Euler lagrange equation

1. Chapter2
TheEuler-Lagrange equation
Inthischapter,wewillgivenecessaryconditionsforanextremumofafunctionofthetype
I(x)=
Z
b
a
F(x(t);x
0
(t);t)dt;
withvarioustypesofboundaryconditions.Thenecessaryconditionisintheformofadierential
equationthattheextremalcurveshouldsatisfy,andthisdierentialequationiscalledtheEuler-
Lagrangeequation.
Webeginwiththesimplesttypeofboundaryconditions,wherethecurvesareallowedtovary
betweentwoxedpoints.
2.1 Thesimplest optimisation problem
Thesimplestoptimisationproblemcanbeformulatedasfollows:
LetF(;;)beafunctionwithcontinuousrstandsecondpartialderivativeswithrespectto
(;;).Thenndx2C
1
[a;b]suchthatx(a)=yaandx(b)=yb,andwhichisanextremumfor
thefunction
I(x)=
Z
b
a
F(x(t);x
0
(t);t)dt: (2.1)
Inotherwords,thesimplestoptimisationproblemconsistsofndinganextremumofafunction
oftheform(2.5),wheretheclassofadmissiblecurvescomprisesallsmoothcurvesjoiningtwo
xedpoints;seeFigure2.1.Wewillapplythenecessaryconditionforanextremum(established
inTheorem1.4.2)tothesolvethesimplestoptimisationproblemdescribedabove.
Theorem2.1.1LetS=fx2C
1
[a;b]jx(a)=yaandx(b)=ybg,andletI:S!Rbea
functionoftheform
I(x)=
Z
b
a
F(x(t);x
0
(t);t)dt:
IfIhasanextremumatx02S,thenx0satisestheEuler-Lagrangeequation:
@F
@
(x0(t);x
0
0(t);t)
d
dt

@F
@
(x0(t);x
0
0(t);t)

=0;t2[a;b]: (2.2)
27

2. 28 Chapter2.TheEuler-Lagrangeequation
PSfragreplacements
a b
t
ya
yb
Figure2.1:Possiblepathsjoiningthetwoxedpoints(a;ya)and(b;yb).
ProofTheproofislongandsowedivideitintoseveralsteps.
Step1.FirstofallwenotethatthesetSisnotavectorspace(unlessya=0=yb)!SoTheorem
1.4.2isnotapplicabledirectly.HenceweintroduceanewlinearspaceX,andconsideranew
function
~
I:X!RwhichisdenedintermsoftheoldfunctionI.
Introducethelinearspace
X=fx2C
1
[a;b]jx(a)=x(b)=0g;
withtheinducednormfromC
1
[a;b].Thenforallh2X,x0+hsatises(x0+h)(a)=yaand
(x0+h)(b)=yb.Dening
~
I(h)=I(x0+h),forh2X,wenotethat
~
I:X!Rhasalocal
extremumat0.ItfollowsfromTheorem1.4.2that
1
D
~
I(0)=0.
Step2.WenowcalculateD
~
I(0).Wehave
~
I(h)
~
I(0)=
Z
b
a
F((x0+h)(t);(x0+h)
0
(t);t)dt
Z
b
a
F(x0(t);x
0
0
(t);t)dt
=
Z
b
a
[F(x0(t)+h(t);x
0
0(t)+h
0
(t);t)dtF(x0(t);x
0
0(t);t)]dt:
RecallthatfromTaylor'stheorem,ifFpossessespartialderivativesoforder2inaballBofradius
raroundthepoint(0;0;0)inR
3
,thenforall(;;)2B,thereexistsa2[0;1]suchthat
F(;;)=F(0;0;0)+

(0)
@
@
+(0)
@
@
+(0)
@
@

F




(0;0;0)
+
1
2!

(0)
@
@
+(0)
@
@
+(0)
@
@

2
F





(0;0;0)+((;;)(0;0;0))
:
Henceforh2Xsuchthatkhkissmallenough,
~
I(h)
~
I(0)=
Z
b
a

@F
@
(x0(t);x
0
0(t);t)h(t)+
@F
@
(x0(t);x
0
0(t);t)h
0
(t)

dt+
1
2!
Z
b
a

h(t)
@
@
+h
0
(t)
@
@

2
F



(x0(t)+(t)h(t);x
0
0
(t)+(t)h
0
(t);t)
dt:
ItcanbecheckedthatthereexistsaM>0suchthat





1
2!
Z
b
a

h(t)
@
@
+h
0
(t)
@
@

2
F



(x0(t)+(t)h(t);x
0
0
(t)+(t)h
0
(t);t)
dt





Mkhk
2
;
1
Notethatbythe0intherighthandsideoftheequality,wemeanthezeromap,namelythecontinuouslinear
mapfromXtoR,whichisdenedbyh7!0forallh2X.

