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Page 1: Ize J. Cálculo de Variaciones

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Page 10: Ize J. Cálculo de Variaciones

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Page 11: Ize J. Cálculo de Variaciones

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Page 186: Ize J. Cálculo de Variaciones

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X ÿ d lFxeDop q1st l u¸%¸ lwvyx&n75x-n o=µ=´ u¸ ljµ ´Iµr o´ u µ o+;vp ý ³ kmlnTKo#lµ8x+o Ý

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¼ ÅWÆ_ÇÈÉ#ʪË̬ÍÎDÏrÅÍЪÑ_ÉÅWÉÍЪҬÓCÓWËÔWÉÅWÉÕÒmÐYÊÒ¬ÓOP ±:7 ­2 * 47<-47.1 #D$& ±B ) ?"=$&4H KJF6F<- #"=,¨'0;¯ ?"`@+, * ,/+ #4 * V"-,/.' ?$-<-,[254²E2#0;D$&,[ ) 4FN ¡ G+ #.I'47"` * <- ?"-,/G ) 0 #"_472#07$&47G+0±6z0;<ªAlK«áµ« o st lµ+³m´°µ ³o+;vyN_"=$-0±"- ZQ7 #+ ?<&4 * ,/æ541"-,[U6+<-07@ * ?.>47"£"-,y$&<-47@A4¦¥-47.10;"£ #3 * #"=6A472?,/01G+ gh) +2#,[0;+ ?"£Z¤6X07<¢6z #G+47æ#0;"?NP|V~ !$#%¢OP,(p ln x Ý5Ý5Ý x&nG7oCVA"=, u lwvyxN¬xL oa ?"£³ª ?D$&0;+2? #"\kml]o¤pq st u lhvyxL¬xN o+;v #"$&4 * B ) i

kml ´3oW!kl]o¬p q st l u ´ u o+;v´ »¼ q st l u= ´ u= T´­ u ´ u ºo+;v´âPl) ³ oG+0;FG+ u= #" * 4Z.S4T$-<-,/æ`l u¸ ¸ o¨V u0 pl uy¸W oCp u m VA $&27NNN g #2$&0+VA"=, X lwÙKxDY½omp st u lhvyxL·´Ù ´ZY$UxL ´¯Ù ´ZY$ o+DvyV+$- #+ ?.S07" iX ÿ lFxY½opqYst l u lwvyxN ´ZYUxNT´Y$Co ´ u lhvyxN´ZYUxN\T´Y$Co ­ºo+;vp ý kl ´ZYo²;V

X ÿ d lFxe;op ý ³ kl`o?l­xNoprq st l u¸º¸ µ5´ uy¸W µ5L ´ u.º¸ µF W´ u µ+ L o+;vp q st l u= ´ u= ´ um ´ u o+;v Ýå£0T$& #.10;"¢B ) ¡¢ u0 p£ u EVz²;4>B ) u 0 p um Vz"- ]:7 `47"#'9 2 * 47<&4T.S ?$- * 41"=,/.1 ?$-<5'9 4G+ * 4Z"= #Q ) +GA4SGF #<-,:T47GA4FNm"=$-0S"= Q; #+ ?<&4 * ,/æ#4`4 gh) F2#,/07+ #"¢Za6z0;<6z #GA47æ?0;"#NWJF472?$&47.S ?D$& \2#0;.10` # ^ '4 * 2 )+* 0FVF$& ?+ #.10;"f * "=,/Q ) ,[ #D$& <= #" )+* $&47G+0+N¤ r¥ m&¦&!$#!%¨§©-D6ªnTª;m/g«=;/$*O9M)3rS¬ *,.MNOg1-RúXlhn+oLFQ;m/0M),.O7/=6<J­MNO7/8-.:<;/=16¡1-)3,547616¡1-Hh©3S-MNi$-)*-)/±nTIFH-)/$*O7/0MN-N8Vs6 ( §r,8nT@-N8l;m/®pS¬ /$,5pO9JWOMD6<JP¯-N8g1-NM),53úXlnFoúXlnTeoa F°«6<36¢Kn]nT<f«$-D±V;$-©²/0O ( F³*-)/0-)pO8Vs

ý ³ úXlnTKo#lµxeµzoa ¡«6<3D6*COV1O]µ ÝU ( §r, ý ³-úlhn7Ko?ljµ8xKµ½oaµ´¶?µK³IF©«6<36*COV1Oµ­FH-)/$*O7/0MN-N8n7@-N8l;/KpS¬ /$,5pO¡JWOMD6<J[· ¥G¸ ¹¦Z%\l4DoOP,8 ? ) +4YGF,/<- ?2#2?,¨'0;µ8VF$& ?+ #.10;" ý ³ úXlhn7Ko?ljµ8xKµ½oRÜ°FV+2?0; ý úXlnTKo-µSprFV+ ?$-0;+2? #"

úXlnTW´Ù=µ½oWúlhn7Kop Ù ³ ý ³ úXlnTeo#lµ8xKµ½o-Û ¼ ´ÂâPlhÙ ³ o

Page 208: Ize J. Cálculo de Variaciones

ÎDÏ;ÏIÒ1Ê£Ò ¶ÍfÓÑÒÂÔËÐ]ÅWÉÍУÆ_ÌyÒ¬Ó ¼ <º #"f+ ?QD4T$-,[:T4`6A47<&4ZÙR" ) ~A2?,/ ?$- #.1 #D$& \6z #B ) ä+0FNl@Aof47<&41µ8VA2#07?µ£6X ?B ) Lä+0+V+$- #+G+<= #.10;"fB ) úXlhn7´Iµ½oR°úXlnTKo Ý » NÖN|cZNå£0T$& #.10;"B ) \"-,"=0 * 47.1 #D$- _$& #F #.10;"f2#0;.10Yd+,/6f'0T$& #"=,/" i

ý ³ úXlhn7Ko?ljµ8xKµ½o½¼!`6A47<-4>µ¾pr #D$-0;+2# ?"Y6X0LG+ #.10;"6+<-0;@+47<ZB ) 6A47<-4U2547G+4µ¯ KJF,["=$- ) ¶Ù?ljµ½o\$&4 * B ) úXlnTa´°Ù=µ½o úXlnTKo> FV6A47<-4­Ù¡v Ù?ljµ½oKNCOL,/H #.Z@A4T<-Q;0+V #G+,/.1 #+"=,¨'0;H,/P~A+,[$&4FV #"=$-0U+0 #"]" ) ~A2?,/ #D$- >6A47<-4Q; ?+ #<-47< ) A4:; ?2#,[+GA47GY2#07.S6 * ?$&4ªG+ n7¢lh+0"- 6 ) #G+ ) "&47< * 47<=Q ) .1 #D$&0G+ R2#0;.16A472?,/GA47G`GF * ^ '4 * 2 )F* 0_G+ a:T47<=,§47":T47<-,/47@ * #"eoKV+4Y.S ?+0;"B ) \"- $& ?+QD47­ #"%$&,[.>472?,/0;F #".I'47"6+<- ?2#,["&47"f"-07@+<- Ù?ljµ½o ÝP|V~ !$#¿x%YOP #4klÀXoªpÁL u lhvyx&nzxLÀyxLÀ$ÃàxLÀ ¸ o+DvzDnzV8G+0;FG+ ÅÄ( ?" ) G+07.S,[+,/04T2#07$&47G+0G+ *6 * 47+0+ND$&07+2# ?"_4 * 2?0;+"=,/G+ ?<&47< X lhÙKxDYAop(kmlzÀyx-ÙÆ_´YÇ_oVA2#07¢Æ`pxÇpr`"-07@+<- 2ÈrÄ_V½$- #+ ?.S07"X ÿ lPxDY½omprqDq  l uÉ lhvyx&nzxLÀ´ZYÇ]xLÀ$ä´YǽÃDxLÀ ¸ ´ZYÇ ¸ oÆ_´ unÉIÊ Æ7ÃR´ uÉ)Ë Æ ¸ o+;vzn

X ÿ d lFxeDop ý ³ kmlzÀXo#lƽxLÇ_op·qDq  ¾ unÉÉ ÆÇ´ uÉÉ)Ê lzÆǽä´ÆÌÃÇo´ uÉÉ)Ë lzÆÇ ¸ ´Æ ¸ Ço´ unÉ)ÊDÉ)Ê ÆÌÃǽÃR´ uÉ)ÊDÉ)Ë lÆÌÃÇ ¸ ´Æ ¸ ǽÃào´ uÉ)ËÉIË Æ ¸ Ç ¸ × ;vzDn07< * 0Z$e47D$-0+VA"-," ) 6X07+ #.10;"fB ) +ÀC #"£³¢² u #"¢lÍý ³ klÀXo#lzÆzxDÆFoWpq;q  ¾ Î unÉÉ ÈÈ+v l unÉ)É)Ê oW ÈȽn l unÉÉIË oCÏÆ ³ ´´ uÉ Ê É Ê Æ ³Ã ´ ¼ unÉ Ê É Ë ÆÌÃÆ ¸ ´ unÉ Ë É Ë Æ ³¸ × DvzDnG+0;FG+ ) "&47.10;" * $& ?0;<- ?.>4_G+ * 4_G+,[:7 #<=Q; #+2?,§4_²Z * d+ ?2ed+0B ) ÐÆZpr"-0;@+<= ÐÈrÄ_N¬OP,+6A47<-4ªÀS"-47@X ?.S07"B ) uÉÉ lwvyx&nXxDÀxDÀrÃàxLÀ ¸ o8ÒÑÑ Ã l uÉÉ)Ê o8¨ÑÑ ¸ l unÉÉIË oa lh6X0;<R Õ¥= #.16 * 0]"-, u +0`GF #6z #+G+ GF fÀ i # #"= 254T"-01+0Y"- \+ #2? #"-,$e4YB ) u "- 541+Í£0YB ) +ÀC"= 54>³KoKVA"-,y47G+ ?.!'4T"" ) 6z0;+ ?.S0;"fB ) * 4Z.S4T$&<=,/æ

qc Î unÉ Ê É Ê uÉ Ê É ËunÉ Ê É Ë unÉ Ë É Ë Ï #"fG+ ~A+,[GA416z0;"=,[$&,:T4FVF ?$-0;+2? #" iý ³ klÀXo#lzƽxLÆPo¤µ´`qDq  lÆ ³Ã ´ZÆ ³¸ o+;vzn Ým #.10;"_B ) * 4>"- ?Q ) +G+4­G+ ?<-,:74TGA4>+0 ?" gh) #<%$& #.1 #D$- 16z0;"=,[$&,:T4S2#073<- #"=6X ?2?$-0C4 * 4>+0;<=.>4«W« V8 ?"G+ ?2#,/<?Vm+0"&4T$-,/" g 4T2# * 4Ud+,/6f'0T$& #"=,/"blh@½o]G+ * $& ?0;<- ?.>4 » N » Vm6z #<-03"#'9a* 03 ?"`2#0;¯<= #"-6z #2$&04 * 4C+07<-.S4Æ°K³ pÓLDÂlzÆL³Ã ´qƳ¸ o+;vzn"=0;@+<- b * #"=6A472?,/0G+ gh) +2?,/0;F #"kÆ #Á`l ÄfoVm2?0;ZÆpU ?ÈrÄl"=,

Page 209: Ize J. Cálculo de Variaciones

¼ » ÅWÆ_ÇÈÉ#ʪË̬ÍÎDÏrÅÍЪÑ_ÉÅWÉÍЪҬÓCÓWËÔWÉÅWÉÕÒmÐYÊÒ¬ÓÆ° p(S #D$&0;F2# #"ÔƱp_2#07+"=$&47D$& ¨pS #È=ÄfoKNOP 6 ) #GF Z6+<-07@A47<¢B ) Z #D$&07+2# ?"#V8"=,À3 ?"`a² ?""-0 */) 2?,¨'0;G+ * 4Z ?2 ) 4T2#,¨'07G+ \ )+* #<#VP$& #F #.10;" ) .'9 F,/.10+N g #2$&0+V kmlzÀ`´ZÆFoW!klÀXop ý ³ kmlzÀo?lƽxLÆPo-Û ¼´q;q  l unÉ)É lwvyx&nzxLÀZ´ÁÀGƽxLÀ$ä´ÂÀ7Æ7Ã;xLÀ ¸ ´ÂÀ7Æ ¸ o unÉ)É lwvyx&nzxLÀyxLÀ$Ã7xLÀ ¸ o=oÆ ³ DvzDnFÛ ¼´ q7q  l=l uªÉÉ Ê uÉÉ Ê oÆÆÌÃR´rl uªÉÉ Ë uÉÉ Ë oÆÆ ¸ ´Õ9Æ l °¶oÕÅÆPÛ ¼ o+;vzDnG+0;FG+ u V 2#0;<=<- ?"-6z0;+G+ ?4 * 0;"47<=Q ) .1 #D$&07"lhvyx&nzxLÀ´À7ÆzxDÀrôÀGÆÌÃDxLÀ ¸ ´ÀGÆ ¸ oKN¬OP, u #"f ³ ²Ä #"¢472?07$e4TG+0+V+:; ?.S07"¢B ) * <- #"=,/G ) 0S ?"£4T2#07$&47G+0Y6X07<fAlK« Æy« oDÂlƳ´¿«Õ9Æ« ³Ko+;vzDnXN¡ dF0;<&4±@+,[ #8VX2#0;.10Ä¿ #"4T2#07$&47G+0+V86z0LG+ #.10;"ª6z0;+ ?< * 0±G+ #D$-<-0G+ ) 3<= #2$½'47+Q )+* 0+Vz²;V82?0;.10>Æbp #¢ÈrÄ_VA6z0PG+ ?.S07"R JL$- #+G+ ?<ÐÆS2?0;.S0Z gh) #<-4ZG+ +Ä ²1$& ?+ #< ) A4 gh) +2?,¨'0;2?0;D$&,/ ) 4blw:; ?<a~AQ ) <-4 » N » o0LG+ ?.S0;"m ?"-2#<=,/@+,[<ÆXlhvyx&nFoP@ÆlAx&n+opx Ãt Æ7ÃLl5Y+x&nFo+Y­lÆÌâ #"m2#07$-,/ ) 0\6X07<6X ?GA47æ#07"eoN0;< * 0_$&47D$&0 iÖÖÖÖ Ælwvyx&nFo ÖÖÖÖ p ÖÖÖÖ q Ãt » Æ7ÃY ÖÖÖÖ v Î q Ãt Y Ï ×³ Î q Ãt Æ ³Ã Y Ï ×³l ) "&47+GF0 * 4ZG+ #"=,/Q ) 4 * GA47GbG+ OP2edØR47<-æ5oN

y

xx ÙÚÛ

y Ω

Ü

ü,/Q+N » N »qDq Â Æ ³ ;vzDn`pq st q þÝ Æ ³ ;vzDnv·q st q þÝ lwv±LoAq st Æ ³Ã l5Y+x&nFo+Y;vzDnv(lRLo ³ q þÝ q st Æ ³Ã Y+Dn+Û ¼ pl¤¯Lo ³ qDq Â Æ ³Ã ;vzDn+Û ¼ Ý07< * 0Z$e47D$-0+VA JP,/"%$& ) A4Y2#07+"=$&47D$& 2Þ`lB ) +01G+ #6z #FG+ GF lÆPo¤$&4 * B ) i

Page 210: Ize J. Cálculo de Variaciones

ÎDÏ;ÏIÒ1Ê£Ò ¶ÍfÓÑÒÂÔËÐ]ÅWÉÍУÆ_ÌyÒ¬Ó ¼ »7»qDq Â Æ ³ ;vzDnvµÞqDq  «Õ9Æy« ³ ;vzn Ý"=$-0S"= *[* 47.S4 * 4@ßàNáâãräå7æßmåßBßàªÇPçÌâWè­éåGê Èà5ND$&0;+2? #" * <- ?"-,[G ) 0± ?"472#0T$e47G+0>6X0;<£Ale« Æ« o  `«Õ9Æ« ³½;vzDnXN£0;< * 0S$&47D$&0±6+47<&4ƽV½$e4 * B ) C« Æy« "- #416z #B ) ä+0+V kmlzÀ`´ZÆFo!kmlÀXoR Ým #<= #.10;"R.I'47"R47G+ * 47D$& £B ) _"-,z·$-,/ #F ) ±:T4 * 0;<m6+<-0;6F,/0]F #QD4T$-,[:70+VP ?D$&0;+2? #"HÀ¢ ?" ) ±.'9 F,/.10+V,/FG+ #6z #+GF,/ #D$- #.1 #D$& \G+ * "=,/Q;F01GF unÉÉ ëÑÑ Ã l uÉÉ Ê oWìÑÑ ¸ l uÉÉ Ë oKNW4T<&4 * 0;"6F<-0;@ * #.S47"_2?0;+G+,[2#,/07A47G+0;"?Vml ? * 254T"-0­ #HB ) * 0;"_Q;<-47G+,[ #D$& ?"\GF * 47"2?0;+G+,[2#,/07+ #""=0;* ,[+ 54 * .1 #D$& _,[+G+ ?6X ?+G+,/ ?D$& #"&oR$& #F #.10;"f * "-,[Q ) ,[ #D$& <= #" )F* $e4TG+0 i

¤ r¥ m&¦!$#% §©-D6]n7k;/«=;/$*OM)3rS¬ *,.MNO1-kmlnFogMNO7/=1<,.M),.O7/=61OÔ«$O73¶í+ lnFoprFîWp » x Ý#Ý5Ý xDïF*.6<J±V;r-Vsý kmlhn7Ko=µ]´

ð ý íªÕlhn7eo=µ±pr¡«6<3D6*COV1O#µ Ý6 (¡ñ /$*O7/0MN-N8Ô8),8n7@-N8l;/KpS¬ /$,5pOÅ1-)JQ;/0M),.O7/=6<J

klnFo8´ ð 5íªÕlnFo*.6<pÅUV,S-)/&JWO-N8@1-YklnFoMNO7/=1<,.M),.O7/=61O2«$O73+í+ lnFo_p $[ ñ /«6<3*,.M);J6<3LFÐ8),Pí+jx#kò*,.-)/0-)/&8-.:<;/=161-)3,547616>1-fh03S-MNi$-)*j-)/nTg*.6<J±V;r-

ý ³ lÕkS´ ð 5íªho?ln7eo#ljµ8xKµ½oRµ´Ð?µ ³-)/$*CO7/0MN-N8kó*,.-)/0-l;m/KpS¬ /$,5pOkJWOMD6<JMNO7/=1<,.M),.O7/=61O-)/nTÌ[U (¡ô 6<36*OV16BM);34761<, Q`-)3L-)/0M),6UVJW-nlhÙ=oNFHMNO7/>n8lDop·n7)F³õkíªÕlhn8lwÙ=o-oprFj-)/$*O7/0MN-N8NF8),klnFo+*,.-)/0-;/KpS¬ /$,5pO9JWOMD6<JMNO7/=1<,.M),.O7/=61O<s

ý ³ l kS´ ð í+wo#lnTKo#lhnL§l;ox&nwlDo-oR Ý

· ¥G¸ ¹¦T%_l4Do¤klnFoL´®ö ð 5íª%lnFo+HlÕkln7-o´ö ð 5íªÕlhn7Ko=o #"fkmlnFo+Ukmlhn7-o"=,=í+ lnFop"íªÕlnTKomp·FN07+,/ ?+G+0>µ±pnn75VA * 47<-Q ) .S ?$-0YG+ * $- #0;<= #.S4 » N » N gh) +2#,[0;A4FNl@Ao¢OL,86A47<-4 ) +4YGF,/<- ?2#2?,¨'0;bn l;oV * 4Z"- #Q ) +GA41G+ #<=,[:T47GA4` #"f+ ?QD4T$-,[:T4FV+ ?$-0;+2? #"klnlhÙ=o-oWÂkmlnTKomp ý ³ l kS´ ð í+wo#lhn7Ko?lnlhÙ=o¶nT5x&nlhÙ=oW¶nTKo-Û ¼´âFlNKn8lwÙ=o¶n7G ³ o

