Recall Taylor s Theorem for a function f(x) in three dimensions with a displacement δx = (δx, δy, δz): δx + δy + δz + higher order terms. = f. δx +.

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1 Chpter 1 Vritionl Methos 1.1 Sttionry Vlues of Functions Recll Tylor s Theorem for function f(x) in three imensions with isplcement δx (δx, δy, δz): so tht f(x + δx) f(x) + f f f δx + δy + δz + higher orer terms x y z δf f(x + δx) f(x) f f f δx + δy + x y z δz + f. δx +. In the limit δx 0 we write f f. x. This result is true in ny number n of imensions. At n extremum ( mximum or minimum) f must be sttionry, i.e. the first vrition f must vnish for ll possible irections of x. This cn only hppen if f 0 there. Note tht if we try to fin the extrem of f by solving f 0, we my lso fin other sttionry points of f which re neither mxim nor minim, for instnce sle points. (This is the sme ifficulty s in one imension, where sttionry point my be point of inflection rther thn mximum or minimum.) If we nee to fin the extrem of f in boune region for instnce, within two-imensionl unit squre then not only must we solve f 0 but we must lso compre the resulting vlues of f with those on the bounry of the squre. It is quite possible for the mximum vlue to occur on the bounry without tht point being sttionry one. 1 R. E. Hunt, 2002

2 Constrine sttionry vlues Suppose tht we wish to fin the extrem of f(x) subject to constrint of the form g(x) c, where c is some constnt. In this cse, the first vrition f must still vnish, but now not ll possible irections for x re llowe: only those which lie in the surfce efine by g(x) c. Hence, since f f.x, the vector f must lie perpeniculr to the surfce. But recll tht the norml to surfce of the form g(x) c is in the irection g. Hence f must be prllel to g, i.e., f λ g for some sclr λ. This gives us the metho of Lgrnge s unetermine multiplier: solve the n equtions (f λg) 0 for x together with the single constrint eqution g(x) c. The resulting vlues of x give the sttionry points of f subject to the constrint. Note tht while solving the totl of n+1 equtions it is usully possible to eliminte λ without ever fining its vlue; hence the moniker unetermine. If there re two constrints g(x) c n h(x) k, then we nee multiplier for ech constrint, n we solve (f λg µh) 0 together with the two constrints. strightforwr. The extension to higher numbers of constrints is 1.2 Functionls Let y(x) be function of x in some intervl < x < b, n consier the efinite integrl F ( {y(x)} 2 + y (x)y (x) ) x. F is clerly inepenent of x; inste it epens only on the function y(x). F is simple exmple of functionl, n to show the epenence on y we normlly enote it F [y]. We cn think of functionls s n extension of the concept of function of mny vribles e.g. g(x 1, x 2,..., x n ), function of n vribles to function of n infinite number of vribles, becuse F epens on every single vlue tht y tkes in the rnge < x < b. 2 R. E. Hunt, 2002

3 We shll be concerne in this chpter with functionls of the form F [y] f(x, y, y ) x where f epens only on x n the vlue of y n its first erivtive t x. However, the theory cn be extene to more generl functionls (for exmple, with functions f(x, y, y, y, y,... ) which epen on higher erivtives, or ouble integrls with two inepenent vribles x 1 n x 2 inste of just x). 1.3 Vritionl Principles Functionls re useful becuse mny lws of physics n of physicl chemistry cn be recst s sttements tht some functionl F [y] is minimise. For exmple, hevy chin suspene between two fixe points hngs in equilibrium in such wy tht its totl grvittionl potentil energy (which cn be expresse s functionl) is minimise. A mechnicl system of hevy elstic strings minimises the totl potentil energy, both elstic n grvittionl. Similr principles pply when electric fiels n chrge prticles re present (we inclue the electrosttic potentil energy) n when chemicl rections tke plce (we inclue the chemicl potentil energy). Two funmentl exmples of such vritionl principles re ue to Fermt n Hmilton. Fermt s Principle Consier light ry pssing through meium of vrible refrctive inex µ(r). The pth it tkes between two fixe points A n B is such s to minimise the opticl pth length B A where l is the length of pth element. µ(r) l, Strictly speking, Fermt s principle only pplies in the geometricl optics pproximtion; i.e., when the wvelength of the light is smll compre with the physicl imensions of the opticl system, so tht light my be regre s rys. This is true for telescope, but not for Young s slits: when the geometricl optics pproximtion fils to hol, iffrction occurs. For exmple, consier ir bove hot surfce, sy trmc ro on hot y. The ir is hotter ner the ro n cooler bove, so tht µ is smller closer to the ro surfce. A light ry trvelling from cr to n observer minimises the opticl pth length by stying 3 R. E. Hunt, 2002

