Explicit differential characterization of PDE systems pointwise equivalent to Y_{X^{j_1}X^{j_2}}=0, 1\leq j_1,j_2\leq n\geq 2

0.25 MB·English

by  Joel Merker

EXPLICIT DIFFERENTIAL CHARACTERIZATION OF
PDE SYSTEMS POINTWISE EQUIVALENT TO Y =0
5 Xj1Xj2
0 1 j ,j n 2.
1 2
≤ ≤ ≥
0
2
JOE¨LMERKER
n
a
J
ABSTRACT. In[Lie1883],asanearlyresult,SophusLieestablishedthatasecondorder
9 ordinary differential equation yxx = F(x,y,yx)is equivalent, through aninvertible
1 pointtransformation(x,y)7→(X(x,y),Y(x,y)),tothefreeparticleequationYXX =
0ifandonlyif therightmemberFisadegreethreepolynomialinyx,namelythereexist
] fourfunctionsG,H,LandMof(x,y)suchthatF canbewrittenas
V
F(x,y,yx)=G(x,y)+yx·H(x,y)+(yx)2·L(x,y)+(yx)3·M(x,y),
C
and furthermore, the four functions G, H, Land M satisfy two secondorder partial
.
h differentialequations:
t 4 2 2 4
a 0=−2Gyy+ Hxy− Lxx+2(GL)y−2GxM−4GMx+ HLx− HHy,
m 3 3 3 3
2 4 2 4
[ 0=−3Hyy+ 3Lxy−2Mxx+2GMy+4GyM−2(HM)x− 3HyL+ 3LLx.

In[M2004a],thistheoremwasgeneralizedtosystemsofNewtonianparticleswithm≥2
2
degreeoffreedom,i.e.withoneindependentvariablexandm≥2dependentvariables
v
(y1,y2,...,ym). Inthispaper, wegeneralize S.Lie’stheoremtothecaseofseveral
7
independentvariables(x1,x2...,xn),n≥2,andonedependentvariabley.Strikingly,
3
as in[M2004a], the(complicated) differential system whichcorresponds totheabove
6
twosecondorderpartialdifferential equations isoffirstorder. Bymeansofcomputer
1
programming,thisphenomenonwasdiscoveredin[BN2002],[N2003]inthecasen=2.
1
In[Bi2003],[M2003],thegeneralcasen≥2washandled.
4
0
/
h
t Tableofcontents
a
1.Introduction ....................................................................................1.
m
2.Completelyintegrablesystemsofsecondorderordinarydifferentialequations .....................9.
3.Firstandsecondauxiliarysystems ..............................................................16.
:
v
i
X 1.INTRODUCTION
§
r This paper, a direct continuation of [M2004a], provides a summarized proof of the
a
followingstatement,labelledasTheorem1.23intheIntroductionof[M2004a]. Allour
functionsareassumedtobeanalytic.
Theorem 1.1. (n = 2: [BN2002], [N2003]; n 2: [Bi2003], [M2003]) Let K = R
or C, let n N, supposen 2 andconsidera≥system of completely integrablepartial
∈ ≥
differential equations in n independent variables x = (x1,...,xn) Kn and in one
dependentvariabley Koftheform: ∈
∈
(1.2) yxj1xj2(x)=Fj1,j2(x,y(x),yx1(x),...,yxn(x)), j1,j2 =1,...n,
Date:2008-2-1.
2000 Mathematics Subject Classification. Primary: 58F36 Secondary: 34A05, 58A15, 58A20, 58F36,
34C14,32V40.
1
2 JOE¨LMERKER
where Fj1,j2 = Fj2,j1. Under a local change of coordinates (x,y) (X,Y) =
7→
(X(x,y),Y(x,y)), this system (1.2)is equivalentto the simplestsystem Y = 0,
Xj1Xj2
j ,j = 1,...,n, if and only if there exist arbitrary functions G , Hk1 , Lk1 and
1 2 j1,j2 j1,j2 j1
Mk1 ofthevariables(x,y), for1 j1,j2,k1 n, satisfyingthetwo symmetry condi-
≤ ≤
tionsG = G andHk1 = Hk1 ,suchthattheequation(1.2)isofthespecific
j1,j2 j2,j1 j1,j2 j2,j1
cubicpolynomialform:
(1.3)
n
1 1
yxj1xj2 =Gj1,j2 + yxk1 Hjk11,j2 + 2yxj1 Lkj21 + 2yxj2 Lkj11 +yxj1yxj2 Mk1 ,
kX1=1 (cid:18) (cid:19)
forj ,j =1,...,n.
