a0 a1 star a2a3a4a5a6a7a8a9a10 4 x21.1Wf¥à Q?2: x21.1Wf¥à Q FWfe?1 aü ^[??f ?11M ¥f ???à Q V[ A? ^f ?à Q¥ w< F ?ìf ? ^·ó?1M x(?l  uW =)¥ ?B ?′üμB?yD-?y? 1x¥f ?:1y = f(x) F !x;y ü ? μB[ wLy(x) é L = Z C ds = Z x1 x0 q 1+y02dx: A ?y(x)?]L9?]'L¥ ?′G ????f ?y(x)7?ML?f ?y(x)-W ¥??G ?1"?1Wf1" ? ?¥ è0? V[ ? è ?>? wL??¥ ? ü ? wL ?%?à 7 3?¥è88V ??? ? ì9?? ò1¥Wf1" F !?( Bf ?"? =¥) ?iB?f ?y(x)μ 6B? ?J[y]D-?5?J[y]1y(x)¥ Wf ? ú¥f ?"?'Wf¥?l×Yèc1 py(x) ?@B?¥H?Hqi O μ ??¥=¨? ??"¥y(x)?1 V |f ? Wf?]?ˉ?f ? è ?g = g(f(x)) ?a?ó?B?x′ ˉ ? ^μB?g′D-? ? -?5A?ó B uW ¥f ?y(x)? ?¤?B?Wf′J[y] (?l]B uW ¥)f ??]Wf′? ??] 1  <?Wf′J[y]Df ?y(x)-W¥G ?1"èè?üf ?y(x)?1M f ? Wf¥? T V[ ^?"¥? ^' ??á ìoK?¨s J[y] = Z x1 x0 F(x; y; y0)dx ?l¥Wf ?¥F ^ ?¥7 ¥X?f ? μ ??¥=¨ ê? ? ?TM f ? ^=íf ?u(x;y)5Wf1 J[u] = ZZ S F (x; y; u; ux; uy) dxdy; ?ux · @u=@x; uy · @u=@y ?÷?1M ¥íf ?9 V[μ ? ?¥?l x21.1Wf¥à Q?3: m`21.1 è è è1 ?m21.1 ? U× ?T¨/B?é?V(x0;y0)? ü ? wLy(x)í ?11? /á?(x1;y1)?5 ?31¥ HW T = Z (x1;y1) (x0;y0) dsp 2g(y0 ?y) = Z x1 x0 p 1+y02p 2g(y0 ?y)dx ü ^y(x)¥Wf? ú1 pM f ?y(x)B?YV ?(x0;y0)?(x1;y1)(Nù5K* ?Galileo Galilei4) è è è2?¥???ù5 !? ? ?@ ¥B ?5? ?¥ ? ?= 12‰?x @u @t ?2 ; ] ?= 12T?x @u @x ?2 ; ?u(x;t) ^?¥?_êM‰ ^?¥L áT ^f ??"?¥HamiltonT¨  S = Z t1 t0 dt Z x1 x0 1 2 h ‰ @u @t ?2 ?T @u @x ?2i dx 9 ^êMu(x;t)¥Wf L = Z x1 x0 1 2 h ‰ @u @t ?2 ?T @u @x ?2i dx ?1Lagrange (Lagrangian)7$f ? 1 2 h ‰ @u @t ?2 ?T @u @x ?2i ?1Lagrange  á x21.2Wf¥′?4: x21.2Wf¥′ ??ìWf′$ ?Wf |′¥A1Hq$ F5íkB/μ1f ?′¥à Q ?ìf ?f(x)x0? |l′ ^·?xx0?# ?íjx?x0j < " H?μ f(x) ? f(x0); 7 ?T?μ f(x) ? f(x0); 5?f ?f(x)x0? |v′ f ?f(x)x0? |′(lv)¥A1Hq ^??¥? ?10 f0(x0) = 0: F V[¨]"¥ZE?lWf¥′ o?M f ?1y(x) HWfJ[y] |l′p¥clü ^?′f ?y(x)# o? íp¥M f ?y(x)+–y(x)?