Introduction to Mathematical Methods of Analytical Mechanics
Henri Gouin
First published 2020 in Great Britain and the United States by ISTE Press Ltd and Elsevier Ltd
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Preface
Theobjectiveofthisbookistoofferanoverviewofgeometricmethodsofcalculus ofvariationsandhowthiscanbeappliedinanalyticalmechanics.Itisfollowedbythe studyofpropertiesofspacesinmechanicalsystemswithafinitenumberofdegrees offreedom.Thebookwasinspired,inpart,bymethodsproposedbyPierreCasal,a formerProfessorattheFacultyofSciencesofMarseillesUniversity.Thesemethods wereconsideredagainanddevelopedduringacoursethatItaughttostudentsinthe thirdyearoftheAppliedMathematicsBachelor’sprogram.
Themathematicaltoolsusedthroughoutthebookarethoseusedinelementary algebra,analysisanddifferentialgeometry.Thebookdoesnotrequiremathematical toolsthatwouldbebeyondthescopeofathird-yearuniversitystudent(readersmay refer,amongothers,totheworksof(Queysane1971,CoutyandEzra1980,Martin 1967,Brousse1968).
Part1
Inthefirstpartofthebook,wepresentgeometricmethodsusedinthecalculusof variations.Freeextremaorextremathatarerelatedtointegralornon-integral constraintsarestudied.Thesemakeitpossibletointroducetheconceptofthe Lagrange multiplier.Aninitialstudyofthe Hamiltonequations1 associatedwith theconceptofageneratingfunctionfollowsfromthesemethods.Researchintothe geodesicsofsurfacesisalsoanaturalapplicationthatusesdifferentialgeometry. Thesemethodsofdifferentialgeometrycanbeextendedtothecalculationofthe variationofcurvilinearintegrals.Twoformsofvariationscanbemadeandexplicitly discussed:thefirstformusestheconceptofvariationofavectorialderivative;the secondformissimilartofindingtheopticalpathfollowedbylightinamediumwith
1Itshouldbenotedthattheseequationsarelikelytohavebeenwrittenby Huygens,andthe symbol H correspondstohisinitials.
avariablerefractiveindex.Thisleadstothestudyof Descartes’laws and isoperimetricproblems. Noether’stheorem,groupsofinvarianceassociatedwith differentialequationsandtheconceptofa Liegroup arenaturalextensionsofthese calculations.Thus, Fermat’sprinciple,associatedwiththeopticalpathfollowedby light,leadstofindingfirstintegrals.Thetoolsusedarerelatedtotensorcalculations thatbringinthestructureofavectorspaceanditsdualspace.
Part2
Thesecondpartofthebookpresentstheapplicationofthetoolsdiscussedearlier tothemechanicsofmaterialsystemswithafinitenumberofdegreesoffreedom. Afterbrieflyreviewingthe d’Alembertprinciple,weintroducetheconceptof Lagrangiandefinedinspace-timebythehomogeneousLagrangian.Wefindthatthe firstintegralinmechanicsisassociatedwithNoether’stheorem.Thereintroduction ofpartialresultsleadstothe Maupertuisprinciple andtotheintroductionof Riemanniangeometry inthecaseoftheconservationofenergy;thisisthefoundation for deBrogliewavemechanics.Theintegrationmethodsforequationsinmechanics areanalyzedusingthe Jacobimethod anditsapplicationintheimportantcaseof Liouville’sintegrability.Thissectioncontinueswiththeconceptsof angular variables and actionvariables forperiodicandmulti-periodicmotions,presentedby Delaunay,whichareespeciallyusefulincelestialmechanics.Thesecondpart concludeswiththestudyofspaceinanalyticalmechanics,includingvariousphase spaces.Theconceptsof dynamicvariables, Liebrackets, Poissonbrackets and Lie algebra arenaturaldevelopmentsofthisstudy.Wethenbrieflyreturntofirstintegrals whenstudyingPoissonbracketswithtwodynamicvariables.Canonical transformationsthatconservetheformofthemechanicalequationsindifferent dynamicvariablesleadtotheconceptof symplecticscalarproduct,whichisthefirst stepinstudying symplecticgeometry.