1 2
We refer the reader to [M2004a] for an extensive introduction and for a more sub-
stantial bibliography. Applying E´. Cartan’s equivalence algorithm (see [G1989] and
[OL1995]formodernexpositions),thegeneralcasen 2oftheabovetheoremhasalso
≥
been establishedin [Ha1937], wherethe constructionofa projectiveconnectionassoci-
atedtoasecondorderordinarydifferentialequationachievedin[Ca1924]wasextendedto
severalvariables. Theorem1.1wasre-discoveredin[BN2002],in[N2003],in[Bi2003]
andin[M2003],thankstopartialparametriccomputationsoftheHachtroudi-Chernten-
sors in n 2 variables, which were described in a non-parametric way in [Ch1975]
≥
(see 1.8below).
§
It may seem quite paradoxicaland counter-intuitive (or even false?) that every sys-
tem (1.3), for arbitrary choices of functionsGj1,j2, Hjk11,j2, Lkj11 and Mk1, is automati-
cally equivalentto Y = 0. However, a strong hidden assumption holds: that of
Xj1Xj2
completeintegrability.Shortly,thiscrucialconditionamountstosaythat
(1.4) Dxj3 Fj1,j2 =Dxj2 Fj1,j3 ,
(cid:0) (cid:1) (cid:0) (cid:1)
forallj ,j ,j =1,...,n,where,forj =1,...,n,theD arethetotaldifferentiation
1 2 3 xj
operatorsdefinedby
n
∂ ∂ ∂
(1.5) D := +y + Fj,l .
xj ∂xj xj ∂y ∂y
xl
l=1
X
Theseconditionsarenon-voidpreciselywhenn 2. Moreconcretely,writingout(1.4)
≥
when the Fj1,j2 are of the specific cubic polynomial form (1.3), after some nontrivial
manualcomputation,weobtainthecomplicatedcubicdifferentialpolynomialinthevari-
ables y labelled as equation(1.25)in [M2004a]. Equatingto zero all the coefficients
xk
of this cubicpolynomial,we obtainfourfamiles(I’),(II’),(III’)and (IV’)offirstorder
partialdifferentialequationssatisfiedbyGj1,j2,Hjk11,j2,Lkj11 andMk1:
n n
(I’) 0=G G + Hk1 G Hk1 G .
( j1,j2,xj3 − j1,j3,xj2 j1,j2 k1,j3 − j1,j3 k1,j2
kX1=1 kX1=1
DIFFERENTIALCHARACTERIZATIONOFYXj1Xj2 =0,1≤j1,j2≤n≥2 3
0=δk1G δk1G +Hk1 Hk1 +
j3 j1,j2,y− j2 j1,j3,y j1,j2,xj3 − j1,j3,xj2
1 1
 + G Lk1 G Lk1+
(II’)  ++ 1221δδjkk1j111,kjX32nn=1j2GG−k2,2j3LLjkjk1222,j−2 211j3δδjkk111 kX2nn=1 GGk2,j2LLkjk322++
2 j2 k2,j3 j1 − 2 j3 k2,j2 j1
 +kX2n=1 HkXk2k=21,1j3Hjk12,j2 −kX2n=1 Hkk21,jkX22=H1jk12,j3.
0= δk2)Hkσ(1) δkσ(2)Hkσ(1) +
j3 j1,j2,y− j2 j1,j3,y
 σX∈S2(cid:16)
(III’)  ++++δδ1221jjkk12σσδδ,((jjkk1221σσ))((,jG22k2))σj(LL12),jjkkj12σσ3k,,X((xx3M11n=jj))331k−−σG(11122k)3−δδ,jjjkk331σσδM((jk223σ))(kLL23)kjkj−G13σσ,,((xxj11δ1jj))jk22,j1σ++2,(1M),jk3kσσ((21))+kX3n=1 Gk3,j2Mk3+
n n
+ 1δkσ(1) Hkσ(2)Lk3 1δkσ(1) Hkσ(2)Lk3+
 ++22112δδjjkkj23σσ1((11)) kkXXk33Xnn3===111HHkjkkk13σ33,,(jj,22j3)3LLkkkjσ3j132(2−)−−1221δ2jkj3δσ1jk(21σ)(1k)Xk3Xn3k=X=31n=1H1 Hkkk3σ3j,k(j,123j2,)2j3LkjLj133kk+σ3(2)!.