μ J[y +–y] ? J[y]: ?ìf ?y(x)+–y(x) 6B?f ?y(x)¥o?íp·¥ ^ 1. j–y(x)j < " 2.μ H?1 pj(–y)0(x)j < " ? ú¥–y(x)?1f ?y(x)¥Ms F V[_vf ?′A1Hq¥?÷E?Wf |′¥A1Hq ?^? > ?R?1L? ? I n¥M f ? (YV%?¥ ? ? y(x0) = a; y(x1) = b; ' –y(x0) = 0; –y(x1) = 0: I nWf¥μ′ J[y +–y]?J[y] = Z x1 x0 h F ?x; y +–y; y0 +(–y)0¢?F(x; y; y0) i dx; x21.2Wf¥′?5: ?f ?¥Ms–y(x)@l H V[|$f ?′f ??íTTaylorZ 7? ^μ J[y +–y]?J[y] = Z x1 x0 ‰h –y @@y +(–y)0 @@y0 i F + 12! h –y @@y +(–y)0 @@y0 i2 F +¢¢¢ dx = –J[y]+ 12!–2J[y]+¢¢¢ ; ? –J[y] · Z x1 x0 h@F @y –y + @F @y0(–y) 0idx; –2J[y] · Z x1 x0 h –y @@y +(–y)0 @@y0 i2 Fdx = Z x1 x0 h@2F @y2 (–y) 2 +2 @2F @y@y0–y(–y) 0 + @2F @y02(–y) 02idx sY ^WfJ[y]¥B)Ms?=)Ms?"ü¤?WfJ[y] |l′¥A1Hq ^Wf¥ B)Ms10 –J[y] · Z x1 x0 h –y @F@y +(–y)0 @F@y0 i dx = 0: |  Ts?¥?=[s?s] H} ?H?Hqüμ –J[y] = @F@y0 –y flfl flfl x1 x0 + Z x1 x0 h –y @F@y ?–y ddx @F@y0 i dx = Z x1 x0 h@F @y ? d dx @F @y0 i –ydx = 0: ??–y¥ ?i?ü V[¤? @F @y ? d dx @F @y0 = 0: ??Z??1Euler–LagrangeZ? ? ^WfJ[y] |l′¥A1Hq¥±s? TB? a ?? ^B?=¨è±sZ? ?WfJ[y] |v′¥ f?9 V[ ? ?1) ?i O9?¤?]"? T¥A1 Hq ?Euler–LagrangeZ? H L= ¨? MsE¥B?×1¥'? ? !`(x) ^x¥ ??f ?·(x) μ ??¥=¨? ? O·(x)flflx=x 0 = ·(x)flflx=x 1 = 0 ?? ?i·(x)Z x1 x0 `(x)·(x)dx = 0 (? ?5Aμ`(x) · 0 è è è3 !é?μ ] ??? ^?q = q(t)?t0; q(t0)???t1; q(t1)? ?¥HamiltonT ¨  ^ S = Z t1 t0 L(t; q; ˙q)dt x21.2Wf¥′?6: ?q?˙q ^ íé??¥<lUS?<l? L = T ? V ^? ?T? ] ?V-μ? 1Lagrange  Hamiltoneee ? ? ?á ?á ìB M(?D  ?¥) V ? ^??? L?¥('? ?D? p %?¥) ^? PT¨ S |′ ?  ?¥) ? V?T¨ S |′¥A1Hq¥s? T?±s? TsY ^ –S = Z t1 t0 h@L @q –q + @L @ ˙q –˙q i dt = 0 ? @L @q ? d dt @L @ ˙q = 0: ó?¥μ ] ???Lagrange L¥ 8? Tü??C ??Newton ?D¥? ?DZ ?? ?B" C) ?Wf J[y] = Z x1 x0 F(x; y; y0)dx ¥ ?èn¥+ y f? FWf?¥F = F(x; y0)?Acy ? H¥Euler–LagrangeZ?