3. 2.1.Thesimplestoptimisationproblem 29
andsoD
~
I(0)isthemap
h7!
Z
b
a

@F
@
(x0(t);x
0
0(t);t)h(t)+
@F
@
(x0(t);x
0
0(t);t)h
0
(t)

dt: (2.3)
Step3.Nextweshowthatifthemapin(2.3)isthezeromap,thenthisimpliesthat(2.2)holds.
Dene
A(t)=
Z
t
a
@F
@
(x0();x
0
0();)d:
Integratingbyparts,wendthat
Z
b
a
@F
@
(x0(t);x
0
0(t);t)h(t)dt=
Z
b
a
A(t)h
0
(t)dt;
andsofrom(2.3),itfollowsthatD
~
I(0)=0impliesthat
Z
b
a

A(t)+
@F
@
(x0(t);x
0
0(t);t)

h
0
(t)dt=0forallh2X:
Step4.Finally,usingLemma1.4.5,weobtain
A(t)+
@F
@
(x0(t);x
0
0(t);t)=kforallt2[a;b]:
Dierentiatingwithrespecttot,weobtain(2.3).ThiscompletestheproofofTheorem2.1.1.
NotethattheEuler-Lagrangeequationisonlyanecessaryconditionfortheexistenceofan
extremum(seetheremarkfollowingTheorem1.4.2).However,inmanycases,theEuler-Lagrange
equationbyitselfisenoughtogiveacompletesolutionoftheproblem.Infact,theexistenceof
anextremumissometimesclearfromthecontextoftheproblem.Ifinsuchscenarios,thereexists
onlyonesolutiontotheEuler-Lagrangeequation,thenthissolutionmustafortioribethepoint
forwhichtheextremumisachieved.
Example.LetS=fx2C
1
[0;1]jx(0)=0andx(1)=1g.ConsiderthefunctionI:S!R
givenby
I(x)=
Z
1
0

d
dt
x(t)1

2
dt:
Wewishtondx02SthatminimizesI.Weproceedasfollows:
Step1.WehaveF(;;)=(1)
2
,andso
@F
@
=0and
@F
@
=2(1).
Step2.TheEuler-Lagrangeequation(2.2)isnowgivenby
0
d
dt
(2(x
0
0
(t)1))=0forallt2[0;1]:
Step3.Integrating,weobtain2(x
0
0
(t)1)=C,forsomeconstantC,andsox
0
0
=
C
2
+1=:A.
Integratingagain,wehavex0(t)=At+B,whereAandBaresuitableconstants.

4. 30 Chapter2.TheEuler-Lagrangeequation
Step4.TheconstantsAandBcanbedeterminedbyusingthatfactthatx02S,andso
x0(0)=0andx0(a)=1.Thuswehave
A0+B=0;
A1+B=1;
whichyieldA=1andB=0.
Sotheuniquesolutionx0oftheEuler-LagrangeequationinSisx0(t)=t,t2[0;1];see
Figure2.2.
PSfragreplacements
0
1
1
x0
t
Figure2.2:MinimizerforI.
Nowwearguethatthesolutionx0indeedminimizesI.Since(x
0
(t)1)
2
0forallt2[0;1],
itfollowsthatI(x)0forallx2C
1
[0;1].But
I(x0)=
Z
1
0
(x
0
0
(t)1)
2
dt=
Z
1
0
(11)
2
dt=
Z
1
0
0dt=0:
AsI(x)0=I(x0)forallx2S,itfollowsthatx0minimizesI.
Denition.ThesolutionsoftheEuler-Lagrangeequation(2.3)arecalledcriticalcurves.
TheEuler-Lagrangeequationisingeneralasecondorderdierentialequation,butinsome
specialcases,itcanbereducedtoarstorderdierentialequationorwhereitssolutioncanbe
obtainedentirelybyevaluatingintegrals.WeindicatesomespecialcasesinExercise3onpage31,
whereineachinstance,Fisindependentofoneofitsarguments.
Exercises.
1.LetS=fx2C
1
[0;1]jx(0)=0=x(1)g.ConsiderthemapI:S!Rgivenby
I(x)=
Z
1
0
(x(t))
3
dt;x2S:
UsingTheorem2.1.1,ndthecriticalcurvex02SforI.DoesIhavealocalextremumat
x0?
2.WritetheEuler-LagrangeequationwhenFisgivenby
(a)F(;;)=sin,
(b)F(;;)=
3

3
,
(c)F(;;)=
2

2
,
(d)F(;;)=2
2
+3
2
.