Part3
Thethirdpartpresentssomeeasyapplicationsofdifferentialequationsto mechanicalsystems.Theconceptof flow inphasespaceleadstothe Liouville theorem,whichisessentialinstatisticalmechanics.Itcorrespondstothe conservationofvolumeinthisspaceandcanbeinterpretedusingthe Poincaré recurrencetheorem.Thesmallmotionsofmechanicalsystemsareanalyzedusingthe specificcaseofthe Weierstrassdiscussion.Theequilibriumpositionsofthesystems associatedwithautonomousdifferentialequationsbringustotheconceptsof Lyapunovstability and asymptoticstability.Thenecessaryconditionsforstabilityare presentedinthecontextofthe LejeuneDirichlettheorem.Theconceptof linearizationofdifferentialequationsintheneighborhoodofanequilibriumposition makesitpossibletostudythesmalloscillationsofdynamicLagrangiansystemsand theirfrequency,aswellasthesystems’disturbances.Thispartendswithadiscussion
onthestabilityofperiodicsystemsandthetopologyofphasespaces,withthe extensionoftheconceptsofLyapunovstability,asymptoticstabilityandthenew conceptof strongstability intheneighborhoodofanequilibriumposition.The Hill and Mathieu equationsallowustoapplytheseconcepts.
Attheendofthebook,thereadercanfindacollectionofexercisesforgeometrical andmechanicalapplications,followedbybibliographicalreferencesandanindex.
IwouldliketothankFrançoiseforreviewingtheproofsandhelpingmewithher valuablecomments.
October2019
HenriG OUIN
Mathematicians,Physicistsand AstronomersCitedinthisBook
JeanleRondd’Alembert(1717–1783),Frenchmathematician,physicistand philosopher.
VladimirIgorevitchArnold(1937–2010),Russianmathematician.
LouisdeBroglie(1892–1987),Frenchmathematicianandphysicist.
Charles-EugèneDelaunay(1816–1872),Frenchmathematician.
RenéDescartes(1596–1650),Frenchmathematician,physicistandphilosopher.
PierredeFermat(1601–1665),Frenchmathematician.
WilliamHamilton(1805–1865),English–Irishphysicistandastronomer.
GeorgeHill(1838–1914),Americanmathematicianandastronomer.
ChristiaanHuygens(1629–1695),Dutchmathematician.
CharlesJacobi(1804–1851),Germanmathematician.
JohannLejeuneDirichlet(1805–1859),Germanmathematician.
SophiusLie(1842–1899),Norwegianmathematician.
JosephLiouville(1809–1882),Frenchmathematician.
AleksanderMikhailovitchLyapunov(1857–1918),Russianmathematician.
EmileMathieu(1835–1890),Frenchmathematician.
PierredeMaupertuis(1698–1759),Frenchmathematician,physicist,astronomer andnaturalist.
AmalieEmmyNoether(1882–1939),Germanmathematician.
HenriPoincaré(1854–1912),Frenchmathematician,physicistandphilosopher.
SiméonDenisPoisson(1781–1840),Frenchmathematicianandphysicist.
BernhardRiemann(1826–1866),Germanmathematician.
KarlWeierstrass(1815–1897),Germanmathematician.
ImportantNotations
:transposeoperationinavectorspace
x, Q :vectorsinavectorspace,representedinbold italics
x , Q ··· :lineartransposedformsofthevectorsinavector space
⎢ ⎣ x1 . . . xn ⎤ ⎥
, ⎡ ⎢
q1 . . . qn
: n-tuplesof Rn writtenascolumnsofthe elementsofthevectors x, Q writteninthe canonicalbasis
[x1 , ,xn ], [q1 , ,qn ] : n-tuplesof Rn writtenasrowsofthelinear formsofthecomponents x , Q ,where Rn denotesthedualof Rn
∂ V
∂ x :linearmapping R definedbytherelation dV = ∂ V ∂ x dx andrepresentedbythematrix
∂v1 (x1 ,...,xn ) ∂x1 ,...,
∂v1 (x1 ,...,xn )
∂xn . . . .
∂vn (x1 ,...,xn )
∂x1 ,...,
∂vn (x1 ,...,xn )
∂xn
⎥ ⎥ ⎥ ⎥ ⎦ , where V ,afunctionof x,isrepresentedbythe columnmatrix
1 . . vn ⎤
˙ a = da dt
:derivativeof a withrespecttotimeinNewtonian notation.Similarly, ˙ Q = dQ dt 1 :identitytensorinavectorspace
grad :gradientinavectorspace
rot :rotationalin R3
CalculusofVariations
Thecalculusofvariationsisdoneusingallmethodsthatallowtheresolutionof extremumproblems.Numerousproblemsinphysicscanbesolvedusingvariational methods.Inmechanics,forexample,an equilibriumposition isonewherethe potentialofforcesappliedtotheconsideredsystemisanextremum.Inoptics,the opticalpath followedbylightisanextremum.Incapillarity, thesurfacesofbubbles anddrops whosevolumeisgivenisthevaluethatmakesthemminimal.Wewillsee thatanon-dissipativemotionisonethatmakesthe Hamiltonianaction anextremum.