ü ^ d dx @F @y0 = 0; ?[ ?'ü V[¤? ?¥ nQs @F @y0 =è C: FWf?¥F = F(y; y0)?Acx ?^£ ü d dx h y0@F@y0 ?F i =y00@F@y0 +y0 ddx @F@y0 ? @F@y y0 ? @F@y0y00 =?y0 h@F @y ? d dx @F @y0 i ; ?[? H¥Euler–LagrangeZ?9 V[μ nQs y0@F@y0 ?F =è C: ü??2T?¨? è21.3? ?TLagrange L?Act5μ ˙q@L@ ˙q ?L =è C; x21.2Wf¥′?7: ?ü ^ ?  o? / ?ù?=íf ?¥ f? !μ=íf ?u(x; y); (x; y) 2 SN$  V[?lWf J[u] = ZZ S F(x; y; u; ux; uy)dxdy: ˉ ???u(x; y)S¥H?? ¥ ?′ó?' uflfl?%?: n5? ?19 ? J[u+–u]?J[u] = ZZ S F (x; y; u+–u; (u+–u)x; (u+–u)y)dxdy ? ZZ S F(x; y; u; ux; uy)dxdy = ZZ S h –u @@u +(–u)x @@u x +(–u)y @@u y i F dxdy + 12! ZZ S h –u @@u +(–u)x @@u x +(–u)y @@u y i2 F dxdy + ¢¢¢ ; ? ^WfJ[u] |′¥A1Hqü ^Wf¥B)Ms10 –J[u] = ZZ S h –u @F@u +(–u)x @F@u x +(–u)y @F@u y i dxdy = ZZ S h@F @u ? @ @x ? @F @ux · ? @@y ?@F @uy ·i –udxdy + ZZ S h @ @x ? @F @ux –u · + @@y ?@F @uy –u ·i dxdy =0: ?¨ TZZ S ?@Q @x ? @P @y · dxdy = Z ? ? Pdx+Qdy · ; | Q = @F@u x –u; P = ? @F@u y –u; ü ?|  ?¥2T?1 –J[u] = ZZ S h@F @u ? @ @x @F @ux ? @ @y @F @uy i –udxdy + Z ? h ? @F@u x dx+ @F@u y dy i –u: x21.2Wf¥′?8: ? H?Hquflfl?%? V? –uflfl? = 0;  T· ?=[¥Ls10 ?[ –J[u] = ZZ S h@F @u ? @ @x @F @ux ? @ @y @F @uy i –udxdy = 0:  ?¨–u¥ ?i?ü V[?  ?¥$f ?B?10 @F @u ? @ @x @F @ux ? @ @y @F @uy = 0; ?ü ^=íf ? f?/Wf J[u] = ZZ S F(x; y; u; ux; uy)dxdy |′¥A1Hq¥±s? T(Euler–LagrangeZ?) ü??2T?¨? è21.2??¥???ù5 ü¤? PT¨  S = Z t1 t0 dt Z x1 x0 1 2 h ‰ @u @t ?2 ?T @u @x ?2i dx |′¥A1Hq @2u @t2 ? T ‰ @2u @x2 = 0; ?? ^? E=c?¥?¥???Z? [ Bíf ??íf ?¥Wf′ù5??K? M f ? ?H?  |?′y7M f ?¥Ms ?H? B?10 ??Wf′ù5?1%? ?%?H?¥Wf′ù5 ? ?ù5 ?D  ^Ke?¥ ?7 ?? ^t ? K訥 / ?[Bíf ?1 è92B/Ms¥+He? ?E5 1. n5??Ms ^f ?yé?¥? ??1M x ?[Ms ??±s±  ? V?DQ? –dydx = d(–y)dx'–y0 = (–y)0: 2.Ms ?9 ^B?L? ? –(fiF +flG) = fi–F +fl–G; ?fi?fl ^è ? x21.2Wf¥′?9: 3.