ds xˆ h dx xˆ h dx xˆ h dx.

Transcription

1 Lecture : Legedre Poyomias I (See Chapter i Boas) I the previous ectures we have focused o the (commo) case of d differetia equatios with costat coefficiets However, secod order differetia equatios with NON-costat coefficiets do arise i physics, especiay whe we cosider fuctios of 3 spatia dimesios, ie, whe we cosider fieds that satisfy partia differetia equatios Famiiar eampes are eectric ad magetic fieds, gravitatioa fieds ad the fieds that describe partices whe we quatize them We wat to discuss a eampe of that situatio ow This eampe is caed the Legedre equatio, which arises whe we cosider Lapace s equatio i 3-D with spherica coordiates (eg, reevat for ay probem described by a potetia, away from the sources, as i eectromagetism) Reca that i spherica coordiates we have r r si 0 r r r r si r si () Note, i particuar, the etra factors that appear both iside ad outside of the differetias due to the curviiear coordiates (see Chapter 0 i Boas, ad reca the discussio ast quarter ad the begiig of this quarter i Lecture 3, especiay i Appedi A) To uderstad how this factors arise cosider a ifiitesima ie eemet writte i terms of ifiitesima steps i each coordiate directio with appropriate coefficiets empoyig the otatio of Chapter 0 (these coefficiets carry the reevat iformatio to esure that each differetia eemet is a ie eemet), ds ˆ h d ˆ h d ˆ h d I rectaguar coordiates this form is particuary simpe, () h ds d ˆ ydy ˆ zdz ˆ : y h 3 z h3 (3) Thigs become more iterestig i curviiear coordiates For spherica coordiates we have Physics 8 Lecture Witer 009

2 r h ds rdr ˆ ˆ rd ˆ r si d : h r 3 h3 r si (4) I terms of these coefficiet factors the gradiet operator ooks ike U U U U ˆ ˆ ˆ 3 h h h3 3 (5) Note that the deomiator i each term is the correspodig eemet of distace from Eq () So i rectaguar coordiates we have the famiiar form U U U U ˆ yˆ zˆ, y z (6) whie i spherica coordiates we fid the more compicated epressio U ˆ U ˆ U U rˆ r r rsi (7) The correspodig divergece ad Lapacia operators are epressed as ad h h V h h V h h V V h h h 3 3 V r Vr si V, r r r si r si hh3 U hh 3 U hh U U h h h 3 h h 3 h 3 3 U U U r si r r r r si r si (8) (9) Physics 8 Lecture Witer 009

3 I each case we have dispayed the form with the epicit coefficiet factors, which coud be used to fid the same operators i other coordiates (eg, rectaguar or cyidrica), ad the specific spherica coordiate epressio As we have discussed earier the coefficiet factors i the umerator (ad deomiator) arise because, for curviiear coordiates, the divergece free basis vectors have the form (see eercise 09: i Boas) ˆ ˆ ˆ hh3 hh 3 hh 3 0 (0) Hece, before takig the divergece of a vector, we shoud be thikig of epressig it as V V ˆ ˆ V ˆ V 3 3 ˆ ˆ ˆ h h h h h h 3 h h V h h V h h V () Dottig the previousy defied gradiet operator (with aother factor of a h k i each term) resuts i the above epressio for the divergece, where these speciay defied basis vectors have zero divergece ˆ ˆ ˆ V h h V h h V h h V h h h h h h h h V h h V h h V h h h () A simiar argumet appies for the Lapacia operator For referece we shoud aso cosider the same epressios i cyidrica coordiates, Physics 8 Lecture 3 Witer 009

