Copyright © 2012 University of Cambridge. Not to be quoted or reproduced without permission. es that sense xs n are and exist . ieetaiiyadfis re conditions order function first a If and Differentiability 2.1 o on h rbes ti o opeesmay o learni For summary. complete a not books is the t and It intended lectures syllabus, the the problems. of of combination the parts of doing version for expanded an is Below Introduction 1 R ∇ aitoa rbesfrfntoson functions for problems Variational 2 the to corrections problems and starred errors end, send the Please at are supervision. tre problems for detailed The tended more much give course.) books this three last for (The used. be should l o-eovectors non-zero all iery n h eiaiei h iermpon map linear the is derivative the and linearly, ierin linear em .. Scn re eesr conditions) necessary order (Second 2.2.1 Lemma em .. 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R ) ∂x i = n f ∂ sannpstv rs.nnngtv)sym- non-negative) (resp. non-positive a is h i ( 2 ∂x P ie by given x f → oa iiu o aiu)o a of maximum) (or minimum local A lim = ) j i n o a For . =1 there. s .Ti en tcnb approximated be can it means This 0. fasainr point stationary a If x i e t i Df → } P tet hnpossible/necessary than atments en oedffiutadntin- not and difficult more being C 0 xntto n terminology and notation fix o where mi drs above. address email ij ( 2 t x − R P o convexity for UP) lso variations of ulus A function )( 1 n ( ij uga n Mandelstam and ourgrau galtemtra some material the all ng h f v = ) e ( i x 1 v j + (1 = > ∇ t f e f x ( (resp. 0 i , x ( ) n-negative). fa of 0 x + , . . . , − ) · h f f h ( ) hc is which , x ∈ )etc. 0) − )which )) ≥ x C C f r nthe in )for 0) ( 2 ( x R ( R ) C n n − ). 1 ) Copyright © 2012 University of Cambridge. Not to be quoted or reproduced without permission. subset A Convexity 2.3 maximum). ∂ function f x hnvri a eie o 0 for i.e. be can it whenever any em .. Cneiy eesr n ucetscn orde second sufficient and necessary (Convexity: 2.3.3 Lemma poin stationary particular, In solution. one unique. than more no have f em .. Scn re ucetconditions) sufficient order (Second 2.2.2 Lemma P em .. Cneiy rtodrconditions) order first (Convexity: 2.3.1 Lemma sconvex is em 2.4.1 Lemma hr exists there f em .. Src ovxt:sffiin eododrcondi order second sufficient convexity: (Strict 2.3.4 Lemma if that implies this corollary, a As osdrahypersurface a multipliers Consider Lagrange 2.4 conditions) order first convexity: (Strict 2.3.2 Lemma ovxif convex h matrix the sacrlay hsipista if that implies this corollary, a As n h eeaie on inequality Young generalized the and . eedeTransformGiven Legendre 2.5 holds. result lblminimum. global nyfor only x h function The a multiplier. olw meitl rmtedfiiinof definition the from immediately follows ij − 2 ( ( ( , h vector The . x x y y ij 1 f loti hw that shows this Also o rbeswt eea constraints several with problems For = ) + ) ) + ( t ∂ in x f > ∈ 2 b ) f h ∇ S − a [0 ij sapstv rs.ngtv)smercmti then matrix symmetric negative) (resp. positive a is p : 1 ( f f + S ∂ , ( n any and x R ⇐⇒ and 1 = x ( ](rmr eeal ti ovxo ovxsubset convex a on convex is it generally more (or 1] 2 uhta hsspeu sfiie h eedetasomi auto is transform Legendre The finite. is supremum this that such : + ) ∂ x b ⊂ n f λ , R i λ ) ij g · → n R · ) α h n ( x ∇ ∈ v ( x ( n Let ( → r xmlso ucin hc r ovxbtntsrcl c strictly not but convex are which functions of examples are , y i x ∂ f steLgag umne ucin h number The function. augmented Lagrange the is R x ) v R is 2 a rank has ) ( j − = ) > t x f R > a t eedetransform Legendre its f is ij convex ) ∈ x uhthat such 0 is · ( ) ∈ ∇ ≤ x ( [0 ∀ y sawl-nw pca ae(e exercises). (see case special well-known a is 1 o all for convex ) x g , C C 0 C . ( − ])Further 1].) ≥ 2 x 1 (resp. = ( ffrany for if x ) 0 R ovxfntosleaoetertnetplanes. tangent their above lie functions convex / l ) n consider and ) { ∀ k∇ n < t < x o all for x x ) ∇ if , hnif Then . . g ∈ y h f ≥ ( ( x ((1 R x f x ⇐⇒ 0 ) λ , n o l vectors all for ) k and 1 ∈ x x sasainr on faconvex a of point stationary a is − f f : , 0 = ) seeyhr omlto normal everywhere is 6= ( C y g x { t scalled is ( ) 1 g y + ) ( in x x f ( α ∇ g , R 0 = ) ¯ ¯ x } + h g C f S = n α l ⇐⇒ g ( ( 6= where ssrcl ovx h equation the convex, strictly is ) =1 x t a aiu rs.mnmm at minimum) (resp. maximum a has ( = x n any and 2 y f p λ , ) ) } y ∗ suete r needn i h es that sense the (in independent are they assume , ) tityconvex strictly f ∇ − = ) . ≤ ≥ sgvnby given is where ∗ ( h inequality The . Affine ∇ (1 h p v f f ( f · x ( − ( uhthat such ( t x y x f x λ , ∈ g If )) t ) ) ) = ) − ∈ ∇ − ucin,ie ucin fteform the of functions i.e. functions, ∈ f [0 · f ( ( P , C x x sfrsrcl ovxfntosare functions convex strictly for ts x C g ]tepit(1 point the 1] ∈ f + ) f 2 ( S C − 1 sasrc oa iiu (resp. minimum local strict a is f λ p ( ( fteaoeieult sstrict is inequality above the if ( . ∈ R ( α v y C R