Page 211: Ize J. Cálculo de Variaciones

¼ » ¼ ÅWÆ_ÇÈÉ#ʪË̬ÍÎDÏrÅÍЪÑ_ÉÅWÉÍЪҬÓCÓWËÔWÉÅWÉÕÒmÐYÊÒ¬Ólh²;41B ) 2íªÕlnlhÙ=o=oprZ²nT£ ?" ) 6 ) $-0S2?<5'9 $&,/2?0`G+ `k>´ ö ð 5íªhoKNW4T<&4`ÙR6z #B ) äF0+VAn8lwÙ=oW¶nlDop n lDo Ù´ÁâFlwÙ=o

lÕkmlhn8lwÙ=o-o!klnTKo-o=ÛàÙ ³ p ý ³ lÕkS´ ð í+o?lnTeo#lhnL x-nL o=Û ¼ ´ÂAl » o"- ?<5'9 4YF #QD4T$-,[:70+N» NÖN|cZN

åU'0T$& #"= ±B ) 7VW2#0;.10C ?¶ * 2547"=0UG+ * ^ '4 * 2 )+* 0 ?E:T47<-,/47"\:T47<=,§47@ * #"?V * 46A4T<=$& Ul@½oG+ * Ú #0;<= #.S4 » N ¼2#07<-<- ?"-6z0;+G+ 4 *÷ *C-N8)*H1-ªJ6Å8-.:<;m/=16>1-)3,547616­ø¤<- ?"=$&<=,/+Q7,' #FG+0;"- 4 * 2#0;ॠ) $-0SGF \GF,/<- ?2#2?,/0;+ ?"$& #LMQ; ?$- #"]lGA4TGA47"f6X07<n lDo=oa4 * 47"d+,[6X ?<-" ) 6X ?<=~A2?,/ #"¶íªÕlhn+op Ýçbèù ú ð1õ3ò3÷5ö¢÷#ð1õUîñ ñô®ûHö¢÷#îõ±ìî_ñýüUøZíøî£ëbìíîï ð1ñîõ ù#ø õUð1í¬ï ø

þ »OP #41 *8gh) +2#,[0;A4 * kmlnFopq st u lhvyx&nzx&n o+;vyV+2?0;­n8lPop ]VAnlopr>N u #"£\³5N¤OF4T@X ?.S0;"B ) ý ³ kmlhn+o?ljµ8xKµ½op q st l uy¸=¸ µ ³ ´ ¼ u¸ µ½µ ´ u µ ³ o+;v ÝOF47@z #.10;" B ) ) +4 2?0;+G+,[2#,5'0; + #2? #"-47<-,/4 6+47<&4 B ) n7 "= 54 ) .'9 +,/.10 #" B ) ý ³#klnTKo#lµxeµzoa FVF6X ?<-01B ) #<- ?.S07"f$- #+ ?<f2#0;+GF,/2#,[0;+ ?"¢F #2# ?"&47<=,§47"fB ) \,/D:;0 */) 2?<- #­ * ,/D$& ?Q;<&47FG+0"-0 * 47.1 #D$& TNÚ #+ ?.S0;"f6+<=,/.1 #<=0 ) A4Y2#0;+GF,/2#,5'0;b+ #2? #"-47<-,/4Y6+47<&4 ) ­.C'9 +,/.10 i¤ r¥ m&¦¡©#!Å%¨¯ÿn-.:-)/=1<3L- ( §r,DnT¶-N8;/¡pS¬ /$,5pOÐJWOMD6<J^F«6<3D6¶J6Ô/0O73)pÅ6¤`LF-)/$*CO7/0MN-N8 u lhvyx-n7¦lwvXox&n lhvXo=oÅ«6<36\vó-)/HAxe&C[

· ¥G¸ ¹¦%>O ) 6z0;+Q;47.S07"£B ) u Ür> # ) C6 ) D$&0±v½B ) Z6z0LG+ #.10;"£$-0;.S47<#Vz6X07<£2?0;D$&,/ ) ,[GA47G8V #ElAxeoKN¤D$&07+2# ?" u ÜRÞ_ # ) ­,/D$& ?<=:T4 * 0­ v½¬Z´¤x-v½´´y2#07$- #+,[G+01 #E Axe&NOP #4>µ ) A4 gh) +2#,5'0;b2#0;­Q;<+'4T~A254 i

a x − 0

h

bx +0α αx +x − xε ε

ε2

0 0 0ü,/Q+N ¼ N »

Page 212: Ize J. Cálculo de Variaciones

ÎDÏFÏÅÍУÑ_ÉÅWÉÍfУҬÓÓËÔWÉÅWÉÕÒmÐYÊÒ¬ÓCÇzÆffÆÒ1Ê£Ò ¶ÍfÓÒmпÌÆ(ÐZÍHÔ ­ÆZ ¼ »^ 0;¯«µ «Tp#V¢Üó´¤NmD$&0;+2? #"ý ³ kmlnTo#lµxeµzoprq Ãà l u¸%¸ µ ³ ´ ¼ u¸ µ½µ+7´ u µF ³ o+;vv ¼ wW´R ¼ Þ ÍG+0;FG+ w ?" ) A42#07$&46A47<-4 u¸%¸ V 6+47<&4 u¸ # Axe&jNC0;< * 0b$e47D$&0C #"=$-0U"- ?<A'4CF #QD4T$-,[:70C"-, #"ª" ) ~A2?,/ ?$- #.1 #D$& `6z #B ) ä+0+N£0;<_0T$&<&416A47<%$& k st lµ+³´°µ ³o+;vv ¼ ´ ¼ DÍ_$&,[ #+GF Z4±12 ) 47FG+0$&,[ #+GF 41PN_«µW« p·.S4àJ«µW«K´.>4¦J]«áµ «vy´-³f$e4T.Z@+,' ?±$&,[ #+G+ 4`PNmå£07$- #.10;"B ) µ­ #"¢Z¤"A'0 * 06z0;<6A47<%$& #"?N¬ #<=0>6z0PGF #.10;"f476+<=0¨JP,[.>47<µ­6z0;< äµ #ZKV+$e4 * B ) ­«áµ1 äµW« vDͨNm ?"- \2547"=0 i

ý ³ kmlhn7Ko?l äµ8x äµ½omp ý ³ kmlnTo#lµxeµzo´ ý ³ kln7eo#l äµYÁµ8x äµ]´Âµzo Ý^ 0;.10­«áµ_ 䵫 v Í Vy«µf´ 䵫 v «áµ 䵫 ´ ¼ «áµ« v #VP6+47<&4¤6z #B ) ä+0FVF * $P' #<-.1,/F0 ý ³ kmlnTKo#l äµµ8x äµ]´Âµ½o ?"¢472#07$&47G+0Y6z0;<¶w V+² * <- #" )+* $&47G+0Y #":R'4 * ,/G+0`6A4T<&4 äµ8N» NÖN|cZN

P|V ¥ ©#!%fOP 2#07+"-,[G+ #<-41 *8gh) +2#,[0;A4 *kmlnFop q st u lwvyx&nXx-n x&n o+Dv2#07n1²n G+47G+0;"f #S² ?C¨N

4DoOP, u #"£ ³ VA2#4 * 2 )+* 4T< ý ³ klnFo#ljµ8xFoN@½omOP, u ?"¬ ²`n7 * 0 ?"¬$&47.Z@+,' #l"=,/ ?.S6+<= 2#,[ #<=$-0"-, u¸%¹ ¹Ö¸=¹ ¹ ¾p·²`n7m ?" ) 16 ) D$-02?<5'9 $-,/2?0oV;6F<-0;@A4T<B ) iý ³ kmlnTKo#lµ8xKµ½o¬p·q st lhµ ³ ´Â`µF ³ ´ÂµF ³ o+;v2#07

·p uy¸=¸ l u¸%¸ ¹ o T´rl uy¸=¸ ¹ ¹ oÕ p uy¸=¹ ¸%¹ l uy¸=¹Ö¸=¹ ¹ o ¼ u¸%¸=¹ ¹¿p uy¸=¹ ¹|¸%¹ ¹2¨o_<=0;@A47<B ) YSlhvXo ± #" ) +4b2?0;+G+,[2#,5'0;U+ ?2# #"-47<-,/46A47<&4±$- #+ ?< ) .C'9 +,/.10+N1ljOL,>lwv½oÜ(FV$&07.>47<µlwvXomp l-lwv>¶v½oÕ³¬=³o%³¢"-,a« vS¶v½;«v#V½`"-,a« v±Ev½;«¼eo Ý oP|V ¥ ©#%b * Õ¥= #.16 * 0 » N V+6+<=0;@A47<B ) "=, * 4Z.S4T$&<=,/æ£$&,/ ?+ ) :T4 * 0;<a6+<=0;6+,/0]+ #QD4à$&,[:70 #Elhv½¨x-n7Koa #D$-0;+2# ?"RÀC+0Y6 ) #GF ]"- ?< ) .'9 F,/.10+Nl! ª472? #< ) 2#47.Z@+,[0`G+ :74T<-,§4T@ * ?"aG+ $e4 *g 07<-.S4B ) Z2? #<-2#4G+ blhv½#x&n7Ko * 4>.S4T$-<-,/æ "= ]: ) * :; `G+,/47Q;0;+4 * N ^ ) ,/G+47G+0±2?0; * 4SG+, g ?<- #F2#,§4T@+, * ,/GA47GN£"ÕM2#07Q; #<HÆ12#0;"=0;6z0;<=$- # ) A4]:7 #2#,[+GA47GbG+ 1lwv½¨x&n7eoVF2?0;bQ;<&4TG+,/ ?$- _ #"= #+2?,§4 * .S ?D$& # * 4ZG+,/<= #2?2#,¨'07G+ * :; #2$&0;<¢6+<=0;6+,/0Y4T"-0L2#,§4TG+014 * :T4 * 07<a6+<-076+,/0Y+ ?QD4T$&,:;0+N|o

Page 213: Ize J. Cálculo de Variaciones

¼ » ÅWÆ_ÇÈÉ#ʪË̬ÍÎDÏrÅÍЪÑ_ÉÅWÉÍЪҬÓCÓWËÔWÉÅWÉÕÒmÐYÊÒ¬Ó¡ dF0;<&4C@F,/ #VW #"]2 * 4T<-0CB ) u ¼ C+0 #" ) +4C2#0;FG+,/2?,¨'0;" ) ~A2?,/ ?$- 7V²;4CB ) ± #" ) A4C2?0;+G+,[2#,¨'07* 0L254 *jif* 4"=" ) .S4$#G+ `G+0;".'9 F,/.10;"Zlh # * "- #D$-,/G+0bG+ `B ) ) +0b ?" * 4±2#07$-,/ ) 472?,¨'0;3GF * 07$-<-0o¢+0$&,[ #+ 6z0;<fB ) ' "- ?< ) .'9 +,[.S0&%P6z0;< Õ¥= ?.S6 * 0` #­ * 2#47"-01G+ * 4T"Q; #0LG¤' #"=,/2#47" i

ü,/Q+N ¼ N ¼O ) 6z0;+GF<- #.10;"B ) Y #H6 ) D$&0­2#<#'9 $-,/2#0nT5V8$& ?+ #.10;" u lwvyx&n7àlwvXox&n lhvXo=o2¼(FN1OF47@z #.10;" ?$-0;+2? #"B ) \n7f #"¢ ³ lh6+<-0;6z0;"=,/2#,5'0;ZN » N » oN¤OP, u #"¢ Í #D$-0;+2# ?"ª6z0PGF #.10;" #"=2#<-,[@+,/< i

kmlnT´Iµ½oW!kmlhn7Komprq st l-l u¸%¸ ;v l uy¸D o=o-µ ³ ´ u µF ³ o+;vXÛ ¼ ´ÁAlK«áµ« oÌ?µ ³ c q st l('µ ³ ´)]µ ³ o+;vXÛ ¼ ´ÁAlK«áµW« o?µ ³ G+0;FG+ _,/D$- #Q;<-47.S07"¬6z0;<¤6A47<=$- #"¤ * $P' #<=.S,[+0 uy¸D lµ+³o Vljµ>pr ?b`²SoV8l ?"=$&0]"- _6 ) #G+ _d+472# ?<R6A47<-4gh) +2?,/0;F #"¢`R6z0;<f6X ?GA47æ#07"eoR²*'²)"=0; gh) +2#,[0;+ #"¢G+ vyVA2#07+)ý¼°Y #E½x&eNªlhm #<f * Õ¥= #.16 * 0» N » VA47B ) '9 ?µK³ p" st lµA³W´Iµ ³o+;vyN|oOP #4

úXlµzoprq st l!']µ ³ ´,)µ+ ³ o+;v) gh) F2#,/07A4 * 2 ) 4TG+<A'4T$-,/2?0+N OP,ÂúXljµ½o ¼ 6A47<&4 µ ¾p FV·6F<-0;@A4T<- #.10;"B ) ãG+ ãd+ ?2ed+0úXljµ½oa ¼ ´k st lµ+³´Âµ ³o+DvyVF6A47<&4`2?,/ #<%$e49´¤Nm #"- 2#47"-0kmlhn7´Iµ½oW!kmlhn7KoRx´Ð?µ ³ +lK«µW« oÌ?µ ³ µ´¶?µ ³ Û ¼ x"-,a«áµW« #"f6X ?B ) ä+0 i $- #+ ?.S0;" ) b.'9 +,[.S0FN¡ dF0;<&4>@F,/ #CúlDoap(FVB ) ?<- #.10;"_6z0;< * 01$e47D$&0S:; #<ª@A4¦¥=0±B ) Z2?0;+G+,[2#,[0;+ #"ª> #"ª * ') +,/2?0>.'9 +,[.S0G+ úlµ½oNW4T<&4 ?"=$-0I$& ?+ #.10;"­6+<-,[.S ?<-0ÁB ) E #"%$ ) GF,§47< * 0;"­6 ) D$-0;"C2?<5'9 $-,/2?0;"bG+ EúXljµ½oKN lW4T<- #2?

Page 214: Ize J. Cálculo de Variaciones

ÎDÏFÏÅÍУÑ_ÉÅWÉÍfУҬÓÓËÔWÉÅWÉÕÒmÐYÊÒ¬ÓCÇzÆffÆÒ1Ê£Ò ¶ÍfÓÒmпÌÆ(ÐZÍHÔ ­ÆZ ¼ » ÃB ) U6z0PGF<5'9 4T.S0;"Y2?0;D$&,/ ) 4T<K61 ,5/-³/$,5*;p`V6z #<-0 * 4Q;<&47IG+, g ?<- #F2#,§4H #"±B ) Cúlµ½o1 #" ) gh) +2?,/0TMA4 * 2 ) 47G+<+'4T$&,[2#0+VF2 ) ²;41 ?2 ) 472?,¨'0;­G+ \ )+* ?< #" * ,[+ 54 * N o.0;"a6 ) $-0;"2#<#'9 $-,/2#07"¤G+ úXlµzom"-0;@F<- _ * #"=6A472#,[0` Axe&$/>aà½x&eVF ?"=$&47bGA47G+0;"R6X0;< * 47"¤"-0 *[) 2#,[0;+ ?"4 * 4` #2 ) 472?,¨'0;bG+ \ )+* ?<¢47"-0L2#,/47GA4 iDv l!)]µ o0']µ>pr x µylPompµyl?ompr ÝOP,1) ?"¢ VA2?0;.102) ¼!FVzµ ?"¢ ³ ²"-0 */) 2?,¨'0;G+ i

)µF T´,)ºµFL0'µp· x µlLo¬p·µylompr Ý" ) A4Y #2 ) 472#,5'0; * ,/+ #4 * GF \"= #Q ) +G+010;<-G+ ? i "- */* 47.>4 * 4àVéäåéLâ ÈçÌè43)åGêDâ åéNâzçÌèråGæX6A47<-4klnFoNå£0T$& #.10;"B ) ]µ>c` #"f"-0 */) 2?,¨'0;8NOF47@z #.10;"B ) \ * 6+<=0;@ * #.S4ZG+ :74 * 0;<= #",/F,/2#,/4 * ?" i

l()µF o P5']µ±p· x µylLoprFx£µFhlLo¬pq´$&,[ #+ ) A4') +,/2#41"=0 */) 2#,¨'078NfOP,ÀlhvXo #" * 41"=0 */) 2#,¨'076A4T<&4Å´p » V½ #D$&0;F2# #"?V86z0;< * 4 * ,/+ #4 * ,[GA47G±² * 4) +,[2#,[GA47G8V * 4Z"-0 *[) 2#,5'0;b6A47<-4Å´¶2 ) 4 * B ) ,/ #<-4` #"¶´ÀlhvXoKN ©#!£%Áä"-N8B;/«=;m/$*COMNO7/6;<:61O 1-18),g-E,W8)*-;m/=68O7Jo;$M),$SO7/³IFk/0O*3,54,6<JÐ6<J«=3LO<UVJW-)pÅ6±l()µ o ']µ±pFWµylLomprµyl=äLompr&¯Õä¾p· ( [lå¢0C$&<=,[:L,§4 * B ) ,/ #<= >G+ ?2#,[<`µ ¾c PN o0;< * 4 ) +,/2?,/GA47G #"" ) ~A2?,/ ?$- ±6z #G+,[<`µ lPo¢¾p FV²6X0LG+ ?.S0;"" ) 6X0;F #<µ lLoRp » VA²D4>B ) Z"=,$& #F #.10;" ) +4>"-0 *[) 2#,5'0;­2#0;Uµ lLog¾p FVz #D$&0;F2# #"µlwvXo-Û;µ lLo$-,/ #F G+ ?<-,[:T47G+4 » #bANm #"%$& * ,/@+<=0+V2547GA4\:; #æB ) "= _.1 #+2?,/0;A4 * 4]F0P2?,¨'0;SG+ _6 ) D$-0;"2#0;¦¥ ) QD47G+07"#VF"= " ) 6X0;F \B ) * 4Z ?2 ) 4T2#,¨'07­:T47<=,§472?,/0;A4 * ?"£+0Y"=,/+Q )F* 47<?VF #"¢G+ ?2#,/<B ) )SlhvXof¾pr`6A47<-4ZvU #E½x&e Ý

y

x78 98

ü,/Q+N ¼ N

Page 215: Ize J. Cálculo de Variaciones

¼ » Ä ÅWÆ_ÇÈÉ#ʪË̬ÍÎDÏrÅÍЪÑ_ÉÅWÉÍЪҬÓCÓWËÔWÉÅWÉÕÒmÐYÊÒ¬Óå£0T$& #.10;"¬B ) ª ?> * 2#47"-0ZGF gh) F2#,/07+ #"R`6X0;<m6z #GA4Tæ#0;"?V * 42#07+G+,/2?,¨'0;1G+ ¢2#0;D$-,/ ) ,/G+47G>G+ ;: #,/ ?<-"ÕM$&<-47"-"alÚ #0;<= #.S4£]N PN » oy #"<)µ Ý p·FVT #"G+ #2?,/<µ #"W2?0;D$&,[ ) 4 i* 0;"y6 ) D$&07"2#<#'9 $&,[2#0;"G+ * 4f #2 ) 472#,5'0;:T47<-,/472#,[0;A4 * "-07CZR²7VA2#0;.10=)ý¼°FVF"-0;\³#N¬Ú #F #.10;"f #D$&0;F2# #" i¤ r¥ m&¦©#%맩-D6)ý¼°T-)/3 Axe&zF>) õ' MNO7/$*,5/$;$O8NF½-)/$*CO7/0MN-N8Vs¤úXljµ½oj¼°9«6<36Zµ&¾c ¯MNO7/µylLomprµylo¬p· ( 8),õ98O7JWO¡8),K/0OÅ*,.-)/0-½«=;/$*O8¡MNO7/6;<:61O8k-)/Axe&C[· ¥G¸ ¹¦%O ) 6X0;FQD47.10;"mB ) _Z+0$&,[ #+ ¢6 ) D$&07"¤2?0;ॠ) QD4TG+0;"#VL #D$-0;+2# ?" * 4"=0 */) 2?,¨'0;S4 * 4 #2 ) 472#,5'0;l!)]µFºo ']µ>p·Fx µylLomprFx µFhlPomp » #"$&4 * B ) µlwvXo½¼°` #b§Axe&N07<2#0;D$&,[ ) ,/GA4TG­GF * 4Z"-0 */) 2?,¨'0;2?0;b<- #"=6X ?2?$-0>4 * 0;"GA4T$-0;",/+,[2#,/4 * ?"#V * 4Z"=0 */) 2?,¨'0;ÀlwvXoa4bl()+À o '+ÀHp PV©ÀlLoªp?#V°À lLoªp » V2?0;4g¼¿FV8$&47.Z@+,' #3 #" ?"=$-<-,/2$e47.1 #D$- `6X0;"=,[$-,[:T4> #ÂAxe&jV6A4T<&46z #B ) ä+0 i '¿pl()+À o ÛGÀyN