5. 2.1.Thesimplestoptimisationproblem 31
3.Provethat:
(a)IfF(;;)doesnotdependon,thentheEuler-Lagrangeequationbecomes
@F
@
(x(t);x
0
(t);t)=c;
whereCisaconstant.
(b)IfFdoesnotdependon,thentheEuler-Lagrangeequationbecomes
@F
@
(x(t);x
0
(t);t)=0:
(c)IfFdoesnotdependonandifx0istwice-dierentiablein[a;b],thentheEuler-
Lagrangeequationbecomes
F(x(t);x
0
(t);t)x
0
(t)
@F
@
(x(t);x
0
(t);t)=C;
whereCisaconstant.
Hint:Whatis
d
dt

F(x(t);x
0
(t);t)x
0
(t)
@F
@
(x(t);x
0
(t);t)

?
4.Findthecurvewhichhasminimumlengthbetween(0;0)and(1;1).
5.LetS=fx2C
1
[0;1]jx(0)=0andx(1)=1g.FindcriticalcurvesinSforthefunctions
I:S!R,whereIisgivenby:
(a)I(x)=
Z
1
0
x
0
(t)dt
(b)I(x)=
Z
1
0
x(t)x
0
(t)dt
(c)I(x)=
Z
1
0
(x(t)+tx
0
(t))dt
forx2S.
6.Findcriticalcurvesforthefunction
I(x)=
Z
2
1
t
3
(x
0
(t))
2
dt
wherex2C
1
[1;2]withx(1)=5andx(2)=2.
7.Findcriticalcurvesforthefunction
I(x)=
Z
2
1
(x
0
(t))
3
t
2
dt
wherex2C
1
[1;2]withx(1)=1andx(2)=7.
8.Findcriticalcurvesforthefunction
I(x)=
Z
1
0

2tx(t)(x
0
(t))
2
+3x
0
(t)(x(t))
2

dt
wherex2C
1
[0;1]withx(0)=0andx(1)=1.

6. 32 Chapter2.TheEuler-Lagrangeequation
9.Findcriticalcurvesforthefunction
I(x)=
Z
1
0

2(x(t))
3
+3t
2
x
0
(t)

dt
wherex2C
1
[0;1]withx(0)=0andx(1)=1.Whatifx(0)=0andx(1)=2?
10.Considerthecopperminingcompanymentionedintheintroduction.Iffuturemoneyis
discountedcontinuouslyataconstantrater,thenwecanassessthepresentvalueofprots
fromthisminingoperationbyintroducingafactorofe
rt
intheintegrandof(1.37).Suppose
that=4,=1,r=1andP=2.Findacriticalminingoperationx0suchthatx0(0)=0
andx0(T)=Q.
2.2 Calculus ofvariations: some classicalproblems
Optimisationproblemsofthetypeconsideredintheprevioussectionwerestudiedinvarious
specialcasesbymanyleadingmathematiciansinthepast.Thesewereoftensolvedbyvarious
techniques,andthesegaverisetothebranchofmathematicsknownasthe`calculusofvariations'.
Thenamecomesfromthefactthatoftentheprocedureinvolvedthecalculationofthe`variation'
inthefunctionIwhenitsargument(whichwastypicallyacurve)waschanged,andthenpassing
limits.Inthissection,wementiontwoclassicalproblems,andindicatehowthesecanbesolved
usingtheEuler-Lagrangeequation.
2.2.1Thebrachistochroneproblem
ThecalculusofvariationsoriginatedfromaproblemposedbytheSwissmathematicianJohann
Bernoulli(1667-1748).HerequiredtheformofthecurvejoiningtwoxedpointsAandBina
verticalplanesuchthatabodyslidingdownthecurve(undergravityandnofriction)travelsfrom
AtoBinminimumtime.Thisproblemdoesnothaveatrivialsolution;thestraightlinefromA
toBisnotthesolution(thisisalsointuitivelyclear,sinceiftheslopeishighatthebeginning,
thebodypicksupahighvelocityandsoitsplausiblethatthetraveltimecouldbereduced)and
itcanbeveriedexperimentallybyslidingbeadsdownwiresinvariousshapes.