°¤9 ?ü V[¤?f ?e¥MsE5 –(FG) = (–F)G+F (–G): 4.Ms ??s(±s¥ I ?)9 V[?DQ? – Z b a F dx = Z b a (–F)dx: ?o1ü? T  ¥?s?) ??' V A 5.ˉ?f ?¥Ms ? E5?±s ?? ?M]o1e?1|±sE5?¥odp D?o–p' V è ? –F(x; y; y0) = @F@y –y + @F@y0 –y0: ? ú?i? FM?¥ey ^f ?y¥Ms71M x ^?M?¥ ?[ '??C o(@F=@x)–xp[ ?t ?E5? ?? ? V[x? ? 41w<?íf ?¥ f? FT1??¥Wf′ù5 Wf |′¥A1Hqa'Euler–LagrangeZ?a ?31ó?¥?3Hq/ p3±sZ??μ V ? p¤′f ? F31?iEuler–LagrangeZ?o ^Wf |′¥A1Hqi? ^ sA1Hq ó?¥?3Hq/Euler–LagrangeZ?¥3 V ???B? ? ìo ^′f ?¥?ê ??? 'B(+)?3 ^1 p¥′f ??31éB?F[?Y F? pf ?′¥ f?B"?Y¥ZEμ ? FB? ^°¤1? ? p¤¥3# o?íp¥f ?¥Wf′? Wf′¥?lF[ ? ??ZE? t L¨à ? #?¥9 ? F 6B?ZE ^9 ?Wf¥=)Ms–2J ?T? ? p¤¥3Wf¥=)Ms | ?(μ)′5?3'1′f ?Wf |l(v)??ZE? ?1?eL? ?T=) Ms10531??) ?ú)Ms F L=ù5aa?+Ye??ü ^ó?¥H?Hq/Euler–LagrangeZ?oμB? 3] HVt ? ?D = ? ? ? ??Wf¥′B?i * 1? H p¤¥ ·B3B?ü ^ ?1 p¥′f ? x21.3Wf¥Hq′?10: x21.3Wf¥Hq′ 5íkB/íf ?¥′ù5 F !μ=íf ?f(x; y) ? |′¥A1Hq ^ df = @f@xdx+ @f@ydy = 0: y1dx; dy ?i ?[=íf ?f(x; y) |′¥A1Hq? V[? @f @x = 0; @f @y = 0: F?μ 6B ?=íf ?¥′ù5=íf ?¥Hq′ù5'? ?Hq g(x; y) = C / pf(x; y)¥′? He5  V[?? ?Hq3y = h(x) ?ah ?f(x; y)? ¥y?"  ?Hq′ù5ü?1Bíf ?f(x; h(x))¥ ?Y′ù5 ? |′ ¥A1Hqü ^ @f @x + @f @yh 0(x) = 0: F???2T?μ 6B? ?3y1  ?i?31????y = h(x)¥Vr T7o3 1?? dy dx · h 0(x): ?" .à?A(v ? f?/9? V ?) py = h(x)ü V[°¤? ?Hq±s @g @xdx+ @g @ydy = 0; V7 p dy dx = ? @g=@x @g=@y; ? ^' V|  ?=íf ? |′¥A1Hq? @f @x ? @f @y @g=@x @g=@y = 0:  ?¥) ?? ?? ?^w<?÷?1M ¥íf ?¥ f?? ^ ?" 1M  ? "¥9 T9ü ? ?P F L¨?÷è¨Lagrangeeee000EEE ?) ?íf ?¥Hq′ù5 x21.3Wf¥Hq′?11: è ??  ?¥? ?Hq g(x; y) = C / pf ?f(x; y)¥′ù5ü V[?éLagrangee0?7?lB??¥=íf ?? h(x; y) = f(x; y)??g(x; y): ˉ|x?y A? ^ ?? ?M ?"??=íf ? |′¥A1Hqü ^( ?^ Ah ???ü ??1  ?