úXljµ½omp q1st l!']µ ³ ´,)µF ³ o+;vp q1st l=l()+À½ oÕºµ ³ ÛGÀ`´)]µF ³ o+;vprq1st l!)lÀA§l% ¼ µ½µFÛGÀ]´Iµ ³ ÀAºÛGÀ ³ o´)]µF ³ o+;vl,[D$& #Q7<&47+GF016z0;<6A4T<=$& ?"²GA47G+01B ) µylLompµylomp·DoKNúXljµ½omprq st )SljµFÁµÀAºÛGÀXo ³ Dv Ý07< * 0Y$&47D$&01úXljµ½oм PVX4 * .S ?+0;"¢B ) Zµ p µÀ Û7ÀN¤ ?<-0S2#0;.10±µylPo¬p FV+$& ?+G+<#'9 47.10;"¢µ lLo¬p²6X0;< * 4 ) +,/2?,/GA47G¶4 * 6+<=0;@ * #.S4G+ ±:T4 * 0;<- ?"],/F,/2#,/4 * ?"6A47<&4U ?"&4U ?2 ) 472?,¨'0;GF, g #<= #+2?,§4 * G+ ±6F<-,/.1 #<0;<=G+ #8VzµylwvXocPN D:; ?<-"&4T.S ?$- 7V;"=,LW #"6 ) D$&0¢2#07ॠ) QD47GF0£G+ ¬AVà"- 54¶À]"=0 */) 2?,¨'0;G+ £l!)lÀ o @'+ÀbprHÀlLopqÀlomp·FND$&0;+2? #"

úlzÀXopq st l=l()+ÀA o À`´)lÀA ³ o+;vbprq st l%A)lÀA ³ ´)lÀA ³ o+Dvp·,/D$- #Q;<-47+G+0b6z0;<6A47<%$& #"?NY"=$&0b2?0;D$&<&4TG+,/2? 1 * d+ #2edF0B ) YúXlÀXol¼FN`0;< * 0±$&47D$&0+0b ?"\6 ) D$&02#07ॠ) QD47GF01GF AN¡ dF0;<&4FV"=,fäC ?/AxeàX #"6 ) $-02?0;ॠ) Q;47G+0G+ 1AV2#0;F"-,/GF #<- ?.S07" * 4 g 47.1, * ,§4YG+ gh) +2#,[0;A4 * #"_6+47<&4àM.1 ?$&<=,/æ#47G+0;"6z0;<aÙKVAGA47G+4Y6z0;< iB±lµ8x-Ù=op q1st lhÙC)µ+ ³ ´ÙC']µ ³ ´rl » EÙ=o=µF ³ o+;vbppÙÕúXlµzo´·l » ¶Ù=oAq st µF ³ ;v6A47<-4­$-0PG+0­Ù #° Fx » jVDB±lµx=Ù=o9¼ FVW4 * .S ?+0;"\B ) bµc PNElc_ SdF #2ed+0EB>ljµ8x-Ù=o¡¼ C"-,mÙ`Ü » ²úXljµ½oa FN|o

Page 216: Ize J. Cálculo de Variaciones

ÎDÏFÏÅÍУÑ_ÉÅWÉÍfУҬÓÓËÔWÉÅWÉÕÒmÐYÊÒ¬ÓCÇzÆffÆÒ1Ê£Ò ¶ÍfÓÒmпÌÆ(ÐZÍHÔ ­ÆZ ¼ » .y4` #2 ) 472#,5'0;GF )+* #<6A4T<&4Y *8gh) +2?,/0;+4 * B±lµ8x-Ù=oR #" i

lhÙC)µ ´rl » EÙ=o=µ o p ÙC'µ ÝOP #4>µylhvyx=Ù=o * 4Z"-0 *[) 2#,5'0;bG+ #"%$e4Y #2 ) 472#,5'0;b2#0;­GA4T$-0;"f,/+,[2#,§4 * #"µlAx-Ù=omp·bx µFlAx-Ù=omp »;ÝW4T<&4ZÙmpFV½µylhvyxe;opvY¶AVF²;4YB ) ?­ ?"- 2#47"-0 * 4Z ?2 ) 4T2#,¨'07 #" i µ p·FNm47<&4ZÙmp » V+"-47@X ?.S07"B ) ]µl%äAx » op·PN^ 0;+"=,/G+ ?<- ?.S0;"£ * 2#0;ॠ) $-0±G+ Z6 ) D$&0;"Ylwvyx-Ù=of$&4 * ?"£B ) `µlwvyx-Ù=op i #"_G+ ?2#,[<£v #"ª2#0;ॠ) QD47G+0SG+ >6A4T<&4` * 6+<=0;@ * #.S4FB±ljµ8x-Ù=oKNW4T<&4¶254TGA4E6 ) D$-0lhv½¨x-ÙÕKo1G+ U #"%$& U2?0;ॠ) D$-0 i µÃLlwv½¨x-ÙÕKop µ lhv½¨x-ÙÕKo¾p l"=,ª+0E$- #+GF<5'9 4T.S0;"µylhv½¨x-ÙÕeoZpµ lhv½¨x-ÙÕKo]pPNU07< * 4 ) +,/2?,/GA4TGEG+ * 4"=0 */) 2?,¨'0;E4 * 6+<-0;@ * #.S4GF >:T4 * 0;<= #"\,/+,[2#,/4 * ?"#V #vyV$& ?+G+<#'9 47.10;"µylhvyx-ÙÕolc FV * 02 ) 4 * +0C ?"]6z0;"-,[@ * `²;4CB ) µ lAx-ÙÕop » oNb0;< * 0b$&47D$&0+VW46A47<%$&,[<ªG+ ­l=äzx » o¢$& ?+ #.10;" ) 2 ) <=:T4±vpvlhÙ=oV2?0;µylhvlhÙ=oKx-Ù=oªp PNG £4¨²C #D$&07+2# ?" * 47"_"-,[Q ) ,[ #D$& ?"6z0;"-,[@+, * ,/GA47GF #"R6+47<&4` #"%$e412 ) <%:74PN

x

t0 1

HIJ

A

D

E B

C

Hü,/Q+N ¼ N

K ¦ML 0N %O.4f2 ) <=:T4¢GF #"&4T6A47<- ?2# ¤47D$& ?"G+ */* #QD4T<84 * 4 g <-0;D$- #<&4 i #"=$-0ª+0f6 ) #G+ ¬"- ?<6z0;< * $- #0;<= #.S4G+ * 4 gh) +2#,5'0;b,/.16 * '9 2#,$e4 i 2#0;.10SµÃLlhvyx-Ù=oª¾pFV * 4`2 ) <=:T4YG+ #@z 2#0;D$&,[ ) 47<#NK ¦ML 0P %m"%$&01 #"¢,/.16X07"-,/@ * £²;41B ) +01 #"fG+ * 4 g 0;<-.S4Zvbp vlwÙ=oNK ¦ML K %ãD$&0;F2# #"aa #"R2?0;ॠ) Q;47G+0]GF _Z6A47<-4GB>ljµ8x-Ù=oNm #<=0Z2#0;.10B±ljµ8x-Ù=o½¼IFVP * 47<-Q ) .S ?$-0B ) ) "&47.10;"6A47<-4 * 2#47"-0­Ùp » #":R'4 * ,[G+0²F06z0PGF #.10;"$& #F #<] ?"=$- >254T"-0+Nc_ 1d+ ?2ed+0C"=,úXljµ½ob FVm$e47.16X0L2#06z0PGF<5'9 4T.S0;"$& #F #<1 #"=$- 2#47"-0²D4HB ) * 6 ) D$&0H$& #<=.S,[A4 * G+ #@z ­$- #+ ?<ÙRÜ » V½4 * .S ?+0;"fB ) Sä`pr¨N

Page 217: Ize J. Cálculo de Variaciones

¼ » ÅWÆ_ÇÈÉ#ʪË̬ÍÎDÏrÅÍЪÑ_ÉÅWÉÍЪҬÓCÓWËÔWÉÅWÉÕÒmÐYÊÒ¬ÓK ¦ML %ã47<&4±Ù£p FV * 6 ) D$&0b$& #<=.S,[A4 * #"2#07ॠ) QD47GF0­GF S­6A47<-4*B>ljµ8xeDoKV #"G+ ?2#,[<6A47<&4 * 4 ?2 ) 472?,¨'0;µ pFNm ?<-0` #D$-0;+2# ?"¢µylhvXompvZHZ²S+0Z"- 6 ) #G+ ª2 ) .16 * ,[<¤B ) µlwvXo¤p]6A47<-4vCG+,/"%$&,/D$-0`G+ \ ÝK ¦ML % OP 54¶lAx-ÙÕKo * 6 ) D$-0C$& ?<-.1,/A4 * N±D$&0;F2# #"$& #F #.10;"B ) 1vlhÙ=o_$&,[ #+G+ >4AV2 ) 47FG+0CÙ$-,/ #FG+ 4ÙÕ¨N #<-0HµylwvlhÙ=ox=Ù=o`p Hp µlAx-Ù=oN0;<Z * $& ?0;<- ?.>4CGF * :T4 * 0;<].1 #GF,/0+V JP,/"%$& ¯äv ?$-<- 3²HvlhÙ=o\$&4 * B ) bµÃLlävx-Ù=oZpFV6A47<-4C254TGA4CÙKN0;< * 0$&47D$&032 ) 47+G+0UÙ$-,/ ?+G+ 4ÙÕ¨V$- #+G+<#'9 47.10;"£µÃLlAx-ÙÕeoRp FN #<-0> #D$&0;F2# #"µlwvyx-ÙÕojc¿FV½6z0;< * 4 ) +,/2?,/GA4TGG+ * 6F<-0;@ * #.S4`G+ :T4 * 0;<- ?",/+,[2#,§4 * #"%P ?"=$-0S2?0;D$&<&4TG+,/2? ]µ lAx-Ù=omp » N^ 0;.10±d+ ?.S0;" * ,[.S,[A47G+0Y$&0LG+0;" * 0;"_2#47"-07"#V * 4S2#0;+2 */) "=,¨'0;C #"B ) ­ä ?Á AxeàXF06 ) #GF 1"= #<_6 ) D$&02#07ॠ) QD47GF01GF S"-,úXljµ½oa FN

» NÖN|cZN^ 0; * 0;"GF0;"$& #07<- #.S47"f47D$- #<-,[0;<- ?"$& #F #.10;" i¤ r¥ m&¦©#¿%맩-D6 u -)/b+ÍÌ[6 ( §r,mn7®-N8@;m/ZpS¬ /$,5pO¢JWOMD6<J^Fl-)/H`OT-)/H`ª«$O73ª«$-D16RQ7O8NF«6<3D6CklnFoMNO7/n8lLoNF¤n8lo>161O8NF-)/$*CO7/0MN-N8Vs) c u lwvyx&n7¦lwvXox&n lhvXo=of F þþ à l u oW uy¸ prõF³8),S) ¼!T-)/H Axe&x/0Oki6<õª«=;m/$*CO8MNO7/6;<:61O821-¢-)/H Axeà ÝU ( §r,£n7-N8®8O7Jo;$M),$SO7/ò1-l u o uy¸ p òMNO7/ u lhvyx&nT¦lhvXoKx&n lwvXo-o ¼ µõ8),g/0O&i6<õ®«=;m/$*CO8MNO7/6;<:61O821-¢-)/H Axe&zFH-)/$*O7/0MN-N8n7@-N8l;/pS¬ /$,5pO9JWOMD6<JP¯«6<36J6@/0O73pÅ6­«« ( 1-_kmlnFoV[· ¥G¸ ¹¦% l4Do¬0;< * $- #0;<= #.S4 » N » VP$& ?+ #.10;"aB ) úXlµzoa FVL²±6z0;<R * $& ?0;<- ?.>4 ¼ N » V u FNT.y4')+* $-,/.S46A4T<=$& :L,/ ?+ G+ * 2?0;.1 #D$e47<=,/0Z~A+4 * # * 4Z6+< ) #@+41G+ * $& #07<- #.S4 ¼ N ¼ Nl@Ao0;< * 4d+,/6'07$& ?"-,/"B ) 1+0­dA45²6 ) D$&07"2?0;ॠ) QD4TG+0;"\G+ 1U #I Axe&V ?$-0;+2? #" * 4"-0 *[) 2#,5'0;H4 * 4 #2 ) 472?,¨'0; i l!)]µFºo Fp'µx£µlLo¬p·FxKµFhlPomp » #"6z0;"=,[$&,:T4 #/Axe&jN¬0LG+ ?.S0;"R #"-2?0;Q; ?<д$e476z #B ) ä+0`²±6z0;"-,$&,[:70ZB ) * 4"-0 *[) 2#,5'0;4 * 6+<-0;@ * #.S4 il=l()r ´o-µF o Fp l('rZ´Wo-µ xªµ lLo¬p·FxKµF lLomp »"- #4U$e47.Z@F,' #6z0;"=,[$&,:T4C #/½x&eR²E$e4 * B ) ) x´ ¼ãU #· Axe&N lc_ ¥-47<- ?.S07"`4 *R* #2$&0;<Z6F<-0;@A4T< #"%$e4ÁG+ #6z #+GF #+2?,§4I2?0;D$&,/ ) 4ÁG+ * 47"b"-0 */) 2?,/0;+ ?"GF ?2 ) 472?,/0;F #"­0;<-G+,[A47<=,§47"2?0;r<= #"-6z #2$&0°4 * 0;"2#0L ?~+2#,/ ?D$& #"?N ¡ B ) '9 "- 16 ) ?G+ >6+<=0;@A47<G+,[<- ?2?$e4T.S ?$- S ?+2#07$-<&47FG+0 ) A4b #2 ) 472?,¨'0;+0Cd+07.S0;Q' #+ #46A47<-4ªµ\µ Và2#0;6A47<%$& m+0ªdF0;.S07QW' #+ #4дaljµ \µ oNW©£"&4T+G+0 * 4 g '0;<-. )F* 4aG+ m:T47<-,/472#,5'0;G+ ¬2#0;+"%$e47D$- #" ** #2$&0;<G+ ?@X 6+<=0;@A47<fB ) «µYÁµ «vµÞI´¬N o07<¢ * $& ?0;<- ?.>4 ¼ N ¼ VF$- #+G+<= #.10;"fB )

úVUzljµ½omp qYst l-l!)·Z´o=µF ³ ´rl('·Z´o-µ ³ o+;vC Ý

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ÎDÏFÏÅÍУÑ_ÉÅWÉÍfУҬÓÓËÔWÉÅWÉÕÒmÐYÊÒ¬ÓCÇzÆffÆÒ1Ê£Ò ¶ÍfÓÒmпÌÆ(ÐZÍHÔ ­ÆZ ¼ » º07< * 0Z$e47D$-0SúXljµ½oRµ´¡ st lµ+³´Âµ ³o+DvyN^ 0;.10`²D4`:P,[.S07"B ) \ * <- ?"-,/G ) 0S ? * 4Z JP6+<= #"-,5'0;klnTW´Âµ½oW!kmlnTKop·úXljµ½o=Û ¼ ´ Wf #"-,[G ) 0 #"¢Ale«µW« o st ljµ+³W´Iµ ³o+;vyV+ #D$-0;+2# ?"¢$- #+ ?.S07"¢ * <- ?" )+* $e47G+0FN

» NÖN|cZNP|V~ ©#!%­ * X¥= #<=2#,/2?,/0Y-=eN VF:L,/.10;"B ) *8gh) +2?,/0;A4 *kmlnFop q l » ´Án ³ o DvnlDop·FV+nl » op » V+$&,[ #+ 2?0;.S0°') +,[2#0`6 ) $-0S2?<5'9 $&,/2?0Y4 * 4 gh) +2?,¨'0;bnlhvXopvyN¡ B ) '9 uy¸=¹Ö¸=¹ p ¼ l ¼ n ³m » o#l » ´¯n ³&o Í ÝW4T<&4Yn8lwvXop vyV u¸%¹|¸%¹ p)p N^ 0;.10 uy¸=¸ p u¸%¸=¹ p FV * 4Y #2 ) 472#,5'0;:T47<=,§472?,/0;+4 * "- <- ?G ) 2# ]4 i µ Û ¼ p FV½ #"£G+ #2?,/<?Vz2?0; * 0;"fGA4T$&07"µylDoªp FVµ lDo_p » i µylhvXo_p vyN1å£0­dA45²36 ) D$&07"2?0;ॠ) QD4TG+0;"G+ 1±²3n8lwvXo_p v¶ #" ) H.'9 +,[.S0* 0L254 * 6A47<-4 * 4Z+0;<=.>4C«y« Nå£0T$& #.10;"¤B ) _"-, u +0]G+ #6z #+GF JP6 * '9 2#,$e47.1 #D$& ¢G+ _nzVL #D$&0;F2# #";'¿p·² * 4 ?2 ) 472?,¨'0;S:74T<-,§4T2#,/07A4 *"- ><- #G ) 2# 4El()µ o p Fx2 ) ²D4U"=0 */) 2#,¨'072?0; * 4T"]2#07+G+,/2?,/0;F #"\,/+,[2#,§4 * #"µylLopPxKµ lLop » x ?"µylhvXo£p Ãt*X \ t bX

\Yb [Z+xz²U6z0;< * 0S$e47D$&0µylhvXo+¼¿6A47<&4>vZ¼¿Ax8"=, * 4±2?0;+G+,[2#,5'0;\) ¼¿ #"_:R'4 * ,[GA4FN\"G+ ?2#,/<?V8+0bdA45²U6 ) D$&0;"2#07ॠ) QD47GF0;"_G+ Yb²;V6A47<-4]) ¼¿FV82 ) 4 * B ) ,[ #<_6 ) $-0b2#<#'9 $&,[2#0±G+ *Wgh) +2?,/0;+4 * V #" ) ­.'9 +,[.S0 * 0P2#4 * ? * 4Z+0;<=.>4U«A« N¤c4TG+0`B ) G)µ´Z¼IZ²'¿prFx+"= \$-,/ ?+ B ) úXlµzoRµ´Yq st µF ³ ;vU aµ ³ xG+0;FG+ ]"= ) "A'0 * 4SGF #"-,[Q ) 4 * GA47GG+ Z0;,/+2#47<L' 6A47<-4±µylhvXof2#0;µylPop¿ Ý 0;< * 01$e47D$&0FVX * <= #" )F* $e4TG+0"- 6 ) #G+ :; ?<£G+,[<- ?2?$e4T.S ?$- "-,[6A4T"&47<6z0;< * 07"6 ) D$&0;"¢2#07ॠ) QD47GF0;"#NP|V~ °#%a ) +0`G+ * 0;"¤ #"=$ ) G+,[0;"a6A47<-4Z * 6+<-0;@ * #.S4,["-0;6z #<=,/.¯' ?$-<-,/2?0l¥= ?<-2?,/2#,[0]]N^ºFN » o¤"= */* #Qm'0Y4 ) gh) +2?,/0;+4 * G+ * 4 g 07<-.S4 iklnFopq st lhn] ð l » ´Ân ³ o ׳ o+;vnlLo±p n8l?o>p FV ð ¼FN!0;<S * Ú ?0;<- ?.>4 » N ¼ V ) Â.!'4¦JF,[.S03GF HklnFo`G+47<A'4 ) I.I'4àJP,/.103G+ *6+<=0;@ * ?.>4f2#07+G+,/2?,/0;+47G+0¢0;<-,[Q;,/A4 * V¨2 ) ²D4_"-0 */) 2?,¨'0;\ #" ) `47<=2#0_G+ ¤2;'9 <-2 )F* 0+Và"-,PL]v^v`_¬lL]Lo-Û ¼ Nl ** #2$&0;<¢6 ) ?G+ `6+<=0;@A47<?V84>6A4T<=$&,[<£GF * 41 #2 ) 472#,5'0;UG+ ` )+* #<¢²C * <= #B ) ,/"=,[$-0±B ) ZnlhvXoª¿>B ) ðG+ ?@X "= #<f6X07"-,[$-,[:70 i n lPo½¼IPoKNm ?"=$& 2#47"-0 uy¸=¹Ö¸=¹ p( ð l » ´¶n ³Ko Í׳ªÜ!PV+ #"G+ ?2#,[< * 4]2?0;+G+,[2#,¨'07G+ a. #Q; ?+G+<= 6A4T<&4 ) *b ÈådcmâebBç"= 2 ) .16 * TN^ 0;.10 u¸%¸ p u¸%¸%¹ p·FV * 4Z #2 ) 472#,5'0;­:T47<=,§472?,/0;+4 * "= <- #G ) 2# 4 i

