Table Of ContentDe Gruyter Studies in Mathematical Physics 14
Editors
MichaelEfroimsky,Bethesda,USA
LeonardGamberg,Reading,USA
DmitryGitman,SãoPaulo,Brasil
AlexanderLazarian,Madison,USA
BorisSmirnov,Moscow,Russia
Oleg N. Kirillov
Nonconservative
Stability Problems of
Modern Physics
De Gruyter
PhysicsandAstronomyClassificationScheme2010: 02.30.Oz,02.30.Tb,02.40.Yy,03.65.Vf,
05.45.-a,45.20.-d,45.30.+s,46.40.-f,47.20.-k,47.65.-d.
Author
Dr.OlegN.Kirillov
Helmholtz-ZentrumDresden-Rossendorf
InstitutfürFluiddynamik
AbteilungMagnetohydrodynamik
Postfach510119
01314Dresden
Germany
[email protected]
ISBN978-3-11-027034-1
e-ISBN978-3-11-027043-3
Set-ISBN978-3-11-027044-0
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TothememoryofmyparentsAlevtinaandNikolai
Preface
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Almosthalfacenturyago,theseminalbookbyV.V.Bolotin,Nonconservativeprob-
lems of the theory of elastic stability, came into print [75]. It summarized the expe-
rienceofengineersintheanalysisofnonconservativesystemsaccumulatedsincethe
1920swhenthefirstattemptstotakeintoaccountdissipativeeffectsinrotordynam-
icsandtoexplainsuchdangerous instabilities asaerodynamicflutterandshimmyin
aircraftlandinggearwereundertaken.Despitethemanyexcellentbooksonstability
theorythatwerepublishedduringthenextfiftyyears[22,60,121,127,141,150,162,
168,182,213,239,346,367,369,371,394,400,412,423,437,470,503,518,528,570,
590,626],Bolotin’smonographstillremainstheonlyonedevotedsolelytothemeth-
ods and challenges of nonconservative stability. Its only drawback is that this book
cannottakeintoaccountthedramaticdevelopmentsinmathematics,engineeringand
physicsthathavebeenmadesince1963.
Thebook[75]wasmotivatedmainlybytheproblemsofstabilityofslenderstruc-
turesunderfollowerforcesandofrotatingflexibleshafts.Alreadythesemodelsdeal
with the two fundamental nonconservative perturbations – viscous dissipation and
nonpotentialpositionalforces[332,333].
Last five decades extended considerably the range of applications in which such
nonconservative forces play a crucial role. We mention friction-induced instabilities
causing the flipping of the tippe top [84] and the problems of the acoustics of fric-
tionrelatedtotheexcitationofaudiblevibrationsinbrakesandclutches[268],paper
calenders [92,537], prostheses of hip joints [255] and even in the singing glasses of
BenjaminFranklin’sglassharmonica[6,293].
viii Preface
Nonconservative models appear in modern studies of landslides on gentle slopes
when the constitutive relation of a nonassociated geomaterial, such as loose sand, is
describedbyanonsymmetricmatrix[59,60,117].
Inhydrodynamicsandplasmaphysics,thecounter-intuitivedestabilizinginfluence
ofdissipationonnegativeenergywaves[115,168,446,552]isanimportantingredi-
ent in the theories of boundary layer [349], flow control [111,112,171] and stabil-
ityofwavepropagation[90].Inrotatingfluids,interplayofthenonconservativeand
gyroscopic forces may lead to the paradoxical discontinuous change in the stability
boundary as happens in the case of the baroclinic instability when the Ekman layer
dissipationisinfinitesimallysmall[334,501,563].
Inmagnetohydrodynamics,theVelikhov–Chandrasekharparadox[120,581]occurs
in the theory of magnetorotational instability when for infinite electrical conductiv-
ity of the differentially rotating fluid the limit of the vanishing axial magnetic field
does not trace back to the Rayleigh threshold of hydrodynamics [1,28,199]. Non-
conservativeforcesplayanincreasingroleincelestialmechanics,e.g.,inthemodeling
oftetheredsatellitesystems,satelliteandplanetaryrings[43,44,87].