0= 1δkσ(3),kσ(2)Lkσ(1) 1δkσ(3),kσ(2)Lkσ(1)+
2 j3, j1 j2,y − 2 j2, j1 j3,y
 σX∈S3(cid:18)
(IV’)  ++δδjjkk22σσ,,((33)),,jjkk11σσ((12))Mnxkjσ3(1H) −kk4σ,(δj23jk)3σM,(3)k,jk41σ−(2)δMjk3σ,xk(3jσ)2(,j1k1)σ+(1) n Hkk4σ,(j22)Mk4+
kX4=1 kX4=1
Thesesystems (I’),+(I14I’δ)j,k1σ(,(I1I)I,’jk3)σ(a3n)dkX4(n=IV1’L)kksσ4h(o2u)lLdkj2b4e−di14stδinjk1gσ,(u1i)s,jkh2σe(d3)fkrXo4n=m1tLhekkσ4s(y2)stLekjm34s!(.I),
(II), (III) and (IV) of Theorem 1.7 in [M2004a], although they are quite similar. Here,
the indicesj ,j ,j ,k ,k ,k varyin 1,2,...,n . By S and by S , we denote the
1 2 3 1 2 3 2 3
{ }
permutationgroupof 1,2 andof 1,2,3 .Tofacilitatehand-andLatex-writing,partial
{ } { }
4 JOE¨LMERKER
derivativesaredenotedasindicesafteracomma;forinstance,G isanabreviation
j1,j2,xj3
for∂Gj1,j2/∂xj3. Todeduce(I’),(II’),(III’)and(IV’)fromequation(1.25)of[M2004a],
itsufficestoobservethateverycubicpolynomialequationoftheform
n n n
0 A+ B y + C y y +
≡ k1 · xk1 k1,k2 · xk1 xk2
(1.6) kX1=1 kX1=1 kX2=1
n n n
+ D y y y
k1,k2,k3 · xk1 xk2 xk3
kX1=1 kX2=1 kX2=1
(as for instance (1.25) in [M2004a]) is equivalent to the annihilation of the following
symmetricsumsofitscoefficients:
0=A,
0=B ,
(1.7)  k1
0=Ck1,k2 +Ck2,k1,
0=D +D +D +D +D +D .
k1,k2,k3 k3,k1,k2 k2,k3,k1 k2,k1,k3 k3,k2,k1 k1,k3,k2
forallk1,k2,k3 =1,...,n.
In conclusion, the functions Gj1,j2, Hjk11,j2, Lkj11 and Mk1 in the statement of Theo-
rem 1.1 are far from being arbitrary: they satisfy the complicated system of first order
partialdifferentialequations(I’),(II’),(III’)and(IV’)above.
Our proofof Theorem1.1 issimilar to the oneprovidedin [M2004a], in the case of
systemsofNewtonianparticles,sothatmoststepsoftheproofwillbesummarized.Inthe
endofthispaper,wewilldelineateacomplicatedsystemofsecond orderpartialdiffer-
entialequationssatisfiedbyGj1,j2,Hjk11,j2,Lkj11 andMk1 whichistheexactanalogofthe
systemdescribedintheabstract.ThemaintechnicalpartoftheproofofTheorem1.1will
betoestablishthatthissecondordersystemisaconsequence,bylinearcombinationsand
bydifferentiations,ofthefirstordersystem(I’),(II’),(III’)and(IV’).Beforeproceeding
further,letusdescriberapidlyasecondproofofTheorem1.1,confirmingitsvalidity.
1.8. Confirmationof Theorem 1.1 by the method of equivalence. Let us summarize
thestrategyof[BN2002],[N2003],[Bi2003]. In[Ch1975],inspiredbyM.Hachtroudi’s
thesis[Ha1937],S.-S.Chernconductedtheequivalencealgorithmassociatedtothesys-
tem (1.2). His computationsare essentially equivalentto M. Hachtroudi’s(thoughwith
muchlessinformation),buttheyaredoneinanon-parametric way.Itwasdeeplyknown
toE´lie Cartananditisnowwellknownbymodernfollowers(cf. forinstance[G1989],
[GTW1989],[HK1989],[Fe1995],[OL1995])thatachievingparametriccomputationsin
anequivalenceproblemisanextremelyhardtask,oftendevotedtocomputersnowadays.