4 h ds ˆ d ˆ d zdz ˆ : h, 3 z h3 U U U U ˆ ˆ ˆ 3 h h h 3 3 U ˆ U U ˆ zˆ, z h h V h h V h h V V h h h Vz V V, z hh3 U hh 3 U hh U h h h 3 h h 3 h z U U U U 3 3 (3) I ay case, it shoud ow be cear that o-costat coefficiets aturay arise i physics i 3-D A powerfu too for addressig partia differetia equatios with o-costat coefficiets is (our od fried) the techique of separatio of variabes (Here we are separatig the depedece o the various idepedet variabes, rather tha separatig depedet from idepedet variabes as i Lecture 4) Goig back to spherica coordiates as i Eq (), we start with the separatio of r,, R r (this is ot the theta fuctio discussed variabes Asatz, earier), ad focus (for ow) o the case a costat (o aguar mometum i the z-directio ad hece o depedece) If we substitute this Asatz ito Eq () ad the divide the etire equatio by r,, we fid that Lapace s equatio spits ito two distict terms, oe depedig o r aoe ad oe depedig o aoe (we wi come back to the case of depedece) Note i particuar that such a separated equatio ca be satisfied if ad oy if the sides of the equatio are each equa to the same costat (ie, idepedet of a of the idepedet variabes) We choose to epress this resut as Physics 8 Lecture 4 Witer 009

5 d r d Rr d si d, R r dr dr si d d (4) where we have abeed the costat by for ater coveiece With the (stadard) chage of variabe cos, d d si,, we obtai Legedre s equatio for the poar age ( ) depedece i Lapace s equatio, d d d 0 d d d 0 d d d d 0 d d (5) This ast form is the caoica form (ie, the coefficiet of the highest derivative is uity) The ew ad iterestig feature is the (ie, cos ) depedece of the other coefficiets To uderstad the overa behavior of the soutios to such equatios, we first cosider the questio of where these coefficiet fuctios are siguar (thought of as a aaytic fuctio of a potetiay compe argumet as i Lecture 7) This behavior wi determie the possibe siguarity structure of the soutios I particuar, we see that, based o the siguarity structure of the coefficiet fuctios i Eq (5), 0 is a reguar poit (o siguarities i the coefficiet fuctios), whie are reguar siguar poits (oy simpe poes i the coefficiet fuctios) (Note the ew termioogy!) Sice the coefficiets are we behaved at 0, we epect to be abe to fid soutios of this equatio i terms of Tayor series epasios about 0 However, the siguarities i the coefficiet fuctios te us that we wi have to worry about whether these series epasios coverge or ot at From the stadpoit of the physics, there is othig specia about these poits (just the orth ad south poes,, 0, ) The physicay iterestig soutios wi (typicay) be the oes that are aaytic (we behaved) at = 0 ad, ie, we wat the series epasio that coverges o the fu cosed iterva (ie, our boudary coditios are that we wat fuctios that are we behaved at ) Physics 8 Lecture 5 Witer 009

6 ASIDE: Note that, sice the origia equatio is secod order, we epect aso a secod soutio, which wi have distictive, o-aaytic behavior at We wi retur to this poit at the ed of the ecture As we wi shorty see, this combiatio of a differetia equatio (Legedre) ad boudary coditios (aaytic o the cosed iterva) is typica of our efforts to describe physica systems ad eads to a eigevaue probem As i our earier studies of eigevaue probems i terms of matrices, soutios wi eist oy for specific (discrete) vaues of the parameter (the eigevaue) ad, moduo the (famiiar) questio of ormaizatio, there wi be a uique eigefuctio for each vaue of The correspodig eigefuctios (the Legedre poyomias) wi costitute a compete ad orthogoa set of fuctios of o the iterva (agai simiar to our earier eigevaue probem studies) Thus our efforts to fid basis fuctios of various kids are ow ceary coected to our uderstadig of eigevaue probems We wi aso use this eampe of Legedre poyomias to see how our kowedge of the differetia equatio ca be tured ito kowedge about the properties of the resutig eigefuctios To proceed we defie a power series Asatz (the method of udetermied coefficiets) c, (6) 0 ad substitute ito Legedre s equatio, Eq (5) As usua we isoate each idividua power of to fid (ie, sice the right-had-side vaishes, each idividua coefficiet i the power series must vaish) c c c c c c c c c c 0 Physics 8 Lecture 6 Witer 009