fti nqaiyhlsfrany for holds inequality this if sup( = ) y x g tf n )) C n · 2 ) ) α satisfies ) ( n ) ≥ ( ( 1 · − R tion) x y ( ( λ xy 0 = convex n R x ,adtecorresponding the and ), o any for ) 0 condition) r C λg ) , n scle h Lagrange the called is − 1 ) o all for . ( ≤ and p x y ucinte ti a is it then function tityconvex strictly f · ) ) a n furthermore and , − x > ∈ − ∇ aial convex, matically onvex. Df − t 1 ⇐⇒ ∇ g C 0 ) x x x ( x f f 2 a x , o all for ( , ( ( x y + y ( ) + R x x 0 = ) x . o all for 0 6= ) defined )), n in t = ) f b y ) C ∈ − f ∈ sstrictly is R ∈ 1 x ( y b n y C S b 6= then ⇐⇒ ) and and 2 A . can for ( y ≥ R . n ) Copyright © 2012 University of Cambridge. Not to be quoted or reproduced without permission. equation h ietoa eiaieo h functional the of derivative directional the i . eeaiiso functionals on Terminology: Generalities functionals for 3.1 problems Variational 3 h rm on prime the for nyoedrcina ie ..wl osdrdrcinld directional consider 3.1.1 will Lemma i.e. stud time, will a we at useful: reason direction this For one only topologies. equivalent define spaces agn lns n h urmmgvsback gives supremum the and planes) tangent upfnto na nevl( interval an in function bump A nerto yprsgvs for gives, parts by integration ucin.Ti atas olw rmlma231-js aeth take just of - graph 2.3.1 the lemma below as from expressed lying follows be those also always fact can This functions convex functions. a that implies This ( hntefunction the then eootieteitra (exercise). interval the outside zero n ylma311 eddc that deduce we 3.1.1, lemma by and the hoe 2.5.1 Theorem h upfnto nteitra ( interval the on function bump the ytespeu ( supremum the by h mapping the . ietoa eiaie ffunctionals of derivatives Let Directional 3.2 fteitra ( interval the of Then ntespace the on Proof xml h ia functional differentiabil Dirac and the continuity the example ways, inequivalent many in ′ x 0 = (0) 0 ucinli utafnto nasto ucin.Snespaces Since functions. of set a on function a just is functional A lgtvraino hslmasae htif that states lemma this on variation slight A − y L f 2 ,x ǫ, a g : om(endby (defined norm hsflosuigcniut n upfntos(exercise). functions bump and continuity using follows This DI aihsietclytruhu h interval. the throughout identically vanishes C R 0 2 3 + [ y iiie.Tequantity The minimizer. → ǫ ]( n aihsfor vanishes and ) DI φ C φ V then ) = ) R 0 ∞ Let ,b a, [ of y If ( esot n osdrtefunctional the consider and smooth be : ] ,b a, i L .Tespoto ucini h lsr ftestweei sno is it where set the of closure the is function a of support The ). R g C ( f a ∞ ǫ b φ 1 ∈ = ) g stesaeo mohfntoswoespoti lsdbounded closed a is support whose functions smooth of space the is ) ( sconvex is norm ) f 7→ saconstant. a is ucin with functions y C k φ φ ([ DI + I k ,b a, [ L 2 y f 2 [ δ y y φ k ]) + 0 DI = | ′ ]( f φ ( x φ f ∈ n hwta hsfml snnepy(ic tcnan the contains it (since non-empty is family this that show and , φ φ aetepoet that property the have k ǫφ ′ ∗∗ − R ) L x = ) [ scle h rtvrain n oeie written sometimes and variation, first the called is ) C y dx ∞ 0 | a iiu at minimum a has ] x δy δI − ]( φ = 0 ∞ 0 − ( φ 1 max = ≥ | φ aihsfrec such each for vanishes ( x δy δI f , = ) = ,b a, y 0 scniuu on continuous is (0) ,x ǫ, ) dx )gvnby given 1) . d | ( 2 ¡ ssmtmskona h ucinldrvtv,and derivative, functional the as known sometimes is a .Teecnb osrce ytasaigadscaling and translating by constructed be can These 0. dx ): f ( = ) 0 Z f y I a + y .I otatalnrso nt iesoa vector dimensional finite on norms all contrast In ). | − b ′ φ along ¡ ) f ǫ ( f 3 α − dx safunction a is ) x . y d ) R − f | and a ( u o ihrsett htdtrie by determined that to respect with not but , b y f e g φ dx y 0 = d (1 ( ′ suefrhrthat further Assume . ) x − y − ¢ ( ) x R f ( 1 φ 2 0 = a b y b ) ǫ ′ = ) 2 ′ ( g ) I t ffntoasi oesbl.For subtle. more is functionals of ity x ( ¢ o all for 0 = C [ for x ) rvtvs h olwn em is lemma following The erivatives. y φdx φ dx ( ) = ] h quantity The . R φ β b aiyo ffiefntost be to functions affine of family e x urmmo aiyo affine of family a of supremum a ( ieetaiiyo functionals of differentiability y Assume . ihtetplg determined topology the with ) o all for 0 = 2 x ∈ ) R ffnin a etopologized be can funtions of < dx a C b n xeddwt value with extended and 1 f 0 ∞ 0 = ( ( ,y y y, x, R φ hc spstv in positive is which ) o all for ∈ φ I [ ′ ∈ y y C ) DI min = ] dx 0 ∈ C ∞ 0 [ ∞ ( C y φ safunction a as ,b a, ]( ( 2 ∈ ,b a, φ ( ,b a, ,s that so ), scalled is ) C w (notice ) δI 0 ∞ ∈ ,then ), n-zero. subset V The . ( ,b a, I [ w ) . ] Copyright © 2012 University of Cambridge. Not to be quoted or reproduced without permission. gi.Tefunction The again. aitna form Hamiltonian sup ovxin convex tteunique the at hsdfiiinatmtclygvsgoeiswt paramet a with geodesics gives automatically definition this soitdt h cinfunctional action the to associated ovnini nesod)Te ov h equation the solve They understood.) is convention . arninadHmloinmechanics Hamiltonian equation and The Lagrangian 4.3 theorem). (Noether law conservation second the by ot ewe h w omltos oieta h urmmi supremum the that ensure Notice variables formulations. velocity two the the in between forth Lagrangian the of Convexity ds hseuto steElrLgag qainascae othe to associated equation Euler-Lagrange the is equation This ob h rlnt othat so arclength invariant, the parametrization be is to functional length the Since steElrLgag qainascae to associated equation Euler-Lagrange the is aldteLgaga.Ti sthe is This Lagrangian. the called o atceo mass of particle a for ih asflospaths follows rays Light principle Fermat 4.1 Applications 4 trick. parts by integration an and above mentioned R constant hc r ttoaypit o h eghfunctional length the for points stationary are which sot)Reana erco noe subset open an on metric Riemannian (smooth) A Geodesics 4.2 position. on from g = ij ˙x x ( k ˙ U p i γ ˙ x ˙ ( · notesaeo elpstv symmetric positive real of space the into j t ˙x ) dt vnfor even k ˙x − dt ota natraiedfiiino edsci a is geodesic of definition alternative an that so , h eedetasomto ntevlct aibe gives variables velocity the in transformation Legendre the L steeeeto rlnt along arclength of element the is ˙x ( x ie by given , ˙x : C )fo which from )) 1 > m H iiies-ti a eddcduigtevraino em 3 lemma on variation the using deduced be can this - minimizers p γ steHmloin n ie neuvln omlto f( of formulation equivalent an gives and Hamiltonian, the is oigi potential a in moving 0 dt = g d hc iiie(rmk ttoay h time the stationary) make (or minimize which ij ³ m x ˙ √ i ˙x x ˙ g hsdfie the defines this : dt j d g L lm ij arninformulation Lagrangian ,i hc aeteeuto ipie to simplifies equation the case which in 1, = ³ x ˙ x a ercvrdjs yapyn h eedetransform Legendre the applying by just recovered be can x S ˙ ˙ g j l j ij [ x = ˙ x x m ˙ = ] m j ∂p ∂H ´ ´ ¨x − − j + I , R nfc thlsi nertdform integrated in holds it fact In . 1 2 1 2 ∇ 4 L γ ∂g ∂g p ∂x ∂x ( V ˙ x j and n jk jk V ojgt momentum conjugate , i (4.1) 0 = = i U × ˙x ( x √ ˙ x ) − j c dt n l ⊂ a edrvda h Euler-Lagrangian the as derived be can ) g x [ ˙ ∂x x ∂H x stesedo ih,wihmydepend may which light, of speed the is lm ˙ where , k arcs h edsc are geodesics The matrices. ti osbet hoeteparameter the choose to possible is it j R = ] x 0 = x iainfrwhich for rization j ˙ ˙ n k fNwoinmcais Since mechanics. Newtonian of l x ˙ h osblt fgigbc and back going of possibility the s sa(moh function (smooth) a is C R m . 2 ( kntceeg integral” energy “kinetic g h ento of definition the n 0 = uv o which for curve ij L x ( ˙ x . i x , ˙ j ˙x ) = ) . 2 1 dt function a weesummation (where , g T 2 1 ij m x = ˙ k i I x ˙ ˙x R j f H sstationary- is k γ y 2 = x ′ 1 c H sattained is − − C ds constant 7→ ( 2 R V where , x .)in 4.1) a I curves x , g ( [ p x ij f x L = ) .1.1 is ) y ( = ] x = is ) t , Copyright © 2012 University of Cambridge. Not to be quoted or reproduced without permission. htfrall for that sdisedof instead used where hsi rvdi hoe nscin2 fGladadFomin. and Gelfand of 26 section in 1 theorem in proved is This em .. Sffiin conditions) (Sufficient 5.0.3 Lemma Teei loacrepnignto of notion corresponding a also is (There maximum. weak Then y em .. Ncsayconditions) (Necessary 5.0.2 Lemma o such for ealta if that Recall h eodvariation functional the second Consider The 5 52:i ssffiin that sufficient is it (5.2): Here V o h functional the for xlctapoc odtriigwehr(.)hlsfrsom for holds (5.2) whether determining to approach explicit h eodvrainstse,frsome for satisfies, variation second the that ento 5.0.1 Definition fteSumLovleoperator Sturm-Liouville the of ssmtmscle h eodvrain n denoted and variation, second the called sometimes is where φ where ons˜ points noSumLovleform: Sturm-Liouville into ( 0 ∈ b = ) n h eodvariation second the and suig sawy,that always, as Assuming, V φ ,φ φ, 0 y | ( Q |R| a f p β with a sawa oa minimum. local weak a is ( ( Let . = 0 = ) ∈ x ,y x, φ steqartcpr fteTyo expansion Taylor the of part quadratic the is ′ = ) ǫ < r vlae ihargument with evaluated are ( o dnial eo h iiu ssrc.Teei corre a is There strict. is minimum the zero, identically not x ,b a, k D + φ ∈ V R y 2 k uhta hr sanntiilfunction non-trivial a is there that such ] f a ,y φ, I 0 b k [ C y ,b a, is [ ( φ ′ y etevco pc of space vector the be 1 | y Q φ φ I k ]( ′ function A C ′ ( max = ( : C | and ] b φ = + 2 ,y x, 2 1 en hti hscs h oml o h eodvrainca variation second the for formula the case this in that means ) = ) + e hpe nGladadFomin.) and Gelfand in 6 Chapter see , φ tsle h ue-arneeuto if equation Euler-Lagrange the solves it 1 2 ( ′ | I ¡ ) p φ x [ φ Z I k [ ( − y ) ′ a,b φ 2 [ | x y , = ] D y 2 ¡ f k ) f ) D ] φ yy + ′ C 2 dx D ( > | ( 2 φ I ,y y y, x, 2 1 x y R f ( f φ 2 [ I ( L ,y y y, x, a )and )) n[ on 0 y yy δ < b I x for = ] [ ∈ ssot,Tyo’ homipisthat implies theorm Taylor’s smooth, is ]( y [ f ) = ( y ]( | φ ( ,y y