Geometric degree of nonconservativity: Set of solutions for the linear case and extension to the differentiable non-linear case

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Geometric degree of nonconservativity: Set of solutions for the linear case and extension to the differentiable

non-linear case

Jean Lerbet, Noël Challamel, François Nicot, Félix Darve

To cite this version:

Jean Lerbet, Noël Challamel, François Nicot, Félix Darve. Geometric degree of nonconservativity: Set

of solutions for the linear case and extension to the differentiable non-linear case. Applied Mathemat-

ical Modelling, Elsevier, 2016, 40 (11-12), pp.5930–5941. �10.1016/j.apm.2016.01.030�. �hal-01321974�

Geometric degree of nonconservativity: Set of solutions for the linear case and extension to the differentiable non-linear case

J. Lerbet

^a,∗

, N. Challamel

^b

, F. Nicot

^c

, F. Darve

^d

aIBISC, UFRST-UEVE, 40, rue du Pelvoux CE 1455 91020 Evry Courcouronnes cedex, France

bUniversité Européenne de Bretagne, Université de Bretagne Sud LIMATB -UBS -Lorient Centre de Recherche Rue de Saint Maudé - BP 92116 56321 Lorient cedex, France

cIRSTEA, ETNA – Geomechanics Group, Domaine universitaire, 38042 Saint Martin d’Heres, France

dLaboratoire Sols Solides Structures, UJF-INPG-CNRS, BP 53 38041 Grenoble cedex 9, France

Thispaperdealswithnonconservativemechanicalsystemsasthosesubjectedtononcon- servativepositionalforcesandleadingtonon-symmetrictangentialstiffnessmatrices.Ina previouswork,thegeometricdegreeofnonconservativityofsuchsystems,definedasthe minimalnumberofkinematicconstraintsnecessarytoconverttheinitialsystemintoa conservativeoneisfoundtobe,inthelinearframework,thehalfoftherankoftheskew- symmetricpartofthestiffnessmatrix.Inthepresentpaper,newsresultsarereached.First, amoreefficientsolution oftheinitiallinearproblemisproposed.Second,alwaysinthe linearframework,theissueofdescribingthesetofallcorrespondingkinematicconstraints isgivenandreducedtotheoneoffindingalltheLagrangianplanesofasymplecticspace.

Third,theextensiontothelocalnon-linearcaseissolved.Afourdegreeoffreedomsystem exhibitingamaximalgeometricdegreeofnonconservativity(s=2)isusedtoillustrateour results.Theissueoftheglobalnon-linearproblemisnot tackled.Throughoutthepaper, theissueoftheeffectivinessofthesolutionissystematicallyaddressed.

Introduction

Nonconservativeelasticmechanicalsystemsexhibitseveralparadoxicalmechanicalbehaviors.Destabilizingeffectbyad- ditionalfriction iscertainlythe mostfamousparadox ofthesemechanicalsystemsandhasbeen deeplyinvestigated(see [1–3]forexample).Onelessreportedparadoxicaleffectisthedestabilizing effectbyadditionalkinematicalconstraints.J.J.

Thompsonmentionedthiseffectin[4]but,tothebestofourknowledge,thisparadoxicaleffecthadneverbeensystemat- icallyinvestigatedbeforerecently.Thisparadoxicaleffectledtotheso-calledkinematicalstructuralstability(ki.s.s.)issue:

whenandhowisitpossibletodestabilizebyaddingkinematicalconstraint(s)agivenstablesystem?

Duringthelastfiveyears,inasequenceofpapers([5–9]),weelucidatedthiskinematicalstructuralstability(ki.s.s.)issue forthelineardivergencestabilityofbothconservativeandnonconservativeelasticsystemsaswell.Abigpartoftheseworks arealsorelatedtotheso-calledsecondorderworkcriterionintroducedbyHillintheframeworkofplasticityin1958([10])

∗ Corresponding author. Tel.: +33 169477503.

E-mail address: jlerbet@gmail.com (J. Lerbet).

andindependentlyintroducedandusedintheframeworkofelasticnonconservativesystemsin2004([11]).Themainresult involvesthe symmetric partKs ofthe stiffnessmatrix andthemagnitude ofthe loadparameter aswell but itdoes not dependonthenumberoftheadditionalkinematicconstraints.

