Lagrangian Formalism for nonlinear second-order Ri

arXiv:math-ph/0505024v19May2005LagrangianFormalismfornonlinearsecond-orderRiccatiSystems:one-dimensionalIntegrabilityandtwo-dimensionalSuperintegrabilityJos´eF.Cari˜nena†a),ManuelF.Ra˜nada†b),MarianoSantander‡c)†DepartamentodeF´ısicaTe´orica,FacultaddeCienciasUniversidaddeZaragoza,50009Zaragoza,Spain‡DepartamentodeF´ısicaTe´orica,FacultaddeCienciasUniversidaddeValladolid,47011Valladolid,SpainFebruary4,2008AbstractTheexistenceofaLagrangiandescriptionforthesecond-orderRiccatiequationisanalyzedandtheresultsareappliedtothestudyoftwodifferentnonlinearsystemsbothrelatedwiththegeneralizedRiccatiequation.TheLagrangiansarenonnaturalandtheforcesarenotderivablefromapotential.TheconstantvalueEofapreservedenergyfunctioncanbeusedasanappropriateparameterforcharacterizingthebehaviourofthesolutionsofthesetwosystems.Inthesecondparttheexistenceoftwo–dimensionalversionsendowedwithsuperintegrabilityisproved.Theexplicitexpressionsoftheadditionalintegralsareobtainedinbothcases.Finallyitisprovedthattheorbitsofthesecondsystem,thatrepresentsanonlinearoscillator,canbeconsideredasnonlinearLissajousfiguresKeywords:Nonlinearequations.Lagrangianformalism.Integrability.Superin-tegrability.Riccatiequations.Nonlinearoscillations.Closedtrajectories.Runningtitle:LagrangianFormalismforRiccatiSystems.PACSnumbers:02.30.Hq,02.30.Ik,02.40.Yy,45.20.JjMSCClassification:37J35,34A34,34C15,70H06a)E-mailaddress:[e-mail])E-mailaddress:[e-mail])E-mailaddress:[e-mail]11IntroductionIncestudied,inhiswell-knownbookofdifferentialequations[1],thefollowingequationw′′+3ww′+w3=q(z)andprovedthatithasthegeneralsolutionw=u′/u,whereuisageneralsolutionofthelinearequationofthethirdorderu′′′=q(z)u.ThisequationwasalsostudiedbyDavisin[2]asaparticularcaseofthegeneralizedRiccatiequations(accordingtoDavisthefamilyofthesenonlinearequationswasfirststudiedbyE.Vessiotin1895andG.Vallenbergin1899;see[3,4,5]forsomemorerecentstudiesrelatedwithhigher-orderRiccatiequations).LateronLeachetal[6,7]considertheequation¨q+q˙q+βq3=0(1)andpointoutthat“forβ=1/9islinearizable,possesseseightsymmetriesandiscompletelyinte-grable”andtheyadd“consequently,wecouldexpectthatthisremarkablemathematicalpropertycorrespondstoanimportantphysicaloneappearing(ordisappearing)forthisvaluewhichconse-quentlywouldappearasacriticalone”.Thisparticularβ=1/9equationwasalsoobtainedin[8]inthestudyofnonlinearequationswiththemaximumnumberofsymmetries(see[9],[10],[11],and[12]fortheLiesymmetryapproachtodynamicalsystems).RecentlyChandrasekaretal[13]havestudiedageneralizationofthisequationobtainedasaparticularcaseoftheLienardequation¨x+f(x)˙x+g(x)=0givenbyf(x)=kxandg(x)=(1/9)kx3+λx.AlthoughthisnewequationalsobelongstothegeneralizedRiccatifamilystudiedbyDavisandLeachetal,theymakeuseofatwostepproceduretosolvetheproblem:firstlytheyusetheso-calledPrelle-Singermethod[14,15,16,17]forobtainingasetoftime-dependentintegralsofmotionandsecondlytheyusethistime-dependentfamilyinordertocomputethesolution.Theresultisinterpreted,whenλ0,asan“unusualLi´enardtypeoscillatorwithpropertiesofalinearharmonicoscillator”.Butwewishtocalltheattentiontoonepropertydiscussedinthefinalpartofthepaper(afterfinalizingwiththePrelle-Singermethod):theexistenceofaLagrangiandescription.ThemainobjectiveofthisarticleistodevelopadeeperanalysisofthesenonlinearequationsusingtheLagrangianformalismasanapproach.InfactthestartingpointofourapproachisthefactthattheRiccatiequationbelongstoafamilyofnonlinearequationsadmittingaLagrangiandescription.Thishasinterestingconsequences,themostimportantofthemisthatRiccatisystemsaresystemsendowedwithapreservedenergyfunction.Westudythetwononlinearsystemsfirstlyinonedimensionandthenintwodimensions.Moreoverweprovethatthetwo-dimensionalextensionsarenotonlyintegrablebutalsosuper-integrable.Wenotethatthissituationhascertainsimilaritywiththeone-dimensionalnonlinearoscillatorstudiedbyMathewsetalin[18]thathasbeenprovedtoadmitasuperintegrabletwo-dimensionalversion[19].2Theplanofthearticleisasfollows:InSec.2wepresentaLagrangianapproachtoafamilyofnonlinearequationsthatincludesthesecond-orderRiccatiequationasaparticularcase.Sec.3,thatisdevotedtothefirstnonlinearsystem(‘dissipative’-lookingsystem),isdividedinthreepartscorrespondingtotheone-dimensionalsystem,geometricformalismandsymmetries,andtwo-dimensionalsystemandsuper-integrability,respectively.Sec.4,thatisdevotedtothesecondnonlinearsystem(‘nonlinearoscillator’),alsofirstlystudiesthen=1systemandthenthetwo-dimensionalsystemthat,aswehavepointedout,isalsoendowedwithsuper-integrability.FinallyinSec.5wemakesomecomments.2Lagrangianformalismandsecond-orderRiccatiequa-tionsInthisarticleweshallconsiderthefollowingnonlinearsecond-orderequationy′′+[b0(t)+b1(t)y]y′+a0(t)+a1(t)y+a2(t)y2+a3(t)y3=0,(2)wherewesupposethata30andthetwofunctionsb0,b1,arenotindependentbutsatisfyb0=a2√a3−a′32a3,b1=3√a3.Themoreimportantpropertyofthisequationisthatitcanbetransformedintoathird-orderlinearequationbythesubstitutiony(t)=1pa3(t)v′(t)v(t).Thustheequation(2),thatisthenaturalsecond-ordergeneralizationofthewellknown