y, x, V ]( : max + ,y y y, x, k φ ) ′ − I ) φ φ ,b a, ) ′ [ ≥ ( sa is 2 + ) y − = ) k pφ ≥ strong + ] q C > c C c ( n htteeaen ojgt ons ..teeaeno are there i.e. points, conjugate no are there that and ] φf x ′ 1 x 0 Z ′ 2 + ) ′ 1 rmti olw orsodn alrexpansion Taylor corresponding a follows this From . Assume ) ) Z [ ∀ = ) δ < a DI a,b weak If y φφ ′ dx a ucin with functions 0 b φ + ( b ( , ,y y y, x, ] y ¡ | iiie for minimizer ′ ∈ qφ [ | φ ( φφ ntespace the on f p y φ ǫ ∈ f 5 yy | ( ]( V .Teqartcpart quadratic The ). ′ 2 hr r lognrlcniin hc ensure which conditions general also are There . yy x oa iiie for minimizer local ( ′ 0 φ ′ V x f + ) ( . ( + ) y ′ φ yy ) ,y y y, x, ) ,y x, | | ′ sawa iiu then minimum weak a is φ 2 ∈ − ′ ucetysal fteieult sstrict is inequality the If small. sufficiently ( ′ + ,y y y, x, δ 2 1 | V ( φ 2 h 2 x D ) ′ q I ′ f ) dx + ) ( uhthat such ssc that such is rmti ecnra off: read can we this From . 2 y , y x y I ′ I V e ) ( ( ′ [ ′ ∀ φ ,y y y, x, a + ) y ( ihtenorm the with φ > c φ x of and 0 = ) 2 ]( ′ 2 ¢ )) φ ∈ DI f dx C φ + ) y st aclt h eigenvalues the calculate to is 0 − V ′ ′ 1 2 ′ y [ 0 ) f Lh y ′ ucin with functions . dx ( y − R ]( d I ,y y y, x, ′ y DI φ ( Q and 0 = if ′ f 0 = ) ( y yy | ,y y y, x, pnigdfiiinfor definition sponding [ I ( ǫ < y ∀ b ′ [ ′ ]( ( y > ǫ k 0. = ) ) ,y x, φ ¢ φ + ∀ ( . k DI 0 = ) | φ ′ φ C ) h φ 0 ( ¢ | 0 x ∈ 2 ( ] [ dx ∃ a y ) max = + δ ≥ y , y = 0 = ) ]( V ∀ ( ( 0 ǫ φ | φ ′ a I φ ) h fact The . ( 0 = ) = ) [ x ′ ∈ y > eput be n | ) One )). 2 ] [ ) V a,b such 0 o all for α 0 h ∀ (5.2) ] and (˜ φ and | φ a ). ∈ ( x ) | Copyright © 2012 University of Cambridge. Not to be quoted or reproduced without permission. xml he 1 sheet Example 6 0 For 10. 5 o nielgs h nenlenergy internal the gas, ideal an For 15. 2 rv htteLgnr rnfr fafnto sawy co always of is function transform a Legendre of the transform Find Legendre the 13. that Prove 12. 1 h area The 11. 4 idteLgnr rnfr of transform Legendre the Find * 14. 6 o lc oyrdainteitra energy internal the radiation body black For * 16. .*Poeta elsmercmatrix symmetric real a that Prove * 2. .Gv neapeo ucinwihi titycne u wh but convex strictly is which function a of example an Give 6. .So that Show 7. .Given 5. .*Poe sn h ozn-eesrs rpry u without but property, Bolzano-Weierstrass the using Prove, * 3. .Poeta if that Prove 1. .Gvnafamily a Given 8. .*With * 9. .*Let * 4. lotecntn oueha capacity heat volume constant the also everywhere utb lblmnmm ieacutreapet hwti is this show to counter-example a Give minimum. global a be must o oeconstants some for sacne uciniff function convex a is ievle r positive. are eigenvalues where of fara ymti matrix symmetric real a if scne hntefunction the then convex is est ie h nenleeg e ntvlm)is volume) unit per energy internal the (i.e. density by e)so that show set) matrix. hsway. this h sphere the that show to multipliers Lagrange F sacne ucinof function convex a is i hwta faltinlso ie eiee 2 perimeter given of triangles all of that Show (i) A nlssII). analysis i)Fn i em fteprmtr h ags osbeae o area possible largest the perimeter) the of perimeter. terms given (in Find (ii) rnfr of transform a is F for oueis volume − 11 ( ( 1 ,V T, ,V T, 3 1 U dU T + + A fteeeg est.CluaeteHlhlzfe energy free Helmholtz the Calculate density. energy the of safnto of function a as rmtefruayudrvdi h rtpr ftequestion! the of part first the in derived you formula the from f A C b f = min = ) min = ) n elsymmetric real any − L 22 ∈ : sacntn.Calculate constant. a is 1 α dS T A R (Young). 1 = C > si h rvosqeto,so httefunction the that show question, previous the in as x { n > fatinl ihsides with triangle a of v 2 2 and 0 /y f → ( f ∈ R 0. − ( S S f x L ∈ 2 ( R scne nteuprhl ln ( plane half upper the on convex is R ( ( pdV = ) aeasainr point stationary a have ) U α U A x n C ( en t pgaht be to epigraph its define sup = ) ( ( U x A = ,V S, : 1 ,V S, ,advrf that verify and ), fan ucin nee by indexed functions affine of ) 0 ( 11 a k R T , p v − S A A E a nyoesainr on hc salclmnmm hni then minimum, local a is which point stationary one only has ) k 1 ) ) 0 [ aclt h emot reenergy free Helmholtz the Calculate . Asm h atfo nlssI htacniuu ucinon function continuous a that II analysis from fact the (Assume . 22 x s f V , − 1 = − ( a α s scne subset. convex is U > a , − S T 0 yf L n S T A − S , } α = A ij × ( ( ,adso htisvleis value its that show and ), a tan t supremum.) its attains 0 .[ ). 12 2 y x U U )( ,n R n, α, , − n > ,T P, endon defined 1 scne.