Infact,theso-calledCartanLemma([OL1995],p.26)wasessentiallydevisedbyE´.Car-
tan as a shortcut, alongside his groundbreaking progresses, after some experiences of
explicithandcomputations.Thetrickinthislemmaistobypassthe(oftentoolong)para-
metriccomputations,neverthelesskeepingtrackofsomeimportantinformations. Nowa-
days, tostudyanequivalenceproblem,oneoftenachievesthenon-parametriccomputa-
tionsbyhand,leavingtheexplorationsofparametriccomputationstoacomputer. Inthis
respect,thethesis[N2003]ofSylvainNeutisextremelyinteresting,sinceitimplements
thegeneralequivalencealgorithmasaMaplepackage.
However, computer programs achieving formal algebrico-differential computations
with a general number n of variables are still undevelopped. Consequently, it is in-
teresting to achieve the computations of [Ch1975] in a parametric way, because at the
DIFFERENTIALCHARACTERIZATIONOFYXj1Xj2 =0,1≤j1,j2≤n≥2 5
end, the vanishing of all the invarianttensors that one obtainsin the final e -structure
{ }
would showexplicitelyunderwhich conditionthesystem (1.2)is equivalentto the sys-
tem Y = 0. To our knowledge, this problem is considered as open in the field
Xj1Xj2
of symmetriesofCauchy-RiemannsubmanifoldsofCn, a subfield of the mathematics
area calledSeveralComplexVariables. However,no specialist of the subfield seems to
beawareoftheoldreference[Ha1937],atextonlyread,apparently,byS.-S.Chern,who
wasastudentofE´.CartancontemporarytoM.Hachtroudi.
Let us expose briefly how one obtains the final e -structure attached to the equiv-
{ }
alence problem associated to the system (1.2). Since we have not been able to follow
everythingin the ancienttext [Ha1937], we follow [Ch1975] with the only change that
wedonotintroducetheimaginarynumberi =√ 1inourstructureequationsandalso,
−
wemakesomeminorchangesofsign;in[Ch1975],thecomputationsareconductedover
K = CwiththeideathattheyapplytoComplexAnalysis,buttheydoholdaswellover
K=R,orevenoveranarbitarycommutativefieldofcharacteristiczeroequippedwitha
valuation(inordertoprovideK-analyticfunctions).
Tobeginwith,considerthefollowingfamilyof2n+1initialdifferentialforms:
ω :=dy y dxβ,
xβ
−
β
 X
(1.9) ωωeα ::==ddxyα, Fα,βdxβ.
α xα
e −
Here,theGreekindicesα,β,γ,eδ,ε,ζ,etc.,runXβfrom1ton.Sums n areabbreviated
α=1
as .IntheparagraphprecedingthestatementofTheorem1.7in[M2004a],weexplain
α P
whywecannotusecoherentlytheEinsteinsummationconventionthroughoutthepaper.
P
For the same reason, we shall abandon this conventionin the present paper. However,
we wouldlike tomentionthatin(1.9)aboveandin (1.10), (1.11), (1.12), (1.13), (1.14)
and (1.15)below, the summation conventionapplies coherently, so that the reader may
dropthesumsif(s)heisusedto.
Following [Ch1975], define the local Lie group consisting of (2n+1) (2n+1)
×
matricesoftheform
u 0 0
(1.10) g := uα uα 0 ,
β
 u 0 uu′β 
α α
 
whereuαandu aren 1vectorsclosetothezerovector,whereuαisan ninvertible
α × β ×
matrixclosetotheidentitymatrix,whereuisascalarcloseto1andwhereu′α denotes
β
theinversematrixofuα. Withthisgroup,definetheinitialG-structure,consistingofthe
β
followingcollectionof(2n+1)differentialforms,calledtheliftedcoframe:
ω :=u ω,
·
(1.11) ωωα ::==uuαe·ωωe++Xβ uuαβu′·βωeβω, .
α α· α· β
 e Xβ e
6 JOE¨LMERKER
These forms depend both on the “horizontal” variables (xi,y) and on the “vertical”
(“group”, “fiber”) variables u, uα, u and uα. The introduction of this lifted coframe
α β
maybejustifiedasfollows.