Theproblemsthatwestudyareintroducedindifferentforms:
–Innumericalform:theunknownconsistsofasetofscalarsorfunctions.When theunknownisasetofscalarsorfunctions,thecalculusofvariationsiscarriedout usingelementarydifferentialcalculus.Thisisthecasefor n scalars x =(x1 ,...,xn ), whichisanelementof Rn orforfunctionsoftheform
∈ [t1 ,t2 ] ⊂ R −→ φ(t) ∈ Rn .
–Ingeometricform:theunknownisrepresentedbyasetofpoints,curvesor surfaces.
1.1.Firstfreeextremumproblems
Theunknowniscomposedof n scalars x =(x1 ,...,xn ) ∈ Rn .Wewilldetermine thevaluesof x suchthat a = G(x) becomesanextremum,where G,assumedtobe differentiable,isamappingfrom Rn toarealset R.Thereasoningusedisasfollows:
letuswrite dx = ⎡ ⎢ ⎣ dx1 . . . dxn ⎤ ⎥ ⎦ an n-tupleof Rn ,whichwenamethe variation of x.We canderivethevalueofthedifferentialof G(x),
da = ∂G ∂x1 (x1 , ,xn ) dx1 + + ∂G ∂xn (x1 , ,xn ) dxn , whichcanbewritteninmatrixformas
da = ∂G ∂x1 (x1 , ··· ,xn ) , ··· , ∂G ∂xn (x1 , ··· ,xn ) ⎡ ⎢ ⎣ dx1 . . . dxn
Thecolumnmatrix ⎡ ⎢
canonicalbasisof R
.
representsavectorin Rn givenbyitselementsinthe
.
. 1
⎥ ⎦ .Thevectorswillnotbewritten withanarrow,butinbolditalicletters.Thelinematrix ∂G ∂x1 (x1 , ··· ,xn ), ··· , ∂G ∂xn (x1 , ··· ,xn )
representsalinearform,elementofthedualspaceof Rn ,denotedby Rn (thedual spaceisalsodesignatedby L(Rn , R)).Thislinearformisexpressedinthedualbasis
e1 =[1, ··· , 0] ,..., en =[0, ··· , 1]
ofthebasis e1 , , en andsatisfies ei ej = δij ,where δij istheKroneckersymbol, byalinematrixwith n columns.Thus, designatesthe transposition in Rn assumed tobeEuclidean.Wehave
G (x)= ∂G ∂x1 (x1 , ··· ,xn ) e1 + ··· + ∂G ∂xn (x1 , ··· ,xn ) en and dx = dx1 e1 + ··· + dxn en ,
anditispossibletowrite
da = G (x) dx,
whichiscalledthevariationof a.Forreasonsthatwillbeunderstoodlaterinthebook, wewrite δ x and δa insteadof dx and da,respectively.
D EFINITION 1.1.– a isanextremumat x ifandonlyif,foranyvariation δ x,the variation δa iszero.Thisassertioncanbewrittenas
∀ δ x, δa ≡ G (x) δ x =0
andasaresult, G (x)= 0 ,where G isalinearformof Rn ,whichcanbedeveloped as
∂G
∂x1 (x1 , ··· ,xn )=0, ··· , ∂G
∂xn (x1 , ··· ,xn )=0.
Thedefinitionisacommononeinthecalculusofvariations.Itis,nevertheless, preferabletocallita stationarypoint insteadofan extremum,asitwillbeseeninthe simpleexamplegivenbelow.
E XAMPLE 1.1.–Considerthemapping (x,y ) ∈O⊂ R2 −→ f (x,y ) ∈ R where O isanopensetof R2 .Itisassumedthatat (x0 ,y0 ) ∈O ,f (x,y ) isanextremum.