*So htall that Show * convex. is ) f 0 ( s > 1 f ,V S, ( λ nti formula this In x + arxconsider matrix x − ( then 0 ,b c b, a, ( implies 0 scne n( on convex is ) x = ) A U αnRT si h rvosqeto n eiyta h energy the that verify and question previous the in as b = ) stelretegnau of eigenvalue largest the is ) )( = ) pV = A aclt h rsueadtmeaue(defined temperature and pressure the Calculate . s e ij 6 sgvnby given is C U 2 1 − x ³ = x 0 P gvn t oanas) idteLegendre the Find also). domain its (giving , V P ( E is 3 h c ,V S, 4 ( = nRT )] f ¡ S ij A ij = > x > V , V = ´ ij A U s A 0 x safnto fetoyadvlm is volume and entropy of function a as ) 3 4 h rageo ags rai equilateral. is area largest of triangle the , ,i h es endin defined sense the in 0, T ¢ where ij { ij > 1 ³ = α ( ielgseuto fsae.Calculate state). of equation gas (ideal 1 T x , ,addeduce and 0, v ∂T ∂S x ,y x, CT x CV ,y x, e λ i othat so 0 i U 1 v α z , 2 x safie ubr-d o substitute not do - number fixed a is ( S | αnR j n let and ) A − V ( j : ) 4 : ) ∈ : ) ,V S, ´ S s sup = ) ≥ n omn nteconvexity the on comment and , where 0 n httevleo h pressure the of value the that and 3 1 N > y = z > y − c o nfc nabtayindex arbitrary an fact in (or , − safnto fetoyand entropy of function a as ) ≥ k 1 2 1 nvex. f v ( 1 3 C i .*So htif that Show * 0. x f sn ignlzblt,that diagonalizability, using 0. ( a k s eoddrvtv snot is derivative second ose U A x ih-nldtinl of triangle right-angled a f 1 ( 2 A sasrc oa minimum. local strict a is + x v . inf = ) ij F ovxfntosaiein arise functions convex F ∈ ij o some for ) xy b ⊂ } R sapstv symmetric positive a is A ] = + n = as in false = lopoethat prove Also * . : ≤ k c ∂ v F F R ) α k ij 2 a . ( n ( =1 L − ,V T, f ,V T, +1 1 α ( v > c x x § ( R hwthat Show . x .,i l its all iff 2.2, a .So that Show ). · 2 endby defined ) endby defined ) sconcave. is ) + . ( f A .(After 0. b v ∈ − .Use ). 1 C y b 2 ( for R f λ t ) Copyright © 2012 University of Cambridge. Not to be quoted or reproduced without permission. 8 rt onteElrLgag qainfrtefunctiona the for equation Euler-Lagrange the down Write 18. 9 h rcitcrn rbe ed otesuyo h func the of study the to leads problem brachistochrone The * 19. 7 hwta h ue-arneeuto ftefunctional the of equation Euler-Lagrange the that Show 17. 0 banteElrLgag qainfrtefunction the for equation Euler-Lagrange the Obtain 20. energy ujc otecntan htislnt ean fixed, remains length its that constraint the to subject where o h integral. the for oiiesolution. positive R function al hc ssseddbtentopit ( points two between suspended is which cable qain hc sol eesr odto o minimizer, a solutio for a condition as necessary obtained a we only which solution is cycloid which the equation, prove to used where rt onteElrLgag qainfor equation Euler-Lagrange the down Write eueta oe on for bound lower a that Deduce r ylis sfrteElrLrneeuto for equation Euler-Lgrange the for as cycloids, are n n l ouin hc aif lim satisfy which solutions all find and a h rtintegral first the has ieeta qainwhich equation differential hwta if that Show for h rtpr fteqeto.Gv l h ucin satisfying functions the all Give Euler-Lagran question. the the solves of equation part order first first the this of solution any fvariables of idtefunction the Find o xdvle fboth of values fixed for 1 2 t C 4 [¨ 1 x a > L ,x c, ( curves t )] l ( 0 2 ,v u, dt and sn h arnemlile ehd hwta h uv sa is curve the that show method, multiplier Lagrange the Using . y u nldn eosrto hti samnmzr(o utasta a just (not minimizer a is it that demonstration a including , ( = ) = y I ∈ [ y y = C 0 φ = ] x 2 1 r osat.*Fn neuto for equation an Find * constants. are u y ( n hwthat show and , ( t − ( R f with ) Z x 2 ( aifislim satisfies ) x ) ,y y, 4 + x x 1 2 > ( t ′ n ˙ and ) I f v ) u uhthat such 0 [ x 2 ( I − u ,y y, ) utstsyi re oraieti oe on.So that Show bound. lower this realize to order in satisfy must 1 1 = (1) [ = ] u 1 2 y y = ] ′ Z ssrcl ovxon convex strictly is Z ′ ∂y x I − ) t ∂ − 1 2 1 [ ( dx t u a ′ Z 2 t Z y a x Z tboth at ) mns uhfntosi ,adgv a give and 8, is functions such amongst ] − 0 −∞ f →−∞ , J −∞ 1+ (1 f a 0 = + a x ( = + ˙ [ x ( ,y y, ∞ φ 1 = (1) ,x t, ∞ →−∞ y y = ] c y , 1+ (1 ( 0 and 0 = (0) 1 2 7 cosh y ′ u u ( constant = ) u ′ t ( ′ 2 ′ ) x ( ) 2 − I u , − J x 1 t n lim and 0 = ) y [ x (1 + ( / ˙ φ [ 1 2 sin 2 ′ = µ x φ 2 ( 2 = ) 2 x , t − dx n lim and 0 = ) ) ,slefor solve ], = ] ) x 1 t , ,b a, 2 = (2) 1 / I − x − ¨ 2 2 = y c u x 2 . and ( dx 1 t R x ( cos n ( and ) ) { )) t 0 2 0 and X ( y htmkssainr h integral the stationary makes that ) L, ,v u, dt dx . ¶ ( ( t 4 1 u X h uv sue yauniform a by assumed curve The c φ , = ) l n hwta thsaunique a has it that show and , n ˙ and − 8 + dx = ) y : ) x φ 2 ( → t ,b a, x 2 tional 4 + x n hwta h solutions the that show and + . . > u 2 I → x = ) ∞ eeuto o eie in derived you equation ge > Y [ culyde minimze does actually iiie h potential the minimises ) u 2 = (2) + φ 8. = ] u fteEuler-Lagrange the of n ∞ ′ 0 ( 2 y x ∈ } ) I 2 u .Mk h change the Make 0. 2 = ) 2 1 [ ( y fixed − dx x = ] 2 = ) R 1 4 hwta the that Show . htminimises that , 2 π Ti a be can (This . 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Not to be quoted or reproduced without permission. xml he 2 sheet Example 7 .Cnie h rbe fminimizing of problem the Consider 2. .Oti h ue-arneeutosascae otefunct the to associated equations Euler-Lagrange the Obtain 3. .Cnie h rbe fmxmzn h area the maximizing of problem the Consider 1. .Oti h ue-arneeutosascae otefunct the to associated equations Euler-Lagrange the Obtain 4. .Consider 6. .Tesot functions smooth The 7. .Consider 5. .Let 8. .