Arichsourceofnonconservativeproblemsismodernroboticsandautomaticcon-
trol [41] and biomechanics [216] – for example, the spine is frequently modeled as
a column loaded by a distributed follower force [500]. Of course, traditional areas
such as aerospace engineering and structural mechanics remain one of the biggest
consumersofthenonconservativestabilitytheory[224,240].
Alreadyinthe1960sBolotinemphasizedthatprogressinthenonconservativesta-
bilitytheorydependsondevelopmentsinthetheoryofnonself-adjointoperators.Im-
portantcontributions tothelattermotivatedmainlybymechanicalapplications were
made,e.g.,inthe1940sbyS.L.Sobolev[536]andL.S.Pontryagin[483],inthe1950s
byM.V.Keldysh[261]andM.G.Krein[336,337]andsincethenbymanyotherau-
thors.
Theneedsofoptimaldesignandrationalexperimentplanningrequiredconsidera-
tionofmultiparameterfamiliesofnonself-adjoint boundaryeigenvalueproblems.In
the 1970s, the studies by V.I. Arnold and his co-workers established a sharp corre-
spondencebetween themultiple eigenvalues ofnonsymmetric matricesandgeomet-
ricsingularities ontheboundaryoftheasymptotic stabilitydomainofamatrixfam-
ily[17,18,374,375].Animmediateconsequenceofthisresultistheresolutionofthe
famous Ziegler’s paradox (1952) of destabilization of a reversible system by small
dissipation[622]bymeansoftheWhitneyumbrellasingularity,whichwasdonein-
dependentlybyanumberofauthorsstartingwithO.Bottemain1955–56[80,81].
Sincethe1950s,theconceptofthesymplecticorKreinsignatureofeigenvalueshas
beenwidelyusedinhydro-andmagnetohydrodynamicstodescribewavesofpositive
andnegativeenergy[264,320,390,392,606,607].Inthe1990s,theinfluenceofnon-
HamiltonianperturbationsonthestabilityofHamiltoniansystemsbecameatopicfor
a systematic investigation [391,395] that gave birth to the area of research known
Preface ix
as ‘dissipation-induced instabilities’ [74] – a concept that touches a broad variety of
physicalapplications[333].
Thoughveryexcellent,anetworkoftheseresultsischaoticallyscatteredatpresent
throughout the specialized journals. Many brilliant physical phenomena that could
crown the nonconservative stability theory are almost unknown to the stability the-
orists with a classical mechanical background. On the other hand, achievements of
the theory of nonself-adjoint operators [421,611], the theory of operators in the
spaces with indefinite metric (Krein and Pontryagin spaces) [202], Lidskii–Vishik–
Lyusternikperturbationtheoryformultipleeigenvalues[428],theoryofmultiparam-
etereigenvalueproblems[24,585],modernresultsofappliedlinearalgebra[576]as
wellassingularitytheoryappeartobestillnotfamiliartomanyengineers,physicists
andevenstabilitytheoristsworkingwithnonconservativestabilityproblems.
However, the combination of these approaches of modern applied mathematics
withthecomplexfundamentalnonconservativephenomenaofphysicsandmechanics
seemstobetheonlywaytounderstandthelatterandtocreatearathercompleteand
unifiedconstructivetheoryofstabilityofnonconservativesystemsandofdissipation-
inducedinstabilities.Thatiswhythereisastrongneedforadetailedexpositionwhich
wouldbringtogetherthescatteredresultsofthelastfiftyyearsandwhichwoulden-
deavortounifyandtosystematizeboththeresultsandthemethodsoftreatmentofthe
nonconservativestabilityproblemsofmodernphysics.Thisbookisanattempttofill
thisneed.