ó¥A1Hq) @(f ??g) @x = 0; @(f ??g) @y = 0: ?N V[ p x = x(?); y = y(?); }í?? ?Hq??Lagrangee0?¥ ?′ü V[ p V ?¥′?(x; y) ?T ^÷?1M ¥íf ?9 V[]"1) ? ?T #?? ?Hq9 üo3? ??Lagrangee0' V Cí?Wf¥Hq′ù5 ?T1 pWf J[y] = Z x1 x0 F(x; y; y0)dx H?Hq y(x0) = a; y(x1) = b [#? ?Hq J1[y] · Z x1 x0 G(x; y; y0)dx = C /¥′5 V?l J0[y] = J[y]??J1[y]; ˉ|–y A? ^? ?¥5WfJ0[y]H?Hq/ |′¥A1Hqü ^ ? @ @y ? d dx @ @y0 · (F ??G) = 0: ?N±sZ?aH?Hq[#? ?HqA1 HüV?Yü V[ pLagrangee0¥ ′? = ?0a′f ?y = y(x; ?0)[#M?¥WfJ0[y]¥Hq′ è è è4 pWf I[y] = Z 1 0 xy02 dx H?Hq y(0)μ?; y(1) = 0 ?1 [a¥ZL? ú¥Lagrangee0 - ? B?μ| x21.3Wf¥Hq′?12: ?? ?HqZ 1 0 xy2 dx = 1 /¥′ wL 333?¨  ? í ?¥Lagrangee0E V[¤?A1Hq ? @ @y ? d dx @ @y0 ·? xy02 ??xy2¢ = 0; ' d dx xdydx ? +?xy = 0: (#) NZ?# Q¥H?Hq'?B?'?′ù5 ?¥'?′ ?i = ?2i; ?i ^ ,¨ ?9f ?J0(x)¥ ?i?? ,?i = 1;2;3;¢¢¢ ?zü ^Lagrangee07′f ?ü ^M?¥'?f ? yi(x) = C J0 (?ix): è C V[?? ?Hq?y1 C2 Z 1 0 xJ20(?ix)dx = C 2 2 J 2 1(?i) = 1; ?[ C = p2 J1(?i): ?"ü p ′f ? yi(x) = p2 J1(?i)J0(?ix): ??Lagrangee0¥?éEuler–LagrangeZ?C ??? ? QH?HqF ?B ü?'?′ù57T1'?′ù5 ?¥3'?′?'?f ?μí k ?? úμ ?ù531) ? F?B?ù5?í k?'?f ?? ^′f ? ? V[V/ ?¥Ms9 ? A?H?Hq[#?Nw¤¥ –y flfl fl x=0 μ?; –y flfl fl x=1 = 0: V[ pI[y]¥B)Ms –I[y] = 2 Z 1 0 xy0(–y)0dx; é7 V[ pI[y]¥=)Ms –2I[y] = 2 Z 1 0 x?–y0¢2 dx > 0: y1WfI[y]¥=)Ms? |?′ ?[?t′f ? ( PWf |l x21.3Wf¥Hq′?13: F?=?ù5 ^?í k?'?′?z9ü ^Wf¥′ ? ^y1|Z?(#)e[′f ?y(x)süμ ? Z 1 0 xy2 dx = ? Z 1 0 y?xy0¢0dx = ?y ¢xy0 flfl fl 1 0 + Z 1 0 xy02 dx = Z 1 0 xy02 dx; ? ? ?Hqü ?¤? ? = Z 1 0 xy02 dx: Ka?14??B ?Wf¥Hq′ù5¥e? V[ ò?o>? wL?é B?7 ? |vp¥e S+?ù5yNWf¥Hq′ù5è?1??ù 5(Isoperimetric problem) x21.4±sZ??3ù5?'?′ù5¥Ms? T?14: x21.4±sZ??3ù5?'?′ù5¥Ms? T Wf |′¥A1Hq¥±s? T(Euler–LagrangeZ?) ^è±sZ? ê±sZ ? ??M f ?¥?3Hq2?  ?ü?è±sZ? ê±sZ?