Page 219: Ize J. Cálculo de Variaciones

¼;¼ ÅWÆ_ÇÈÉ#ʪË̬ÍÎDÏrÅÍЪÑ_ÉÅWÉÍЪҬÓCÓWËÔWÉÅWÉÕÒmÐYÊÒ¬Ó

f g

L

ü,/Q+N ¼ N Ãl!)]µFºo Fp·Fx µylLomprFxµ+hlLomp »;Ý07< * 0Z$e47D$-0±µylhvXompx Ãt )SlLo=Û$)>lih+o+h Ý^ 0;.10j)>lLo-Û$)Sl!hFo ¼!PV½µ­ ?"2#<- ?2#,[ #D$& ²±F01d+4¨²>6 ) D$&07"¢2?0;ॠ) Q;47G+0;" i* 4]"=0 */) 2?,¨'0; ?+2#07$-<&47G+4Y ?") ­.!'4¦JF,[.S0 * 0P2#4 * ? * 4`+0;<-.S4U«Ûz« Nmm #<= #.10;"f.!'47"f4TG+ * 47D$& B ) #"¢G+ d+ #2ed+0 ) ­.!'4àJP,[.S0` ? * 4+0;<=.>4C«y« 5N

P|V ¥ ©#¿%fOP #4bklnFopq st Ø lhvyx&nFo#l » ´Án ³Ko&׳;v2#0;­nlLopnlomp·FV Ø lhvyx&nFo¤ FN4Doa<-07@A47<fB ) * 4`2#0;FG+,/2?,¨'0;G+ a. #Q7 #+G+<= "=,/ ?.S6+<= "- 2 ) .S6 * 7N@½o£OP, Ø lwvyx&nFojc » Vz #+2?0;D$&<-47<_ * ') +,[2#0>6 ) D$&02#<#'9 $-,/2#0FVz6F<-0;@A4T<ªB ) Z+0>dA45²C6 ) $-0;"_2?0;ॠ) QD4TG+0;"ª²2#07+2 *[) ,/<?N2¨oOL, Ø lwvyx&n+ofp l » ´InP³eo ׳¨V86+<=0;@A47<ªB ) `n®c ± ?"\"=0 */) 2#,¨'07UG+ * 4S #2 ) 472#,5'0;3G+ Y )+* ?<#N<-0;@A4T<B ) \6A47<&4` ?"&4Y"-0 *[) 2#,5'0;dA45²b6 ) D$&07"¢2?0;ॠ) Q;47G+0;"f"-,8²±"-0 * 01"=,yª`_N ^ 0;+2 */) ,[<#NP|V ¥ ©#lkó%fOP #4bklnFopq t n ͽ;vU2?0;nl;oprFV+nlLoprR¼!FN4Doa+2?0;D$&<&4T<£ * ') +,/2?0Y6 ) D$&012#<#'9 $&,[2#0+N@½o<-0;@+47<B ) F01d+4¨²±6 ) D$&0;"¢2#07ॠ) QD47GF0;"fG+ \FN ^ 0;+2 */) ,[<#NP|V~ ©#¿%nm So pL`¦LÅ ¸ ¦qL ¸ ~P ¥Rr r#^ 0;.10C²D4 * 0C:L,/.10;"] ? * ¥= #.16 * 0CZN N V * 47"]Q7 #0LGR' #"=,/254T"Z # ) A4U" ) 6z #<=~+2#,/ TVm<- #6F<- #"= #D$e4TGA43 #g 0;<=.>4Z6A4T<&47.¯' ?$-<-,/2#4FVF"=0; * 0;".'9 +,[.S07"RG+ *8gh) +2#,[0;A4 *i

úlzÀyxLÆPoprq ÿtsÿlu Iw lhÙ=o8;Ùpq ÿtsÿlu l!B2ÀA ³ ´ ¼ u ÀA ÆDT´ílÆD ³ o ׳ ;Ù2#07\B p Iw É ³ V u p l.w É xNw4v#oV­í p IwEvm ³ lh6X07< * 0S$e4T$-0 u ³ ÜwBgíYV * 41,[Q ) 4 * GA47G­"- ?<5'9 4S"-0 * 0"-,w É ²¢w4v_"-0;­2#0 * ,/+ #4 * ?"R² * 4Z" ) 6z #<%~A2#,[ G+ ?Q; #F #<&4TGA4DoN"=$- Z6+<-07@ * ?.>41 #"_2?0;3G+0;" gh) +2#,[0;+ ?"#V86z #<=0±2#0;.10>"= ]:L,/0± ? * 4Så£07$&41ZNÖÄFN » VGF ?$àlNIw ÉK¹ v ¹ opFN07< * 01$&47D$&0 * 4Y$- #0;<#'9 4Y2#0;<=<- ?"-6z0;+G+,[ #D$& 6A47<&41 #"%$& ]2#47"-03lh$&<-4T$e4TGA41 ? * 4Y"-,[Q ) ,[ #D$& \"- ?2#2#,5'0;½o+0>"= 476 * ,/2#4FNmO ) 6z0;+GF<- #.10;"f6 ) ?"£B ) * 4Z2 ) <=:T4Y2547FG+,/GA4à$e4Y41Q7 #0LGR' #"=,/254Y"- 6 ) #G+ \6A47<-47.S $&<=,/æ54T<a6X07<HÀN ?"=$- 2#47"-0 iúXlzÆFoprq ÉxÉ s l!B ´ ¼ u ÆDT´í+Æ ³ o ׳ À

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ÎDÏFÏÅÍУÑ_ÉÅWÉÍfУҬÓÓËÔWÉÅWÉÕÒmÐYÊÒ¬ÓCÇzÆffÆÒ1Ê£Ò ¶ÍfÓÒmпÌÆ(ÐZÍHÔ ­ÆZ ¼;¼ »$&,[ #+ u v ¹ pl u ´ílÆ oÌIw ׳#V u v ¹ v ¹ p l!Bgír u ³oÌIw Í׳ԼIFNÚ0;.S ?.S07"2#0;.10 Õ¥= ?.S6 * 0 * 2#47"-0­G+ * 4± ?" g ?<&4FV2#0;4B p "- ?A³eÀPV u p PVníãp » ²C" ) 6z0;+Q;47.S07"B ) * 4`2 ) <=:T4`2547+GF,/GA4T$&4Y4YQ7 #0LGR' #"=,/254Y+0Y6A47"-416z0;< * 0`6X0 * 0;"lÀp·Fx_o¬²±B ) * 4Z6A47<&4T.S $&<-,[æ547.10;"6z0;<zy¯ ?¶Fx ¼ _8jNúXlÀ;op q2 x

sl"= # ³ Àf´ÁÀà ³ o ׳ py Ý * ¥= #<=2#,/2?,/0Z]N N ¼ "= 6+,/GF,¨'0]<= #"=0 * :7 #< #"%$& 6+<=0;@ * #.S4FVL47B ) '9 l4 ) +B ) +0Z"= 54` #"= #+2?,§4 * 6A47<-4Z #"=$- Õ¥= ?.S6 * 0o¤6z #GF,/.10;"f4 *8* #2?$-0;<B ) \2#0;.16+< ) #@z \B ) * 4Y"=0 */) 2#,¨'07b4 * 4` #2 ) 472#,5'0;bG+ ] )+* #< ?" iÀ\p ¡ <-2?$&47l-l´]2#07"¨l|yUZÞ?o-o ox 2#0; ´¼! ݲBÞfpwy ¡ <-2?0;"¦l´ 2#07$zÀ oRGF $&4 * .10PG+0`B ) À+l(y omprÀ N³´ #"fG+ $& ?<-.1,/A47GF016z0;<fÀFl|y ³ oprÀ ³ V0Y6X07< * GA4T$-0S,[+,/2?,§4 *i

´plÀ¦ ³ ´ "= # ³ ¼ À Ûo ׳ Û_"- ? ³ À Ý¡ dF0;<&4`@+,[ #) p u~¹~¹ p "- #A³KÀFl"= #A³eÀ¢´ÂÀ ³Ko Í׳¨VA2#0;u~~ p 2#0;" ¼ ÀFl"- ? ³ Àf´ÂÀ ³ o ׳ "- ? ³ À2#07" ³ À+lh"- # ³ Àf´ÁÀ ³ o Í׳u>~~ ¹ p¿£"= #CÀ2#0;"½ÀYÀàwl"= # ³ Àf´ÂÀ¦ ³ o Í׳ Ýc_ * 4Y #2 ) 472#,5'0;CG+ ] )+* ?< i lÀ lh"- ?A³KÀ£´ÂÀ ³o ׳o p "- ?½ÀW2#07"XÀFl"= #A³eÀ£´IÀ ³o ׳_0;@P$& #F #.10;"ªÀ p2#07"½À+l ¼ À ³ ´Á"= # ³ À7o-ÛT"= #½ÀPN^ 4 * 2 )+* 47+G+0

) p(z;"= #½À2#07"½À1À lh"- # ³ Àf´ÁÀ ³ o Í׳ Ý07< * 0Z$e47D$-0'¿p u>~~ l u~~¹ o p l2?0;" ¼ ÀT"= # ³ À ¼ À ³ o?l"- ? ³ Àf´ÂÀ ³ o Í׳ Ýlc_ ¥-47.10;"a2#07.S0Y Õ¥= ?<-2?,/2#,[014 *8* ?2?$-0;<a:; ?<-,[~+2547<f #"%$&0;"¢2'4 * 2 )F* 0;"?N o.y4` #2 ) 472#,5'0;­:T47<=,§472?,/0;+4 * l!)]µ o p'µ"- <- ?G ) 2? ]4 iµ lim2#07$½ÀTÀ o-µ l2#07" ¼ À_ ¼ À ³ ÛT"- ? ³ À;o=µ±prp·"- ? ³ ÀnhlµzÛà"- # ³ À7oÕ T´·ljµ½ÛT"= # ³ À;o Ý^ 0;.10Y"- # ³ À¢¾p·`²bµl(y omprFV * 4Z"-0 */) 2?,¨'0;­4` #"%$e4Y #2 ) 472#,5'0; ?" iµyl|yWopòP"- ?l|yUy oÕ"- ? ³ À Ý * 6+<=,/.1 #<R6 ) D$&0Y2#0;¦¥ ) QD47G+0lµl yoprDo¬ #"6A47<-4 ypy ´q_VF ?b #"- 2#47"-0 i ÀFl yomp_¶À Nm"=$- 6 ) D$-0S ?"¢ * 0;6 ) ?"=$-0±4 * 6 ) D$&01,/+,[2#,§4 * "=0;@+<- * 4Z ?" g ?<&4FNP|V ¥ ©#e %Wf ?$-0;.>4T<¤ ?"=$& ¥= #.16 * 0FVL6z #<=0` #"%$e4:7 #æ6A47<-47.1 ?$&<=,/æ#47<¤6z0;<yalÀ7oªl ?"a.I'47" g '472?, *( oKNP|V~ ©#lkó%+· ¥Ì ¹¦ o ¥Ì ¥ ¦z¹ ¥ r#OP #41nT ) 6 ) $-0S2?<5'9 $&,/2?0`G+ *gh) F2#,/07A4 *ji

Page 221: Ize J. Cálculo de Variaciones

¼;¼;¼ ÅWÆ_ÇÈÉ#ʪË̬ÍÎDÏrÅÍЪÑ_ÉÅWÉÍЪҬÓCÓWËÔWÉÅWÉÕÒmÐYÊÒ¬Ókmlhn+oprq1st u lwvyx&nzx&nL|o+;v2#07nlop ä ²nlLo * ,[@+<- TN u ?"ª Í N07< * 0Z$e47D$-0SnT¢ ?"£"=0 */) 2#,¨'07G+ * 6+<=0;@ * #.S4 i

l u o u¸ p­xfnlop ä>x u lAx&n8lLox&n/lLo-omp·lÚ #07<- #.S4`ZNÖÄFN » oNOP #4YûpTµb #`#Axe&jx¢2#0;µylo¬p·Z² ?µfp «µW« Ý^ 0;.10Y #­ * Õ¥= ?.S6 * 0 » N » VklnFo¤$&,/ ?+ "- ?Q ) +GA4YG+ ?<-,[:T47G+4YGF üA<L' #2ed+ $ iý ³ kmlhn7Ko?ljµ8x+op q st l uy¸=¸ µr´ u¸%¸%¹ µr ´ uy¸=¹Ö¸ µ \´ u¸%¹|¸%¹ µ o+;v Ý07< * 0Z$e47D$-0+VA"-,nT¢ ?" ) 6 ) $-0S2?<5'9 $&,/2?0 i

kln7W´Iµ½oWIklnTKompq st l u¸%¸ µ ³ ´ ¼ uy¸D µzµF7´ u µ+ ³ o+DvXÛ ¼ ´ÂAle«µW« o?µ ³ OP,,[$- #Q;<-47.10;"a6z0;<6A47<%$& #"¢ * $P' #<=.S,[+0 ¼ u¸ µ½µ V+$& ?+G+<= #.10;" iý ³ kmlnTKo#lµ8xKµ½op q st l=l uy¸=¸ Dv l u¸ o-o-µ ³ ´ u µ ³ o+Dv1 uy¸D l½x-n7¦lPoKx&n lLo-o=µ ³ lLo Ý^ * 4T<&47.1 #D$& * 412#0;FG+,/2?,¨'0;G+ n. #Q7 #+G+<= u lwvyx&n7àlwvXox&n lhvXo=o£ >"=,/Q ) :R'4 * ,[GA4FV+²;4±B ) Z6X0LG+ ?.S0;"$&07.>47<µlLo¬p·Y6+47<&4 ) ­" ) @z2#07ॠ) D$&0SG+ ûNOP, ?"-2#<=,/@+,[.S07"

ý ³ klnTKo#lµxeµzopúlµ½opq st lhµF ³ ´ ¼ `µ½µ+7´Á]µ ³ o+;v #D$-0;+2# ?" * 4` #2 ) 472#,5'0;bG+ ] )+* #<6A4T<&4`úXljµ½oV * 4Z #2 ) 472#,5'0;­:T47<=,§472?,/0;+4 * 6+47<&4±kmlhn+oKV+ #"lhµFT´Â`µ½o P°l`µF7´Âµ½o¬p]pl_µ+ºoÕT´rlLo-µl" ) 6z0;+,/ ?+G+0 u # Í ² u ¼°PV+ #D$&07+2# ?"£nT£ ?"£ ³ ²b ?"fG+, g ?<- #F2#,§4T@ * 5oN"fG+ #2?,/< i

l()µFºoÕL0'µ±pr Ý^ 0;) p u V'¿p u¸%¸ l u¸ o N"=$-0\F0 #"m"-0;<=6+<- ?+G+ ?$- ²D4B ) ¢`?Axe&/aàAxe&A #"%$½'42#0;D$- #+,[G+0 #`ûN ¡ B ) '9 "-0 * 0 * 4T"2#0;FG+,/2?,/0TM+ ?"¤G+ g <-07$- #<-4254T.Z@+,§4T i* 4\2?0;+G+,[2#,5'0;YA4T$ ) <&4 * 6A4T<&4 * 4 #2 ) 472#,5'0;1:T47<=,§472?,/0;+4 * #"¬µ ´U`µW« t p·FNu lAx&n7¦lLox&n lLo-o=µFhlLo´ u¸ lAx&n75lLox&n lLo=o-µylLompr Ý

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ÎDÏFÏÅÍУÑ_ÉÅWÉÍfУҬÓÓËÔWÉÅWÉÕÒmÐYÊÒ¬ÓCÇzÆffÆÒ1Ê£Ò ¶ÍfÓÒmпÌÆ(ÐZÍHÔ ­ÆZ ¼;¼ O ) 6z0;+GF<- #.10;"\) ¼ I ? Axe&(% * 4Á #2 ) 472#,5'0; G+ "= #Q ) FG+0 0;<-G+ ? l()µ o p 'µVZµylLoIp #Vµ lPop &ZÛ? l6z0;< * 0$&47D$&0"-4T$&,[" g 472?,/ #FG+0 * 4E2#0;FG+,/2?,¨'0;ÂA4T$ ) <&4 * 6A47<&4 * 6F<-0;@ * #.S4H:T47<=,§4àM2#,[0;A4 * oVA$&,[ #+ ) A4r') +,[2541"-0 *[) 2#,5'0;8N£å£07$- #.10;"ªB ) TV8G+ #@F,/G+0b4 * 4 * ,/+ #4 * ,[GA47G8V * 4S"-0 */) 2?,¨'0;2#0;Eª ?"DÀlhvXoKVA2#0;¢ÀlLop » V$À lLop£`Û?Vl()+À o p'lÀyN ó©#%\ä-)/H½x&eÐ-N8R m| ¸ ¦ on 1-£®8),©Àl=äPomp$[å£0T$& #.10;"B ) Sä>+0YG+ ?6X ?+G+ \G+ a]lh07$&<-4`:7 #æ * 4Z<= * 4T2#,¨'07­µlwvXo¤pDÀlhvXo=oN

y

x

1

ü,/Q+N ¼ N|Ĥ r¥ m&¦©#lkó% ô 6<36¢-)J«=3LO<UVJW-)pÅ61-³Q3LO7/$*-)36@Jo,UV3L-¡-)/Z¡O\ZH«$O73j«$-D16RQ7O8Vs6 ( ý ³ klnTKo#lµxeµzoj¼!F©«6<36`µ-)/>û 8),Põ¡8SO7JWOk8),P/0Oki6<õª«=;m/$*CO8¡MNO7/6;:61O8+1-£-)/ Axe& [U ( §r,Xn79-N8ª;/®pS¬ /$,5pO¡JWOMD6<J^F -)/$*O7/0MN-N8Vs£l uy¸ ¹ o u¸ p·F u¸ ¹ « t p·FynT¦lomp ä9F) c u¸ ¹ ¸ ¹ õ8),1)ý¼I-)/½x&ej/0Oki6<õª«=;m/$*CO8¡MNO7/6;:61O8l1-¢-)/H Axeà[M ( §r,anT-N8>8O7Jo;$M),$SO7/µ1-l u¸ ¹ o u¸ p F u¸ ¹ « t p Hn7¦l?oSp äÅF uy¸ ¹ ¸ ¹ ¼ õ/0Oi6<õ>«=;m/$*CO8MNO7/6;<:61O821-¢-)/H Axe&zFH-)/$*O7/0MN-N8n7@-N8l;/pS¬ /$,5pO9JWOMD6<J-)/±û&F½MNO7/KJ6@/0O73pÅ6]ZV[· ¥G¸ ¹¦%Sl4DoaOP,X+0`dA45²±6 ) $-0;"f2#0;¦¥ ) QD47G+07"aG+ Y # Axe&V+ #D$-0;+2# ?" * 4]"-0 */) 2?,¨'0;>À­G+ 1l!)lÀ o '+ÀprFVÀWlLo¬p » V$À lLo¬p(ª`Û #"¢"-,/ ?.S6F<- 6z0;"-,$&,[:T4Z ?¶Axe&jN