*A natraieapoc o(.) let (7.3), to approach alternative an As * 9. au of value I a n tueu orwieteElrLgag qaina ne an as equation Euler-Lagrange the rewrite you to Euler- function useful the minimizing it solve the find to with may solutions relation series the power on of comment method i and the down Use wrote * multiplier. you (iii) Lagrange the equation of Euler-Lagrange value the satisfies function R i)So htudrteassumption the under for that equation Show Euler-Lagrange corresponding (ii) the down Write (i) o hscntandpolmi aaercform. parametric in problem constrained this for i)* (ii) (i) where uhthat such where i)* (ii) (i) where nmmi o tandin attained not is infimum ffie length fixed of xs a exist eo hwta h olwn he odtosaeequivale are conditions three following the that Show zero. h eodexample? second the (ii) (i) y onaycniin n ujc oteconstraint the to subject and conditions boundary y (iii) onaycniin at conditions boundary inator Asm that (Assume h arnemlile ucinfraimt bantefol the obtain to formalism function multiplier Lagrange the o h rbe fminimizing of problem the for h rgn(hyaegetcircles.) great are (they origin the x iae,adasm h uv on curve the assume and dinates, [ ( ( ( ψ ψ a − t ) I I y = 0 = ) + 1 = ] 2 )= 1) I x [ [ Q dx ∈ u u aifis ( satisfies: [ ( I I u [ = ] = ] t u u u R u S [ [ = ] ) C u u 1. = a I = ] b 2 = = : ∈ = ] = ] 1 − hwta h ouin fteElrLgag qainleon lie equation Euler-Lagrange the of solutions the that Show . I wy I y s1adi tando ucinwihsol egvnexplici given be should which function a on attained is and 1 is R R R [ [ R ntjs piecewise just (not u u .B considering By 1. ( R y y y R ( ( 2 R ± ( ( −∞ |∇ 2 3 R = ] a = ] ( 2 1 + 1 2 a b ,y x, ,x t, → b dx b )=1 yconsidering By 1. = 1) u ( (det ∞ eacrewihi osrie oleo h sphere the on lie to constrained is which curve a be y ( R ): pu u t 2 pu py l ( ( sntietclyzr,adthat and zero, identically not is R | safnto on function a is ) R R safnto on function a is ) − n(i)i non-zero.) is (iii) in u = 2 ψ ′ − − ′ ′ 1 2 +1 2 t 2 2 ) + Du ′ 1 1 F n det and , ′ R + − + + (1 ( − 0 e ( 2 xy u ) 2 π qu c qu p xψ − dxdy qy u ( x ˙ ( ( ′ u ) x x u )) 2 2 ) y dxdy ) ) 2 2 ) ) ) = x 2 2 dxdt = 2 dx q , dx/ dx ) dx ˙ + S u , 2 − x 2 ( y Du . dx y x n ec hwta mns uhfntosteminimum the functions such amongst that show hence and , ) hti h au of value the is What . for ssainr at stationary is , y λwy C R dxdt , 2 and ) ǫ a ) ( I b with 1 en h aoindtriat hti nsa about unusual is What determinant. Jacobian the means 2 1 x [ ucinfrwihti au sattained? is value this which for function ) wu y x dt = ) ; R ( S R , = ] x rt onadsleteElrLgag equations Euler-Lagrange the solve and down Write . 2 2 2 w 2 nteset the in ) y where , y dx, and , I ¨x ( sgvnas given is ( arctan rtn1 arctan [ x = x ψ xψ R + ) = ) = ] ssainr amongst stationary is k y ≥ 8 k ( ,φ θ, ( ˙x x ˙x x k x/ǫ r rsrbdo [ on prescribed are 0 | ) k yn nteset the in lying ) /ǫ x F 2 R 2 2 dt −∞ | u 2 1 + esadr nlsgvnb peia coor- spherical by given angles standard be x hwta h min the that show and → hwta inf that show ∞ R > w = S (7.3) 0 = mns uvsstsyn h constraint the satisfying curves amongst 0 φ 2 ¡ π of ψ as 0 R y = ( c a Q ′ x b 2 n( in 0 amongst C r ie mohfunctions. smooth given are y wu φ [ ˙ + y 1 ( − ]? θ x 2 x tfor nt ucin uhthat such functions .So httelnt integral length the that Show ). y dx 2 ,b a, oigElrLgag equation Euler-Lagrange lowing → x ψ ˙ ) ionals ionals 2 constant; = dt y ota ota h denom- the that so that so ) ¢ S ∈ + C C dx hscntandproblem. constrained this C S bandi i) Hr you (Here (ii). in obtained ,b a, ′ nlsdb a by enclosed ∞ uto for quation 1 2 of 1 I i,fra appropriate an for (i), n arneeuto n(i), in equation Lagrange y S functions mns ucin with functions amongst [ ∈ ucin aifigthe satisfying functions ,with ], ucin aifigthe satisfying functions y 2 ti osbet write to possible is it piecewise S .So htthis that Show 0. = ] ′ = I l.Vrf htthis that Verify tly. [ { y x .De there Does 0. = ] w ln through plane a : k y o identically not f x ( y C x = closed k 1 and 1 = (1) 1 satisfying ) 1 = e functions x 2 2 } ψ curve Use . ( x )). Copyright © 2012 University of Cambridge. Not to be quoted or reproduced without permission. 8 For 18. 7 For 17. 2 o h rcitcrn rbe,so htteei uniq a is there that show problem, brachistochrone the For * 12. tr minimum the that show problem, brachistochrone the For * 11. 0 ban(.)b osdrn aitoso h curve the of variations considering by (7.3) Obtain * 10. 3 na pia eimfiln h ein0 region the filling medium optical an In 13. 5 With * 15. 6 o h eghfntoa o uvsi h plane the in curves for functional length the For 16. 