Withoutlossofgenerality,itcanbeassumedthat (x0 ,y0 )=(0, 0) and f (x0 ,y0 )=0.Thepoint (0, 0) correspondstoamaximumof f (x,y ) ifandonlyif thereexists r ∈ R suchthatforany (x,y ) satisfying 0 ≤ x2 + y 2 ≤ r 2 ,wehave f (x,y ) ≤ 0.Similarly,thepoint (0, 0) willcorrespondtoaminimumof f (x,y ) if andonlyifthereexists r ∈ R suchthatforany (x,y ) satisfying 0 ≤ x2 + y 2 ≤ r 2 , wehave f (x,y ) ≥ 0.Assumethat f belongstothe C 2 class;intheneighborhoodof (0, 0),then f (x, 0) and f (0,y ) areextremaat (0, 0).Hence, fx (0, 0)=0 and fy (0, 0)=0.Iftheseconditionsaresatisfied,thesecond-orderTaylor–Young formulaimplies f (x,y )= 1 2 αx 2 +2 βxy + γy 2 + x 2 + y 2 ε(x,y ), where lim ε(x,y )=0 when (x,y ) −→ (0, 0),with α = ∂ 2 f ∂x2 (0, 0),β = ∂ 2 f ∂x∂y (0, 0),γ = ∂ 2 f ∂y 2 (0, 0)
Consequences:if β 2 αγ< 0,wehaveaminimumwhen α> 0 andamaximum when α< 0;if β 2 αγ> 0,thequadraticform αx2 +2 βxy + γy 2 isnotdefined, andthereisnoextremumfor f (x,y ),butthereisasaddlepoint.If β 2 αγ =0,we cannotarriveatanyconclusion;forexample,inthecasewhereallsecondderivatives arezeroat (0, 0) andifathirdderivativeisnon-zero,thereisneitheramaximumnora minimum.Itisstillimportanttonotethatthedefinitionforanextremumisassociated withthecondition G (x)= 0 ,whichinallcasesiscalledthestationaritycondition.
E XAMPLE 1.2.–Considerinaplanethreepoints A, B and C .Wewishtofindapoint M intheplanesuchthat l (M ) ≡ MA + MB + MC isofminimumlength.
Giventhat l (M ) iscontinuousandgreaterthanzero,thelength l (M ) doesadmit alowerlimit.Giventhatitispossibletolimitourselvestoacompactdomaininthe plane(suchasthedomainboundedbyadiskwhoseradiusissufficientlylarge),the minimumisobtained.
Letuscalculatethevariationdenotedby δl (M ): δl (M ) ≡ δMA + δMB + δMC . Wefirstcompute δMA;letuscarryoutthecalculususingorthonormalaxeswhose originis A;for M withthecoordinates x and y , r = MA = x2 + y 2 .Wederive
δMA = x x2 + y 2 δx + y x2 + y 2 δy =grad r δx δy where gradr= uA with uA =
Thebipoint AM correspondstothevectoroftail A andhead M .Thecomputation iscarriedoutinthesamewayforthepoints B and C ,andweobtain
δl (M )=0 ⇐⇒ uA + uB + uC = 0.
Thepoint M belongstothearcthatisabletosupport AB withtheangle 2 3 π .This isthesamefor BC andfor AC .Inorderforthearcstohaveacommonpoint,itis essentialthat |ABC | < 2 3 π ;noangleintriangleABCcanbelargerthan 2 3 π .The processforfindingthepointMisrepresentedinFigure1.1.
InthecasewhereoneoftheverticesintriangleABCcanhaveananglegreaterthan 2 3 π ,thepoint M cannotbefoundatanypointintheplaneexcept A, B or C .Indeed, weknowthatanextremumexistsandthat A, B and C arethepointsforwhich l (M ) isnotdifferentiable;atthesepoints,thecalculusgivenaboveisnotvalid.Itmustalso benotedthatthereexistsasinglesolutionandthissinglesolutionisaminimum.
Figure1.1. Representationofthetriangle ABC inthecasewherenovertexcanhave ananglegreaterthanorequalto 2 3 π .Thepoint M correspondstotheminimum distance MA + MB + MC .Foracolorversionofthefiguresinthischapter,see www.iste.co.uk/gouin/mechanics.zip
1.2.Firstconstrainedextremumproblem–Lagrangemultipliers
1.2.1. ExampleofLagrangemultiplier
T HEOREM 1.1.–Let A bealinearmappingfrom Rn to Rq andlet B bealinear mappingfrom Rn to Rp .Regardlessofthevector V in Rn , B V =0 implies AV =0 isequivalent:thereexistsalinearmapping Λ,from Rp to Rq ,calledtheLagrange multipliersuchthat A =ΛB
P ROOF.–Asthepropertyisanequivalence,itmustbeformulatedasanecessaryand sufficientcondition.