ý ³ kmlhn7Ko?ljµ8xKµ½omprq st l!)]µF ³ ´']µ ³ o+;v1 u¸ lLo-µ ³ lLopq st l()µF ³ ´rl!)lÀA oÕµ ³ ÛGÀXo+;v1 u¸ µ ³ « tp qYst )>lµ+ÀA µzÛ7Ào ³ ;v>l!)lÀA µ ³ Û7ÀZ´ u¸ µ ³ o¨« t

Page 223: Ize J. Cálculo de Variaciones

¼;¼ ÅWÆ_ÇÈÉ#ʪË̬ÍÎDÏrÅÍЪÑ_ÉÅWÉÍЪҬÓCÓWËÔWÉÅWÉÕÒmÐYÊÒ¬Óp qYst )Sljµ À µzÛ7Ào ³ Dvyxl6z0;< * :74 * 0;<aGF lÀ­²À #PoKNÚ #+G+<= #.10;" ý ³#kmlhn7Ko?ljµ8xKµ½o½¼°14 * .S ?+0;"fB ) ]µ pqÀ µzÛ7ÀN ^ 0;.10Sµylo¤p·FVA #D$&07+2# ?"µ l?o¬prFV² * 4!') +,/2#4Y"=0 */) 2#,¨'07 #"£µ>cFN

D:; ?<-"&4T.S ?$- 7Vz"-,yª ?"f2#0;ॠ) QD47G+01G+ \AVA #D$&0;F2# #"¶ÀC6z #<%$& ?+ #2? ]4`ûV+2?0;ý ³ klnTKo#lzÀxDÀoWpq st l!)lÀA ³ ´'lÀrÀXo+;vS uy¸D À ³ lLopq st l!)lÀA ³ 0)+ÀA ³ o+;vS l()+ÀAà´ uy¸D ÀXoÀ« tG+0;FG+ ª"= ) "A'0 * 4 ?2 ) 4T2#,¨'07]'lÀ±p l()+À o ²1"- ª,/D$& ?Q;<A'06X0;<¤6A47<%$& #"?Nm©£"-47+G+0 * 47"¬2#0;+GF,/2#,[0;+ ?"R,/F,/2#,/4àM* ?"6A47<-4ÅÀyVF$- #+ ?.S0;" ý ³ kmlnTKo#lzÀyxLÀXoprFV * 0`2 ) 4 * 2#0;D$&<-47G+,[2# * 4`d+,[6f'07$& ?"-,["#NOP,2?0;+"-,[G+ #<-47.10;"f47d+0;<-4 i

úXljµ8x-Ù=op qYst lhÙ?l]µ ³ ´ ¼ `µ½µF7´Á_µ+ ³ oy´rl » ¶Ù=oÕµF ³ o+;vyxr"=0;@+<= \û ÝåU'0T$& #"= B ) TVA #­ * Ú #0;<= #.S4 ¼ N ¼ VF"- d ) @+,/ ?<&4Y6X0LG+,[G+0 ) $&, * ,[æ547<¤úlµ8x-Ù=oR # */) Q;47<G+ aB±lµx=Ù=o Ý^ 0;.10úXlµx » oj¼°FVL #D$&07+2# ?"RúXljµ8x-Ù=o½¼°6A47<-4Zµ¾cFV²D4\B ) ª6A4T<&4\Ùp·FVL"-,zúXljµ8xeDop ?$-0;+2? #"µ cFVzµylhvXomprµylo¬p·FN¢lOP,úXljµ8x » oR Y #D$-0;+2# ?"£úXlµx=Ù=oj¼°`6A47<&4ZÙRÜ » ²­µ¾cFN|o.y4` #2 ) 472#,5'0;­:T47<=,§472?,/0;+4 * 4T"-0L2#,§4TGA4Y4YúXlµx=Ù=oR #" il!)]µFºo FpÙC']µHx µyl?omprPx Ù%ZµlLo´)]µF§lLompr Ý"=$&41 ?2 ) 472?,¨'0;­2#072?0;+G+,[2#,/07+ #",[+,/2?,§4 * #" iµylLomp » x µFhlPompfÙ%`Û$)$&,[ #+ ) A4 ') F,/254Z"=0 */) 2#,¨'07Cµylwvyx-Ù=oNW4T<&4­ÙpãFVµylhvyx&Do ?"$&4 * B ) 3l()µ o pãFV #"G+ #2?,/<)]µ p2?0;+"%$e47D$& TV6z #<-02#0;¶µ lAxeDopãFV$& ?+ #F0;"_µ cFV½µylwvyxeDomp » Nmå¢01d+4¨²b6 ) D$&0;"f2#07ॠ) QD47GF0;"fG+ \>6A47<-4`Ùp·FN¡ dF0;<&4±@+,[ #8V"=,mµylhv½¨x-Ù=o¢p l ?"\GF #2#,[<ªv½\ #"2#07ॠ) QD47GF0G+ 1­6A4T<&4úXljµ8x-Ù=o-oKV86z0;< * 4 ) +,/2?,/GA47GCG+ * 4"-0 *[) 2#,5'0;8VmµÃLlwv½¨x-Ù=o¾p²E6X0LG+ ?.S0;" JP6+<= #"&4T<1 * 2?0;ॠ) D$&0G+ >6 ) D$&0;"`2?0;ॠ) Q;47G+0;"Z2?0;.10 ) 2#07ॠ) D$&0SG+ 2 ) <%:74T"£G+ * 4 g 07<-.S4`vlwÙ=oNRmD$-0;+2# ?"ª"=,¬ä> #" ) C6 ) D$&012#0;ॠ) QD47G+01G+ ½VA6+47<&4`Ù¬p » V$& ?+ #.10;" * 4T""-,/Q ) ,/ ?$- #"f6z0;"-,[@+, * ,/GA47GF #" i

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ÎDÏLÏÅÍУÑ_ÉÅWÉÍfУҬÓÓËÔWÉÅWÉÕÒmÐYÊÒ¬ÓCÇzÆffÆLÆffÉ ÆÓbÔËÐZÅWÉÍÐªÒ¬Ó ¼;¼ Ãx

t0 1

A

D

E B

C

tt

0 1ü,/Q+N ¼ N c_ Õ¥-4T.S0;"]4 *R* #2$&0;<`4TA4 * ,[æ547<\254TGA4U2547"=0U²E:7 #<`B ) +,[+Q ) +03 ?"Z6z0;"-,[@ * Ul6A47<-43 i úXljµ8x-ÙÕKo`p ²D4±B ) Y #"ª2#0;ॠ) QD47G+0G+ Z­6A4T<&4> ?"- gh) +2?,/0;+4 * %z6A47<-4]B i µylAx-Ù ofpFVX6X0;<£2#0;D$-,/ ) ,/G+47G8V6z #<=0µylAx-Ù=omp » oNm #.10;"a$&47.Z@+,' #bB ) "=,úXljµ8x » oR FVF #D$&0;F2# #" * 4I') +,[254Z6z0;"-,[@+, * ,/GA4TG± #"\ä`p¦N¬"=$-0S6F< ) ?@A4Ulh@½oNl25oOP,+0 dA45² 6 ) D$-0;"2?0;ॠ) Q;47G+0;" G+ 6+47<&4úXljµ½o $e47.16z0P2?0 * 0;" d+4¨² 6A47<&4 úXljµ½or st ljµ+³´Iµ ³o+;v6A47<&4jf6X ?B ) ä+0+N¬ * <- ?"=$-0SG+ * 47<-Q ) .S ?D$&0Y #"¢ #D$&07+2# ?"£2?0;.S0147D$- #"#N

» NÖN|cZNçbè ú ð1õ3ò3÷5ö¢÷#ð1õUîñãñô®ûHö¢÷#îõ±ìî_ñ üUøZíø \øZím÷#ø]ñ ó;ô3õ3ö£÷#ð1õUîñîõ ù#øõUð1ímï ø þ »^ 0;+"=,/G+ ?<- ?.S0;"¢47dF0;<&4` *8gh) +2#,[0;A4 *i

kln x Ý5Ý5Ý x&nG7omp qYst u lhvyx-n x Ý#Ý5Ý x&n7Lx&nL x Ý5Ý5Ý x&nL o+;v2#07n7ÕlLop·¶VAn7Õlomp·¶N u #"¢+ͨN¤4Z:L,/.10;" #­ * ¥= #.16 * 0 » N ¼ B ) ikml ´3oW!kl]o¬p·q st l u ´ u ­ o+;v´3»¼ qYst l u= ´ u= T´­ u ´ u ºo+;v

Page 225: Ize J. Cálculo de Variaciones

¼;¼ Ä ÅWÆ_ÇÈÉ#ʪË̬ÍÎDÏrÅÍЪÑ_ÉÅWÉÍЪҬÓCÓWËÔWÉÅWÉÕÒmÐYÊÒ¬Ó´AlK« « o qYst l ´ o+Dv Ý

¡ 6 * ,/2#47+G+0` * $& ?0;<- ?.>4 » N » :; #.10;"¢B ) "-,n #" ) ­.'9 +,/.10Z ?$-0;+2? #"ý ³ kml]o#lxN3oR ²YB ) ª"=, ý ³#kmlZo#lxNUoRµ´¶)K³ # ) >6 ) D$-0Z2#<#'9 $&,[2#0l>V #D$-0;+2# ?"H¿ #" ) >.'9 +,[.S0 * 0P2#4 * VD2?0;<- ?"-6z #2$&0±4 * 4Z+0;<-.S4U«« Nc_ .S47+ #<-4Y47f'4 * 0;QD4]$& ?+ #.10;" * 4Z2?0;+G+,[2#,5'0;b+ #2? #"&4T<-,§4 i

¤ r¥ m&¦¿©#!%= ³ m on¥ ¡Ì#ñ /®;/®pS¬ /$,5pOgJWOMD6<J½IFJ6ÅpÅ6<*3),¢QÔ8),5p"S-)*3,.MD6 um lwvyxN±lwvXoxN lhvXo=of/0O¡*,.-)/0-f476<JWO73L-N8«=3NON«=,.O8f/0-¤£:6<*,54<O8V[· ¥G¸ ¹¦%Å|V ¥ #SOP, um $&,/ ?+ ) C:T4 * 0;<f6+<=0;6+,[0>+ ?QD4T$&,:;0S #Uv½5VXG+,§47Q70;A4 * ,/æ54T< um 2? #<=254G+ Yv½\²3$-0;.S47< ) H,/+2?<- #.1 #D$-0® * 0L254 * ,/æ#47G+0±2? #<-2#4­GF 1v½\²3 # * 4G+,/<= #2?2#,¨'07HGF * :; ?2?$&07<6+<=0;6+,[02#07<-<- ?"-6z0;+G+,[ #D$& TNO ) 6z0;+GF<- #.10;"fB ) a) p u ?"¢GF ?~A+,[GA4Y6X0;"=,[$-,[:T4² ?"=$ ) G+,§47<= #.10;"f *8gh) F2#,/07A4 *jiúXl3op·q st l l´ ÔT´ ´ )2­ºo+;vG+0;FG+ \¿p u= V+·p u­ NW4T<&4_6z0PG+ ?<6+<=0;@A47<B ) úl3oj¼°FV76A47<&4Ô ¾c FV7+ #2? #"-,$e47.10;"W:7 #<B ) Hp·_ ?" * ') +,/2?0_.'9 +,[.S06A47<-4`úl3oR"-07@+<- \Z# Axe&p/Ca¨ Axe&VA0S ?CZa6z0;<6z #GA4Tæ#0;"?N.0;"f6 ) D$&07"¢2?<5'9 $&,/2?0;"G+ úXl3oKVF$&,[ #+ ?

ý úXlUopr6A47<-4Z$&0LGA49 #­ * ?"-6A4T2#,/0FV+ #"¢G+ #2?,/< iq st l l´ fT´ ´ )2­ºo+;vlGF0;+G+ "- ) "A'0 p V¥) p¦) oKN ^ 0;.10q) #"1+0"-,/FQ )+* 4T<#V½ #"> ³ lX¥= #<=2#,[2#,/0HZN N ¼ oNO ) 6z0;+GF<- #.10;"fB ) u #"¢+Í5VA6X0LG+ ?.S0;" ?D$&0;+2? #"¢,/D$& ?Q;<&4T<6X0;<6+47<=$- #" iq1st l2 ´¯f ;v l 3oW ;v l()l o-o+;vp·6A47<-4Z$&0LGA49 #­ * ?"-6A4T2#,/0FN

.y4` #2 ) 472#,5'0;­:T47<=,§472?,/0;+4 * ?"£ ?D$&0;+2? #" i

Page 226: Ize J. Cálculo de Variaciones

ÎDÏLÏÅÍУÑ_ÉÅWÉÍfУҬÓÓËÔWÉÅWÉÕÒmÐYÊÒ¬ÓCÇzÆffÆLÆffÉ ÆÓbÔËÐZÅWÉÍÐªÒ¬Ó ¼;¼ l!)2 o lh° o lrE o p2#07®lLop"loprPN

lU * 2#47"-0>G+ `Zf6X07<£6z #GA4Tæ#0;"?V * 4T"£2?0;+G+,[2#,/07+ #"£G+ ]2?0;D$&,/ ) ,[GA47GG+ G: #,/ ?<-"%$&<&4T"-"ª,[.S6 * ,/2#47bB ) #"¢ZR²7VA2#0;.10=) #"fG+ ~A+,/G+416z0;"-,$&,:74PV #"¢³5N o ¿©#!%z§+,53L-)pO8Ô±V;r-ªä-N8~ ¸ | ¸1 ¦ o 1-a>8),°J6@-NM);6M),$SO7/476<3,6M),.O7/=6<J*,.-)/0-;/=6Å8O7Jo;rM),$SO7/¯ /0O¡,1-)/$*,.MD6<p-)/$*C-¢ ( ¶lhvXo¡MNO7/>lLomp·F¶l=äomprF -N8£\³[OP #47µ x Ý5Ý5Ý xNkLV¶ï·"-0 *[) 2#,[0;+ ?"±2#0;ó¡ lLoCp PVÐ lPop lFx Ý5Ý5Ý x » x Ý5Ý#Ý xeDo­l » # * 4H,M7' #"-,[.>4gh) +2?,¨'0;AoN]l ^ 0;U ?"=$&47"_2#07+G+,/2?,/0;F #"#V * 47"¢"-0 */) 2?,/0;+ ?"ª"=0;34 ) $-0;.I'4T$&,[2547.1 #D$& * ,/+ #4 * .1 #D$& ,[+G+ #6z #LMG+,[ #D$& ?"¢2? #<=254SGF AV+:7 #<f2 ) 4 * B ) ,/ ?< * ,[@+<-0`G+ ?2 ) 472?,/0;F #"fG+, g ?<- ?+2#,/4 * ?"#N|oOP #4 p l x Ý5Ý5Ý xNkDo * 4E.S4T$&<=,/æ­2 ) ²;47"b2#0 *[) .S+47">"=0;x x Ý5Ý5Ý xNkPN("- */* 4T.>4 * 4qbå&¨ê⪩« ä<è­ßå¬bBàLè­¨ åGæmG+ * "=,/"%$& #.S4FN>mD$-0;+2# ?"]$&0LGA4b"-0 *[) 2#,5'0;H4 * 4b ?2 ) 4T2#,¨'07:T47<-,/472#,[0;A4 * 2#0;lLop FV lLo£p ´¬V8"- ` ?"-2?<-,/@z i lhvXo¢p lhvXo´¤V8²;4±B ) S lLo£pú­²C6X0;< * 4 ) F,/2#,[GA47G34 * 6+<=0;@ * ?.>4SG+ :T4 * 07<- #",[+,/2?,§4 * #"?Ndzçê¶æzç2¨ åGè­¨CçäàNáªéVçÌè®Ìä7ãåßç9ßàªá⯢á ÈçÌæzç2áDâG+ $zl=äLomp·FN¤ r¥ m&¦Z¿©#Wó% 6 ( §r,) -N8k1-i-³/$,16g«$O8),5*,5476>õ u -N8_lÍ)FÔ-)/$*CO7/0MN-N8Vs¢úXl3o¶¼r«6<36¡ ¾cF8),Põ¡8SO7JWOk8),P/0Oki6<õª«=;m/$*CO8¡MNO7/6;:61O8+1-£-)/H Axe& [U ( §r,­ -N8Ô;/pS¬ /$,5pO2JWOMD6<J^F-)/$*O7/0MN-N8Vs°) /0Og*,.-)/0-Ô476<JWO73L-N8³«=3NON«=,.O8R/0-.:6<*,54<O8+-)/CAxe&Fml u. o u¸ prFõ2/0Ofi6<õЫ=;/$*CO8ªMNO7/6;:61O8R1-¤®-)/bAxeà[2¯ ô 3LO<UN6<3L-)pO8R6<J :O+p S678nQ;$-)3*-ªMNO7/3N-N8C«$-NM)*CO6¢-N8)*C-½«=;m/$*CO<s¶8),úXl3oR T-)/$*O7/0MN-N8+/0Ogi6<õ+«=;/$*O8¡MNO7/6;<:61O8l1-£-)/Axeà[ (M ( §r,1) -N821-i-³/$,16+«$O8),5*,54767FÒ-N8ª8O7Jo;$M),$SO7/1-lJ678k-NM);6M),.O7/0-N8g1- ñ ;JW-)3lõ@/0O¡i6<õª«=;m/$*CO89MNO7/M£6;<:61O821-£-)/ Axe&zFÐ-)/$*O7/0MN-N8¶¨-N8+;/KpS¬ /$,5pO9JWOMD6<Jn1-kmlZok-)/J6/0O73pÅ6­«W« [l * 4å¢07$e4±FN » GA4T<- #.10;"07$-<&4±G+ ?.S07"=$&<-472#,5'0;8V8@+47"&47G+4b ? * 4S$- #0;<#'9 4±G+ `2#47.S6z0;" i G+ ` ?"=$& Y.10PGF0 ** #2$&0;<_6z0LG+<A'4GA47<="- Y2 ) #D$e4bG+ YB ) * 0;"4T<-Q ) .1 #D$-0;"B ) 16+47<- ?2# #H"-,[.S6 * #"ª # ) A4G+,/.1 #F"-,¨'07+0 * 0±"-0; ?3.!'47"ªG+,/.1 #+"=,/0;F #"ª²B ) Y6+<- ?"- #D$&47+G+0 * 4T"_2#0;"-47"G+ `.S47+ ?<&4 ) 6z0P2?0±G+,["=$&,[D$e4l ? * #"=6A472#,[0G+ gh) +2?,/0;F #" 6X0;<£6X ?GA47æ?0;"_² ) "-47+G+0.I'47" * 4S$& ?0;<-,/4>G+ Z ?2 ) 4T2#,/07+ #"G+, g #<= #+2?,§4 * #"eo) +016 ) ?G+ */* #QD47<a4 ) A41GF #.10;"=$-<&472?,¨'0;b.I'47" * #QD47D$- 0S@+,[ #­4`<- ?" )+* $e47G+07"¢.I'47"Q; ?+ #<-4 * ?"eoKN· ¥G¸ ¹¦"%Ál4DoO ) 6z0;+Q;47.S07"\B ) 1+0dA45²6 ) D$&0;"2#0;¦¥ ) QD47G+07"\ ?°Axe&jV #"]GF #2#,[<G+ ?$zlwvXo¾p 6A47<-4ZvU #E½x&eNOP #4]² ) A41.S4T$-<-,/æª"-0 *[) 2#,5'0;G+ * 6F<-0;@ * #.S4 i