Bydualitytotheki.s.s.issue,weinvestigatedin[12]theissuetoconvertby(judicious)additionalkinematicconstraintsa nonconservativesystem^intoaconservativeone.Thisissueleadstotheconceptofgeometricdegreedofnonconservativity of^.Calculationsshowthatd=sisthehalfsoftherankroftheskew-symmetricpartKa(p)(thatisalwaysevenr=2s).

Inasecondstage, abuildingofthejudicious additionalkinematicconstraintsC₁,...,C_s∈(Rⁿ)^{s hasbeenproposed thanks} totheeigenspacesE_−λ2

i,i=1,...,softhesymmetricmatrixK_a²(^p)^whosetheeigenvalues−

λ

²1,...,−

λ

²s arealldouble:each C_imaybe chosenineachdistinct E_−λ2

i.Itisworthnotingthat,forbothissues,themechanicalsystem ^isapproximated byitslinearfirstorderapproximationatagivenequilibriumconfigurationqe.Thatmeansthat^{isdescribedbythemass} matrixMandthestiffnessmatrixK.Ifpisaloadparameter,thenK=K(^p)^.Thenon-symmetryofK(p)(namelyK=K_sor Ka=0)isthenthesignatureofthenon-conservativenatureofthemechanicalsystem^{.Inourpreviousworks,thesource} of the nonconservativity lies in external forces like follower forces actingon elastic system. Hypoelasticity mayalso be anothermechanicalframework leadingtoasimilar mathematicalproblem.Thereexists abroadliteraturecoveringhypoe- lasticity(seeforexample[13–16]).

Inthispaperwe are concernedby finding thecomplete solutionofthe linearcaseandby thegeneralization andthe extensiontothenon-lineardifferentiablecaseabouttothelatterissue.Wethenusethelanguageofanalyticmechanics.Ina firsttime,wereinvestigatethelinearcasebyusingthelanguageofexteriorp-formsandespeciallyexterior1-and2-forms.

Thatallow usto moredeeply highlightthe issueof effectivenessof thecalculation of thesuitable kinematic constraints converting the systeminto a conservative one. Thatalso allow us to investigatethe issueof building the set of all the solutions andto illustrate the geometrical meaning of thesesolutions. To do it, the language of symplectic geometry is systematicallyused.Thatalsosuggeststhegoodwayfortacklingthenon-linearcase.

Thus,inasecond time,we tacklethenon-linearproblemwithappropriate notationsandespeciallythanksto thelan- guageofdifferentialp-forms.Weaccuratelyfocusonthelinkwiththelinearcase.Inathirdstep,thesolutionisproposed byextendingtothenonlinearcasetheconceptofgeometricdegreeofnonconservativityandyieldingageometricmeaning tothecorrespondingnon-linearconstraints.Inthelastpart,theissueofthecalculationoftheappropriatenon-linearcon- straintsisinvestigated.Theproblemofaglobalsolutioninrelationship withthetopologyoftheconfigurationmanifoldis onlyevokedbyjustsettingtheconvenientgeometricframeworkofvectorbundles.Afourdegreeoffreedomsystemcalled theBigonisystem(see[12,17])iscontinuouslyusedthroughoutthepapertoillustratethegeneralresults.

1. Thelinearcase

Inwhatfollowswe referto[12].Weonlyrecall thatforthelinearframework,dynamicequation oftheunconstrained system^read:

MX¨+KX=0, (1)

withKany(namelynon-symmetric)matrixandMsymmetricpositivedefinite.Kisthestiffnessmatrixofthesystemand Mitsmassmatrix.Becauseofthenonconservativityofthe positionalforcesactingon

σ

^{,K isany.Theminimumnumber}

ofkinematicconstraintsallowingtoconvertthesystemintoaconservative one(withacorrespondingsymmetricstiffness matrix)isthegeometricdegreeofnonconservativityof^{.(1)isdeducedfromtheLagrange equationbytheusualprocess} oflinearizationaboutanequilibriumconfiguration.