⇐ If V ∈ Ker B ,then ΛB V =0 and V ∈ Ker A
⇒ Reciprocally,assumethat Ker A⊂ Ker B , Ker A beingavectorsub-space of Rn .Thereexistsanadditionalvectorsub-spaceEof Ker B suchthat Rn =Ker B E .Thus, B|E isanisomorphismfromEtoIm B⊂ Rp andis invertible;and B|E 1 isalinearmappingfromIm B to E .Letuswrite Λ1 = A B|E 1 ,amappingfromIm B to Rq .Wecanwrite Rp = Im B E , where E isanadditionalvectorspaceofIm B relativeto Rp .Let Λ2 beanarbitrary linearmappingover E (wecanconsider,forexample,thelinearmappingwithzero values).Thus,
∀ V , V ∈ Rp , V = V 1 + V 2
Thisdecompositionisunique.Letuspositthat Λ isalinearmappingfrom Rp to Rq suchthat Λ(V )=Λ1 (V 1 )+Λ2 (V 2 ).Therefore, A =ΛB .Indeed,forany W , beingavectorin Rn ,itispossibletowrite W = W 1 + W 2 ,where W 1 ∈ Ker B and W 2 ∈ E ,thedecompositionbeingunique.Ontheonehand, A B|E 1 B (W 1 + W 2 )= A(W 2 ) since B|E 1 B istheidentityover E and B (W 1 )=0.Onthe otherhand, A(W )= A(W 1 + W 2 )= A(W 1 )+ A(W 2 ) and Ker A⊂ Ker B impliesforany W 1 ∈ Ker A, A(W )= A(W 2 ).Finally,foranyvector W in Rn , A(W )=ΛB (W ),i.e. A =ΛB .
1.2.2. Applicationtotheconstrainedextremumproblem
Let G beadifferentiablemappingfrom Rn to R and F beanotherdifferentiable mappingfrom Rn to Rp .Wewishtofindvaluesfortheelement x belongingto Rn suchthat a = G(x) beanextremum,knowingthat x satisfiesthecondition F (x)= 0 (called aconstraint ).Wemustwrite: δa =0 notforall δ x butforall δ x verifyingthe condition F (x) δ x =0.Wewrite,
∀ δ x, δ x ∈ Rn ,F (x) δ x = 0 =⇒ G (x) δ x =0 with F (x)=0,
Onthebasisof Rn and Rp ,letuswrite F usingcolumnmatrices x =
x1 . xn
1 (x1 ,...,xn ) . fp (x1 ,...,xn )
∈ Rp and F (x)=0 ⇐⇒
p (x1 ,...,xn )=0 .
TheJacobianmatrixofthe derivative of F (x) isrepresentedby
1 (x1 ,...,xn ) ∂x1 ,...,
F (x)=
1 (x1 ,...,xn )
n . . . .
p (x1 ,...,xn )
1 ,...,
p (x1 ,...,xn )
Thecondition F (x) δ x = 0 iswritteninmatrixformas
∂f1 (x1 ,...,xn )
∂x1 ,...,
∂f1 (x1 ,...,xn ) ∂xn . . . . . .
∂fp (x1 ,...,xn )
∂x1 ,..., ∂fp (x1 ,...,xn ) ∂xn
wherethefinalcolumnismadeupof p lines.
Theconditionfortheextremumof G canthenbewritteninmatrixformas
∂G(x1 ,...,xn )
∂x1 ,..., ∂G(x1 ,...,xn )
ApplyingTheorem1.1,thereexistsaLagrangemultiplier Λ,whichisalinear mappingfrom Rp to R suchthat
G (x)=Λ F (x)with F (x)=0.
Onthebasisof Rp ,wewrite
Λ=[λ1 ,...,λp ] or λi ∈ R,i ∈{1,...,p}
Thefollowingthreepropertiesareequivalent:
(A) ∀ δ x, δ x ∈ Rn ,F (x) δ x = 0 =⇒ G (x)δ x =0 with F (x)= 0;
(B) G (x)=Λ F (x) with F (x)= 0;
(C) ∀ δ x,δ x ∈ Rn and ∀ δ Λ,δ Λ ∈L(Rn , R),δ [G(x) Λ F (x)]=0.
Wehavedemonstratedthat (A) isequivalentto (B).Property (C) showsthatfinding aconnectedextremumisthesameasfindingafreeextremum,butwithrespectto n + p variables,thepositionoftheextremumbeingmadeupoftheelements x and Λ. TheintroductionofaLagrangemultiplierleadstothefreeingoftheconstraint.This propertyresultsfromTheorem1.1,andwecanwritethat G(x) Λ F (x)= b,where b isthenewfunctiontostationarize.