PSfragreplacements
A(0;0)
B(x0;y0)
gravity
y0
x0
x
y
Figure2.3:Thebrachistochroneproblem.
Toposetheprobleminmathematicalterms,weintroducecoordinatesasshowninFigure2.3,
sothatAisthepoint(0;0),andBcorrespondsto(x0;y0).Assumingthattheparticleisreleased
fromrestatA,conservationofenergygives
1
2
mv
2
mgy=0,wherewehavetakenthezero
potentialenergylevelaty=0,andwherevdenotesthespeedoftheparticle.Thusthespeedis
givenbyv=
ds
dt
=
p
2gy,wheresdenotesarclengthalongthecurve.FromFigure2.4,wesee
thatanelementofarclength,sisgivenapproximatelyby((x)
2
+(y)
2
)
1
2.

7. 2.2.Calculusofvariations:someclassicalproblems 33
PSfragreplacements
y
x
s
Figure2.4:Elementofarclength.
Hencethetimeofdescentisgivenby
T=
Z
curve
ds
p
2gy
=
1
p
2g
Z
y0
0
v
u
u
t 1+

dx
dy

2
y
dy:
Ourproblemistondthepathfx(y);y2[0;y0]g,satisfyingx(0)=0andx(y0)=x0,which
minimizesT,thatis,todeterminetheminimizerforthefunctionI:S!R,where
I(x)=
1
p
2g
Z
y0
0

1+(x
0
(y))
2
y
1
2
dy;x2S;
andS=fx2C
1
[0;y0]jx(0)=0andx(y0)=x0g.Here
2
F(;;)=
q
1+
2

isindependentof
,andsotheEuler-Lagrangeequationbecomes
d
dy

x
0
(y)
p
1+(x
0
(y))
2
1
p
y
!
=0:
Integratingwithrespecttoy,weobtain
x
0
(y)
p
1+(x
0
(y))
2
1
p
y
=C;
whereCisaconstant.Itcanbeshownthatthegeneralsolutionofthisdierentialequationis
givenby
x()=
1
2C
2
(sin)+
~
C;y()=
1
2C
2
(1cos);
where
~
Cisanotherconstant.Theconstantsarechosensothatthecurvepassesthroughthepoints
(0;0)and(x0;y0).
PSfragreplacements
(0;0)
(x0;y0)
x
y
Figure2.5:Thecycloidthrough(0;0)and(x0;y0).
Thiscurveisknownasacycloid,andinfactitisthecurvedescribedbyapointPinacircle
thatrollswithoutslippingonthexaxis,insuchawaythatPpassesthrough(x0;y0);seeFigure
2.5.
2
Strictlyspeaking,theFheredoesnotsatisfythedemandsmadeinTheorem2.1.1.Notwithstandingthisfact,
withsomeadditionalargument,thesolutiongivenherecanbefullyjustied.

8. 34 Chapter2.TheEuler-Lagrangeequation
2.2.2 Minim um surface area of rev olution
Theproblemofminimumsurfaceareaofrevolutionistondamongallthecurvesjoiningtwo
givenpoints(x0;y0)and(x1;y1),theonewhichgeneratesthesurfaceofminimumareawhen
rotatedaboutthexaxis.
Theareaofthesurfaceofrevolutiongeneratedbyrotatingthecurveyaboutthexaxisis
S(y)=2
Z
x1
x0
y(x)
p
1+(y
0
(x))
2
dx:
Sincetheintegranddoesnotdependexplicitlyonx,theEuler-Lagrangeequationis
F(y(x);y
0
(x);x)y
0
(x)
@F
@
(y(x);y
0
(x);x)=C;
whereCisaconstant,thatis,
y
p
1+(y
0
)
2
y
(y
0
)
2
p
1+(y
0
)
2
=C:
Thusy=C
p
1+(y
0
)
2
,anditcanbeshownthatthisdierentialequationhasthegeneralsolution
y(x)=Ccosh