4 aitnsPicpei plcbeas othe to also applicable is Principle Hamilton’s * 14. qainadtescn aito xlcty n hwta th that show and explicitly, variation positive. second the and equation h eodvrain o h case the For variation. second the arneeuto.Cmuetescn aito of variation second the Compute equation. Lagrange up rmtesatn on (0 point starting the from cusp) distance a and level same the at saslto.Fn h eodvrain ie()acniino condition a (i) Give variation. second the Find solution. a is inl nert tadso httersligsltosaeg are solutions resulting the that show and it integrate tional, h eodvraini oiie n i)acniinwihen which condition eigenvalue. a negative (ii) one and least positive, at is variation second the F h cino rvt ihu rcin h ceeaindu acceleration The friction. without gravity of action the y hr h lcrcfield electric the where hc i on lie which is eoiyo atceo rest-mass of particle a of velocity potential ihteLrnzfactor Lorentz the with hwta h ah flgtry ntemdu r parabolic. are medium the in rays ( at light medium of the paths enters the that Show otefunctional the to na lcrmgei ed h prpit hieo Lagra of choice appropriate The field. electromagnetic an in where qain,wt hscoc of choice this with equations, ( ′ b l 0 .Wiedw h soitdElrLgag qain and equation, Euler-Lagrange associated the down Write 0. = (0) [ = ) φ I I = ] [ [ y y y = ] = ] E β 0 R ( and A hwta h tagtline straight the that show 1+sin + (1 h < R R 0 a 0 ( S b 1 x ( B ³ stegets au of value greatest the is ) 2 y t , 1 2 ′ n requiring and si h rvosqeto,oti h ue-arneequati Euler-Lagrange the obtain question, previous the in as 2 n ie etrpotential vector given a and ) y I + 2 ′ 2 [ A θφ y + = ] 4 ′ 2 ) F − dx E ) γ ( 1 2 R x y (1 = dθ = 0 ( ) with E , ´ c banteElrLgag qainascae oti func- this to associated equation Euler-Lagrange the Obtain . ∇ )adlae ta ( at it leaves and 0) ( dx 2 L y L A − − dǫ = ) d il h qaino motion of equation the yield , = dt 0 , y d v with I B )t on ( point a to 0) ( l ( − 2 [ α x − ( kx a x /c 2 m pr s(2 is apart m (1 ǫ = ) ) m ǫ ∂ = 0 = ( ∂t dxdt. 2 0 at 0 = ] 0 0 A t ) − y ) 0 γ ) − n charge and y 2 c 0 = 0 = (0) y v ≡ c n h antcfield magnetic the and ,y α, 2 ky 1 4 = 0 = ) = / γ tando h a path. ray the on attained h < y < 9 2 − k ) Ti ie w fMxelsequations). Maxwell’s of two gives (This β n where and , 1 x x y 1 ( ky / b q 0 rt onteslto fteEuler-Lagrange the of solution the down write ( ( ǫ 2 + πl/g t t = ) ( ( 0 + ) + ) x o vr smooth every for 0 = E (1 qA ,Y X, onn ( joining ) (0 x relativistic + I A h pe flgtis light of speed the , 0 ) − β [ ǫ ǫ y 0 q , < k < 1 y ( z z v 1.Asm that Assume (1). ) then 0), / + x = ] ky n h ue-arneeuto and equation Euler-Lagrange the find eo h trigpoint. starting the below ) ( ( nfilsdtrie yagvnscalar given a by determined fields in 2 × t t t , x I ) ) q fraba oigo ieunder wire a on moving bead a (for 0 k ( x .Vrf htteEuler-Lagrange the that Verify ). v B ) at R t , fteform the of ) a stepsto and position the is · b ) 1 1+ (1 , A ogaiyis gravity to e y ,α a, /h yaiso hre particle charged a of dynamics 0 , ngian ue h eodvrainhas variation second the sures eaco yli wtota (without cycloid a of arc ue ) n hwta ti positive. is it that show and vltm ewe w points two between time avel . etcircles. reat o( to ) eodvraini strictly is variation second e B n y ′ 2 F = ) hwas ht faray a if that, also Show L ′′ 2 1 × ∇ 0 hc nue that ensures which (0) ,β b, dx [ z t, ( hwthat show F with , x t ovsteEuler- the solves ) ). ( A t ∈ g ) .) . , C v ˙x n associated ons ( y 2 = t ( ( ]is )] R a = ) y ˙x satisfies ) ( 0 t ( sthe is ) x α 0 = ) and Copyright © 2012 University of Cambridge. Not to be quoted or reproduced without permission. diinlquestions Additional 8 1 i osdrtefunctional the Consider (i) 11. 0 rt onteElrLgag qainascae to associated equation Euler-Lagrange the down Write 10. .*Consider 5. .Fn h aiu oueo etnua aallppdi parallelopiped rectangular a of volume maximum the Find 3. .Fn h aitna bandvateLgnr transforma Legendre the via obtained Hamiltonian the Find 8. .*hwta if that *Show 4. good are exams methods recent from questions following The 1. .Fn h aitna o h eaiitcdnmc facha a of dynamics relativistic the for Hamiltonian the Find 9. .A o aypit in points many how At 2. .Consider 7. .Cnie h rao ufc bandb oaigacurve a rotating by obtained surface a of area the Consider 6. n compute and ih eiaieat derivative right f h nevl iea xml facne ucindfie onl defined function convex a of example an Give interval. the hti h oanof domain the if that derivative D hwta o any for that show lim hwthat show i)Nwcnie h aethat case the consider Now (ii) equation. h ihteraiycniin ¯ conditions reality the with ucin ihzr mean: zero with functions eist n h ievle fteascae Sturm-Liouvill associated the of eigenvalues the find to series R x nsetII. sheet in aeismnmmvle hwta hslatvleis value least this that Show value? minimum its take /I1E 06 //Aad4/II/16B. an and 1/II/14D 2/I/5A 2008 2006: etc: 4/II/16E, equations Euler-Lagrange multipliers, eedetasomto oteLagrangian the to transformation Legendre hna ela h ue-arneeyto