1.1. Effectivinessofthesolutionproposedin[12]

Inintroduction,we alreadyrecalledthealgebraicmeaningofthe geometricindexordegree ofnonconservativity:this thehalf s ofthe rankr=2s ofK_a andthe distinct constraints,viewed asvectorsof Rⁿ, can be chosen in thes distinct eigenspacesE_−λ2

i,i=1,...,sof K_a². We now questionthe effectiveness of the buildingof the constraintsas proposed in [12].Todoit,we usethespectral theoremforK_a².Whatdoesmeantheeffectivenessforthespectral theorem?Theusual proof isdone by inductiononthe dimensionof thespace.Forinitializing theinductionreasoning, theD’AlembertGauss theoremis usedforfinding an eigenvalueof thecharacteristicpolynomial of K_a² andthis theoremisnot effectivein the sensewhereonlyanumericalmethodmayleadto(anapproximationof)theeigenvalues.So,withthesetools,thesolution ofthe linear caseitself is not effective.Remark howeverthat theconstraints are also the criticalpointsof the Rayleigh quotientRassociatedwithK_a² andthat onlythe eigenspacesareinteresting andnot theeigenvalues−

λ

²i,i=1,...,s.The useofRayleigh quotientisthen especiallyrelevant andtheconstraints maybe evaluated bysuccessive minimizationsof R(^X)=−^X_X^TKT^2aX^X.ByMinimaxtheorem,theconstraintsarealsothesolutionsof

dimminF=k max

X∈F\{⁰}R

(

^X

)

,

fork=1,...,navoiding by thiswaythe useof D’alembert–Gausstheorem.However, thisminimization process gives no analyticexplicitresult.

1.2. Thelanguageofexteriorp-forms

Now,before addressing ina followingstep thenon-linear case, we still focuson thelinear casewiththehelp ofthe exterior p-forms:welookthematrixK_a(skewsymmetricpartofthestiffnessmatrix) nolongerasthe matrixofa linear map ofRⁿ butasthematrixof anexterior 2-formon Rⁿ.We indifferentlynote E=Rⁿ andE^∗ its dualspace, thevector spaceofthelinearformsonE.Thus,let

φ

^{theexterior2-formdefinedonE}=Rⁿ by:

φ (

u,

v )

=u^TKa

v

, ⁽²⁾

afteridentifyingavectoru=(^u1,...,un)^ofRⁿwiththecolumnvectoru=

⎛

⎝ u1

.. . un

⎞

⎠ofMn1(R)^{.Thankstoabasictheoremof}

linearalgebra(see[18]forexample),thereisabasisB=(^e1,...,en)^ofRⁿandanumberr=2s≤nsuchthat

φ

(^e2i−1,e_2i)=

−

φ

(^e2i,e_2i−1)=1fori≤sand

φ

(^ei,e_j)=0fortheothervaluesofiandj.Inthedualbasis(^e∗₁,...,e^∗_n)^of(^e1,...,en),the form

φ

^reads:

φ

=e^∗₁∧e^∗₂+...+e^∗_2s−1∧e^∗_2s. (3) According to [12], we have to find out a subspace H of Rⁿ such that

φ

(^u,

v

)=0

∀

^u,

v

∈H and the constraints, viewed now aslinear formsC₁,...,C_s∈E^∗, then belong to H^⊥, theorthogonality being then understoodin the sense of duality.

ChoosingeachconstraintC_iinthesubspace<e^∗_2i−1,e^∗_2i>spannedbye^∗_2i−1 ande^∗_2iinE^∗leadstothewantedresult.Indeed, suppose tosimplifythat C_i=e^∗_2i−1 forall i=1,...,s andthat Gis thevector subspacespannedby (^e∗_2i−1)1≤i≤s. Letu,

v

∈ H=G^⊥wherethebidualisidentifiedwiththespaceitself.Thus,byuseof(3),ifu,

v

∈H,e^∗_2i−1∧e^∗_2i(^u,

v

₎=e^∗_2i−1(^u)^e∗2i(

v

₎− e^∗_2i(^u)^e∗_2i−1(

v

)=0−0=0andthus

φ

(^u,

v

)=0.

