcle t pot? α d du U U U U U F Smll o U d U.

28 P 7 - ectos, Felds d Mtces Semeste 8 α α F α F α α α F ) ( hs s the D geelto o oulom s Lw: potetl v U F Eecse: Fd the gdet o u ) ( u.8 hgg vles e.g. Supposg ou e gve ucto ), ( d decde tht ou wt to wo Pol coodtes (the th tes).e. s Mght wt (e.g.) get t susttutg c o c use ch ule s eoe. h ule tells us tht: clculte d dectl om (,) s M lso eed (o othe devtves e.g. ) s (,)

29 P 7 - ectos, Felds d Mtces Semeste 9 ould go the othe w,.e. om ucto o (,). he we mght eed,,,. g g g s Let s evlute them. Need to wte (,) ( ) ( ) ( ) Need ct ), ( ct Note: ecuse deet vles e eg held ed. e.g. (,) Evlute (Impled tht s ed). ( ) s s s s

30 P 7 - ectos, Felds d Mtces Semeste chec susttuto. s ( s ) How do we compute the gdet pol coodtes? Gve g(,), wht s g? g g g emptg, ut wog. dg o deve t popel we ll cosde the dectol devtve g. u. ds u We ow tht g. Gol s to gue out d. û ds d d uds d d u So g. u d ds dg ds d ds u d d LHS s ds ds ut we c use the ch ule to smpl RHS. g g dg d d d dg g d g ds d ds g g, d Hece g g g d ds d ds d ds e.g. om ove.

31 P 7 - ectos, Felds d Mtces Semeste ( )... α e.g. oulom s lw U, α costt. u α F U.9 Le Itegls How do ou clculte the legth o cuve dmesos? d l d d d lm Legth o cuve om to δl dl δ l ll. ts ( deotes the cuve om to ) dl ( d) ( d ) ( d) d d d So legth d d d d d d d dl e.g. lculte the legth o the cuve deed h ( ) h s ves om to. Legth d sh d h [ sh ] sh uve m e deed pmetcll,.e. ( s) ( s), ( s), (eg. s could e tme)

32 P 7 - ectos, Felds d Mtces Semeste dl d d d dl s s ds d ds d ds d ds Emple: lculte the mss o ccul hoop o dus whose mss pe ut legth p ( ) d Smll pece, dl dl ( d) ( d ) d dlρ dρ d dl ρ mss o elemet dlρ dlρ dρ ( ) hole ( ) d d, s, d d π d. d d d d s π d π π dl d Mss dlp π ( ) d π. Le Itegls Ivolvg vecto elds Emple: Wte dow omul o the wo doe oce ( ) ptcle tht moves om to log some cuve. F() d F whch cts o dw wo doe F s ptcle moves om to d. otl wo doe F dw dw F. d heeoe otl Wo F. d

33 P 7 - ectos, Felds d Mtces Semeste Emple: Wht s the wo doe ) log the cuve W W ( )(. d d ) d d d ( d d ) d d F gog om (,) to (,) ltetve w s to wte evethg tems o ethe o. Sce : d dw ( ) W ( d d( ) d ( d d) ( ) d ) log the pth show the gue. D ( d ) W d How do we hdle the lmts? Need to cosde ech o the pts o the pth septel..e. W W W W D ( d ) W d d sce W d ( d ) W D d, theeoe d W D d ( d ) W D d, theeoe d W D d So W- s eoe. D [ ]

34 P 7 - ectos, Felds d Mtces Semeste Emple: lculte the tegl dl whee s ccle o dus ceted o the β(, ) og d (β costt) c dl s d sd d d dl d β dl dl β ( s ) ( s ) π Quc w: β dl π d πβ βd πβ d β d d βd

35 P 7 - ectos, Felds d Mtces Semeste. Mtces. Itoducto We led met mtces. Emple: le equtos,,,,.., m [,...] II I I... etc e wtte whee M M M M m m... m ( ( m ) mt) Mtces e used. I qutum mechcs. o desce smmet Mt lge s le vecto lge o ddto d sutcto..e., c e.g. 6. Mt Multplcto s emple, suppose,,,,...,,,,,..., So s p mt d, ( (,,,..., m )) s m mt. Wht s? ( ( ) ( )) ut wht does t me? I compoets p p So Let s wte out eplctl. p M M M M (... )

