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 PDF
Preview 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
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 Thesesystems (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 forallk1,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 Thesestrudcϕdtψαur==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 beverified(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)
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