IfC_iisanyin<e^∗_2i−1,e^∗_2i>asimilarproofashereafterfordifferentialformsmaybeusedandisnotreproduced.

The effectiveness ofthe building ofthe constraints isnow brought back to theone ofthe basis B=(^e1,...,en)^.The proof isagaindone byinductiononthedimensionnofE (seeforexample[18]pp 30–31).Thisproofiseffectiveandthe following paragraph will highlighthow it is performing on an example.Before dealingwiththe example,an interesting issueistocharacterizeallthesolutions.

1.3. Setofsolutions

For describing the set of solutions, usual concepts of symplectic geometry are used. To do it, we first brought back theissueinthe usualframework ofsymplectic geometry.Theexterior 2-form

φ

^doesnotnecessarily definea symplectic structureonRⁿbecauseithasnotnecessarilyamaximalranknamely

φ

^maybedegenerate.Forinstance,thatnecessarily occurswhen n isodd. Letthen Fbe the kernelof

φ

^.Then (Rⁿ/F,

φ

˜) ^{isa 2s}-dimensionalsymplectic vector space where

φ

˜ ^is canonically defined by

φ

˜(^u¯,

v

¯₎=

φ

(^x,y) ^{withx (resp, y) anyvector ofthe class u¯ (resp.}

v

^¯). Remark that thanks to the canonical scalar product (.

|

^{.) on} Rⁿ, one could choose the orthogonal F^⊥ of F for the scalar product as “canonical”

supplementary spaceof F inRⁿ and

φ

F^⊥ the restriction of

φ

^{to F⊥. Then} (^F⊥,

φ

F^⊥) ^{is also a 2s}-dimensional symplectic vector spacesandthere arethree possiblemeanings fororthogonalityinF^⊥:duality, scalarproduct,and

φ

-orthogonality.

Howeverthescalarproducthasnomeaningon(Rⁿ/F,

φ

˜)^andonly

φ

-orthogonalityandorthogonalityfordualitykeepuseful on(Rⁿ/F,

φ

˜)^.Moreover,whengeneralizingthereasoningfromvectorspacestomanifolds,thenaturalEuclideanstructureof Rⁿ does not exist in the tangent and cotangent spaces. It is then more judicious to avoid the use of this structure. The orthogonalforthedualityofanysubspaceGwillthedenotedbyG^⊥.

If Wis any subspace of Rⁿ/F the

φ

-orthogonal or symplectic orthogonal of W isthe vector subspace ofRⁿ/F noted W^⊥φ definedby:u¯∈W^⊥φ⇐⇒

φ

˜(^u¯,

v

¯₎₌0

∀ v

^¯_∈W.Because

φ

˜ ^isnon-degenerate,themap

φ

^:Rⁿ/F→(Rⁿ/F)^{∗definedby}

φ

(^u¯)(

v

^¯₎=

φ

˜(^u¯,

v

¯₎

∀

^u¯,

v

¯∈Rⁿ/F isanisomorphism(canonical)andthen

φ

(^W⊥)=W^⊥φ foranyWsubspaceofRⁿ/F.

A subspaceLof Rⁿ/F iscalledLagrangian ifL=L^⊥φ.It isalso oftencalleda Lagrangian plane eventhough dimL=s. Astraightforwardcalculationshowsthatthedualbasis ofanybasisofanyLagrangiansubspaceisasolutionoftheinitial problem. Moreaccurately,ifL₁ isa Lagrangianplane then thereisa Lagrangiansupplementary spaceL₂ ofL₁ inRⁿ/F.If (^e2i−1)1≤i≤s(resp.(e_2i)_1≤i≤s)isanybasisofL₁(resp.L₂),thenthedualbasis(^e∗_2i−1)1≤i≤sof(^e2i−1)1≤i≤sinL^∗₁isafamilyof constraintssolutionofthemechanicalissue.Infactthisprocessrealizesanysolution.Thus,by thisprocess ,thesetofall thesolutionsoftheproblemisinabijectiverelationshipwiththesetofthebasesofallLagrangianplanesofRⁿ/Fbutthe geometricalmeaning ofthesolutionsliesin thesetofLagrangianspaces noted⁽

φ

^{) of}(Rⁿ/F,

φ

˜)^.(

φ

^{) hasbeendeeply}

investigatedespeciallyinrelationshipwiththetheoryofMaslovindex(see[19]forexampleforahighlightingpresentation oftheconstruction ofthisindex).⁽

φ

^{)isa s(s+}2¹⁾ dimensionalsubmanifoldoftheGrassmanniannmanifoldofalls-planes ofRⁿ/FandcalledtheLagrangianGrassmanniannof(Rⁿ/F,

φ

˜)^.