4 close to the ro, n so bens ppropritely. The light seems to the observer to come from low ngle, leing to virtul imge (n hence to the mirge effect). Hmilton s Principle of Lest Action Consier mechnicl system with kinetic energy T n potentil energy V which is in some given configurtion t time t 1 n some other configurtion t time t 2. Define the Lgrngin of the system by L T V, n efine the ction to be S t2 t 1 L t ( functionl which epens on the wy the system moves). Hmilton s principle sttes tht the ctul motion of the system is such s to minimise the ction. 1.4 The Clculus of Vritions How o we fin the function y(x) which minimises, or more generlly mkes sttionry, our rchetypl functionl F [y] f(x, y, y ) x, with fixe vlues of y t the en-points (viz. fixe y() n y(b))? We consier chnging y to some nerby function y(x) + δy(x), n clculte the corresponing chnge δf in F (to first orer in δy). Then F is sttionry when δf 0 for ll possible smll vritions δy. Note tht more nturl nottion woul be to write F rther thn δf, since we will consier only the first-orer chnge n ignore terms which re secon orer in δy. However, the nottion δ is tritionl in this context. 4 R. E. Hunt, 2002

5 Now δf F [y + δy] F [y] f(x, y + δy, y + (δy) ) x { f(x, y, y ) + f f δy + y y (δy) f(x, y, y ) x } x [using Tylor expnsion to first orer] { } f f δy + y y (δy) x [ f y δy ] b + { f y δy ( ) } f δy x x y [integrting by prts] { f y ( )} f δy x x y f(x, y, y ) x since δy 0 t x, b (becuse y(x) is fixe there). It is cler tht δf 0 for ll possible smll vritions δy(x) if n only if This is Euler s eqution. ( ) f f x y y. Nottion f/ y looks strnge becuse it mens ifferentite with respect to y, keeping x n y constnt, n it seems impossible for y to chnge if y oes not. But / y n / y in Euler s eqution re just forml erivtives (s though y n y were unconnecte) n in prctice it is esy to o strightforwr orinry prtil ifferentition. Exmple: if f(x, y, y ) x(y 2 y 2 ) then f y 2xy, f y 2xy. Note however tht /x n / x men very ifferent things: / x mens keep y n y constnt wheres /x is so-clle full erivtive, so tht y n y re ifferentite with respect to x s well. 5 R. E. Hunt, 2002

6 but Continuing with the bove exmple, ( ) f 2y, x y Hence Euler s eqution for this exmple is ( ) f x y x (2xy ) 2y + 2xy. 2y + 2xy 2xy or y + 1 x y + y 0 (Bessel s eqution of orer 0, incientlly). Severl Depenent Vribles Wht if, inste of just one epenent vrible y(x), we hve n epenent vribles y 1 (x), y 2 (x),..., y n (x), so tht our functionl is F [y 1,..., y n ] f(x, y 1,..., y n, y 1,..., y n) x? In this cse, Euler s eqution pplies to ech y i (x) inepenently, so tht ( ) f f x y i for i 1,..., n. y i The proof is very similr to before: δf n i1 { f δy f δy n + f y 1 y n y 1 (δy 1 ) + + f } y n (δy n ) x n i1 { f δy i + f } y i y i (δy i ) x { f ( )} f y i x y i δy i x using the sme mnipultions (Tylor expnsion n integrtion by prts). It is now cler tht we cn only hve δf 0 for ll possible vritions of ll the y i (x) if Euler s eqution pplies to ech n every one of the y i t the sme time. 6 R. E. Hunt, 2002