¥?3ù 5 ?Wf¥Hq′ù5 A1Hq?C??? (Lagrangee0) ?? Q H?Hq2?  ?ü?±sZ?'?′ù5 ?B?|ù? ?¥Qù5 ??|±sZ?¥?3ù5'?′ù5?1Wf¥ ′Hq′ù5? a ??|±sZ?¥?3ù5'?′ù5¨Ms? yV ? è è è5è±sZ?H′ù5 d dx ? p(x)dydx ? +q(x)y(x) = f(x); x0 < x < x1; (#) y(x0) = y0; y(x1) = y1 (z) ¥Wf? T'sM?¥Wf ?H?Hq(z)/ |′¥A1Hq'1(#) 333; ?Wf′A1Hq¥±s? Tü ^Z?(#) * 1??Z?B? ?1 Z x1 x0 ‰ d dx ? p(x)dydx ? +q(x)y(x)?f(x) –y(x)dx = 0: C¥ù5ü ^1ü  TP ?? Bs¥Ms???s$f ?¥?=a ?[ ^? ?^ LC¥Z x1 x0 q(x)y(x)–y(x)dx =12– Z x1 x0 q(x)y2(x)dx; Z x1 x0 f(x)–y(x)dx =– Z x1 x0 f(x)y(x)dx: X?f ?q(x)?f(x) ^Dy(x)¥Msí1¥yNMs9 ?? ? ì? ^è  ?$f ??¥?B[ V[s?sZ x1 x0 d dx ? p(x)dydx ? –y(x)dx = p(x)dydx–y(x) flfl flfl x1 x0 ? Z x1 x0 p(x)dydx d(–y)dx dx =? Z x1 x0 p(x)dydx– dy dx ? dx =? 12– Z x1 x0 p(x) dy dx ?2 dx; ?¨? –y(x)flflx 0 = –y(x)flflx 1 = 0ü  ?¥2T8?  ?ü¤? Z x1 x0 ‰ d dx ? p(x)dydx ? +q(x)y(x)?f(x) –y(x)dx = ?– Z x1 x0 ( 1 2 " p(x) dy dx ?2 ?q(x)y2(x) # +f(x)y(x) ) dx = 0: x21.4±sZ??3ù5?'?′ù5¥Ms? T?15: ?ü a üZ?(#)B?ü ^Wf J[y] = Z x1 x0 ( 1 2 " p(x) dy dx ?2 ?q(x)y2(x) # +f(x)y(x) ) dx |′¥A1Hq è è è6 ê±sZ??3ù5 r2u(r)+k2u(r) = ?‰(r); r2 V; u(r)flfl§ = f(§) ¥Ms? T 333 V[? ?_v è5¥SE I ns ZZZ V h r2u+k2u+‰(r) i –udr; ?$f ??¥a [μ ZZZ V k2u–udr = 12– ZZZ V k2u2dr; ZZZ V ‰(r)–udr =– ZZZ V ‰(r)udr: ?$f ??¥?B[531?¨Green?B T[#H?Hq–u(r)flfl§ = 0 ZZZ V r2u–udr = ZZ § –uru¢d§ ? ZZZ V ru¢r?–u¢dr = ?12– ZZZ V ?ru¢2dr: yNeZ?ü?1 – ZZZ V ‰1 2 £?ru¢2 ?k2u2??‰u dr= 0: ? a üe ?¥?3ù5ü?N?H?Hq u(r)flfl§ = f(§) / pWfZZZ V ‰1 2 £?ru¢2 ?k2u2??‰u dr ¥′ù5 è è è7 ê±sZ?¥'?′ù5 r2u(r)+?u(r) = 0; r2 V u(r)flfl§ = 0 x21.4±sZ??3ù5?'?′ù5¥Ms? T?16: ¥Ms? T 333 n5 V[|'ù5 A? ^ è6¥+ y f?yNN'?′ù5ü?N?Wf J[u] = ZZZ V n£ ru(r)?2 ??£u(r)?2 o dr  QH?Hq u(r)flfl§ = 0 /¥′ù5÷éB?ü'?