]l()n²¢T´Á ²_o ;´¯a²¢7´Ân²·pr

Page 227: Ize J. Cálculo de Variaciones

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07<`2#0;D$&,[ ) ,/GA4TG2#0;¯<- ?"-6z #2$&04 * 0;"ZG+4T$&0;"`,[+,/2?,§4 * #"?VW"-,;` ?"`6X ?B ) Lä+0+VmG+ $³²`lwvXoZ #"Z:; ?2#,[+0G+ G+ $zlhvXoR²;VA2? #<=254YG+ \½V * 47"2#0 */) .1A47"G+ n²ZlhvXoa"=0; * ,/+ #4 * .1 #D$- _,/+GF #6z #+G+,[ #D$& ?"#N¡ dF0;<&416A47<&4Y6+<=0;@A47<¢B ) úXl3oм1B ) #<- ?.S07"£ ?"-2#<=,/@+,[<f * ,/D$& ?Q;<&47FG+0S2?0;.10 ) 2 ) 4TG+<&47GF03l2?0;.10 # ) A4YG+,[.S ?+"-,5'0;½oNW47<-4Y ?"=$-0>GA47<= #.10;" * "=,/Q ) ,/ #D$- 47<-Q ) .S ?$-0 iOP #4´ ) A4Y.>4T$-<-,[æ_"-,[.¯' $&<-,[254FVP #D$&07+2# ?" iq st l ´ 3oÕK;vpr\pq st l ´ro ´ ¼ ´ o+Dv²D4YB ) 2lLopq¶lomprZ²*´ pw´ N

¡ GF #.I'47"2#0;.10]) pµ) ?"G+ ?~AF,/GA46z0;"-,$&,[:T4FV) p·¶ ¹¸ ¶G+0;+G+ n¶ #"0;<%$&0;Q70;A4 * V2#07.S6 ) #"=$&4G+ * 0;"f:; #2$&0;<= #"6+<=0;6+,[0;"£GF F)1V½²+¸· #"_GF,§47Q;07A4 * Nf0PGF #.10;"ªG+ ~A+,[<ª #D$-0;+2# ?"º)`׳Ôc¶H¹¸ ׳ ¶N^ * 4T<&47.1 #D$& 1l!)`׳eo#l()Z׳op)YVM)Z׳£ ?"¢"=,/.¯' ?$-<-,/2#4Z6X07"-,[$-,[:T4FND$&0;+2? #" i

úXl3opq st l-l() ׳ o ) ׳ T´ ¼ l() ׳ o ) ׳ ´£ ´ ´ro´ ¼ l!) ׳ o ) ׳ ´ l3o+;vG+0;FG+ b47Q;<= #QD47.10;"]U4UúXl3o²¶G+0;FG+ ±"- ) "A'03 * d+ #2ed+0B ) B¢zÔ pÓ z¶­­xW"=,/ #FG+03 #"%$e4')+* $-,/.S4 KJF6F<- #"=,¨'0; ) ­_') .1 #<-0Z²±6X0;< * 0Z$e47D$&0Y,[Q ) 4 * 4`" ) $-<&47+"=6 ) ?"=$e4PN

úXl3op q1st [l!) ׳ ´,) ׳ l|´ ´Á o3o l!) ׳ ´,) ׳ l|´ ´Á o3o]l() ׳ l|´ ´Á o3o l!) ׳ l|´ ´¯ o3o8´ ´ ´ l¦;v Ý

Ú #+G+<= #.10;" ) ­2 ) 47GF<&47G+01"-, i l(´ ´Âl°l|´ ´¯ o ) l(´ ´¯ oUop·6A47<-4Z$&0LG+0ÅENm"¢GF #2#,[< i

´ ´Âp l|´ ´ÁoC) l|´ ´Á olh²;41B ) G´qbp´¿o0Y@+,[ # i´r´») ´ E¹) ´ ´ l) o ´Âr) p Ý

Page 228: Ize J. Cálculo de Variaciones

ÎDÏLÏÅÍУÑ_ÉÅWÉÍfУҬÓÓËÔWÉÅWÉÕÒmÐYÊÒ¬ÓCÇzÆffÆLÆffÉ ÆÓbÔËÐZÅWÉÍÐªÒ¬Ó ¼;¼ º"=$&41 ?" ) A41 ?2 ) 4T2#,¨'07G+, g #<= #+2?,§4 * GF Wf,/2?254T$-,jNW4T<&4Y<- #"=0 * :7 #< * 4`"- #"=2#<=,/@z G´ pw¼j² RG+0;+G+ ¼²½²¿"=0;­6X0;<fG+ $& ?<-.1,/A47<?ND$&0;+2? #" i

´ pw¼ ² ¼n² ² ² Ýlc_ ?<-,:74T+G+0²º² ¤pú½xA"- 0;@P$&,/ ?+ 1l² -o p(º² ² ² Ý o¾ )+* $&,[6 * ,[2547+GF0 * 4Z ?2 ) 472?,¨'0;­G+ aWf,/2#2#4T$&,l6z0;< * 4ZG+ ?<- ?2edA4Do6X0;<²("= 0;@F$&,[ #+ i "- \0;@F$-,/ #F i¼ ¼j² ² 0¼j² ) ¼r) ¼ 0¼j² l¹) o ²Á´Ân² ) ²p¼\) ¼´·lrE¹) o² ¼F² l²¢7´,) ¼ ´rlh¹) o ²o¬p· Ý

07+QD47.10;"¼¿pA)Sl²¢7´rl) o ²_o¬p(]l()n²£7´¯ ²_o Ý

D$&0;+2? #" i¼\) ¼°´·lr) o²prp(]l()n² o ² ² ´Áº² ´¯¹) ²Á´Ân² ¹) ²pZl!)F²¢o l oÕe²Á´rl° o²¢7´Ân²pZl!)F²¢7´Á ²_oÕ7´Áº²£7´ÁF²·p· Ý

"fG+ #2?,/<#V¿² #"f"-0 *[) 2#,5'0;bG+ * 4Z ?2 ) 472?,¨'0;b:T47<-,/472#,[0;A4 * ND$&0;+2? #"_6+47<&416X0LG+ #<£<- ?Q;<- ?"&47<=+0;"#VX+ #2? #"=,[$e4T.S0;"f:; ?<_B ) * 41 ?2 ) 4T2#,¨'07C:T47<-,/472#,[0;A4 * $-,/ ?+ ) +4S"-0àM*/) 2#,¨'07½²]V½,[:7 #<%$&,/@ * _ ?¶Axe&jVA$e4 * B ) ´ p¼j² mp(]l()n² ´¯¶S²_o² "= 54Y"-,[.¯' $&<-,[254 NÚ0;.S ?.S07"T²Á2?0;.10ª4 * 6F<-,/F2#,/6F,/0G+ * 4f6+< ) #@A4&%7+07$& ?.S07"B ) ´ "&4à$&,/" g 472# ¬4 * 4 #2 ) 472#,5'0;\G+ Wf,/2?254T$-, i

´rPpw´») ´ ´·l-lh¹) o´ ´´ l) o lr¶) o-ol<= #Q;<= #"-47+G+0;"= d+47"=$&4 * 4Z #2 ) 472#,5'0;½oN^ 0;.10ZYVP) ¶VÀ)("=0;±"-,[.¯' $&<=,/254T"#VÀ´qb #"¬$e47.Z@F,' #S"-0 *[) 2#,5'0;>GF ª #"%$e4] ?2 ) 4T2#,¨'07lh$-0;.>4T+G+0\" )$&<-47+"=6 ) #"%$e4DoKN ¡ G+ #.I'47" ?"=$e4Y #2 ) 472?,¨'0;b$&,[ #+ ) A4`"=0 */) 2?,¨'0;') F,/254`"=,"- GA42´ lLoN

Page 229: Ize J. Cálculo de Variaciones

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´ lLopA)>lhú´) o%úPÛ$ p(A)¢Û$mlÁ´Á omp´ lLo Ý07< * 0¢$&47D$&0@´ pw´ NWúXl3o8 #"W ?D$&0;+2? #"W£"A'0 * 0¢"-,p)2 pZl(´ ´] oNW #<-0£2#0;.10RlLomp·FV* 4I') F,/254`"=0 */) 2?,¨'0; ?"¶pFNå£0T$& #.10;" ?$-0;+2? #"ªB ) "=,+0Sd+4¨²±6 ) D$&0;"¢2#07ॠ) QD47GF0;"fG+ \S #E½x&eVA$e47.16z0P2?0 * 0`dA47@F<A'4`6A47<-4

úXlµzoW q st l ´ o+;Ù"-,R ?"¤$e4T±6X ?B ) ä+0ZB ) )½-úY"-,/Q ) £6z0;"=,[$&,:T4\G+ ~A+,[GA4FNc_ * d+ ?2ed+0`B ) ª ?"=$- gh) +2?,/0;A4 * GF #@z ª"- ?<6z0;"-,$&,[:70+VF2?0;+2 */) ,[.S0;" * 4Z6+< ) #@A4YG+ Sl25o6+47<&4 gh) +2?,/0;F #"fB ) "=0;`R6X07<6X ?GA47æ#07"#NW4T<&4±6z0PG+ ?<ª6+<=0;@A47< * 0;"¢2#0;D:7 #<-"=0;"+06z0PG+ ?.S07" ) "-47<_ * 47<-Q ) .S ?D$&0±G+ ]2#0;D$&,[ ) 472#,5'0;UB ) `G+,5'0± *<- ?" )+* $e47GF0H6+47<&4Tï p » NÁ g #2$&0+V¤"-,fG+ ?$z±lhvXoZG+ ?$- #<=.S,[A4 * 0;"`6 ) $-0;"12#0;ॠ) QD47G+0;"Y2?0;.S0" ) "2# ?<-0;"?V+ # ) 6 ) D$&0`2#07ॠ) QD47GF0­ä1+0`F #2# ?"&47<=,§47.1 #D$- 1lG+ $zl=äLo-o ¾p·FVP #"G+ ?2#,/<RB ) +0`6X0LG+ ?.S0;"<- ?6+<- ?"- ?$&47<_ * 2?0;ॠ) D$&0>G+ ]6 ) $-0;"ª2?0;ॠ) Q;47G+0;"?VX6A4T<&4S * 6F<-0;@ * #.S41GF #6z #+G+,[ #D$& ]G+ * 6A47<+'47.S $&<=0ÙKVL2#0;.102 ) <=:T47"?N¢l0;<¤ ¥= #.16 * 0 ?±AVPGF ?$zlPomprPVlÂÁ#Ù%lLo-o prFN|oZ0;<¬07$&<-4\6+47<=$- 7V"-,z #"=6X ?2#,[~PM254T.S0;"½lhvXo¤$&4 * B ) ll=äompFV * 07"a2# ?<-0;"fG+ llwÙKx-vXo¤"- ?<5'9 478V+ ?­ * .S ¥=0;<aG+ * 0;"a2#47"-0;"?V$ï32 ) <%:T47"²+0 ) +41"=0 * 4FN¡ 6F<-0¨:; ?2edA47.10;" ?"=$&4­G+,~A2 )F* $e4TGH6+47<&4GA47< ) <= #" )F* $e4TG+0b.!'47"Q7 #+ ?<&4 *i * $& ?0;<- ?.>4G+ * '9 +GF,/2# ]G+ ¾0;<="- TNªlc47<- ?.S07"07$&<-4YGF #.10;"=$-<&472?,¨'0;b # * 4`å¢07$e4F+N » V ) "&47+GF0 * 4]$- #0;<=,§4ZG+ 2#47.S6z0;"?N o * #"-6+472#,[0]û G+ gh) +2#,[0;+ #"a`¬6z0;<R6z #G+47æ#0;"R #D$&<= `²>¨V+2#07$-,/ ) 47"¤²S2?0;BlPopqlomp·FV2#07+"-,[G+ #<-47.S07" *8gh) +2?,/0;+4 *ji

úXl3op q1st l l´ ÔT´ | ´ Ã)2­ºo+;v2#07½)SlhvXoRG+ ~A+,[GA416z0;"=,[$&,:T4FVF "=,/.¯' ?$-<-,/2#4FVFVM)YV½"-0;ZKN.y4`G+ #<=,[:T47GA4`G+ \üA<L' ?2ed+ ?$£G+ úXl3oR ?"¢GF0;":; #2? #" * 4 g 0;<=.>4Z@+, * ,[+ 54 *i

úXl­xNopqYst l l ´ f­T´ ´ Ä)lºo+;v ÝW4T<&4Y2547GA49Þ_ ?$-<- \S²b£G+ ?~AF,/<- ?.S07" * 4 g 0;<-.S4Z@+, * ,/F 54 *i

ú Ý l­xNomp q Ýt l 2 ´ fT´ | ´ l)l o+;v

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Page 232: Ize J. Cálculo de Variaciones

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Page 234: Ize J. Cálculo de Variaciones

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Page 236: Ize J. Cálculo de Variaciones

ÎDÏLÏÅÍУÑ_ÉÅWÉÍfУҬÓÓËÔWÉÅWÉÕÒmÐYÊÒ¬ÓCÇzÆffÆLÆffÉ ÆÓbÔËÐZÅWÉÍÐªÒ¬Ó ¼ )SlÞ?o ä lÞ?oVR²;4¶B ) \)>lÞ#o` #">,/D:; ?<=$&,[@ * TN0LG+ #.10;">2?0;.16 * $e47<1 #"%$& U2?0;ॠ) D$&0EG+ C:; ?2?$-0;<- ?"±6X07<:; ?2?$-0;<- ?"R´ é \ Ý b x Ý#Ý5Ý xN´°PVF0;@F$- #+,[ #+GF0 ) A4`.>4à$&<-,[æ£,[D:; #<%$&,[@ * _2 ) ²D4T"zÚ8lÞ?oR6+<-,[.S ?<&47"¤2#0 *[) .1A47"a"=0;)SlÞ?o ä lÞ?oN.y4.S4T$&<=,/æa,[:7 #<="&4\$- #+GF<&42#0;.10aÚ8lÞ?o6+<-,[.S ?<-0;"<= #+Q * 0;+ ?" * 0;":7 #2$&0;<= #" x Ý5Ý#Ý xN é \ Ý b$e4 * #"B ) 2 )SlÞ?o ä lÞ?opê)¨NOP #4@Þ #D$&<= 2Þª²Þl NROL 549gTlhvXo ) A4 gh) F2#,¨'07Elh #ëÍ Î oa³f$e4 * B ) i

gàlhvXo³cY6A47<-4`v®vµÞjxNgàlÞKop·FxNgàlÞ#op"2 ÝWJL$& #FG+ #<= #.10;" ä+£²l£2#0;.101Y ?D$&<- 2Þ£²BÞ N ® @F$- #+ ?.S0;"f ?$-0;+2? #"¢ * ?.S ?D$&0;"fG+ û Ý ¹ Nú Ý ¹ l a xN+?opú Ý l a xN+oR #"¢+ #QD4à$&,[:T4ZG+ ~A+,[GA4FNú Ý ¹ l a x ä+?opú Ý l a x ä+o¬p]p ú Ý ¹ l ä a x älo Ýú Ý ¹ l a xNgop" ÝÝ l 2 a ´ Ô a ´ Э a ´ )l a o+;vl6z0;< * 47"G+ ?~++,/2?,/0;+ ?"fG+ l a ²goN

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Î ú Ý l a xN+o D )l a lÞ#oALa )2 lÞ?o ¼ ú´-³eú Ý ¹ l a xN2?o ϲD4YB ) \ú Ý ¹ l ä a Û¬mD a x älÛ$L2#omp¿ ¼ ê a ´-³eú Ý ¹ l a xN2#oKN^ * 4T<&47.1 #D$& TVX6+47<&42ª6X ?B ) ä+0+VX #"-4>.>4à$&<-,[æ"= #<+'4>+ #Q;4T$&,:741G+ ?~++,/GA4PNª0;< * 0S$e4T$-0 * 0;"Rî&lÞ?o´ÚlÞ?o:; ?2?$-0;<- ?"aB ) _d+ ?.S0;"¬ #F2#0;D$&<-47G+0Z"=0; * ,/+ #4 * .1 #D$& ,[+G+ ?6X ?+G+,/ ?D$& #"¤²S6z #<%$& ?+ #2? #b4\û Ý ¹ VP ?"RG+ ?2#,[<îelÞ oaîelÞ?oy´Ú8lÞ#oKNÚ #+ ?.S0;"f ?$-0;+2? #"Ôî&lÞ ompxî&lÞ?oy´Ú8lÞ#oa6A47<-4Z$&0LG+0ÅÞ ?$-<- 2Þª²BÞNl N^ 0;.10]d+ ?.S07"¤ ?"-2#07Q;,/G+0 * 4" ) @zG+,[:L,["-,¨'07Þ x Ý5Ý5Ý xNޤɢVPG+ £$&4 * .S0LG+0\B ) ª+0]dA45²S.I'47"¬G+ ) ±"&4 * $&0\6X07<" ) @+,/D$& ?<=:T4 * 0UÞjxNÞNl jVA²b2?0;.S01254TGA4S6 ) $-0±2#07ॠ) QD47GF0SG+ ]AVz ?D$&<- gÞ²¢ÞNl V½6+<-0LG ) 2? ) "&4 * $&0+V

Page 237: Ize J. Cálculo de Variaciones

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» NÖN|cZNÈH o ¦T~ ¥G¸ ¹¦ o ° =¥ ¦ o m+Å o %¤4d+ #.10;"_6F<-0;@A4TG+0l4Do¢²¯l@½oKN ^ 0;.10dA45² ) Hª') .1 #<=0>~A+,$&0G+ `6 ) $-0;"2#0;¦¥ ) QD47G+07"#Vy6X0LG+ ?.S0;" #"=2#0;Q7 #<Þ x Ý#Ý5Ý xLÞÉfVT$&4 * ?"B ) R+,[+Q ) +0fG+ ¬ *[* 0;""= 54 ) ]6 ) $-0£2?0;ॠ) Q;47G+0¢G+ Rª²_B ) R #>Þ xLÞl +0dA45²D4.I'47"WGF ) Y"&4 * $&0_G+ jîelÞ?oalh #"G+ ?2#,/< ) Y6 ) D$&02?0;ॠ) QD4TG+0G+ fLoNÚ ?+G+<- ?.S07"Pîelo¬pxî&lÂyoT´ öl"-4 * $-0;"eo¤²l2#07.S09îelÂomp·Dojî&lo¤p ö Ú8lÞNhoKN