7 1.5 A First Integrl In some cses, it is possible to fin first integrl (i.e., constnt of the motion) of Euler s eqution. Consier f x f f + y x y + y f y (clculting f( x, y(x), y (x) ) using the chin rule). Using Euler s eqution, x f x f ( ) x + f y f + y x y y f x + x ( y f y ) [prouct rule] so tht ( f y f ) f x y x. Hence, if f hs no explicit x -epenence, so tht f/ x 0, we immeitely euce tht f y f y constnt. (Note tht f hs no explicit x-epenence mens tht x oes not itself pper in the expression for f, even though y n y implicitly epen on x; so f y 2 y 2 hs no explicit x-epenence while f x(y 2 y 2 ) oes.) If there re n epenent vribles y 1 (x),..., y n (x), then the first integrl bove is esily generlise to f if f hs no explicit x-epenence. n f y i y i1 i constnt 1.6 Applictions of Euler s Eqution Geoesics A geoesic is the shortest pth on given surfce between two specifie points A n B. We will illustrte the use of Euler s eqution with trivil exmple: geoesics on the Euclien plne. 7 R. E. Hunt, 2002

8 The totl length of pth from (x 1, y 1 ) to (x 2, y 2 ) long the pth y(x) is given by L B A B A l 1 + B A x2 + y 2 ( ) 2 y x2 x 1 + y 2 x. x x 1 Note tht we ssume tht y(x) is single-vlue, i.e., the pth oes not curve bck on itself. We wish to minimise L over ll possible pths y(x) with the en-points hel fixe, so tht y(x 1 ) y 1 n y(x 2 ) y 2 for ll pths. This is precisely our rchetypl vritionl problem with f(x, y, y ) 1 + y 2, n hence f y 0, The Euler eqution is therefore ( ) y 0 x 1 + y 2 f y y 1 + y 2. y 1 + y 2 k, constnt. So y 2 k 2 /(1 k 2 ). It is cler tht k ±1, so y is constnt, m sy. Hence the solutions of Euler s eqution re the functions y mx + c (where m n c re constnts) i.e., stright lines! To fin the prticulr vlues of m n c require in this cse we now substitute in the bounry conitions y(x 1 ) y 1, y(x 2 ) y 2. It is importnt to note two similrities with the technique of minimising function f(x) by solving f 0. Firstly, we hve not shown tht this stright line oes inee prouce minimum of L: we hve shown only tht L is sttionry for this choice, so it might be mximum or even some kin of point of inflection. It is usully esy to confirm tht we hve the correct solution by inspection in this cse it 8 R. E. Hunt, 2002

9 is obviously minimum. (There is no equivlent of the one-imensionl test f (x) > 0 for functionls, or t lest not one which is simple enough to be of ny use.) Seconly, ssuming tht we hve inee foun minimum, we hve shown only tht it is locl minimum, not globl one. Tht is, we hve shown only tht nerby pths hve greter length. Once gin, however, we usully confirm tht we hve the correct solution by inspection. Compre this ifficulty with the equivlent problem for functions, illustrte by the grph below. An lterntive metho of solution for this simple geoesic problem is to note tht f(x, y, y ) 1 + y 2 hs no explicit x-epenence, so we cn use the first integrl: const. f y f y 1 + y 2 y y 1 + y y 2, i.e., y is constnt (s before). The Brchistochrone A be slies own frictionless wire, strting from rest t point A. Wht shpe must the wire hve for the be to rech some lower point B in the shortest time? (A similr evice ws use in some erly clock mechnisms.) Using conservtion of energy, 1 2 mv2 mgy, i.e., v 2gy. Also l v t, so t x2 + y y 2 x. 2gy 2g y 9 R. E. Hunt, 2002