′? A? ^Lagrangee0 * 1??Wf′ù5??N ?Wf J[u] = ZZZ V £ru(r)?2 dr   ? QH?Hq?? ?Hq('?f ?¥BB?Hq) J1[u] · ZZZ V £u(r)?2 dr= 1 /¥Hq′ù5 ? 4 ?3?t'?f ??zü ^Wf¥′f ?7'?′?z ^Wf¥′??W fJ[u]¥=)Ms –2J[u] = 2 ZZZ V £r?–u(r)¢?2 dr ?1? ?[Wf¥′ ^l′?tl′?¥Kl?? ?ü ^'?′ù5¥Kl' ?′ x21.5 Rayleigh–RitzZE?17: x21.5 Rayleigh–RitzZE FMsEt ?D?¥?¨ V[s1 ??1¥Z ? FB??¨ ^T1't ?? p¥V ??y ? V[¨Hamiltone ?  e ? ?¥?y í ? ?D"d(é?aé?F¢¢¢¢¢¢)¥ ? ? V[¨Fermate ? í ?;Loé?¥.l ?? ? ¥Q ?|  ?9 V[¨Ms¥?y í ?èH? .à±4 ì0¥??? t ?D¥?ts|?| ¥té?¥ò?+?? T¥'? píB è?1? V[ V ?1ò1¥Wf′ù5 MsE¥???¨ μ×1¥ ? ?il ? V[ Pá ì÷dB1 3té W?¥ ? V[ Pá ì÷ZL1VX?¥t ? 5×_?¥ 5× ?Z FMsE¥?=??¨5 ^8C ?¥ L¨N′ ?1 p3 8¥t ?ù54 B?? ¥ 2 m  D5á ì ü H ˉ ? ^8? P¨±sZ? ? í?tt ?ù5?8èoμ  ?¥ù5 ? ?ú ? p3 ?¥ L=ù5?aao ?¤?í ?3MsE¥$ üy ? ? L¨¥í ?3E L !μB?B?¥'?′ù5 LX = ?‰X; fi1X(a)+fl1X0(a) = 0; fi2X(b)+fl2X0(b) = 0: p3¥??9 ^5(cμ??? ?¥)è±sZ?¥Y3 ?a} ?H?Hq?'?′ ?'?f ?μ ? V ?B? V ? ^è±sZ?? ?^ p3 6B? V ? ^?31¨è±s Z?) ?3E? ? pè±sZ?¥3B? a ?"   ?Xü ?¥f ??? 4·? ?¤?'?′¥ ?Vr T' P^ X00(x)+?X(x) = 0 ?"Ke?¥Z?μ ??¥ ?L?í13sinp?x?cosp?xB?¥? ? ?H?Hq /9íE'?′¥A üVr T * 1?B?¥'?′ù5? ú¥ ? 4ü VX7? MsEü1á ì4  p3'?′¥í ?ZE F¨Rayleigh–RitzZEí ? p3'?′ù5¥' ± ^ ^ ? n5ü'?′ù5?1Wf¥Hq′ù5 ? ?aB?¥f ? bW? p3y7üù5??f ?¥Hq′ù5 x21.5 Rayleigh–RitzZE?18: Fo1ê4¥f ? bW(?N'?′ù5) ^?!¥e5 9 V[@ú ?1/í'?′ ¥ú ?′ FV L¨¥? Aü ^1ê4B?ozp¥f ? bW( L=  ^B?f ?? )BZ ?L ?9 ?BZ ?? ?@ y1a@ú ?1 p¤'?′¥í ?′ F?ü1 pf ??  μ'?f ? ?1 p¥?1'+?1 pá ì Y5Vt ? ? ?D ?'?f ?¥?éT ?¥ ? è è è8 p'?′ù5 1 x d dx xdydx ? +?y(x) = 0; y(0)μ?; y(1) = 0 ¥Kl'?′ 333??'?′ù521.3?¥ è4?Xü) ?V? H) ?¥ ^Wf I[y] = Z 1 0 xy02 dx H?Hq y(0)μ?; y(1) = 0 ?? ?Hq I1[y] · Z 1 0 xy2 dx = 1 /¥Hq′ù5 ?¥Euler–LagrangeZ?ü ^ 1 x d dx xdydx ? +?y(x) = 0: C¨Rayleigh–RitzZE ?í ? p3??Wf¥Hq′ù5 Y5á ì?'?f ?