Let ^{(s) the Lagrangian} Grassmanniann manifold of the usual R-symplectic vector space (C^s,

ω

) ^{with its canonical} symplectic structure. This manifold maybe explicitly describedby andidentified with the set Us(s) of unitary symmet- riccomplexmatricesofMs(C)^{.ALagrangianplaneL}∈^{(s)isidentifiedwiththematrixU}L∈U_s(s)bythefollowingway:

x∈L←→x=U_Lc(^x)^{where c(x) is theconjugatecolumn vectorof x}∈C^s.Ifu isa symplectomorphismfrom (C^s,

ω

) ^onto (Rⁿ/F,

φ

˜),then(

φ

)=u((^s))^{whichachievesthecompleteandexplicit}descriptionofthesetofsolutionsnamely⁽

φ

^).

Notealsothat thereisanexplicitrepresentationofmatricesofUs(s).IfUbelongstoUs(s),thenU=X+iY withX,Ytwo

Fig. 1. n d.o.f. Bigoni system.

realsymmetric matrices ofsize s. Because of U is unitary, then X²+Y²=I_s andXY=YX.Thus there isa commonba- sisofdiagonalizationof XandYandaR-orthogonal matrixO sothat X=O^TDiag(^xi)^{O andY}=O^TDiag(^yi)^O.Wededuce thatx²_i +y²_i =1foralli=1,...,sandwe mayparametrizethe problemby x_i=r_icos

α

i andy_i=r_isin

α

i.Wededucethat U=O^TRVOwithR=diag(^ri)^andV=diag(^eiαj)^{.Remarkalsothat similartoolsliketheLagrangian planesmaybe usedto} investigateother objects like thespectrum of Hamiltoniansystems (see forexample [20]). Although some developments seemtobe closetoeachother,theoriginalandunexpectedfactofourowninvestigations liesintheuseofthesetoolsof symplecticgeometryandalgebrainanon-hamiltonianframework.

1.4.Theexample

Aspreviouslyannouncedintheintroduction,weillustratetheseresultswithafourdegreeoffreedomsystem.Wetackle thefourdegree offreedom non-linearZieglersystemwithcompletefollowerforcesateach jointlikeinFig.1withn=4. ThissystemiscalledBigonisystemin[12]becauseoftheexperimentaldevice proposedby thisauthorin[17].Thissame casehasbeenhandledin[12]inthe linearframeworkandisalso usedhereinordertocompare bothlinearapproaches (andalso to illustrate hereafter the non-linear case).In this section, namelyin the linearcase, the linearization isdone abouttheuniqueequilibriumposition

θ

=0_R4.

Theusualcaseofauniquefollowerforceattheextremity(namelytheusualZieglersystem)isnotveryinterestingsince thegeometricdegreeofnonconservativityisthenreducedto1(see[12])withanobvioussolution

θ

⁴=0.Inthiscase,the directionoftheexternalforceremainsconstantandthisforcebecomesconservative!.Accordingtothepreviousnotations,it meansthatdimF=2,s=1andthattheLagrangianspacesareonedimensionalsubspaceofthetwodimensionalsymplectic space(R⁴/F,

φ

˜)^{.Findingtheone}dimensionalLagrangiansubspacesofthissymplecticspaceisequivalenttofindthesetof linearkinematicconstraintssuchthatwhenthesystemundergoesoneoftheseconstraints,itbecomesconservative.