1.3.Thefundamentallemmaofthecalculusofvariations
Inthecasewheretheunknownisanelementofafunctionalspace,wearetruly dealingwiththecalculusofvariations–thisisanextensionofdifferentialcalculus. Indeed,intheprecedingsections,wehaveusedthefollowinglemma:
Given A belongingto L(Rn , R),forany V belongingto Rn , AV =0 implies A = 0.
Itispossibletoproposethefollowinggeneralization:
Letusconsider E ,thesetof p-timedifferentiablerealfunctionsdefinedby
[t0 ,t1 ] ⊂ R −→ ψ (t) ∈ Rn .
If p =0,functionsaresimplycontinuous.Wemayadd ψ (t0 )= 0 and ψ (t1 )= 0. Theset E hasthestructureofarealvectorspace.Let φ beamapping
[t0 ,t1 ] ⊂ R −→ φ(t) ∈L(Rn , R) ≡ Rn , then φ,definedfrom R tothedualof Rn ,isassumedtobecontinuousover [t0 ,t1 ].
Letuswrite G (ψ )= ˆ t1 t0 φ(t)ψ (t)dt.Thismappingdefinedover E islinearasfor allscalars λ1 ,λ2 andforallfunctions ψ1 ,ψ2 ,
L EMMA 1.1.–Let φ beacontinuousmappingfrom [t0 ,t1 ] to L(Rn , R) and ψ a continuousmappingfrom [t0 ,t1 ] to Rn ;then,
Foranyψ, ˆ t1 t0 φ(t) ψ (t)dt =0 impliesφ = 0
P ROOF.–Itisenoughtodemonstratethelemmafor n =1.Letusassumethrough reductioadabsurdum thatthereexists t2 ∈ ]t0 ,t1 [ suchthat φ(t2 ) =0 (e.g.the φ(t2 ) valueisstrictlypositive).Since φ(t) iscontinuous,thereexists [t2 ,t3 ] includedin ]t0 ,t1 [ suchthatfor t belongingto [t2 ,t3 ] ,φ(t) > 0. Bychoosing ψ (t)=[(t t2 )(t3 t)]p+1 for t ∈ [t2 ,t3 ] and ψ (t)=0 for t/ ∈ [t2 ,t3 ],wewould have
ˆ t1 t0
φ(t)ψ (t)dt = ˆ t3 t2 φ(t)[(t t2 )(t3 t)]p+1 dt> 0,
whichleadstoacontradiction.Thislemmacaneasilybeextendedwhen t2 takesthe valueofoneofitslimits t0 or t1 andcanbeappliedinthecasewhere ψ (t0 )= ψ (t1 )=0 (aswasthecaseintheproof).
L EMMA 1.2.– duBois-Raymond’slemma. φ beingacontinuousmappingfrom [t0 ,t1 ] to L(Rn , R) and ψ beinganapplicationwithacontinuousderivativefrom [t0 ,t1 ] to Rn suchthat ψ (t0 )= ψ (t1 )= 0;then,
Foranyψ, ˆ t1 t0 φ(t) ψ (t) dt =0 impliesφ = C , where C isaconstantlinearmappingfrom Rn to R.
Then, ψ belongsto D1 [t0 ,t1 ]
P ROOF.–Itisenoughtodemonstratethelemmafor n =1;thiscaneasilybe extendedto n> 1.Letthereal c definedby ˆ t1 t0 (φ(t) c) dt =0.Letuswrite
ψ (t)= ˆ t t0 (φ(u) c) du. Wehave ψ (t) ∈ D1 [t0 ,t1 ].Accordingtothehypotheses, ˆ t1 t0
(t) c) ψ (t) dt =
(t1 ) ψ (t0 ))=0
However, ψ (t)= φ(t) c;hence, ˆ t1 t0 (φ(t) c)2 dt =0,whichimplies φ(t)= c
Letusnotethatwehavewritten G (ψ )= ˆ t1 t0
φ(t) ψ (t) dt; G isalinearmapping from E to R;consequently, G isalinearfunctionalof E oranelementinthedual vectorspace E ∗ .Theparallelwithsection1.2iscomplete.