10 The time tken to rech B is therefore T [y] 1 2g xb y 2 y x n we wish to minimise this, subject to y(0) 0, y(x B ) y B. integrn hs no explicit x-epenence, so we use the first integrl 1 + y const. 2 y 1 + y 2 y y y 1 + y 2 y y 2 y 1 + y 2 We note tht the 1 y 1 + y 2. Hence y(1 + y 2 ) c, sy, constnt, so tht c y y or y y y x. c y Substitute y c sin 2 θ; then sin 2 θ x 2c 1 sin 2 sin θ cos θ θ θ 2c sin 2 θ θ c(1 cos 2θ) θ. Using the initil conition tht when y 0 (i.e., θ 0), x 0, we obtin x c(θ 1 sin 2θ), 2 y c sin 2 θ which is n inverte cycloi. The constnt c is foun by pplying the other conition, y y B when x x B. 10 R. E. Hunt, 2002

11 Note tht strictly speking we shoul hve si tht y ± (c y)/y bove. Tking the negtive root inste of the positive one woul hve le to x c(θ 1 2 sin 2θ), y c sin 2 θ, which is exctly the sme curve but prmeterise in the opposite irection. It is rrely worth spening much time worrying bout such intriccies s they invribly le to the sme effective result. Light n Soun Consier light rys trvelling through meium with refrctive inex inversely proportionl to z where z is the height. By Fermt s principle, we must minimise l. z This is exctly the sme vritionl problem s for the Brchistochrone, so we conclue tht light rys will follow the pth of cycloi. More relisticlly, consier soun wves in ir. Soun wves obey principle similr to Fermt s: except t very long wvelengths, they trvel in such wy s to minimise the time tken to trvel from A to B, B A where c is the (vrible) spee of soun (comprble to 1/µ for light). Consier sitution where the bsolute temperture T of the ir is linerly relte to the height z, so tht l c, T αz + T 0 for some temperture T 0 t groun level. Since the spee of soun is proportionl to the squre root of the bsolute temperture, we hve c αz + T 0 Z sy. This les once gin to the Brchistochrone problem (for Z rther thn z), n we conclue tht soun wves follow pths z(x) which re prts of cyclois, scle verticlly by fctor 1/α (check this s n exercise). 1.7 Hmilton s Principle in Mechnicl Problems Hmilton s principle cn be use to solve mny complicte problems in rigi-boy mechnics. Consier mechnicl system whose configurtion cn be escribe by number of so-clle generlise coorintes q 1, q 2,..., q n. Exmples: A prticle with position vector r (x 1, x 2, x 3 ) moving through spce. Here we cn simply let q 1 x 1, q 2 x 2 n q 3 x 3 : there re three generlise coorintes. 11 R. E. Hunt, 2002