Theforcesystemisthennowsetupby p=(^p1,...,p₄)^{(see Fig.1).TheskewsymmetricmatrixKa(p)reads(see[12])} (obviouslyalltheelastictermshavingasymmetricinputinthesystemarenotinvolvedinthismatrix):

Ka

(

^p

)

=1 2

⎛

⎜ ⎜

⎝

0 p2 p3 p4

−p2 0 p3 p4

−p3 −p3 0 p4

−p4 −p4 −p4 0

⎞

⎟ ⎟

⎠

^,

meaningthatif(

1,...,

4)^{isthecanonicalbasisof}R⁴ then

φ

=1

2

(

^p2

1^∗∧

2^∗+p3

^∗1∧

3^∗+p3

2^∗∧

3^∗+p4

1^∗∧

4^∗+p4

2^∗∧

4^∗+p4

3^∗∧

4^∗

)

.

Here,det(^Ka(^p))=p²₂p²₄showingthat

φ

^{isnotdegeneratewhenp}2p₄ =0whichisnowsupposed.Thus,withtheprevious notations,F=

{

⁰

}

^andR⁴/F=R⁴and(R⁴,

φ

)^becomesafourdimensionalsymplecticspace.

Wewanttofindabasis(^e1,...,e₄)^sothat

φ

^=e∗1∧e^∗₂+e^∗₃∧e^∗₄. ⁽⁴⁾

Letuschoose

e^∗₁=

^∗1,e^∗₂= p2

2

2^∗,e^∗₃=

1^∗+

2^∗+

3^∗,e^∗₄= p3

2

3^∗+p4

2

4^∗. ⁽⁵⁾

Then(4)holdsandforexampletheconstraintsx₁=0,x₁+x₂+x₃=0convertthesystemintoaconservativeone.

WefocusnowonthesetofallsolutionsnamelyhereonthesetofLagrangianplanes.

Let

J=

⎛

⎜ ⎜

⎝

0 1 0 0

−1 0 0 0

0 0 0 1

0 0 −1 0

⎞

⎟ ⎟

⎠

^,

the matrix of the R-symplectic four dimensional vector space (C²,

ω

) ^{in its canonical basis}

v

1=(¹,0),

v

2=(ⁱ,0),

v

3= (⁰,1),

v

4=(⁰,i)^{.Let A be thematrix ofa} symplectomorphismu from(C²,

ω

) ^onto (R⁴,

φ

)^{in their respectivecanonical} basis. Thenthe relation

ω

(^x,y)=

φ

(^u(^x),u(^y))^{forall x},y∈C² leadstothe usualrelationJ=A^TKaA.ButifQ denotesthe change-of-basismatrixtopassfrom(

i^∗)^to(^e∗_i)^thenP=(^QT)^−1isthe correspondingchange-of-basismatrixtopassfrom (

i)to(e_i)andtheformula(4)thenreadsJ=P^TKaP.(5)meansthat

Q=

⎛

⎜ ⎜

⎜ ⎜

⎜ ⎜

⎝

1 0 1 0

0 p2

2 1 0

0 0 1 p₃

2

0 0 0 p4

2

⎞

⎟ ⎟

⎟ ⎟

⎟ ⎟

⎠

.

ItfollowsthatA=(^QT)^{−1 and}calculationsgive:

A=2

⎛

⎜ ⎜

⎜ ⎜

⎜ ⎜

⎜ ⎝

1 0 1 0

0 1

p2

0 0

−1 −1 p2

1 0

p3

p₄ p3

p₂p₄ −p3

p₄ 1 p₄

⎞

⎟ ⎟

⎟ ⎟

⎟ ⎟

⎟ ⎠

. ⁽⁶⁾

Itnowremainstoparametrizethe3(= ^2×2³)dimensionalGrassmaniann^{(2)ofLagrangianplanesof}C²whichisdoneas aboveinthegeneralcasethroughaparametrizationofUs(2).

Let U=

u1 u2

u3 u4

∈Us

(

²

)

.