¥ 3 ^ ?" A? ?@H?Hq-???? μ  } ?(1 I 1$)yN V¨[ T?  yn(x) =fi1?1?x2¢+fi2?1?x2¢2 +fi3?1?x2¢3 + ¢¢¢+fin?1?x2¢n; n = 1;2;3;¢¢¢ ?/í'?f ? n5 |í ?¥'?f ?y2(x)'  T? | - [} ?Wf#? ?Hq ¤ I[y2] = Z 1 0 xy022 dx=fi21 + 43fi1fi2 + 23fi22; I1[y2] = Z 1 0 xy22 dx = 16fi21 + 14fi1fi2 + 110fi22 = 1: x21.5 Rayleigh–RitzZE?19: ? V[ A? ^fi1?fi2¥=íf ?¥Hq′ù5A1Hq ^ @(I ??I1) @fi1 = 2fi1 + 4 3fi2 ?? ?1 3fi1 + 1 4fi2 · =0; @(I ??I1) @fi2 = 4 3fi2 + 4 3fi1 ?? ?1 5fi2 + 1 4fi1 · =0: ?? ^1?fi1?fi2¥} ?Z?Fμd ,3¥ sA1Hq ^ flfl flfl flfl flfl 2? ?3 43 ? ?4 4 3 ? ? 4 4 3 ? ? 5 flfl flfl flfl flfl = 0; ' 3?2 ?128?+640 = 0: 3-¤ ? = 643 § 83p34: ? ?ó¥? ^?¥l′ 21.3?21.4??Xü ?£VKl¥l′ü??Kl¥'?′ ? ú¤?¥? ?o ^'?′ù5¥Kl'?′¥í ?′ ˉ?1 = 5:7841¢¢¢ ; ??ú ?′ ?1 = (2:4048¢¢¢)2 = 5:7831¢¢¢ ¥Mμ??2£10?4M?1'?f ?¥í ?3 ^ ˉy1(x) =fi1?1?x2¢+fi2?1?x2¢2; fi1 =2 q 12?33 p 2=17 = 1:6505676¢¢¢ ; fi2 = q 80?230 p 2=17 = 1:0538742¢¢¢ : 1 Dú ?3 y1(x) = p2 J1(?1)J0 (?1x) T1??^9 ? ¢ = Z 1 0 £y 1(x)? ˉy1(x) ?2xdx =2?2 Z 1 0 y1(x) ˉy1(x)xdx =2 ‰ 1? h4p2fi1 ?31 + 8p2fi2 ?31 ? 8 ?21 ?1 ·i =1:66£10?5: x21.5 Rayleigh–RitzZE?20: m`í ?3Dú ?3¥1? m`í ?3Dú ?3¥μ ??[ T/í??o |  ['?′?'?f ?ü ?r???ú? ^¥ ? 7 |ùs ¥?N V[ ARayleigh–RitzZE¥ ?? >1B?z¥í ?ZE V[X^ ?T |¥[ ?÷¤?¥ú?÷ú FV  ?¥9 ? V[ A?¨Rayleigh–RitzZE Ho ? p¤K?¥+?'?′¥í ?′'?′¥? ?? P¨¥/íf ??¥? ? ? "M] F? ^?¨Rayleigh–RitzZE p3'?′ù5¥B?+? L=?¨?i??y 1Rayleigh–RitzZEo ? p¤μK?'?′7?? ?¥ L¨N′y1μ? ù5o3 1 pKl¥ ???'?′  AB/  ?¤?¥?=?'?′ ˉ?2 = 36:883¢¢¢ ; ??ú ?′ ?2 = 30:471¢¢¢ ; x21.5 Rayleigh–RitzZE?21: -W¥μè?V20%1 1 p¤@ú ?¥?=?'?′? ?A?9F/íf ??¥? ??? ?A?[??$ 9é¥9 ? 1}N L¨?÷z¥÷E ^ p3B??¥Wf Hq′ù5 ??e ?¥WfHq′ù5¥μYo? ?"??B?'?′?o1e ?¥WfHq′ù5??F B???Hq Z 1 0 y(x) ˉy1(x)xdx = 0 ' V?"???¥WfHq′ù5?Kl¥'?′? ?ü ^e ?¥?=?'?′