12 A penulum swinging in verticl plne: here there is only one generlise coorinte, q 1 θ, the ngle to the verticl. A rigi boy (sy top) spinning on its xis on smooth plne. This requires five generlise coorintes: two to escribe the position of the point of contct on the plne, one for the ngle of the xis to the verticl, one for the rottion of the xis bout the verticl, n one for the rottion of the top bout its own xis. The Lgrngin L T V is function of t, q 1,..., q n n q 1,..., q n, so ( S L t, q1 (t),..., q n (t), q 1 (t),..., q n (t) ) t. This is functionl with n epenent vribles q i (t), so we cn use Euler s eqution (with t plying the role of x, n q i (t) plying the role of y i (x)) for ech of the q i inepenently: ( ) L L t q i q i for ech i. In this context these equtions re known s the Euler Lgrnge equtions. In the cse when L hs no explicit time-epenence, the first integrl (from 1.5) gives us tht L n i1 q i L q i constnt. It is frequently the cse tht T is homogeneous qurtic in the q i, i.e., it is of the form n i1 n ij (q 1,..., q n ) q i q j j1 where the coefficients ij o not epen on ny of the generlise velocities q i or on t, n V lso oes not epen on the velocities or time so tht V V (q 1,..., q n ). Then it cn be shown tht L n i1 q i L q i (T V ) 2T (T + V ), i.e., the totl energy E T + V is conserve when there is no explicit time-epenence. This fils however when the externl forces vry with time or when the potentil is velocity-epenent, e.g., for motion in mgnetic fiel. 12 R. E. Hunt, 2002

13 A Prticle in Conservtive Force Fiel Here L 1 2 m(ẋ2 1 + ẋ ẋ 2 3) V (x 1, x 2, x 3 ); hence the Euler Lgrnge equtions re t (mẋ 1) V x 1, t (mẋ 2) V, x 2 t (mẋ 3) V, x 3 or in vector nottion (mṙ) V, t i.e., F m where F V is the force n r is the ccelertion. Two Intercting Prticles Consier Lgrngin L 1 2 m 1 ṙ m 2 ṙ 2 2 V (r 1 r 2 ), where the only force is conservtive one between two prticles with msses m 1 n m 2 t r 1 n r 2 respectively, n epens only on their (vector) seprtion. We coul use the six Crtesin coorintes of the prticles s generlise coorintes; but inste efine the reltive position vector, n r r 1 r 2, R m 1r 1 + m 2 r 2, M the position vector of the centre of mss, where M m 1 + m 2 is the totl mss. Now ṙ 1 2 Ṙ + m 2 M ṙ 2 n similrly m ) 2 (Ṙ + M ṙ m ) 2. (Ṙ + M ṙ Ṙ 2 + m2 2 M 2 ṙ 2 + 2m 2 M Ṙ. ṙ ṙ 2 2 Ṙ 2 + m2 1 M 2 ṙ 2 2m 1 M Ṙ. ṙ. Let r (x 1, x 2, x 3 ), R (X 1, X 2, X 3 ), n use these s generlise coorintes. Then L 1 2 M Ṙ 2 + m 1m 2 2M ṙ 2 V (r) 1 2 M(Ẋ2 1 + Ẋ2 2 + Ẋ2 3) + m 1m 2 2M (ẋ2 1 + ẋ ẋ 2 3) V (x 1, x 2, x 3 ). The Euler Lgrnge eqution for X i is therefore t (MẊi) 0, 13 R. E. Hunt, 2002

14 i.e., R 0 (the centre of mss moves with constnt velocity); n for x i is ( m1 m ) 2 t M ẋi V, x i i.e., µ r V where µ is the reuce mss m 1 m 2 /(m 1 + m 2 ) (the reltive position vector behves like prticle of mss µ). Note tht the kinetic energy T is homogeneous qurtic in the Ẋi n ẋ i ; tht V oes not epen on the Ẋi n ẋ i ; n tht L hs no explicit t-epenence. We cn euce immeitely tht the totl energy E T + V is conserve. 1.8 The Clculus of Vritions with Constrint In 1.1 we stuie constrine vrition of functions of severl vribles. The extension of this metho to functionls (i.e., functions of n infinite number of vribles) is strightforwr: to fin the sttionry vlues of functionl F [y] subject to G[y] c, we inste fin the sttionry vlues of F [y] λg[y], i.e., fin the function y which solves δ(f λg) 0, n then eliminte λ using G[y] c. 1.9 The Vritionl Principle for Sturm Liouville Equtions We shll show in this section tht the following three problems re equivlent: (i) Fin the eigenvlues λ n eigenfunctions y(x) which solve the Sturm Liouville problem ( ) p(x)y + q(x)y λw(x)y x in < x < b, where neither p nor w vnish in the intervl. (ii) Fin the functions y(x) for which F [y] is sttionry subject to G[y] 1 where G[y] (py 2 + qy 2 ) x wy 2 x. The eigenvlues of the equivlent Sturm Liouville problem in (i) re then given by the vlues of F [y]. 14 R. E. Hunt, 2002