BecauseUissymmetric,u₂=u₃ andbecauseUisunitary,thefollowingthreeindependentrelationshold:

|

^u1

|

²+

|

^u2

|

²= 1,

|

^u4

|

²+

|

^u2

|

²=1, u₁c(^u2)+u₂c(^u4)=0. Then

|

^u1

|

=

|

^u4

|

^{and we} parametrize the problem by u₁=cos

α

^eiα1,u₄= cos

α

^eiα4, u₂=u₃=sin

α

^{eiα2. The third relation u}1c(^u2)+u₂c(^u4)=0 then reads cos

α

^sin

α

(^{ei(α1−α2)}+e^i(α2−α4))=0. Genericallythat leadsto

α

2=^α1+₂^α4+(^2k+1)^π2:theparametrizationisgivenby

α

^,

α

1,

α

4 andthecorrespondingmatrix U(

α

^,

α

1,

α

4)reads:

U

( α

^,

α

^1,

α

⁴

)

=

cos

α

^{eiα1 sin}

α

^ieiα1+2α4

sin

α

^{ieiα1+2α4 cos}

α

^eiα4

.

Avector

v

₌₍x,y)∈(²)^ifandonlyif

x

y =

cos

α

^eiα1

(

−1

)

^ksin

α

^ieiα1+2α4

(

−1

)

^ksin

α

^{ieiα1+2α4 cos}

α

^eiα4

c

(

^x

)

c

(

^y

)

^{, (7)}

whichgives

x=cos

α

^eiα1c

(

^x

)

+

(

−1

)

^ksin

α

^ieiα1+2α4c

(

^y

)

y=

(

−1

)

^ksin

α

^ieiα1+2α4c

(

^x

)

+cos

α

^eiα4c

(

^y

)

. ⁽⁸⁾ Writing

v

=(^x,y)=4

i=1x_i

v

iinthecanonicalbasisoftheR-symplecticfourdimensionalvectorspace(C²,

ω

),thenc(

v

₎= (^c(^x),c(^y))=4

i=1(−1)^i+1xi

v

_iand(8)thenreads:

X=

(

^x1x₂x₃x₄

)

^T∈kerB

( α

,

α

1,

α

4

)

, with

B=

⎛

⎜ ⎜

⎜ ⎜

⎜ ⎜

⎜ ⎜

⎜ ⎝

cos

α

^cos

α

^{1−1 cos}

α

^sin

α

¹

(

−1

)

^k+1sin

α

^sin

α

¹⁺

α

⁴

2

(

−1

)

^ksin

α

^cos

α

¹⁺

α

⁴

2 cos

α

^sin

α

^{1 −cos}

α

^cos

α

¹⁻¹

(

−1

)

^ksin

α

^cos

α

¹⁺

α

⁴

2

(

−1

)

^ksin

α

^sin

α

¹⁺

α

⁴

2

(

−1

)

^k+1sin

α

^sin

α

¹⁺

α

⁴

2

(

−1

)

^ksin

α

^cos

α

¹⁺

α

⁴

2 cos

α

^cos

α

^{4−1 cos}

α

^sin

α

⁴

(

−1

)

^ksin

α

^cos

α

¹⁺

α

⁴

2

(

−1

)

^ksin

α

^sin

α

¹⁺

α

⁴

2 cos

α

^sin

α

^{4 −cos}

α

^cos

α

⁴⁻¹

⎞

⎟ ⎟

⎟ ⎟

⎟ ⎟

⎟ ⎟

⎟ ⎠

.

(9) Theseequations define a plane P=P(

α

,

α

¹,

α

⁴) ^ofR⁴ and L(

α

,

α

¹,

α

⁴)=A(^P(

α

,

α

¹,

α

⁴)) ^{withA givenby (6) isthen the} Lagrangianplaneof(R⁴,Ka)^definedby

α

^,

α

1,

α

4.Itistheparametrizationofthesetofallsolutionsofourproblem.

This“symplectic” solution issimpleincomparisonwiththeone proposedin[12]thatinvolvedtediouscalculationsfor thecalculationofonlyonesolution.Thismethodallows usmoreovertogetthesetofallsolutionsaswell.Thusitshows thatthiswayisstronglymoreefficientthanthewayusingthespectraltheoremforK_a²byavoidingthecalculationsofthe eigenvaluesandtheeigenvectorsofK_a².Suchamoreefficientmethodofbuildingthesetofconvenientconstraintsconverting thenonconservativesystemintoaconservativesystemiscertainlyagood(atleastabetter!)wayfortheextension tothe nonlinearcasewhichisnowtackledinthefollowingsection.