7 c c c c 0 a c c (7) Note how we have redefied the subscripts o the coefficiets of some of the terms i the third ie ( i the first term of the first ie, where iitiay the = 0 ad = terms vaish, so that c c ad ) i order to isoate a sige power of i the third ie (previous page) To satisfy the equatio i the secod ie power-by-power, the coefficiet of the power must vaish for a ad thus the origia coefficiets c must i tur satisfy the recursio reatio i the ast ie (kowig c aows us to cacuate c ) The fact that jumps by i the recursio reatio tes us, as epected o both geera ad symmetry grouds, that we have two idepedet kids of soutios Oe ivoves oy eve powers of ad is thus eve i The other ivoves oy odd powers ad is odd i We have m m m, m (8) c c eve odd m0 m0 Oce we kow c 0 a terms i the eve series are kow from the recursio reatio, whie the odd series is fuy specified by c ad the recursio reatio Net cosider the behavior of the series as The famiiar ratio test says that, usig the recursio reatio, m cm m m m m m m m m c m m m m m m m m m (9) Thus i the imit we vioate the usua test for covergece (due to the absece of the m m term) ad the series wi diverge uess the ifiite series is trucated, ie, uess it is actuay a fiite series Thus, if we demad a soutio that is we behaved at the edpoits of the iterva, we must demad that the series trucates This resut is, i tur, easiy arraged as a eigevaue costrait If, istead of beig arbitrary, we require that is a iteger 0, the if foows from the recursio c, or c 0, Thus for the reatio for the coefficiets that Physics 8 Lecture 7 Witer 009

8 eigevaue, with a iteger vaue ( 0), the correspodig eigefuctio is a poyomia of degree (highest power) These are the much oved Legedre poyomias We kow from the form of the recursio reatio for the coefficiets that for odd (eve), the correspodig poyomia ivoves oy odd (eve) powers of ad is thus a odd (eve) fuctio of The ormaizatio of the poyomias is fied (historicay ad arbitrariy) by the costrait P (0) Note that it foows immediatey from the odd/eve properties of the poyomias that P We ca use the ormaizatio costrait ad the recursio reatio for the coefficiets to easiy fid the first few poyomias: 0 : P c, 0 0 : P c, P c P, c 6 P c P c c : 0 3, c0 P 3, c P c P c c : 3, 3 c 3 P3 5 3, c 0 c :, P4 c0 0, c c P4 c0 6 4 P () The behaviors of these poyomias as fuctios of are iustrated i the foowig figures Physics 8 Lecture 8 Witer 009

9 P0 P P P P4 - The other Legedre poyomias ca be (aboriousy) costructed by foowig this ie of cacuatio, but we wat to deveop more powerfu meas to study these fuctios, usig the toos we have eared i this course Here we wi use what we have recety eared to obtai a variety of represetatios for ad isights ito the properties of the Legedre poyomias (see especiay the eampes i Appedi B) By simpe iteratio of the recursio reatio for the coefficiets we fid the foowig cosed epressio for the poyomias, Physics 8 Lecture 9 Witer 009

10 P, 0!!! ()! where the symbo stads for the argest iteger i (ie, = m both for = m ad for = m+) We recogize the form of the terms as arisig from takig derivatives,! d! d, (3) which resuts i a biomia series that we ca sum (ote the simpificatio of the upper imit o the sum after we factor out the derivatives) Thus after some reorgaizatio we fid P d d 0!! d!! d 0!! (4) d! d a biomia series This formua, Rodrigues formua, is easy to use i terms of Leibiz s rue, which says to thik of appyig mutipe derivatives to a product i terms of a biomia of two derivatives, oe actig o each factor of the product, ad the epad the biomia Physics 8 Lecture 0 Witer 009

11 d d d d d U d V U V U V d d d U V U V d d d (5) d d d U V U V d d d With the idetificatio U, V we ca easiy check the vaues of the poyomias at the ed poits For the case, the resut wi come oy from terms with o factors of, which eads uiquey to the ast term i the epasio above where a of the derivatives act o V, d P!! d! (6) At the opposite edpoit the oy ozero cotributio is the first term i the epasio where a of the derivatives act o U, d P!! d! (7) We ca proceed to ear more about the Legedre poyomias by ivokig our kowedge of aaytic fuctios Cauchy, combied with Rodrigues, aows us first to epress the Legedre poyomia as a cotour itegra about the poit i the compe z pae, ie, is iside C, ad the to take the derivatives iside the itegra to fid d d z P dz! d! d i z z dz i z C, C (8) Physics 8 Lecture Witer 009