x+C1
C

: (2.4)
Thiscurveiscalledacatenary.ThevaluesofthearbitraryconstantsCandC1aredetermined
bytheconditionsy(x0)=y0andy(x1)=y1.Itcanbeshownthatthefollowingthreecasesare
possible,dependingonthepositionsofthepoints(x0;y0)and(x1;y1):
1.Ifasinglecurveoftheform(2.4)passesthroughthepoints(x0;y0)and(x1;y1),thenthis
curveisthesolutionoftheproblem;seeFigure2.6.
PSfragreplacements
x0 x1
y0
y1
Figure2.6:Thecatenarythrough(x0;y0)and(x1;y1).
2.Iftwocriticalcurvescanbedrawnthroughthepoints(x0;y0)and(x1;y1),thenoneofthe
curvesactuallycorrespondstothesurfaceofrevolutionifminimumarea,andtheotherdoes
not.
3.Iftheredoesnotexistacurveoftheform(2.4)passingthroughthepoints(x0;y0)and
(x1;y1),thenthereisnosurfaceintheclassofsmoothsurfacesofrevolutionwhichachieves
theminimumarea.Infact,ifthelocationofthetwopointsissuchthatthedistancebetween
themissucientlylargecomparedtotheirdistancesfromthexaxis,thentheareaofthe
surfaceconsistingoftwocirclesofradiusy0andy1willbelessthantheareaofanysurface
ofrevolutiongeneratedbyasmoothcurvepassingthroughthepoints;seeFigure2.7.

9. 2.3.Freeboundaryconditions 35
PSfragreplacements
x0 x1
y0
y1
Figure2.7:Thepolygonalcurve(x0;y0)(x0;0)(x1;0)(x1;y1).
Thisisintuitivelyexpected:imagineasoapbubblebetweenconcentricringswhichare
beingpulledapart.Initiallywegetasoapbubblebetweentheserings,butifthedistance
separatingtheringsbecomestoolarge,thenthesoapbubblebreaks,leavingsoaplmson
eachofthetworings.Thisexampleshowsthatacriticalcurveneednotalwaysexistinthe
classofcurvesunderconsideration.
2.3 Freeboundary conditions
Besidesthesimplestoptimisationproblemconsideredintheprevioussection,wenowconsiderthe
optimisationproblemwithfreeboundaryconditions(seeFigure2.8).
PSfragreplacements
a b
t
Figure2.8:Freeboundaryconditions.
LetF(;;)beafunctionwithcontinuousrstandsecondpartialderivativeswithrespect
to(;;).Thenndx2C
1
[a;b]whichisanextremumforthefunction
I(x)=
Z
b
a
F

x(t);
dx
dt
(t);t

dt: (2.5)
Theorem2.3.1LetI:C
1
[a;b]!Rbeafunctionoftheform
I(x)=
Z
b
a
F

x(t);
dx
dt
(t);t

dt;x2C
1
[a;b];
whereF(;;)isafunctionwithcontinuousrstandsecondpartialderivativeswithrespectto
(;;).IfIhasalocalextremumatx0,thenx0satisestheEuler-Lagrangeequation:
@F
@

x0(t);
dx0
dt
(t);t


d
dt

@F
@

x0(t);
dx0
dt
(t);t

=0;t2[a;b]; (2.6)

10. 36 Chapter2.TheEuler-Lagrangeequation
togetherwiththetransversalityconditions
@F
@

x0(t);
dx0
dt
(t);t




t=a
=0and
@F
@

x0(t);
dx0
dt
(t);t




t=b
=0: (2.7)
Proof
Step1.WetakeX=C
1
[a;b]andcomputeDI(x0).ProceedingasintheproofofTheorem2.1.1,
itiseasytoseethat
DI(x0)(h)=
Z
b
a

@F
@
(x0(t);x
0
0(t);t)h(t)+
@F
@
(x0(t);x
0
0(t);t)h
0
(t)

dt;
h2C
1
[a;b].Theorem1.4.2impliesthatthislinearfunctionalmustbethezeromap,thatis,
(DI(x0))(h)=0forallh2C
1
[a;b].Inparticular,itisalsozeroforallhinC
1
[a;b]suchthat
h(a)=h(b)=0.ButrecallthatinStep3andStep4oftheproofofTheorem2.1.1,weproved
thatif
Z
b
a