1.4.Extremumofafreefunctional
Let G beacontinuouslydifferentiablemappingfrom Ω × [t0 ,t1 ] to R,with Ω ⊂ Rn .Thismappingisdenotedby
(Q,t) ∈ Ω × [t0 ,t1 ] −→ G(Q,t) ∈ R andwesaythatGisa generatingfunction.Let φ beacontinuousmappingfrom [t0 ,t1 ] to Rn .Weposit
a = ˆ t1
G(φ(t),t)dt,
t0
andwewrite a = G (φ);thus, G isafunctionalofthe φ-functionsandbelongsto A(A(R, Rn ), R)
Let ψ beanothercontinuousmappingfrom [t0 ,t1 ] to Rn .Asthescalar x isreal, φ + xψ isacontinuousapplicationfrom [t0 ,t1 ] to Rn and
G (φ + xψ )= ˆ t1 t0 G φ(t)+ xψ (t),t dt.
Forthegiven φ and ψ ,wewrite g (x)= G (φ + xψ ).Thus,themapping g isalinear mappingfrom R to R.Forthegiven φ, ψ and t,wewrite f (x)= G φ(t)+ xψ (t),t . Thefunction f canbederivedat x =0,i.e.
f (x)= f (0)+ xf (0)+ xε(x) where lim x→0 ε(x)=0. Consequently,as ∂G φ(t),t ∂ Q isalinearmappingfrom Rn to R,
(x)= G
)+
Q φ(t),t ψ (t) +x ψ (t) ε xψ (t) , where lim x→0
xψ (t) =0, and,throughintegration
OROLLARY 1.1.–
whichcorrespondstothederivativeof g at 0.
(t),t
(t)dt + xε1 (x),
D EFINITION 1.2.– a = ˆ t1 t0 G φ(t),t dt isanextremumforthecontinuousfunction
φ from [t0 ,t1 ] to Rn ifandonlyifforany ψ continuousmappingfrom [t0 ,t1 ] to Rn ,
g (0) ≡ ˆ t1 t0 ∂G
(t)dt =0.
Wewrite δa = g (0),whichiscalled thevariationof a relativetothecontinuous functions φ from [t0 ,t1 ] to Rn .Whenthereisnoambiguity,wewrite
a = ˆ t1 t0 G(Q,t) dt, and,analogouswiththeprevioussections, ψ (t) isdenotedby δ Q(t) or,moresimply, δ Q Wewrite
g (0)= δa = ˆ t1
FromLemma1.2andCorollary1.1,itfollowsthat
C OROLLARY 1.2.– a = ˆ t1 t0 G Q,t dt isanextremumforthecontinuousfunction
Q from [t0 ,t1 ] to Rn if ∂G(Q,t) ∂ Q = 0.
Whenthereisnoambiguity,wewrite ∂G ∂Q for ∂G(Q,t) ∂Q .
1.5.Extremumforaconstrainedfunctional
Thesearchforthefunction t ∈ [t0 ,t1 ] −→ φ(t) ∈ Rn maysatisfycertain relationshipscalledconstraints.Wecomeacrosstwotypesofconstraints.
1.5.1. Firsttype:integralconstraint
D EFINITION 1.3.–Anintegralconstraintisarelationshipintheform
ˆ t1 t0 F (φ(t),t) dt = b, [1.1]
where b isagivenrealand F isacontinuouslydifferentiablemappingfromanopen set Ω of Rn × [t0 ,t1 ] to R.
Wecangeneralizethedefinitionforamappingfrom Rn × [t0 ,t1 ] to Rp .The generalizationcorrespondsto p integralconstraints.Findingthemappings φ which willmake ˆ t1 t0 G(φ(t),t)dt extremumandwhicharesubjecttocondition[1.1]results intheshortenedform:
∀ ψ continue,ψ ∈A(R, Rp ),
≡
,t
Wecansimplywrite: δb =0=⇒ δa =0.
(t)dt =0.
T HEOREM 1.2.–With δb and δa beingtwolinearfunctionalsdefinedoverthe continuousmappings φ from R to Rn ,withvaluesin R, δb =0=⇒ δa =0
isequivalenttothereexistingaLagrangemultiplier,i.e.alinearmappingfrom R to R suchthat δa = λδb.Wecanwrite
t1 t0
Q (φ(t),t) ψ
Thus, λ isarealconstant.
Wearebroughtbacktothesearchforthefreeextremumof
Lemma1.1ofthecalculusofvariationsimpliesthefollowingtheorem.