In it an effort is made to present the subject of nonconservative stability from the
modernpointofviewascompletelyaspossiblewithintheallottedspace.Itpresents
relevantmathematicalconcepts,bothalreadyfamiliarandthenewonesforthissubject
aswellasrigorousstabilityresultsandnumerousclassicalandcontemporaryexamples
from mechanics and physics. The book is substantially based on the results of the
author;althoughbynecessityitcontainssomeresultsofotherauthorswithoutwhich
it is impossible to create a self-consistent exposition. It is hoped that this book will
serve the present and prospective specialist in the field by acquainting him with the
currentstateofknowledgeinthisactivelydevelopingfield.
Thebook has 12 chapters. After anumber of examples accompanied by ahistori-
caloverviewintheIntroduction,thefirstsixchaptersdealwiththefinitedimensional
nonconservativesystems,whiletherestofthebookisdedicatedtotheinfinitedimen-
sional ones. Naturally, the first part of the book contains fundamentals of the theory
andmoregeneralresultsbecauseofthewidevarietyofmathematicaltoolsavailable
in finite dimensions. The center of gravity in the second part is shifted to studies of
concretephysicalproblems.Allchapterscontainillustrativephysicalexamples.
I would like to express my warmest gratitude to all the colleagues and collabo-
rators whose support, friendly advice, encouraging discussions and fruitful joint re-
search were among the main inspirational factors driving my work on this project:
Abd Rahim Abu Bakar, Sergei Agafonov, Vadim Anischenko, Teodor Atanack-
ovic, Vladimir Beletsky, Carl Bender, Michael Berry, Noël Challamel, Gengdong
x Preface
Cheng,RichardCushman,FelixDarve,BarbaraDietz,MikhailEfroimsky,Yasuhide
Fukumoto,GunterGerbeth,ValentinGlavardanov,YuanxianGu,Eva-MariaGraefe,
Samvel Grigorian, Uwe Günther, Peter Hagedorn, Hanns-Ludwig Harney, Daniel
Hochlenert, Norbert Hoffmann, Igor Hoveijn, Wolfhard Kliem, Anthony Kounadis,
Yuri Leschinski, Alexei Mailybaev, Vadim Marchenko, Jerrold Marsden, Maxim
Miski-Oglu, OliverO’Reilly,HuajiangOuyang, PauliPedersen,DmitryPelinovsky,
AlexandraPerlova,KarlPopp,AchimRichter,IngridRotter,FlorianSchäfer,Guido
Schneider, Alexander Seyranian, Sergei Sorokin, Gottfried Spelsberg-Korspeter,
FrankStefani,FerdinandVerhulst,AlexanderZevin,andMiloslavZnojil.
IlearnedalotfromthelecturesandseminarsgivenduringmystudiesattheMoscow
Institute of Physics and Technology in 1989–1995 by Alexander Abramov, Oleg
Besov,BorisFedosov,VictorGalactionov,VictorLidskii,BorisRauschenbach,Vic-
tor Zhuravlev, and other brilliant professors and lecturers of my Alma Mater whose
enthusiasticworkinthe‘stormy1990s’inRussiawasanimpressiveexampleofano-
blededicationtostudentsandtheprofessionthatdeeplyinfluencedmyformationasa
scientist.
I thank the Russian Foundation for Basic Research (RFBR, Russia), the National
NaturalScienceFoundationofChina(NSFC,China),INTAS(EU),theCivilianRe-
search and Development Foundation (CRDF, USA), the Alexander von Humboldt
Foundation (Germany), Deutsche Forschungsgemeinschaft (DFG, Germany), Saxon
Ministry of Science (Germany) and the Japan Society for the Promotion of Science
(JSPS,Japan)fortheirmostvaluablefunding.
IamindebtedtomywifeKseniaandtomydaughterMarinaandsonNikolaiwho
generouslygiftedmetime,support,andunderstanding.
Dresden,12April2012