@F
@
(x0(t);x
0
0(t);t)h(t)+
@F
@
(x0(t);x
0
0(t);t)h
0
(t)

dt=0 (2.8)
forallhinC
1
[a;b]suchthath(a)=h(b)=0,thenthisimpliesthatheEuler-Lagrangeequation
(2.6)holds.
Step2.Integrationbypartsin(2.8)nowgives
DI(x0)(h)=
Z
b
a

@F
@
(x0(t);x
0
0(t);t)
d
dt

@F
@
(x0(t);x
0
0(t);t)

h(t)dt+ (2.9)
@F
@
(x0(t);x
0
0
(t);t)h(t)




t=b
t=a
(2.10)
=0+
@F
@
(x0(t);x
0
0(t);t)




t=b
h(b)
@F
@
(x0(t);x
0
0(t);t)




t=a
h(a):
Theintegralin(2.9)vanishessincewehaveshowninStep1abovethat(2.6)holds.Thusthe
conditionDI(x0)=0nowtakestheform
@F
@
(x0(t);x
0
0
(t);t)




t=b
h(b)
@F
@
(x0(t);x
0
0
(t);t)




t=a
h(a)=0;
fromwhich(2.7)follows,sincehisarbitrary.Thiscompletestheproof.
Exercises.
1.Findallcurvesy=y(x)whichhaveminimumlengthbetweenthelinesx=0andtheline
x=1.
2.Findcriticalcurvesforthefollowingfunction,whenthevaluesofxarefreeattheendpoints:
I(x)=
Z
1
0
1
2

(x
0
(t))
2
+x(t)x
0
(t)+x
0
(t)+x(t)

dt:

11. 2.3.Freeboundaryconditions 37
Similarly,wecanalsoconsiderthemixedcase(seeFigure2.9),whenoneendofthecurveis
xed,sayx(a)=ya,andtheotherendisfree.Thenitcanbeshownthatthecurvexsatises
theEuler-Lagrangeequation,thetransversalitycondition
@F
@
(x0(t);x
0
0
(t);t)




t=a
h(a)=0
atthefreeendpoint,andx(a)=yaservesastheotherboundarycondition.


PSfragreplacements
a ab b
ya
yb
tt
Figure2.9:Mixedcases.
Wecansummarizetheresultsbythefollowing:criticalcurvesfor(2.5)satisfytheEuler-
Lagrangeequation(2.6)andmoreoverthereholds
@F
@
(x0(t);x
0
0
(t);t)=0atthefreeendpoint:
Exercises.
1.Findthecurvey=y(x)whichhasminimumlengthbetween(0;0)andthelinex=1.
2.ThecostofamanufacturingprocessinanindustryisdescribedbythefunctionIgivenby
I(x)=
Z
1
0

1
2
(x
0
(t))
2
+x(t)

dt;
forx2C
1
[0;1]withx(0)=1.
(a)Ifalsox(1)=0,ndacriticalcurvexforI,andndI(x).
(b)Ifx(1)isnotspecied,ndacriticalcurvexforI,andndI(x).
(c)WhichofthevaluesI(x)andI(x)foundinpartsaboveislarger?Explainwhy
youwouldexpectthis,assumingthatxandxinfactminimizeIontherespective
domainsspeciedabove.
3.Findcriticalcurvesforthefollowingfunctions:
(a)I(x)=
Z
2
0

(x(t))
2
(x
0
(t))
2

dt,x(0)=0andx


2

isfree.
(b)I(x)=
Z
2
0

(x(t))
2
(x
0
(t))
2

dt,x(0)=1andx


2

isfree.
4.DeterminethecurvesthatmaximizethefunctionI:S!R,whereI(x)=
Z
1
0
cos(x
0
(t))dt,
x2SandS=fx2C
1
[0;1]jx(0)=0g.WhatarethecurvesthatminimizeI?