12 which is ofte caed Schäfi s itegra formua Our et step is to make a specific choice for the cotour C, a choice attributed to Lapace Let C be a circe of radius about the poit (admittedy umotivated iitiay, but remember o soap operas back the!) O such a circe we have i z e,0 i dz e id z id z e e i i i i i i e z e e e cos cos (9) Substitutig these epressios ito Eq (8) we obtai Lapace s itegra formua for the Legedre poyomias, z cos P z id i z C P d cos d cos 0 0 (30) This ast resut becomes particuary usefu whe we cosider defiig a geeratig fuctio for the Legedre poyomias At the cost of itroducig this ew cocept, the geeratig fuctio, we wi obtai some very smart but azy resuts I geera, a geeratig fuctio is a fuctio of two variabes such that the coefficiets of the Tayor series epasio i oe of the variabes are iterestig fuctios of the other variabe I our case we wat h, h P P h, (3) 0! h h0 Physics 8 Lecture Witer 009

13 With this goa i mid we ca use the previous itegra epressio to obtai a sum we ca perform, ie, a geometric series h, h d cos 0 0 d h 0 0 d cos 0 h cos (3) To perform the itegra we go back to compe cotour itegratio otatio with the defiitios (the cotour is the uit circe), dz i,cos z z z e d iz dz h, i z z h z z z dz i z h z h h (33) The quadratic epressio i the deomiator wi vaish at two poits i the compe z pae, ie, there are two poes, z h h h h h h h h h h h h h h h, z z z z (34) Note that, sice the product of the magitudes of the two compe umbers is uity, we are guarateed that oe of the poes is iside of our uit circe cotour ad the other oe is outside Thus we ca evauate the itegra i terms of the residue at the Physics 8 Lecture 3 Witer 009

14 iside poe (you shoud covice yoursef that the cocusio that oy oe poe cotributes is idepedet of the precise vaues of the variabes h ad, whie here we thik about sma h ad z as the poe iside), h, dz iresz z i h z z z z z i i h z z h h h h h h (35) To test our uderstadig of this epressio cosider usig it to evauate the Legedre poyomias at the edpoits ad the origi, h,, h P h P h h h 0 0 h, h h h h 0 0 P h P, m h,0 h 0 0 P h h m 0 m! 0 : odd P 0 : eve m m (36) To appreciate this ast epressio we eed to reca the properties of the Gamma fuctio (see Sectio 3 i Boas), Physics 8 Lecture 4 Witer 009

15 !, iteger (37) We ca aso use the geeratig fuctio to derive recursio reatios betwee the poyomias For eampe, if foows from the form of the geeratig fuctio i Eq (35) that h, h h 3 h h h h h h, (38) Usig the series defiitio of the geeratig fuctio, Eq (35), ad this epressio, Eq (38), we ca obtai two epressios for the same fuctio, h, h ad h h h h h P 0 h h h P h h h P 0 0 h P h P P h, h h h h h h P h, h P h P 0 0 (39) Equatig the coefficiets of the same power of h i each of the fia epressios, we fid (see Eq 58a i Boas) that (this is for, for 0 we have the trivia resut P P0) P P P P P P P P 0 (40) Physics 8 Lecture 5 Witer 009

16 Aother usefu reatio (see Eq 58d i Boas ad the et Lecture) arises from performig the foowig (currety umotivated) operatios o the geeratig fuctio i Eq (35) We have h 3 h h h h h h h h h, h h h h h h h h P h P, h 0 0 (4) ad h h h 3 3 h h h h h h h h h h h h h P h h 0, h P h P h P h P (4) Thus Eqs (4) ad (4) are represetatios of the same quatity ad we ca agai equate the coefficiets of h i the equatios This yieds the usefu idetity P P P P, 0,, 0 (43) which we ca quicky cofirm agrees with the epicit epressios for sma i Eq (): P dp d d d, 0 0 3P 3 dp d dp d d 3 d, etc This resut is very usefu for performig itegras of the P sice we ca chage Physics 8 Lecture 6 Witer 009