tstse h addi the satisfies it eqyation Euler-Lagrange the as f well as then adepit twihtesraeo vanishing of surface the which at point, saddle osrainlaw conservation ue-arneequation. Euler-Lagrange tan function y h aua onaycniinfor condition boundary natural the hc r osqec favrainlpolmi hswyar way this in problem variational a equa of Euler-Lagrange consequence order a second are the which for conditions boundary 1 2 y ( 2 ) g n + b ′ : /a − ij ( ? = ) x − f ,y b, R x → f ˙ 2 1 ( 3 i x ( ( + + x ˙ √ x ( utdfie sfiiefrall for finite is defined just ) β → ∞ j b )) y 2). ) f − I bu the about y , 2 u > h , u D R [ /b : I ( y V ( ′ x [ R osdrvrain fteform the of variations Consider . + = ] ( x y 2 ( 2 = ) b 3 f arctan 4 = ) = ] f x dǫ + )=0 oehrwt h nta odto hsgvstecor the gives this condition initial the with Together 0. = )) d ( → smaincneto assumed). convention (summation ) r o-eraigi.e. non-decreasing are 0 ( : x R I z a y ) [ b 2 ,b a, x R y R π φ ′ x /c ≡ f f a ǫ b ( i aclt h eodvrain n i) s h ehdo method the use (ii)* and variation, second the Calculate (i) . hwta if that Show . ∈ ] y y ( x is 0 = | f 2 ) ′ ,y y y, x, lim ai.Wiedw nitga o h ra n ov h associat the solve and area, the for integral an down Write -axis. u ǫ 1 f R =0 ( − 1. = → ( x , ,y y y, x, x R sol nitra httesm stu for true is same the that interval an only is the h hwta if that show ; f = ) 2 3 → R R e x , = ′ − x −∞ ostefunction the does 0+ + ) u scne h n-ie ieec quotients difference one-sided the convex is I φ dx π 3 saslto ihbudr odtoslim conditions boundary with solution a is ′ 0 π constant [ n ) = ) x ,h> < u dx u ( X = ] | = . ,f u, with h n ( x 0 |≤ r one eo for below bounded are ) 4 1 with ) u φ y N dx ( R − x x I r ie yfiiesm fexponentials: of sums finite by given are ∈ − x ( y n + u [ 1 4 h u π ( , = 0 = n C π holds. + o ovxfnto ihdomain with function convex a for xssin exists ) a = ] f y e ¡ ¯ y 2 = ) n φ ( inx u x 10 a 2 minimizes ssc htti szr o l such all for zero is this that such is x x 2 2 4 = = ) R ( f , h − + B L α R ) f ¡ − x = but fu − + ≤ 2 1 ,y α, y φ 3 4 π n π |∇ ( ǫ ) − ¢ x φ ( ∞∪ −∞ stnett obecn fsemi-angle of cone double a to tangent is f − and dx x y = ) m x u ( ( = ) ( b ( x | x I 0 b k 2 = ) ) sntfie.A usual As fixed. not is ) where c 2 f0 if ) dx − amongst 0 2 x N I < γ R y [ 3 u X gu rt onteEuler-Lagrange the down Write . | − ( β yconsidering By . n > h − n oiieitgr hwthat Show integer. positive any = ] x − 1 |≤ where , ¢ h < x − + .So lothat also Show 3. dx u info h Lagrangian the from tion N gdpril yapyn the applying by particle rged 3 operator. e y //D 07 //Eand 3/I/6E 2007: 2/I/5D, d R ota h ih derivative right the that so 0 qA x ǫφ f srbdisd nellipsoid an inside nscribed and −∞ + C o rciewt Lagrange with practice for where 1 = n in onayconditions Boundary tion. ≤ inlbudr condition boundary tional n[0 on y ∞ ( e called e 0 − 1 x inx k y − ucin with functions )where )) f 1 2 f eueta h right the that Deduce . x ( x u 1 q x sasot function smooth a is r el2 real are with ) ′ x v B 2 x 2 nitro on of point interior an φ +(1 · , →−∞ ∞ A steui alin ball unit the is x natural ( φ , h R o hc the which for ) etnme of number rect x − hc appears which = ) φ y − f like u cos ( l ∈ ( a ( sasmooth a is π φ x l = ) h φ htis What . for ) periodic - and 0 = ) C u − y f hnthe then a one has 0 ) ∞ 1 ( power f Show . dx a ( α ( f = ) > l ,b a, ( L and and x ed + = ), α 0 Copyright © 2012 University of Cambridge. Not to be quoted or reproduced without permission. J Hn:lo pCno diagonalization.) Cantor up look (Hint: n hwta mns eune uhthat such sequences amongst that show and i) AtrMtosadAayi I xedyu rueti ( in argument your Extend II] Analysis and Methods (ii). [After in (iv)* obtained you one the u s opeino h qaet hwta h iiu of minimum the that show to square the of completion Use inf h ttoaycniinstse ymnmzr for minimizers by satisfied condition stationary the point I ii*Uetedrc ehdt rv h xsec faminimiz a of existence the prove to method direct the Use (iii)* hwthat show [ ∞ n hwta h orsodn function corresponding the that show and , u v 2 = ] okudrteasmto that assumption the under Work . ∈ C u N πJ hc samnmzr,i.e. , minimizer a is which J N N J [ v [ N u as ] where ] sbuddblw n let and below, bounded is α ∞ → u ( = hwta hr sasbeunewihcnegst limit a to converges which subsequence a is there that Show . J u N 1 u , [ u 2 = ] u . . . , J n X N N =1 f [ n u ) n sgvnb nasltl ovretFuirseries. Fourier convergent absolutely an by given is inf = ] 11 2 ∈ | u C { P n u N u | 2 α n ∞ v =1 ovsteElrLgag qaini (i). in equation Euler-Lagrange the solves } and ∈ − α ∞ C =1 f n ¯ N n 2 J u J | N u easqec uhthat such sequence a be n N n htti iiie stesm as same the is minimizer this that , − [ | v 2 .Fnly eueb considering by deduce Finally, ]. f < n J u ¯ ∞ N n satie o oeunique some for attained is hr soeta minimizes that one is there rfor er i)t h case the to iii) J N sflos First follows. as J N N [ u + = α ] → ∞