Itistheveryfirststepofthemethodofequivalencetocheckthatthereexistsatrans-
formation (xi,y) (xi,y) of the completely integrable system y = Fα,β in the
xαxβ
7→
coordinates(xi,y)toanothercompletelyintegrablesystemy =Fα,β inothercoor-
xαxβ
dinates(xi,y)ifandonlyif thereexistfunctionsu,uα,u ,uαandvα of(xi,y),which
α β β
dependonthefunctions(xi,y),ontheirpartialderivativeswithrespectto(xi,y),onthe
yxl andontheFγ,δ,suchthatthefollowingmatrixbidbentibtyhbolds: b
ω u 0 0 ω
(1.12) ωα = uα uα 0 ωβ ,
   β  
ωeα ubα 0 vαβ ωeβ
 e   b b  e 
e b α b e
wherethe(2n+1)differentialforms ω,ω ,ω aredefinedsimilarlyasin(1.9),inthe
α
barredcoordinates.Ofcourse,itisund(cid:16)erstoodtha(cid:17)tinthelefthandsideofthismatrixiden-
tity,thevariables(xi,y)arereplacedbeytheeirevalueswithrespecttothevariables(xi,y)
(inthelanguageofmoderndifferentialgeometry,oneusuallyspeaksof“pull-back”).By
amorecarefulexaminationoftheexplicitexpressionsofthefunctionsu,uα,u ,uαand
α β
β
vα intermsoftheFγ,δ,oneobservesthatvβ =uu′ ([M2003]). Thus,thecollectionof
β α α
differentialforms (ω,ωα,ω ) is definedmodulomultiplicationby a mbatrbix obf funbctions
α
obf (xi,y) which is of the specific form (1b.10). bBbased on this preliminary observation,
the generalprocedureofthe equivalencemethod([G1989], [OL1995])associatesto the
system(1.2)theliftedcoframe(1.11).
Applyingtheexteriordifferentialoperatordtoω,toωα andtoω andabsorbingthe
α
torsion,itisshownin[Ch1975]thattheinitialstructureequationsmaybewrittenunder
theform
dω = ωα ω +ω ϕ,
α
∧ ∧
α
 X
(1.13) dωα =Xβ ωβ ∧ϕαβ +ω∧ϕα,
dω = ϕβ ω +ω ϕ+ω ϕ ,
 α Xβ α∧ β α∧ ∧ α
whereϕ,ϕα,ϕα andϕ aremodifiedMaurer-Cartanforms. Wenoticethatthereareno
β α
torsioncoefficienttonormalize.Infact,thechoiceofvα :=uu′αmadeinadvanceabove
β β
correspondstohavingachievedafirstnormalizationimplicitely.
SincethedimensionoftheLiesymmetrygroupofthesystem(1.2)isalwaysfiniteand
infactboundedbyn2+4n+3([Ha1937],[CM1974],[Ch1975],[Su2001],[GM2003]),
accordingto the generalprocedureof the methodof equivalence([G1989], [OL1995]),
one simply has to prolongthe initial G-structure. One couldalso applythe E´lie Cartan
involutivitytesttodeducethatitisnecessarytoprolong.
Now,wesummarizetheremainderof[Ch1975]veryrapidly. Afteroneprolongation,
twonormalizationsandonesupplementaryprolongation,thefinal e -structureisofthe
{ }
DIFFERENTIALCHARACTERIZATIONOFYXj1Xj2 =0,1≤j1,j2≤n≥2 7
followingform:
dω= ωα∧ωα+ω∧ϕ,
α
 X
(1.14) ddddωϕϕωdϕαβααα=====XXXX+αββρ ωωXωϕασβααβ∧∧∧∧STϕϕωϕββααβραββσαγ+·++·−ωωδωωβXαρβαα∧∧∧X∧γωϕωωϕσααγω,+++∧γXϕ∧ω21γαϕ∧+γRϕ+ωαβαQγ,ω∧αβ·βψω·∧,ω∧ϕ∧ωαγω++βX+Xγγ ϕTγββαL∧γαϕ·βωαγ·ω∧+∧ω12γωδ+ββα+·ω∧ψ,
β γ β β
X X X X
Thesestrudcϕdtψαur==eeXX+++qαβuϕXϕaXXβt∧∧βγioϕψnϕQRsαβα+αγβαi−∧nβ2·cϕX·ωoXβωβrααpβ+∧oϕ∧ϕrωαβ21aαωtβ∧ωeγ∧+α8ϕ+ϕ∧βXfαaα21+ψ.m,XiH12βliαωesQα·ωωβα∧,∧·ψωωω,α∧α,ωω+βαX,+αϕX,KβϕααβP,·αωϕβα∧·,ωωϕα∧α+,ωψβ+ofdiffer-
entialforms,oftotalcardinalityn2+4n+3,togetherwith8invarianttensorsSασ,Rα ,
βρ βγ
Tαγ, Qα, Lαβ, P , H and K havingsome specific indexsymmetriesthat we shall
β β αβ α α
notuse.