36 P 7 - ectos, Felds d Mtces Semeste 6 he cell s the ow tmes the colum whee the ow d colum meet t the cell. N: Multplcto ol deed the ume o ows s equl to the ume o colums. e.g.: 7 Multplcto s: o-commuttve ssoctve: ( ) ( ) Dstutve: ( ) e.g. whch mt cts o vecto to gve the sme vecto? hs s the ut mt δ ( s stlsed ) ( ). ).( e.g. useul qutum mechcs s the commutto o two mtces, d [ ],. spose spose s deed to e the mt whch s oted swppg ows d colums. e.g. 6 6 I compoet otto: e.g. tspose o colum vecto s ow vecto. ( ) Note s w o tg the vecto poduct

37 P 7 - ectos, Felds d Mtces Semeste [Wht s ( )?] ( ) - ese poduct I ( ) Poo: ( ) ( ) ( ) [I dces e epeted the summto mpled]. Ivese We hve met. Let us toduce the mt such tht Note does NO lws est. How do we d? We use the esult ( det ) δ s the co-cto ( th ) o Poo: osde, δ det deto o detemt. det Fo etc.e. ust the deto o det Now cosde, δ LHS s ust the detemt o mt wth equvlet ows whch we ele sw to e eo. 7

38 P 7 - ectos, Felds d Mtces Semeste 8 e.g.,. ( cse) Hece: ( ) ( ) det det δ ( ) det det Ivese tspose o the mt o coctos (dvded the detemt) N: Ivese does ot est det sgul mt. Emple: Fd the vese o det 6 det hec: 6 6 det

39 P 7 - ectos, Felds d Mtces Semeste 6. Specl Mtces Smmetc: Hs to e smmetc out the dgol. dgge. smol ( ( *) ) dgge. smol s hemt. s the hemt cougte. Othogol:.e. s s othogol. s e.g. v v v v. v v ( v v v ) v v v v v v Rv (Othogol mt) v. v v v ( Rv ) ( Rv ) v R Rv v v Legth o v s vt ude othogol tsomto. I dgge s ut..6 Egevlues d Egevectos c to sstem o msses d stgs to loo t dmcs. Recll F F ( ).e.: F F Newto s d lw F && m F && ( ) 9

40 P 7 - ectos, Felds d Mtces Semeste Geell moto s complcted. ut we c loo o solutos o dete equec ( oml modes ). to d solutos o the om ( ) ( ) t t t t ω ω, (, costts) m t t m t t ω ω ω ω ω ω ω wte s: m K ω K hs s EIGENLUE equto. s the egevecto, whle mω s the egevlue. Let s solve the geel cse. whee s ume. ( ) Homogeous le equto. Fo teestg (ot uque) solutos we eque ( ) det Gves us the egevlues,,,,..., N, ( s NN mt) ( ) det leds to uque ut tvl soluto. Fo ech egevlue we eed the coespodg vecto..e. we must solve ( ) s clled EIGENEOR. Note: sce RHS s deed ol up to ovell cto. e.g.: 6 - d the egevlues d egevectos o. o get egevlues we must solve ( ) det..e. 6 ( )( ) 6 ± Egevlues e,. hec: sum o egevlues tce o mt dd the up, d chec tht the equl to the dgol elemets o the mt (.e. 6. o get egevecto coespodg to, ( ) ( ) 6 Usg the st equto:

41 P 7 - ectos, Felds d Mtces Semeste ( ) ( ) c e thg.e. om o s ot ed. Note, could get om the d equto.e. ( ) - t gvees the sme swe. Fo 6 d c d c Hece ( ) ( ) c c d c chec: Susttute c : ( ) 6 e.g. clculte the egevlues d egevectos o: Wt,.e. ( ). o get : ( ) ( ) [ ]( ) 9 Solutos: ±, Egevlues. hec wos. () o get egevectos: ( ) c Let c heeoe d c S: Ut egevectos:

42 P 7 - ectos, Felds d Mtces Semeste ( ) c c e thg! ( ) c c c So: hese e ll othogol.

43 P 7 - ectos, Felds d Mtces Semeste.7 Rel Smmetc Mtces Note tht o oth d the pevous ems () Egevlues wee REL. () Egevectos wee ORHOGONL (e.g. () d () occued ecuse d e el (.e. * ) d smmetc (.e. ) () d () hold o ll el d smmetc mtces..8 Noml Modes We c ow sh o solvg the coupled spgs polem. We hd: m K ω, t ω K to get egevlues ( ) m m m m m, ω ω ω ω ω hese e the gul equeces o the oml modes. he coespodg egevectos e: coespods to the msses movg phse. coespods to the msses movg tphse.