» NÖN|cZNçbèCì íô3õ±ìWðSñöfð1õîô+ïYøZòUð1ñÌð ö¢ø]ï üEð1ñöfîõ±ìíø]ù#îñ¢è± #"%$e4Z"= #2#2?,¨'0;±:7 #<= #.10;" ) +4Z.>47F #<&4\2#0;D:7 #+,[ #D$& G+ £2547<-472?$- #<=,/æ54T< * 07"¤6 ) $-0;"a2?0;ॠ) QD4TG+0;"lh6X07<* 0`.1 #F0;"f * 6+<=,/.1 #<-0DoNWf #2#07<-G+ ?.S0;"fB ) 6A47<-41 *8gh) +2?,/0;A4 *

kl]o¬p q st u lhvyxNmxN o+;v2#07k±lLo¬p ]V<±lompr>V ) 2547.16X0¢2# ?D$&<&4 * Và2#0; g 0P2?0¢ ?l½x-oVT #" ) A4 g 47.S, * ,/4j±lwvyxÞá x Ý5Ý5Ý xÞáV;oG+ 6 ) D$-0;"£2?<5'9 $&,/2?0;"lh2#0;­"-0 * 0űlLomp_oR$e4 * B ) * 4T".>4T$-<-,[2# #"um lhvyxNlDoxLlDo-ol ÈAn ÈáN o l½x&Fx Ý5Ý5Ý xeDol ÈAn7È­áN o lhvyxeFx Ý5Ý5Ý x&Do2#07UvZ¼ AV8"-0;,[:7 #<%$&,/@ * #"?N\ * 2547+GF,/GA4T$-04± JL$-<- #.10 #"+>lhvyxePx Ý5Ý5Ý xeDoN ¡ B ) '9 " ) 6X0;F #.10;"B ) um lhvyxNlwvyxeDoKxN lhvyxeDo=of ?"ªG+ ~A+,[GA4S6z0;"-,$&,:74PNlhm ?< * 41GF ?~A+,[2#,5'0;]N|ÄFN ¼ N|o]W¢ ?2#0;<=G+ #.10;"ªB ) \$&0TMGA4 g 47.1, * ,§4G+ "=0 */) 2?,/0;F #"]2?0;±l½xá x Ý5Ý5Ý xÞáVDo`p]V¢lÈAn Û<È­áN#oCàlAxeFx Ý#Ý5Ý xe;o\,/D:7 #<=$-,/@ * 7V ?" ) 254T.S6z02# ?$-<&4 * l!. #.S4>ZNÖÄFN oª6A47<&4±v2# ?<-254bG+ YANZÚ47.Z@+,' #d+ ?.S0;"_:L,["=$&0bB ) Y6A47<-4>vE:7 #2#,[+0­4"-,/ ?.S6F<- > KJP,/"=$- ) 2#47.S6z0C2? #D$&<-4 * lh"-0 *[) 2#,5'0;EG+ * 4C ?2 ) 472?,¨'0;EG+ ] ª4T.S, * $&07PMCñ;472?0;@+,§o£²EB ) ±"= 6 ) ?G+ \2#0;+"%$&< ) ,/< KJF6 * '9 2?,[$&47.S ?D$& i +07$e4`ZNÖÄFN N

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Page 238: Ize J. Cálculo de Variaciones

ÎDÏòLÏIÇ\ËÐYÊRÍfÓÅÍкóXËôÆ_ÑYÍfÓ0õ·ÅWÆf CÇ¢ÍfÓEÅÒmÐYÊfÆ_ÌyÒ¬ÓÏ ¼ º· ¥G¸ ¹¦Z%`N¬OP #49lhvyxÞá x Ý5Ý5Ý xá`;o¤ * 2547.16z0`2# #D$-<&4 * V+2#0;¢>lhvyxeFx Ý5Ý5Ý x&DoPcòz¨lhvXoR * 2547+GF,/GA4T$-0Y4 JL$-<- #.10+N¤OP #4Å­özlhvXo * 4Z.S4T$-<-,/æ`lÈAnGÛGÈáN?o#lwvyxeFx Ý5Ý#Ý xe;oN^ 0;.10 þþ à l u lhvyxN>lhvyxÞ÷moxL lwvyxÞ÷mo-o¤p u lhvyxNlwvyxÞ÷moxN lwvyxÞ÷mo-o6A47<&4`$&0LG+0±÷V½6z0PGF #.10;"£GF, g #<= #PM2#,/47< #"%$e47"f ?2 ) 472?,/0;F #"¢2#0;­<- ?"-6z #2?$-0>4=áVy²0;@F$- #+ ?.S07"#VA #½÷¶p i;v l u öª´ u \ö omp u= ­ö£´ u­ \ö #"fG+ ?2#,[< i

lh ­öª´)2ö oÕFp2­ö_´¯f\ö0Y@+,/ ?8Vl!)2ö oÕlh° oö lr¶ o­öZp #"fG+ ?2#,[<¢2#47GA4`2#0 */) .1A4`G+ lö1 #"f"-0 */) 2?,¨'0;G+ * 4Y ?2 ) 4T2#,¨'07b:T47<=,§472?,/0;+4 * lw:; #< * 4YGF ?~A+,[2#,5'0; N » oN^ 0;.10T±l½x÷¬o`p ]Vö½lAxeDoYpU²ö #" ) +4.S4T$-<-,/æS"-0 *[) 2#,5'0;¶G+ * 4C #2 ) 472?,¨'0;:T47<-,/472#,[0;A4 * V,/D:7 #<=$-,/@ * C6A4T<&4Ev :; ?2#,/F0l6z #<-0GF,/"=$-,/D$&0Do1G+ 3·l ?" ) A4¶.S4T$&<=,/æ gh) +GA4T.S ?$&4 * oKVÐö ö lLoS ?"$e4T.Z@+,' ?3"-0 *[) 2#,5'0;32#07:T4 * 0;<_ú #6A47<-4±" ) G+ #<=,[:T47GA4PNk­öXlwvXoªp lhvXo ö lLo1lh6X07< * 4 ) F,/2#,[GA47G½oG+0;FG+ blhvXo\ #" * 4.>4T$-<-,[æ gh) FGA47.1 #D$e4 * B ) ±²D4Ud+ ?.S07" ) "&4TG+0+N&ö½lwvXo #"`"=,/+Q )F* 47<\"-,R²"A'0 * 0U"=,lwvXo * 0` #"?Vz ?"fG+ #2?,/<f # * 0;"R6 ) D$&0;"¢2#07ॠ) QD47GF0;"fG+ \½N ¡ G+ #.I'47" #"%$&0;"¢6 ) D$-0;"f+0SGF #6z #+G+ ?CG+ * 4g 47.1, * ,§4llhvyxÞá x Ý5Ý5Ý xá`;oN » NÖN|cZN

P|V ¥ kn#!&%b * ¥= #.16 * 0]G+ * 47"aQ7 #0LGR' #"=,/254T""-0;@+<= ) A4` #" g #<&4`G+ <-47G+,[0 » V+ Õ¥= ?.S6 * 0 ¼ N V+"= #F2#0;D$&<+'0 ) A4 g 4T.S, * ,§4G+ \6 ) D$&07"¢2?<5'9 $&,/2?0;" iÀFl|y¤xN´Wop ¡ <=2?$&47 l-l´]2#0;"¨l|yUy ´ ¡ <-2#2?0;" l´ 2#07$zÀ o=o-o 2#07®´p lÀ ³ ´ "- ? ³ ¼ À Û¬Do ׳ "= # ³ À Ý<-0;@+47<B ) Ñ ~Ñ U p X l´oÕ"- ?yl|yEy o6A4T<&42?,/ ?<=$e4 gh) +2?,¨'0; X l´oKN07< * 0­$&47D$&0C * 6F<-,/.1 #<6 ) D$&0G+0;FG+ TÑ ~Ñ U pr` #"f6A47<-4ypwy ´0_N£lc_ ?"&47<=<-0 *[* 47<¤ * 2#07"- #F0SG+ * 4Y" ) .S4DoN

P|V ¥ kn#ò%±<-07@A47<\B ) * 4 g 47.1, * ,§4>nnø"´v ?" ) E2547.16X0­6A47<-4 * 6+<=0;@ * #.S4bGF * Õ¥= #.16 * 0¼ N » Nù ¦Ekn#!%\0LG+ #.10;"GA4T<¢0T$&<&4`G+ ?.S07"=$&<-472#,5'0;G+ * $- #0;<= #.S4 N ¼ lh+0YG+ * $- #0;<= #.S4YGF * '9 +GF,/2# 5oNO ) 6z0;+Q;47.S07"¢6F<-,/.1 #<=0SB ) gz£ #" ) U.'9 +,[.S0 * 0P2#4 * N ^ 0;+"=$-< ) ²;47.10;" ) C2547.16X012# ?D$&<&4 * >lhvyxÞ÷moV2#07 g 0L2#0> ?l½x-oV½2?0;.S01 # * 4Y+07$e4`ZNÖÄFN NVX ?"£G+ ?2#,[<#V * 4°') +,/2#41"=0 */) 2#,¨'078VA6A47<-41vU:7 #2#,[+04S½Vz4 *"-,["=$- #.S4 i

lhvXompú]lhvXo ±lPomp·

Page 239: Ize J. Cálculo de Variaciones

¼ D ÅWÆ_ÇÈÉ#ʪË̬ÍÎDÏrÅÍЪÑ_ÉÅWÉÍЪҬÓCÓWËÔWÉÅWÉÕÒmÐYÊÒ¬Óú¢hlwvXomp"í>lwvyxNmxÞú_o úZlPomp÷C´ lLo Ý¡ B ) '9* 4`.>4T$-<-,[æG+ * 4`+0T$e4ZZNÖÄFN ?" * 4Z,/G+ ?D$&,/G+47G²BÞ¢ #"fn lLoNûú°ö½lLompú½NOP,HÞS #"Y * 6F<-,/.1 #<]6 ) D$&02#07ॠ) QD47GF0HGF b½Vm2?0;ÞUܨVm #D$-0;+2# ?"­ö½l;o] #"Y"-,[+Q )+* 47<] ?Þ¦N¯OP #4Ò ) I:7 #2?$-0;<±6+<=0;6+,[0EG+ ®özlÞ¦xeDoKVa2#07özlÞàxe;oCÒ p PNOP #4 ä¶lwvXopë­ö½lhvyxe;oCÒmV6A47<-4Hv v ÞàVälwvXoRp¿S6A47<-41v"ÞàN¢D$&0;+2? #" ä¶lhvXo6X ?<=$& ?+ #2? `41û¶Vz2?0;.10>­ö½lhvyxeDo #" ) A41.>4T$-<-,[æ\"=0 */) 2#,¨'07G+ * 4¯ #2 ) 472#,5'0;·:T47<=,§472?,/0;+4 * Vf$e47.]@+,' ? * 0Á ?" älwvXoV ? ½xLÞ7R²°WÞàxe&N0;< * 0¯$e4T$-0ÂúXl ä3oEpä lÞ#o?ü) ä ´¯ ä Ý p·FN^ 0;.10`úXl3oRZ #bûV+ ?D$&0;+2? #" ä #" ) b.'9 +,/.10]GF \úS"-0;@F<- û i G+ ?@X "&4à$&,/" g 472# ?< * 4Z2?0;+G+,[2#,¨'07G+ \2#0;D$-,/ ) ,/G+47GbG+ a: #,[ #<="=$&<-47"-"]<) ä ´¯¶ ä Ý pPVA6X0;< * 0Y$&47D$&0 ä lÞ#ompò ö lÞ¦xeDoÒprFN¤ ?<-0 #D$-0;+2# ?" ä¶lhvXo³cFV76X0;< * 4 ) +,/2?,/GA47GG+ * 4ª"-0 *[) 2#,5'0;Z4 * 6+<=0;@ * ?.>4¢G+ R:T4 * 0;<- ?"W,/F,/2#,/4 * ?"y #ÅÞàN ^ 0;.10ä lLo`p ö lAxeDoÒ¬V² ö lAxeDo ?"`,/D:; ?<=$&,[@ * TV #D$&07+2# ?"=ÒpFV * 0U2 ) 4 * #" ) A432#0;D$&<-47G+,[2#2#,5'0;8Nå£0T$& #.10;"B ) \ #"%$& \47<-Q ) .S ?D$&0 gh) +2?,/0;A4Z$&47.Z@+,' #l"=0;@+<- ûHoR<= # ?.S6 * 47æ54T+G+0ÅözlwvyxeDoR6z0;<flhvXoKNm #.10;"a$&47.Z@+,' #B ) "-,úXlUoj¼!FV+ * 6 ) D$&01£$&47.Z@+,' # ?"¢ * ,[.S,[A47G+02#0;.10`6 ) D$&012#07ॠ) QD47GF0`G+ 3lh"-,+0YdA47@F<5'9 4 ) #bû $e4 * B ) \úXl3opDoN * ,/D:; ?<-"=0:L,/ ?+ SGF * "-,[Q ) ,/ ?D$& `<- ?" )+* $e47GF0­²G+ * 4­4T6 * ,[25472?,¨'0;3G+ * 0;"<- #" )+* $&47G+0;"G+ * 4b"=,/Q ) ,/ #D$- "- ?2#2?,¨'0;8N

¤ r¥ m&¦»k# % §r,z) -N81-i-³/$,16¢«$O8),5*,54767F u -N8>+ÍVFl½-N8®;/«=;m/$*COZM)3=S¬ *,.MNO 1-l]oNF-)/$*CO7/0MN-N8VsýlOgi6<õR«=;/$*O82MNO7/6;:61O8f1--)/UAxe&8),õ28SO7JWOl8),©½¡-N8 «6<3*C-+1-f;m/TMD6<pÔ«$OÅMN-)/$*36<JMNO7/+Q`OMNO-)/l]0#xN½àl\0eo=oNF°«6<36*OV1Oм!F°«$-D±V;r-0²/0O<[· ¥G¸ ¹¦%Så£0T$& #.10;"£6+<-,[.S ?<-0>B ) `2?0;.10>½ #"³_6z0PG+ ?.S07"ª JL$- #+G+ ?<+z2?0;.10r') +,[254S"=0 */) 2#,¨'07G+ * 4T" #2 ) 472#,[0;+ #"fG+ \ )+* #<f2#0;­2#07+G+,/2?,/0;F #"н¦lLoxL lPoR ?D$&<- ²bANOP, z #"6A47<=$- >G+ ) ¶2#47.S6z0­2# #D$-<&4 * 2#0; g 0P2?0C #°lS`#xN½¦l1eo-o #D$&07+2# ?"Z+0UdA45²H6 ) D$-0;"2#07ॠ) QD47GF0;"G+ ``, ?Á `#xe&jN@¤4±B ) ` #"%$& `d+ ?2ed+0b #",[+G+ #6z #FG+,/ ?$- `G+ * 2547.16X0FV86X0LG+ ?.S0;"" ) 6X0;F #<fB ) * 2547.16z0` #"¢"-0 *[) 2#,5'0;G+ i

lhvXompú]lhvXo >l&o¬pxz¨l]0&oú¢hlwvXomp"í>lwvyxNmxÞú_o ú]l5eompw÷C´\ l]0eo Ý^ * 4T<&47.1 #D$& * 4b"=0 */) 2#,¨'07>lhvyxÞ÷Wx&o ?"G+, g #<= #+2?,§47@ * Y #5?N±07< * 0b$&47D$&06A47<&4b254TGA4­v¼ C~T¥=0+V­ö½lhvyxeFx&Dop * ,[.özlhvyx&Fxeo_ #"\,/D:7 #<=$-,/@ * 7NSÚ ?+ #.10;" ) E2#47.S6z0b2# ?$-<&4 * "&4 * ,/ ?+G+0bG+ ClAxN½¦lLo=o2#07®­ö½lhvyxe;o,[D:; #<%$&,[@ * _6+47<&4`$&0LG+0ZvyVzSÜIv®v ¨N D:; ?<-"&4T.S ?$- 7VX * .S,["-.10147<-Q ) .S ?$-0S6F< ) ?@A4SB ) gözlhvyx&Fxeoa #"¢,/D:7 #<=$-,/@ * 6A47<&4YvT¼ ´´6A47<-4$&0LG+0FÐv¨l´oNm #<=012?0;.10 * 0;"¤6 ) D$&07"2#0;ॠ) QD47G+0;"R+0`6 ) ?G+ #b #"%$e47<2? #<=254bl * #.S4 N » oVP6X0LG+ ?.S0;"$&07.>47<³´U$&47S6z #B ) ä+0B ) 1«á´´YHlReo¨«+Ül.`GA47G+0\ #S *½* ?.>4 N » oKVXl6z0;<m Õ¥= #.16 * 02´ vxFÛ ¼²Y$&0;.S47<= #.10;"Ðv°.1,/l!eÛ<´¬x+Û ¼ oKN0;< * 0$e4T$-0Z+0Zd+4¨²16 ) $-0;"a2?0;ॠ) QD4TG+0;"RGF ¢E¤ #£\?xe&²B>lhvyxÞ÷Wx&o ?" ) 2#47.S6z0`2# ?D$&<&4 * N

Page 240: Ize J. Cálculo de Variaciones

ÎDÏòLÏIÇ\ËÐYÊRÍfÓÅÍкóXËôÆ_ÑYÍfÓ0õ·ÅWÆf CÇ¢ÍfÓEÅÒmÐYÊfÆ_ÌyÒ¬ÓÏ ¼ »» NÖN|cZN

P|V~ kn#!%fOP #41úXlhn+opx tþ l » ´¯n ³Koe׳ÛTn>DnXV+2?0;bn8lDop·@¼!FVAnlLopq´ ¼!FN.y4` #2 ) 472#,5'0;GF )+* #<GA4 i

nPn ´ » ´Án ³ p]pln ³ o Û ¼ ´ »7Ý07< * 0Z$e47D$-0+VA2#0; * 4T"2#0;+GF,/2#,[0;+ ?",/+,[2#,§4 * #" in ³ ´¯v ³ p ¼ Þev`´Á ³ '0 n ³ ´·lhvSZÞ#o ³ pxÞ ³ ´Â ³ Ýl47<&4 * 4Z"=0 */) 2#,¨'07BÞfp t x U x þ x³ tÚ #+ ?.S0;"R #D$&0;F2# #" ) A4 g 47.1, * ,§4_G+ £2;'9 <=2 )+* 0;"#VL2#07±2# ?$-<-0;"R #HlÞ¦xeDo²S6A47"&4T+G+0Z6z0;<lFxeLo i nlhvXoplD³´ ¼ ÞKv>Ev½³oe׳¨N

y

x

α ÿ

ü,/Q+NFN »^ 0;.10 Ñ ¸Ñ Ý lwvyxNÞ?o¬p°vXÛTnK¾pr6A47<&4\v¾prFVP+0ZdA45²S6 ) D$&07"2#0;ॠ) QD47G+0;"¤G+ Z ?H Fx&DX"-,0´ ¼°FN"%$e4g 47.1, * ,§4Y #" ) 2547.16z02# #D$-<&4 * 2?0; g 0L2#0 ?!lFxeLoZlw$e47.Z@F,' #3"= `6 ) ?G+ 1:7 #< #"%$&0bG+,[<- #2$e47.1 #D$- Z²D4B ) \6X0;<2#47GA4`6 ) D$-0Ulwvyx&n+oKVFv¼!`²±n>¼!FVA6+47"&4 ) +0`²"+'0 * 0 ) ­2;'9 <-2 )F* 0oKN¡ GF #.I'47"#V u p n ?l » ´7³o Í׳R¼!FVF6z0;< * 0]$e47D$-0 * 4]"=0 */) 2?,¨'0;> #" ) b.'9 +,[.S0 * 0L254 * ? * 4]+07<-.S4ZKNå£0T$& #.10;"B ) \"-,y]prFV * 4Y ?2 ) 4T2#,¨'07­GF )+* #<$&,[ #+ _$&47.Z@+,' # ) A4Y"-0 */) 2?,¨'0;v ³ ´Án ³ p ¼ ÞKv Ý ?<-01n plÞR¶vXo=ÛTnXV » ´Án ³ pxÞ ³ Ûàn ³ ²q t l » ´¯n ³Koe׳n ;vbp"Þ q t ;v¼ ÞKv>Ev ³ p »¼ * 0;Q½lwvl ¼ Þ¤¶vXo o¨« t p