Applying the exterior differential operator d to these 8 families of structure equa-
tions(1.14), oneverifiesthatthe seveninvarianttensorsRα , Tαγ, Qα, Lαβ, P , H
βγ β β αβ α
andK areinfactfunctionallydependentonthefundamentaltensorsSασ,namelythey
α βρ
arecertaincoframederivativesoftheSασ.Inthe(verysimilar)contextoftheequivalence
βρ
problemassociatedwitha Levinon-degeneratehypersurfaceofCn+1, thiscomputation
was achievedby S.M. Webster in the Appendixof [CM1974]; in the precise contextof
theequivalenceproblemassociatedtothesystem(1.2),thiscomputationisnotachieved
in[Ch1975],butsee[BN2002],[Bi2003],[N2003]and[M2003]fordetails. Forus,the
precisenatureofthisfunctionaldependencedoesnotmatter.
In fact, it is only in the computer science thesis [N2003] that the complete explicit
parametriccomputationofthe8familiesofdifferentialformsω,ωα,ω ,ϕ,ϕα,ϕα,ϕ ,
α β α
ψ together with the 8 invarianttensors Sασ, Rα , Tαγ, Qα, Lαβ, P , H and K is
βρ βγ β β αβ α α
achieved, in the case n = 2 and with the help of Maple. Althoughthe task is really of
impressivesize,theauthorofthispaperhastheprojectofachievingmanuallythegeneral
computationforn 2variables([M2005]). In [Ha1937], the parametriccomputations
≥
areachievedcompletelyonlyinthecasen = 2. Accordingtotheauthor’sexperience,it
8 JOE¨LMERKER
appearsthatthetensorialformalismhelpstoshortenimportantlythesizeoftheelectronic
computationsandthatthecasen 2isnotmuchmoredifficultthanthecasen = 2,as
≥
onemayalreadyobservebyreading[M2004a]. Thisiswhytheproject[M2005]seems
to be an accessible task. A similar project would be to compute in a parametric way
thestructureequationsobtainedin[Fe1995]fortheequivalenceproblemassociatedtoa
systemofNewtonianparticles;thecomputationsarequitesimilar,indeed,aswellasthe
computationsofthispaperasquitesimilartothecomputationsof[M2004a]. Thiswillbe
achievedinthefuture,iftimepermits.
Atpresent,fortunately,thereisakindof“miracle”:theinterestingtensorSασ appears
βρ
essentially at the beginning of the computation of the final e -structure. Thus, it is
{ }
notnecessarytogouptotheendofthealgorithminordertodeducethefinalparametric
expressionofSασ,andespeciallytocharacterizethemaximallysymmetricsystems(1.2),
βρ
i.e. thoseforwhichallthe8tensorsSασ,Rα ,Tαγ,Qα,Lαβ,P ,H andK vanish.
βρ βγ β β αβ α α
In[M2003],weobtained:
1
Sασ =δσ u−1u′δ uαFγ,δ +
 βρ ρ n+2 γ δ ε β ε yxγyxε!