12. 38 Chapter2.TheEuler-Lagrangeequation
2.4 Generalization
TheresultsinthischaptercanbegeneralizedtothecasewhentheintegrandFisafunctionof
morethanoneindependentvariable:ifwewishtondextremumvaluesofthefunction
I(x1;:::;xn)=
Z
b
a
F

x1(t);:::;xn(t);
dx1
dt
(t);:::;
dxn
dt
(t);t

dt;
whereF(1;:::;n;1;:::;n;)isafunctionwithcontinuouspartialderivativesoforder2,
andx1;:::;xnarecontinuouslydierentiablefunctionsofthevariablet,thenfollowingasimilar
analysisasbefore,weobtainnEuler-Lagrangeequationstobesatisedbytheoptimalcurve,that
is,
@F
@k
(x1(t);:::;xn(t);x
0
1(t);:::;x
0
n(t);t)
d
dt

@F
@k
(x1(t);:::;xn(t);x
0
1(t);:::;x
0
n(t);t)

=0;
fort2[a;b],k2f1;:::;ng.Alsoatanyendpointwherexkisfree,
@F
@k

x1(t);:::;xn(t);
dx1
dt
(t);:::;
dxn
dt
(t);t

=0:
Exercise.Findcriticalcurvesofthefunction
I(x1;x2)=
Z
2
0

(x
0
1
(t))
2
+(x
0
2
(t))
2
+2x1(t)x2(t)

dt
suchthatx1(0)=0,x1


2

=1,x2(0)=0,x2


2

=1.
Remark.Notethatwiththeaboveresult,wecanalsosolvetheproblemofndingextremal
curvesforafunctionofthetype
I(x):
Z
b
a
F

x(t);
dx
dt
(t);:::;
d
n
x
dt
n
(t);t

dt;
foroverall(sucientlydierentiable)curvesxdenedonaninterval[a;b],takingvaluesinR.
Indeed,wemayintroducetheauxiliaryfunctions
x1(t)=x(t);x2(t)=
dx
dt
(t);:::;xn(t)=
d
n
x
dt
n
(t);t2[a;b];
andconsidertheproblemofndingextremalcurvesforthenewfunction
~
Idenedby
~
I(x1;:::;xn)=
Z
b
a
F(x1(t);x2(t);:::;xn(t);t)dt:
Usingtheresultmentionedinthissection,wecanthensolvethisproblem.Notethatweeliminated
highorderderivativesatthepriceofconvertingthescalarfunctionintoavector-valuedfunction.
Sincewecanalwaysdothis,thisisoneofthereasonsinfactforconsideringfunctionsofthetype
(1.28)wherenohighorderderivativesoccur.
2.5 Optimisation infunction spacesversusthatinR
n
Tounderstandthebasic`innite-dimensionality'intheproblemsofoptimisationinfunctionspaces,
itisinterestingtoseehowtheyarerelatedtotheproblemsofthestudyoffunctionsofnreal

13. 2.5.OptimisationinfunctionspacesversusthatinR
n
39
variables.Thus,considerafunctionoftheform
I(x)=
Z
b
a
F

x(t);
dx
dt
(t);t

dt;x(a)=ya;x(b)=yb:
Hereeachcurvexisassignedacertainnumber.Tondarelatedfunctionofthesortconsidered
inclassicalanalysis,wemayproceedasfollows.Usingthepoints
a=t0;t1;:::;tn;tn+1=b;
wedividetheinterval[a;b]inton+1equalparts.Thenwereplacethecurvefx(t);t2[a;b]gby
thepolygonallinejoiningthepoints
(t0;ya);(t1;x(t1));:::;(tn;x(tn));(tn+1;yb);
andweapproximatethefunctionIatxbythesum
In(x1;:::;xn)=
n
X
k=1
F

xk;
xkxk1
hk
;tk

hk; (2.11)
wherexk=x(tk)andhk=tktk1.Eachpolygonallineisuniquelydeterminedbytheordinates
x1;:::;xnofitsvertices(recallthatx0=yaandxn+1=ybarexed),andthesum(2.11)is
thereforeafunctionofthenvariablesx1;:::;xn.Thusasanapproximation,wecanregardthe
optimisationproblemastheproblemofndingtheextremaofthefunctionIn(x1;:::;xn).
Insolvingoptimisationproblemsinfunctionspaces,Eulermadeextensiveuseofthis`method
ofnitedierences'.Byreplacingsmoothcurvesbypolygonallines,hereducedtheproblemof
ndingextremaofafunctiontotheproblemofndingextremaofafunctionofnvariables,and
thenheobtainedexactsolutionsbypassingtothelimitasn!1.Inthissense,functionscan
beregardedas`functionsofinnitelymanyvariables'(thatis,theinnitelymanyvaluesofx(t)
atdierentpoints),andthecalculusofvariationscanberegardedasthecorrespondinganalogof
dierentialcalculusoffunctionsofnrealvariables.

14. 40 Chapter2.TheEuler-Lagrangeequation