15 (iii) Fin the functions y(x) for which Λ[y] F [y] G[y] is sttionry; the eigenvlues of the equivlent Sturm Liouville problem re then given by the vlues of Λ[y]. Hence Sturm Liouville problems cn be reformulte s vritionl problems. Note the similrity between (iii) n the sttionry property of the eigenvlues of symmetric mtrix (recll tht it is possible to fin the eigenvlues of symmetric mtrix A by fining the sttionry vlues of T A/ T over ll possible vectors ). The two fcts re in fct closely relte. To show tht (ii) is equivlent to (i), consier δ(f λg) δ Using Euler s eqution, F λg is sttionry when i.e., (py 2 + qy 2 λwy 2 ) x. x (2py ) 2qy 2λwy, x (py ) + qy λwy, which is the require Sturm Liouville problem: note tht the Lgrnge multiplier of the vritionl problem is the sme s the eigenvlue of the Sturm Liouville problem. Furthermore, multiplying the Sturm Liouville eqution by y n integrting, we obtin using the constrint. Hence ( y x (py ) + qy 2) x λ λ ( y x (py ) + qy 2) x [ ypy ] b + wy 2 x λg[y] λ (py 2 + qy 2 ) x [integrting by prts] (py 2 + qy 2 ) x F [y], using pproprite bounry conitions. This proves tht the sttionry vlues of F [y] give the eigenvlues. 15 R. E. Hunt, 2002

16 There re two wys of showing tht (ii) is equivlent to (iii). The first, informl wy is to note tht multiplying y by some constnt α sy oes not in fct chnge the vlue of Λ[y]. This implies tht when fining the sttionry vlues of Λ we cn choose to normlise y so tht G[y] 1, in which cse Λ is just equl to F [y]. So fining the sttionry vlues of Λ is equivlent to fining the sttionry vlues of F subject to G 1. The secon, forml wy is to clculte δλ F + δf G + δg F G F + δf G δf G F δg G 2 ( 1 δg G ) F G [using Tylor expnsion for (1 + δg/g) 1 to first orer] (gin to first orer). Hence δλ 0 if n only if δf (F/G) δg; tht is, Λ is sttionry if n only if δf Λ δg 0. But this is just the sme problem s (ii); so fining the sttionry vlues of Λ is the sme s fining the sttionry vlues of F subject to G 1. In the usul cse tht p(x), q(x) n w(x) re ll positive, we hve tht Λ[y] 0. Hence ll the eigenvlues must be non-negtive, n there must be smllest eigenvlue λ 0 ; Λ tkes the vlue λ 0 when y y 0, the corresponing eigenfunction. But wht is the bsolute minimum vlue of Λ over ll functions y(x)? If it were some vlue µ < λ 0, then µ woul be sttionry (minimum) vlue of Λ n woul therefore be n eigenvlue, contricting the sttement tht λ 0 is the smllest eigenvlue. Hence Λ[y] λ 0 for ny function y(x). As n exmple, consier the simple hrmonic oscilltor y + x 2 y λy subject to y 0 s x. This is n importnt exmple s it is goo moel for mny physicl oscillting systems. For instnce, the Schröinger eqution for itomic molecule hs pproximtely this form, where λ is proportionl to the quntum mechnicl energy level E; we woul like to know the groun stte energy, i.e., the eigenfunction with the lowest eigenvlue λ. Here p(x) 1, q(x) x 2 n w(x) 1, so Λ[y] (y 2 + x 2 y 2 ) x. y2 x 16 R. E. Hunt, 2002