(1.15)  +++δδδρβαασ nn++111X22 XXXγγ XXXδδ XXεε uuu−−−111uuu′′′δρδβδuuuαεσσεFFFyyγγγxx,,,γδδγδyyxxεε!!++
β n+2 ρ ε yxγyxε!−
It may beverified(c−−f.X(cid:0)aδγlρσsoδXβα[δB+i2Xδ0ερα0X3δXγ]βσ)ζ(cid:1)tXh·uδa −t(t1Xhnuεe+′δρgeu1n)σζ1e(urna′lγβ+tue2nαε)sFoXryγγx,SεδβyαXxρσδζ.isua−c1eFrtyγax,iγδny“xδte!ns−orial
rotation” of its value “at the identity”, namely its value when the matrix (1.10) is the
identity;itsufficestoputu := 1,uα := δα andu′α := δα in(1.15)above. Next,itmay
β β β β
be verifiedthatthe vanishingof the valueofSασ at the identity(whichis equivalentto
βρ
thevanishingofthegeneralSασ)providesasystemofsecondlinearsecondorderpartial
βρ
differentialequationsinvolvingonlythepartialderivativesFγ,δ . Finally,oneverifies
yxεyxζ
([Bi2003])thatthislastsystemisequivalenttothefactthattheFj1,j2 areofthespecific
cubic polynomialform (1.3). In conclusion, this provides a second (very summarized)
proofofTheorem1.1.
Intheremainderofthepaper,weshallnotspeakanymoreoftheequivalencemethod
and we will focus on describing precisely how to establish Theorem 1.1 using S. Lie’s
originaltechniques,asin[M2004a].
1.17. Acknowledgment. We are very grateful to Sylvain Neut and to Michel Petitot
(LIFL, University of Lille 1), who implemented the E´lie Cartan equivalence algorithm
and who discovered the validity of Theorem 1.1 in the case n = 2, and also of Theo-
rem1.7of[M2004a]inthecase m = 2. Withoutthesediscoveries,we wouldnothave
DIFFERENTIALCHARACTERIZATIONOFYXj1Xj2 =0,1≤j1,j2≤n≥2 9
pushedourmanualcomputationsuptotheveryendoftheproofofTheorem1.1(andof
Theorem1.7in[M2004a]).Wealsoacknowledgeinterestingexchangesaboutthemethod
ofequivalencewithCamilleBie`che,SylvainNeutandMichelPetitot.
1.18. Closing remark. After Theorem 1.1 was established, namely after the refer-
ences[BN2002],[N2003],[Bi2003]and[M2003]werecompleted,wediscoveredatthe
mathematicallibraryoftheE´coleNormaleSupe´rieureofParisthatMohsenHachtroudi,
anIranianstudentofE´lieCartan,alsoobtainedaproofofTheorem1.1([Ha1937],p.53),
sixty seven years ago. However, his computations follow the method of equivalence,
in the spirit of his master, whereas we conduct ours in the spirit of Sophus Lie. Also,
in[Ha1937],M.Hachtroudionlyreferstotheabbreviatedformulation(1.4)ofthecom-
patibilityconditions,andonedoesnotfindtheretheexplicitandcompleteformulationof
thesystems(I’),(II’),(III’)and(IV’).Finally,thecombinatorialformulasthatweprovide
in(2.10),in(2.14)in(2.15),in(2.22),in(2.23),in(2.31)andin(2.35)belowseemtobe
newintheLietheoryofsymmetriesofpartialdifferentialequations(seealso[M2004b]).
2.COMPLETELYINTEGRABLE SYSTEMSOF
§
SECOND ORDER ORDINARY DIFFERENTIALEQUATIONS
2.1.Prolongationofapointtransformationtothesecondorderjetspace. LetK=R
orC,letn N,supposen 2,letx=(x1,...,xn) Knandlety K. Accordingto
themainas∈sumptionofThe≥orem1.1,wehavetocons∈ideralocalK-a∈nalyticdiffeomor-
phismoftheform
(2.2) xj1,y Xj(xj1,y), Y(xj1,y) ,
7−→
whichtransformsthesys(cid:0)tem(1.(cid:1)2)tot(cid:0)hesystemY = 0(cid:1),1 j ,j n. Without
Xi1Xi2 ≤ 1 2 ≤
loss of generality, we shall assume that this transformation is close to the identity. To
obtainthepreciseexpression(2.35)ofthetransformedsystem(1.2),wehavetoprolong
the above diffeomorphismto the second order jet space. We introduce the coordinates
xj,y,y ,y onthesecondorderjetspace. Let
xj1 xj1xj2
(cid:0) (cid:1) n
∂ ∂ ∂
(2.3) D := +y + y ,
k ∂xk xk ∂y xkxl ∂y
xl
l=1
X
bethek-thtotaldifferentiationoperator.Accordingto[BK1989],forthefirstorderpartial
derivatives,onehasthe(implicit,compact)expression:
Y D X1 D Xn −1 D Y
X1 1 1 1
···
. . . .