T HEOREM 1.3.–Thecontinuousfunctions φ from [t0 ,t1 ] to Rn ,satisfying relationship[1.1]andwhichmakes“integral a”extremum,simultaneouslysatisfythe tworelations:
Wealsousetheresultinsimplifiedformas
Q (Q(t),t)= λ ∂F
Thesepropertiesgeneralize,forfunctionalswithrealvalues,theresultsobtained insection1.4.
P ROOF.–Wedemonstratethepropertyinthecaseofamapping φ from R to R.We caneasilygeneralizethedemonstrationtothemappingswithvaluesin Rn .Letus write a = ˆ t1
(x(t),t)dt and b = ˆ
(x(t),t)dt,
where f and g aretwocontinuouslydifferentiablemappingsfrom R2 to R and x(t) denotesacontinuousmappingfrom R to R.Thus, δa = ˆ t1 t0 ∂g ∂x (x(t),t)δxdt and δb =
(x(t),t)δxdt.
Letuswrite J (δx)= ˆ t1 t0 ∂g ∂x (x(t),t) δxdt and K (δx)= ˆ t1 t0 ∂f ∂x (x(t),t) δxdt.
Theterms J and K aretwolinearmappingsdefinedover δx mappingsfrom R to R.Thus,
∀ δx,δx ∈ C 0 (R, R), K (δx)=0=⇒J (δx)=0
Let δx1 beagivencontinuousmappingfrom R to R suchthat K (δx1 ) =0 and δx isanyothercontinuousmappingfrom R to R.Letuswrite δx2 = δx μδx1 , where μ ∈ R andsuchthat K (δx2 )=0.Thisimpliesthat K (δx μδx1 )=0 or μ = K (δx)/K (δx1 ) ∈ R.Accordingtothehypothesis,weobtain K (δx2 )=0=⇒ J (δx2 )=0.Fromthisresult,itcanbederivedthat
J (δx2 )= J (δx) μ J (δx1 )=0=⇒J (δx2 ) = J (δx) −K (δx) J (δx1 ) K (δx1 ) =0
Letuswrite λ = J (δx1 )/K (δx1 ); λ isascalarthatisgivenby δx1 .Hence, J (δx)= λ K (δx)
C OROLLARY 1.3.–Thethreepropositionsareequivalent:
(A) ∀ δx, K (δx)=0=⇒J (δx)=0;
(B ) ∃ λ ∈ R suchthat ∀ δx,δx ∈ C 0 (R, R), J (δx) λ K (δx)=0;
(C ) ∃ λ ∈ R, aconstantscalar,calledtheLagrangemultiplier,suchthat
J = λ K
Wealsowrite δa λδb =0.
1.5.2. Secondtype:distributedconstraint
Let F beamappingfrom Ω × [t0 ,t1 ] to R,where Ω isanopensetof Rn and [t0 ,t1 ] asegmentof R.Wewishtofindtheextremaof a = ˆ t1 t0 G(Q(t),t)dt suchthat,for anyvalue t of [t0 ,t1 ],wehave F (Q(t),t)=0.Theconstraint F (Q(t),t)=0 is calledadistributedconstraint.
Wedemonstratethatweareledtointroduceamultiplierdenotedby Λ,afunction definedforeachvalueof t belongingto [t0 ,t1 ],i.e.a Λ-multiplierfunctionof t.
T HEOREM 1.4.–Wehavetheequivalenceofthethreepropositions:
(A)thereexistsacontinuousmapping Q from [t0 ,t1 ] to Ω satisfying F (Q(t),t)=0 suchthatforanycontinuousapplication δ Q from [t0 ,t1 ] to Rn and forany t of [t0 ,t1 ],
∂F
∂ Q (Q(t),t) δ Q =0=⇒ δ ˆ t1 t
(Q(t),t)dt =0;
(B)thereexistsacontinuousmapping Q from [t0 ,t1 ] to Ω andthereexistsa continuousmapping Λ from [t0 ,t1 ] to R suchthatforanycontinuousmapping δ Q from [t0 ,t1 ] to Rn andanycontinuousmapping δ Λ from [t0 ,t1 ] to R, δ ˆ t1 t0 G(Q(t),t) Λ(t) F (Q(t),t) dt =0;
(C)thereexistsacontinuousmapping Q from [t0 ,t1 ] to Ω andthereexistsa continuousmapping Λ from [t0 ,t1 ] to R suchthat ∂G
∂ Q (Q(t),t) Λ(t) ∂F ∂Q (Q(t),t)=0 and F (Q(t),t)=0