Page 241: Ize J. Cálculo de Variaciones

¼ ¼ ÅWÆ_ÇÈÉ#ʪË̬ÍÎDÏrÅÍЪÑ_ÉÅWÉÍЪҬÓCÓWËÔWÉÅWÉÕÒmÐYÊÒ¬Ó #"fG+ ?2#,[<#VA *8gh) +2#,[0;A4 * F01 ?"=$A'41G+ ~A+,[G+0S6+47<&4` * 2#47+G+,[GA4T$-0S4`.'9 +,[.S0FNP|V~ kn#%­ * Õ¥= ?.S6 * 0 ¼ Nü+VP$-<&4T$&47.10;" * 0;"a6+<=0;@ * ?.>4T"aG+ g <-07$- #<-4 * ,/@F<- ª ?­½V */* ?QD47+GF0`4* 4Z ?2 ) 4T2#,¨'07­:T47<=,§472?,/0;A4 *ji

l!)]µ o p'µ µylo¬p·Fx&_µ lLo´Á`µylLompr Ý¡ dF0;<&4`@+,[ #­"-,8$& ?+ #.10;" ) A4 g 47.1, * ,§4\G+ "-0 *[) 2#,[0;+ #"GF il u¸ ¹ o P u¸ prPx uy¸ ¹ lAx&nlAxÞá¦oKx&nL/l½xáào=oprFx·n8l¦xáàop·G+ ?<-,[:T47FG+0 * 4\ #2 ) 472#,5'0;±G+ _ )F* #<R2#0;±<= #"=6X ?2?$&014Fá_0;@P$& #F #.10;"aB ) _nm ?""-0 *[) 2#,5'0;>GF * 4\ #2 ) 472#,5'0;:T47<-,/472#,[0;A4 * lh2#0;.10±4T$- #"&oNc_ ?<-,[:T47FG+0 * 4S2?0;+G+,[2#,¨'07CG+ ]$-<&47F"=:; ?<-"-4 * ,[GA47G i uy¸D ny´ u n p(FVX ?"G+ ?2#,/<y£n ´G)n5« t p·PNn8 #""-0 *[) 2#,5'0;GF * 4f ?2 ) 4T2#,¨'07\:T47<=,§472?,/0;A4 * 2#0;\" ) "W2#07+G+,/2?,/0;F #"yG+ g <-0;D$& ?<&4FN¡ GF #.I'47"Yl ¥= #<=2#,/2?,/0o * 07"ª6 ) D$-0;"2#0;¦¥ ) QD47G+07"_G+ `AV ?3 * "- #D$-,/G+0G+ * 4>G+ ~A+,[2#,¨'07 ¼ N ¼ V"=0;3 JF472KM$e4T.S ?$- * 0;"6 ) $-0;"fG+0;+GF \n#lwvyxeDoprPN

çbè ú ð1õ3ò3÷5ö¢÷#ð1õUîññô®ûHö£÷?îõ±ìîñãîõ ù#ø õUð1ímï ø þ m #<= #.10;"`~AA4 * .S ?D$& b2'0;.10H #!4 * Q ) F0;"`2547"=0;"1"- ­6 ) #GF ) "-47< * 432#0;D:; KJP,/GA47GIG+ u #ú 6A47<-46+<=0;@A47<B ) ±+0U"=0 * 47.1 #D$- > * 6 ) D$-032#<#'9 $&,[2#0zZ ?" ) .'9 +,[.S0b ? * 4­+0;<-.S4C l ?"]G+ ?2#,[<]6A47<-4« "½7« 6z #B ) äF0oY6z #<=0H$&47.Z@+,' #Á # * 43+0;<=.>4H l ?"SG+ ?2#,[<S6A4T<&4 ?° 2?0;« qz7« 6z #B ) ä+0 i¬* 47"G+ ?<-,[:T47G+47"G+ l 6 ) #GF #C"= #<f. ) ²±G+, g ?<- #D$- #"fG+ * 47"GF 2½KoKNO ) 6z0;+GF<- #.10;"3B ) z ?"H6+47<=$- G+ ) A4 g 47.1, * ,/4ÁGF "-0 *[) 2#,[0;+ #"3GF * 47"U ?2 ) 472?,/0;F #"G+ Á )+* ?<±lwvyxÞá x Ý5Ý5Ý xÞáV7o£p >lhvyxÞ÷moVX$e4 * B ) @½¦lwvXo£p >lhvyxeDoKV²K­ö½lhvyx&Do£ #",[D:; #<%$&,[@ * Z6+47<&4>v #ÁAxe&l6z0;<` ¥= #.16 * 0"-,a$-0;.S47.S07" ) Á2547.16X02? #D$&<-4 * 2#0; g 0P2?0 # l±#xN½¦leo-oKN!O ) 6X07+G+<- ?.S07"47G+ ?.!'4T"]B ) úãp lhvyxÞ÷mo #" ) 2547.16X0 JF472$&0 i 2#07.S0®öU ?"Z,/D:; ?<=$-,/@ * S ?$-0;+2? #"`6A47<-4C254TGA4lhvyxLZo\:7 #2#,[+04¯lhvyxLz¦lwvXo-oKV6X0LG+ #.10;",/D:; ?<=$&,[<F÷ 2#0;.10 gh) F2#,¨'07¶G+ >N * 2547.16X0C #"Z ?$-0;+2? #"ú p lwvyxÞ÷flhvyxN`o=o±p¦úZlhvyxNZo­lw:; ?< * 43"= #2#2?,¨'0;ÁZNÖÄFNÖÄDoN ^ 07.S0H" ) 6 ) "-,[.S0;"1 * 2#47.S6z0 JF472$&0 #D$-0;+2# ?"

úXlop q u lwvyxN¬xúZlhvyxN]o-o+;v+0YG+ ?6X ?+G+ ]GF * 254T.S,[+0` #D$&<= G+0;"f6 ) D$&07"£ ?qÍ Î ²>$& #F #.10;" * 4 g '0;<=. )+* 4ZGF ¢, * @X ?<=$ ikl]oW!kml½Kopq st B±lhvyxL¬xÞú¢xNºo+DvG+0;FG+ aB±lhvyxL¬xÞú¢xN oR #" * 4 gh) +2?,¨'0;bG+ @: #,[ #<-"%$&<-47"-"f² #"%$½'4YGA47GA4`6z0;< i

B>lhvyxNmxÞú£xL o¬p u lwvyxN¬xL oW u lwvyxN¬xú_oW u lhvyxL¬xÞúªo l úªo Ý"GF #2#,[<#V #H2547G+46 ) D$-0G+ @±lwvXo"= S2?0;.16A47<&4B lhvXo_2#07 * 4G+,[<- ?2#2#,5'0;G+ * 2#47.S6z0úZlhvyxLlhvXo=olh:7 #<f~AQ ) <-41ÃPN » oN

Page 242: Ize J. Cálculo de Variaciones

ÎDÏÏÅÍУÑ_ÉÅWÉÍfУҬÓÓËÔWÉÅWÉÕÒmÐYÊÒ¬ÓCÒmпÌÆÐ]ÍHÔ Æ( ¼

Y

X

Y(x)0

Y(x)

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Page 243: Ize J. Cálculo de Variaciones

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» NÖN|cZNå£0T$& #.10;"¢ #U6+47<=$-,/2 )F* 47<B ) "-,+0>dA45²­6 ) D$-0;"ª2?0;ॠ) QD4TG+0;"¢G+ Z± ?¯ Axe&alh² u #"£,/D:; ?<=$&,[@ * 5oV6z0;< * $& ?0;<- ?.>4+N ¼ V½] ?"6+47<=$- SG+ ) 2547.16X0­2# ?D$&<&4 * 2?0; g 0L2#0­ #!lY#xN½¦l1`eo=oV6X0;< * 0$e4T$-0±By¼·Y²$& ?+ #.10;" ) ') +,[2#01.'9 +,/.10 * 0L254 * # * 41F0;<-.S4SZKNf"=$-0S$- #<-.1,/+4 * 416+< ) #@+4SG+ *$& ?0;<- ?.>4 N ¼ ) "&4T+G+012547.16X07"#NP|V~ ©#!&%a * .!'4àJP,[.S0Z6+47<&4Z * 6+<-07@ * ?.>4Z,["-0;6z #<=,/.¯' ?$&<=,/2?0] #F2#0;D$&<-47G+01 #b * Õ¥= #.16 * 0 ¼ N ¼ ?") ­.!'4¦JF,[.S0] # * 4`+0;<=.>4` V+² ?"¢Q * 0;@A4 * NP|V~ ©#q%F.y4Q; #0LG¤' ?"-,/2#4"-0;@+<= * 4± ?" g #<-4­ ?" ) .'9 F,/.10S6A4T<&4 * 4±+07<-.S4 "-,fl(y xeÀ oª ?") +,[G+014­l|y ³ xeÀ ³ o6z0;< ) ­254T.S,[+0`B ) 2 ) @+<-41.1 #F0;"fG+ .S ?G+,§4Z ?" g #<-4FNP|V~ ©#¿%m * .C'9 +,/.10 #+2?0;D$&<-47G+0_ ?] * ¥= #.16 * 0z+N » #" ) Z.'9 F,/.10aQ * 07@A4 * # * 4f+0;<=.>4¢ NP|V ¥ ©#!T%<-0;@A4T< ) b<- ?" )+* $e47GF0`47f'4 * 0;Q;06A47<&4Z * 6+<=0;@ * #.S4]G+ g <=0;D$& #<-4 * ,[@+<- £G+ * Õ¥= #.16 * 0+N ¼ NP|V ¥ ©#%"=$ ) G+,§4T<¢ * 6+<-07@ * ?.>4ZG+ * Õ¥= ?<-2#,[2#,[0 ¼ N Nå£0T$& #.10;"B ) * 4Z6z0;"-,$&,:P,[GA47GG+ aB ?"¢$&47.Z@+,' #+ ?2# #"-47<-,/4 i¤ r¥ m&¦\©#% §r,0½¦lwvXog-N8Ô;/®pS¬ /$,5pO2JWOMD6<J^F -)/J69/0O73pÅ6 F0«6<36>-)J°-N8C«6M),.O¡1-Q;/0M),.O7/0-N8Zj«$O73j«$-D16RQ7O8NF-)/$*O7/0MN-N8¥B±lhvyxLz¦lwvXoxN lhvXoKxÞ´¿oR 9«6<36Å*COV1O´ -)/+Í Î ¯ u x u x u 8O7/&MNO7/M£*,5/$;rO8 ( [ u lhvyxN½¦lwvXoxÞ´¿og-N8g1-i-³/$,16@/0O9/0-.:6<*,547698), u -N8£\³IF°«6<36*COV1O´ -)/\Í Î [· ¥G¸ ¹¦% 0PG+ ?.S07"¢ ?"-2?<-,/@F,/< i u lhvyxL¬xN oW u lhvyxN½#xL oR2?0;.S0

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Page 244: Ize J. Cálculo de Variaciones

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Page 245: Ize J. Cálculo de Variaciones

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Page 246: Ize J. Cálculo de Variaciones

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Page 247: Ize J. Cálculo de Variaciones

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Page 248: Ize J. Cálculo de Variaciones

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ü,/Q+NmÃLN á GF #.I'47"f2#0;.10 Ñ ¸ \ à _ Ý bÑ Ý p » ¶2#0;"½À¢´Þ"= #½À­Ñ ~Ñ Ý lh²;41B ) À` #" gh) +2?,¨'0;bG+ vC²±G+ lÞ#oKNc_ * 4`<= * 4T2#,¨'078VÞTlÀFlhvyxNÞ?oW"- ?½À+lhvyxLÞ#o=oW¶vbp·PVA0;@F$& ?+ #.10;"?VAG+ #<=,[:T47+G+0Y2?0;<= #"=6X ?2?$&0S4ÅÞ iÀ Ý p¿ZlÀE"- #zÀ7o-Û<ÞTl » 2?0;"½À;o Ý07< * 0Z$e47D$-0Sn Ý pl ¼ ¼ 2#0;"AÀÀà"- ?½À;o-ÛFl » 2#0;"½À7oNOP,6z0;+ ?.S0;" u lÀ7op ¼ ¼ 2#07"AÀ_ÀT"= #½ÀLV u lÀ;omp "- #½À_¶À2#0;"½À¯"- \47 )+* 4Y"=,yÀp$e4TfÀY ?"fG+ #2?,/<#V6A47<-4ªÀf #\§Fx ¼ _¬# # ) ]"+'0 * 0¢6 ) D$&0ªÀ5 #D$-<- û_S² _Û ¼ lQ7<&4T~A2#47<y$e47fÀ²À;oVT6A47<-4¢ * 2 ) 4 * 2#0;"½À5£Ü!FV"- ?½À5Ü FNmD$-0;+2# ?"#V u lÀ5o¡¼ FN ^ 07.S0 u lDo]p u l ¼ _op FV$& ?+ #.10;" ) ¶"A'0 * 0C KJP$-<- ?.S0G+ u lÀ;oª ?D$&<- 1±² ¼ _V ?HÀ5R% u lÀ7o2¼(b6+47<&4bÀ #§Px ¼ _¬yNZm #.10;" #D$&07+2# ?"B ) 1n Ý ¼(FVy6A47<-4bÀ #`/Fx ¼ _mºNRl(Wf #2?0;<-G+ ?.S07"B ) "-0 * 47.S ?D$& ¤$- #+ ?.S07" ) Y47<-2?0G+ 2#,[2 * 0;,/G+ TVT ?"G+ #2?,/<À£ #"=$A'4 # Px ¼ _8N o * 2547.16z01B ) 0;@P$& #F #.10;"f #"f2# ?$-<&4 * 2?0; g 0L2#0Y ?¶lFxeDoKN ¡ G+ ?.!'4T"

u p n ׳ l » ´& ³ o Í׳ ¼°W4T<&4`6X0LG+ ?<2#0;+2 */) ,[<RB ) 0;@F$- #+ ?.S07" ) b.C'9 +,/.10]Q * 0;@A4 * VP ? * 4Z+0;<=.>4] V+GF #@z #.10;" #"%$ ) G+,/47< *gh) +2?,/0;+4 * 2# ?<-2#41G+ nYp·FVF²D4YB ) 47d'9 u #"¢"-,[+Q )+* 47<#N0;<f * .10;.1 #D$&0Z$- #+ ?.S07"¢B ) i

Page 249: Ize J. Cálculo de Variaciones

¼ Ã7 ÅWÆ_ÇÈÉ#ʪË̬ÍÎDÏrÅÍЪÑ_ÉÅWÉÍЪҬÓCÓWËÔWÉÅWÉÕÒmÐYÊÒ¬ÓB>lhvyx&nlhvXoKx'Alhvyx-n8lwvXo-oxDÇo ¼I6A47<-4¡Ç ¾p(Alhvyx-n8lwvXo-oR²±v¼IFN

©£"-47<- ?.S07"¢ * "-,[Q ) ,/ ?D$& 47<=Q ) .1 #D$&0+N°Wf #2?0;<-GF #.10;"fB ) * 4`,/D$& ?Q;<&4 * ,/D:T47<-,/47D$& _G+ a ¢, * @z #<%$ iúXl)8omp·q lÕÔCDvZ´ è nFo #"f,[+G+ #6z #FG+,/ ?$- GF * 254T.S,[+0*U ?$-<- \G+0;"f6 ) D$-0;"fG+ * 6 * 47+0­lh:7 #<fZNÖÄFNÖÄDoN¡ B ) '9 p è \l » ´+ ³ o ׳ n ׳ V+2?0; è p u p,7n ׳ l » ´& ³ o ׳ N¬0;< * 0Y$&47D$&0+Vp(¢n ׳ l » ´& ³ o ׳ x » ´- ³ pl » n è ³ o #"fG+ ?2#,[<#V

p(Zl » n è ³ o ׳ n ׳ Ý¡ dF0;<&4Y6A47<-412 ) 4 * B ) ,[ #<f2 ) <=:T4SnlhvXoR2?0;CnlDompFVAnlLo¬p·]VA$-0;.S ?.S07" * 0;"f"-,[Q ) ,[ #D$& ?"£2#47.S,[+0;".²¶ä`´-/V ) F,/ #FG+0 * 0;"6 ) $-0;"]l(#x&nT¦l(&o-oR²HlAx&_olh:7 #<~AQ ) <-4>ÃLN|ÄDoKNÚ #+ ?.S0;"fúXlompúXleä8o´¯úXl0/yompq t u lwvyx&n75lhvXox-n lwvXo-o+;vyN¤OP 54k-lnFop" t u lhvyx&nzx&n o+DvyNÚ #+ ?.S0;"k&lnFo!k-lnTKomp(k-lnFoúXlop(k-lnFoW¶úXlKäoWúXl0/yop q t B>lhvyx&nlhvXoKx'Alhvyx-n8lwvXo-ox-n lhvXo=o+DvSúXl0/yoG+0;FG+ ) "&47.10;" * 4 g '07<-. )+* 4ZG+ a £, * @z #<=$lh:; ?<ZNÖÄFN|Ä;oN

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Page 250: Ize J. Cálculo de Variaciones

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Al(#x-n+op DnlhvXo;v p nÀ À;v p "- ?½À;Û+l » 2?0;"½À;o Ý^ 0;.10!« è «v n ׳5V« úXl0/o¦«v« ¸ \ b¸ u \ b n ׳zDn« p ¼ « nl(eo&׳ª°n7¨l(eo&׳«|NED$&07+2# ?" * ,/.¯úXl0/yoYpFV2 ) 47+G+0£$-,/ ?+G+ `4±PN ¡ GF #.I'47"]k-lnTKof$-,/ #FG+ `4kln7eo¢2 ) 47+G+0£$-,/ ?+G+ `4±PVz²;4±B ) Z:P,[.S07"£B ) TV6A47<-4 * 4`2#,[2 * 07,/G+ TV * 4Z,[D$& #Q7<&4 * ?"f2#0;D:; ?<-Q; ?D$& 7ND$&0;+2? #"kmlhn+oRkln7oKV+n7¢ #" ) ­.'9 +,[.S0±lL') +,[2#0o¤6+47<&4 * 4`+07<-.S4` N

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nlhvyxÒWoWp·32#0;"=dy vl2#07"-dzÒWo=Û?¯´Òy§Û2#0;"=d;Òlh:7 #<¢ * ¥= #.16 * 0Y]N ¼ N ¼ oKN^ 4 * 2 )+* 47<an76]²6+<-07@A47<B ) 6A47<-4 ) A4`G+ * 47"a"=0 */) 2#,/07+ #"0;@P$& #F,/GA47"f * Õ¥= ?<-2#,[2#,[01]N ¼ N » VF"- 0;@F$-,/ #F ) U.'9 F,/.10 * 0L254 * 6A4T<&4 * 41F0;<-.S4> lh #"=$-0±G+ ?6X ?+G+ ZG+ * "=,/Q;F0>G+ ÒWoKN ^ 4 * 2 )+* 4T< u NªOL, p ]VlpnlLoKx&p·nlDo=oVFd+472# ?<¢ * #"%$ ) G+,[0S2?0;.16 * $&0+NP|V ¥ ©#e% ^ 0;.16 * $e47<¤$&0LG+0;" * 0;" ¥= #.16 * 07"a² Õ¥= ?<-2#,[2#,[0;"G+ * 47"F07$e47"?N

Page 251: Ize J. Cálculo de Variaciones

¼ à ¼ ÅWÆ_ÇÈÉ#ʪË̬ÍÎDÏrÅÍЪÑ_ÉÅWÉÍЪҬÓCÓWËÔWÉÅWÉÕÒmÐYÊÒ¬Ó!2#07+2 *[) "-,5'0;I,[:L,$e47.10;"14 *¢* #2$&0;<S4:70 * :7 #<±4 * ? #< * 4H,[D$&<-0LG ) 2?2#,5'0;²U¥ ) æ?QD47<>6z0;<¯' * .S,["-.10+N¡ GF #.I'47" * 0,/D:L,[$&47.S07"m4\2?0;.S6 * ?$&47<" ) "¤2?0;+0L2#,[.S,[ #D$&0;"?V;:L,/ #FG+0 ) >@ ) ? * ,/@+<=02#0;.10 * G+ fÚ<=0 ) $%M.S478V+4`G+,:; ?<-"-,~A254T< * 07" #b * ?"=$ ) G+,/0YG+ 6+<=0;@ * #.S47"a2#07b:T47<=,§47"¤:74T<-,§4T@ * ?"#VF ? ?"-6z #2?,§4 * ? *8* ,[@+<-0G+ ºWf 98D$&07<=²L"#VF²±4]6+<=0 gh) +G+,[æ547< * 0;"¤2#0; * #"%$ ) GF,/0`G+ _GF #"-,[Q ) 4 * GA47G+ ?"R:T47<-,/472#,[0;A4 * #"?VL<= g ?<- #F2#,§4T"\ à²H » TN

Page 252: Ize J. Cálculo de Variaciones

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Page 253: Ize J. Cálculo de Variaciones

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Page 254: Ize J. Cálculo de Variaciones

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