17 P ito derivatives of other poyomias, ad itegras of derivatives are simpe (for the smart but azy) As a eampe of how we ca use the geeratig fuctio for the Legedre poyomias i physics we appy it to the famiiar r potetia, which appears i E&M ad gravity ASIDE: This famiiar r form arises because it is the behavior of the Gree s fuctio for the ihomogeeous Lapace equatio (Poisso s Equatio) G r, r r r G r, r 4 r r (44) We ca verify that this epressio has the correct behavior by usig our kowedge of vector aaysis, r sphere rˆ ˆ 4 r 4 r 4 r 4 r 0 r 0, 4r 4r r rˆ d d 4r 4r sphere rˆ 4 r d 4 r 4 r sphere rˆ (45) These properties are precisey those of the 3-D Dirac deta fuctio Now cosider a observer at poit r ad a poit charge (or mass) at the poit R i some coordiate system, Q Q Q V r, R Gr, R, (46) 4 R r 4 R r rrcos Physics 8 Lecture 7 Witer 009

18 where is the age betwee r ad R For r R (the far regio ) we epad i powers of h R r (sice it is ess tha ) usig Eq (35) V r, R Q 4 r R r R r cos 0 0 Q R P 4 r 0 r cos (47) I the opposite imit, r R, we simpy epad i terms of powers of h r R Fiay cosider the potetia outside of a cotiuous charge distributio, R, of 3 fiite etet Agai we have r R Q d R R ) ad write ( R R 3 3 Vout r d R d R 4 0 R r 4 0 R r rrcos 4 r 0 r 0 3 d R R P R cos We recogize this epressio as the usua muti-poe momet epasio i a coordiate system with the z-directio aog r The case 0 is the tota charge, whie is the dipoe momet, etc Physics 8 Lecture 8 Witer 009 (48) We cose this discussio by otig that, as epected, there is a secod soutio to Legedre s equatio, which, uike the poyomias above, is poory behaved for sma vaues of, but is we behaved for arge (where is apparety o oger iterpreted as the cosie of a rea age) Thus the secod soutio does ot satisfy the boudary coditios we imposed above (i fact, the secod soutio has ogarithmic brach poits at ) Reca that for arge vaues (ote that, sice the P are just poyomias, there is o particuar probem with takig ), we epect P The secod soutio, sometimes writte as (see eercise :4 i Appedi B) dz Q P, (49) z P z

19 behaves asymptoticay ike Q (or stricty its rea part does) This differece betwee the asymptotic behaviors of the two soutios to a secod order homogeeous equatio is geeric Oe soutio is typicay we behaved for sma vaues of the argumet but ot arge vaues, whie the secod soutio ehibits the opposite behavior Eampes of these secod soutios (see Appedi B) are, with the aaytic structure chose by Boas, Q Q 0, (50) Note that these fuctios ehibit (ogarithmic) brach poits (siguarities) at where the usua Legedre Poyomias are reguar (by costructio) Due to these brach poits we shoud be carefu to defie the Q i the compe pae, ie, where do we put the brach cut? A covetioa choice is to put the brach cut aog the rea ais betwee z ad z With this choice the fuctio Q z is rea aog the rea ais z, but compe for z This choice correspods to the forms Q Q 0 z z z, z z z, z (5) which have the advertised asymptotic behavior Q aog the rea ais The epressios ehibited i Boas ad i Eq (50) evauated aog the rea ais for ca be thought of as the average of Eq (5) o the two sides of the brach cut or simpy the rea part Aterativey we ca thik of Eq (50) as defiig aaytic fuctios that have the brach cuts from z to z (aog the positive rea ais) ad from z to z (aog the egative rea ais) I this case the asymptotic behavior picks up a imagiary part, the sig of which depeds o whether we are above or beow the cut ad i which directio we take the asymptotic imit Physics 8 Lecture 9 Witer 009