(2.4)  .. = .. ..   .. ,
···
Y D X1 D Xn D Y
 Xn   n ··· n   n 
     
where ()−1 denotes the inverse matrix, which exists, since the transformation (2.2)
·
is close to the identity. For the second order partial derivatives, again according
to[BK1989],onehasthe(implicit,compact)expressions:
Y D X1 D Xn −1 D Y
XjX1 1 1 1 Xj
···
. . . .
(2.5)  .. = .. ..   .. ,
···
Y D X1 D Xn D Y
 XjXn   n ··· n   n Xj 
     
10 JOE¨LMERKER
forj =1,...,n. LetDX denotethematrix D Xj 1≤j≤n,whereiistheindexoflines
i 1≤i≤n
andjtheindexofcolumns,letY denotethecolumnmatrix(Y ) andletDY be
X (cid:0) (cid:1) Xi 1≤i≤n
thecolumnmatrix(D Y) .
i 1≤i≤n
By inspecting (2.5)above, we see that the equivalencebetween (i), (ii) and (iii) just
belowisobvious:
Lemma2.6. Thefollowingconditionsareequivalent:
(i) thedifferentialequationsY =0holdfor1 j,k n;
XjXk
≤ ≤
(ii) thematrixequationsD (Y )=0holdfor1 k n;
k X
≤ ≤
(iii) thematrixequationsDX D (Y )=0holdfor1 k n;
k X
· ≤ ≤
(iv) thematrixequations0=D (DX) Y D (DY)holdfor1 k n.
k X k
· − ≤ ≤
Formally,inthesequel,itwillbemoreconvenienttoachievetheexplicitcomputations
startingfromcondition(iv),sincenomatrixinversionatallisinvolvedinit.
Proof. Indeed,applyingthetotaldifferentiationoperatorD tothematrixequation(2.4)
k
writtenundertheequivalentform0=DX Y DY,weget:
X
· −
(2.7) 0=D (DX) Y +DX D (Y ) D (DY),
k X x X k
· · −
sothattheequivalencebetween(iii)and(iv)isnowclear. (cid:3)
2.8.Anexplicitformulainthecasen = 2. Thus, wecanstarttodevelopeexplicitely
thematrixequations
(2.9) 0=D (DX) Y D (DY).
k X k
· −
Init,somehugeformalexpressionsarehiddenbehindthesymbolD . Proceedinginduc-
k
tively,westartbyexaminatingthecasen=2thoroughly.Bydirectcomputationswhich
requireto be clever, we reconstitutesome 3 3 determinantsin the four(in fact three)
×
developedequations(2.9). Aftersomework,thefirstequationis:
X1 X1 X1 X1 X1 X1
x1 x2 y x1 x2 x1x1
0=y X2 X2 X2 + X2 X2 X2 +
x1x1 ·(cid:12) x1 x2 y (cid:12) (cid:12) x1 x2 x1x1 (cid:12)
(cid:12)(cid:12) Yx1 Yx2 Yy (cid:12)(cid:12) (cid:12)(cid:12) Yx1 Yx2 Yx1x1 (cid:12)(cid:12)
(cid:12)(cid:12)(cid:12) Xx11 Xx12 (cid:12)(cid:12)(cid:12)Xx1(cid:12)(cid:12)(cid:12)1y Xx11x1 Xx12(cid:12)(cid:12)(cid:12) Xy1
(2.10) +y 2 X2 X2 X2 X2 X2 X2 +
x1 · (cid:12) x1 x2 x1y (cid:12)−(cid:12) x1x1 x2 y (cid:12)
 (cid:12)(cid:12) Yx1 Yx2 Yx1y (cid:12)(cid:12) (cid:12)(cid:12) Yx1x1 Yx2 Yy (cid:12)(cid:12)
 (cid:12)(cid:12)(cid:12) Xx11 Xx11x1 Xy1 (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)
+y X2 X2 X2 +
x2 ·−(cid:12) x1 x1x1 y (cid:12)
 (cid:12)(cid:12) Yx1 Yx1x1 Yy (cid:12)(cid:12)
(cid:12) (cid:12)
 (cid:12)(cid:12) (cid:12)(cid:12)