17 We cn solve this Sturm Liouville problem exctly: the lowest eigenvlue turns out to be λ 0 1 with corresponing eigenfunction y 0 exp( 1 2 x2 ). But suppose inste tht we in t know this; we cn use the bove fcts bout Λ to try to guess t the vlue of λ 0. Let us use tril function y tril exp( 1 2 αx2 ), where α is positive constnt (in orer to stisfy the bounry conitions). Then Λ[y tril ] (α2 + 1) x2 exp( αx 2 ) x. exp( αx2 ) x We recll tht exp( αx2 ) x π/α n x2 exp( αx 2 ) x 1 2 π/α3 (by integrtion by prts). Hence Λ[y tril ] (α 2 + 1)/2α. We know tht Λ[y tril ], for ny α, cnnot be less thn λ 0. The smllest vlue of (α 2 + 1)/2α is 1, when α 1; we conclue tht λ 0 1, which gives us n upper boun on the lowest eigenvlue. In fct this metho hs given us the exct eigenvlue n eigenfunction; but tht is n ccient cuse by the fct tht this is prticulrly simple exmple! The Ryleigh Ritz Metho The Ryleigh Ritz metho is systemtic wy of estimting the eigenvlues, n in prticulr the lowest eigenvlue, of Sturm Liouville problem. The first step is to reformulte the problem s the vritionl principle tht Λ[y], the Ryleigh quotient, is sttionry. Seconly, using whtever clues re vilble (for exmple, symmetry consiertions or generl theorems such s the groun stte wvefunction hs no noes ) we mke n eucte guess y tril (x) t the true eigenfunction y 0 (x) with lowest eigenvlue λ 0. It is preferble for y tril to contin number of justble prmeters (e.g., α in the exmple bove). We cn now fin Λ[y tril ], which will epen on these justble prmeters. We clculte the minimum vlue Λ min of Λ with respect to ll the justble prmeters; we cn then stte tht the lowest eigenvlue λ 0 Λ min. If the tril function ws resonble guess then Λ min shoul ctully be goo pproximtion to λ 0. If we wish, we cn improve the pproximtion by introucing more justble prmeters. The fct tht Λ[y] is sttionry with respect to vritions in the function y mens tht if y tril is close to the true eigenfunction y 0 (sy within O(ε) of it) then the finl clculte vlue Λ min will be very goo pproximtion to λ 0 (within O(ε 2 ) in fct). If the inclusion 17 R. E. Hunt, 2002

18 of further justble prmeters fils to significntly improve the pproximtion then we cn be resonbly sure tht the pproximtion is goo one. Note tht if the tril function hppens to inclue the exct solution y 0 s specil cse of the justble prmeters, then the Ryleigh Ritz metho will fin both y 0 n λ 0 exctly. This is wht hppene in the exmple bove. An lterntive to clculting Λ[y tril ] n minimising it with respect to the justble prmeters is to clculte F [y tril ] n G[y tril ], n to minimise F subject to G 1. These proceures re equivlent, s we showe t the strt of this section. Higher eigenvlues [non-exminble] Once we hve foun goo pproximtion y 0 tril to y 0, we cn procee to fin pproximtions to the higher eigenvlues λ 1, λ 2,.... Just s λ 0 is the bsolute minimum of Λ[y] over ll possible functions y, so λ 1 is the bsolute minimum of Λ[y] over functions which re constrine to be orthogonl to y 0. (Recll tht y 1 is orthogonl to y 0 in the sense tht wy 0y 1 x 0.) Hence, to estimte λ 1 we procee s before but choose our new tril function y 1 tril in such wy tht it is orthogonl to our previous best pproximtion y 0 tril. This process cn be continue to higher n higher eigenvlues but of course becomes less n less ccurte. 18 R. E. Hunt, 2002