2. Setupofthenon-linearissue.Notations.Linkwiththelinearframework 2.1. Non-linearissueandnotations

Wethen consider nowa mechanical discrete system ^{so that its} configuration manifoldisa C^∞ realn dimensional manifoldand (^q=(^q1,...,qⁿ),U) ^denotes generically a local coordinatesystem. That means that U is an open set ofM andthereisa function

φ

^:U⊂M→Rⁿ (C^∞) withforall m∈M,

φ

(^m)=q.We supposethat thesystemissubjected toa positionalforcesystem^{sothatisdescribedbya}differential1-formQonMwhoselocalexpressionin(q,U)is:

Q= n

k=1

Q_,k

(

^q

)

^dqk. ⁽¹⁰⁾

Accordingto[18],theexact vocablefordescribing ^{isasemi-basic1-form.Thatmeansthat isdescribedbya1-form} (oraPfaff form)

ω

onthetotalspaceTMofthetangentbundle

τ

^Msuchthat

ω

^{isontheimageofthecanonicalvertical}

operator.Thelocalexpressionofasuchsemi-basicformisthen

ω

= n

k=1

ω

,k

(

q,q˙

)

^dqk. (11)

Thankstothepositionalpropertyoftheforces,

ω

,k(^q,q˙)^{onlydependsonqmeaningonlyoftheprojectionTM}→Mofa pointofTMandmaybeviewedasafunctiononthebasisMofthetangentbundle.Then,(11)takestheform(10).

Ifthereisnoambiguity,weomittheforcesystem^{andwriteQinsteadQ}.Anypdimensional(C^∞real)submanifold NofMmaylocally in(q,U) bedescribedby afamily (^f1,...,f^n−p) ^ofn−pindependent(C^∞ real)functionsdefinedon

φ

^{(U)sothatforallq}∈

φ

^(U),m=

φ

⁻¹(^q)∈N∩

φ

⁻¹(^U)⇔ f¹(^q)=...=f^n−p(^q)=0.ThemechanicalsystemwhoseNisthe configurationspaceiscalledasubsystemCof ^andfunctions(^f1,...,f^n−p)^{arecalledthelocal}(non-linear)expressions oftheconstraintsC.WeindifferentlynoteC orN.

Thepositionalforce system^{actingon issaid} conservativeifthereisafunctionhonMsothat inanycoordinate system(q,U):

Q_,k

(

^q

)

=

∂

^h

( φ

⁻¹

(

^q

))

∂

^{qk ,}

which isequivalentto Q=dhon Morthat Q islocallyexact onM. Itimpliesthat Q isa closeddifferential 1-formand thendQ=0onM.Locally,byuseofPoincaré’stheoremandafterhavingchosenanappropriatecoordinatesystem,wemay supposethereciprocalpropertytrueoneachUoftheatlascoveringM.Asmentionedabove,theglobalissueinvolvingthe topologyofMisoutofthescopeofthispaper.In[18],thevocable“conservative” meansonlythatthesemibasicPfaff form Qisclosed.Theword“Lagrangian system” isreservedforthecasewherethisformisexactintheframeworkofpositional forcesystem.Inthispaper,onlythelocalextensiontothenon-linearcaseisinvestigated.

Theissueisthenthefollowing:isthereasubmanifoldNofMsothatthephysicalaction^onN isconservative?Ifso, findallthepossibleNwiththehighestpossibledimensionpsothatthenumberofconstraintsisthesmallestpossible(be- causeeverysubsystembuiltbyaddingkinematicalconstraintstoaconservativesystemisagainaconservativesystem!!!).

2.2. Linkwiththelinearframework

Wefocusnowonthelinkwiththelinearframeworkof[12].Supposethatthereisaconfigurationqe(equilibriumconfig- urationof^{)suchthatthelinear}approximationisused.Letx=q−qe=(^x1,...,xn)^{.ThatmeansthatQ}isapproximated byitsTaylorexpansiontothefirstorderQ˜leadingto:

Q˜=Q˜

(

^x

)

⁼ n

k=1

Q_,k

(

^qe

)

⁺ n

=1

∂

^Q,k

(

^qe

)

∂

^{q x}

dx^k. (12)

Thatmeansthatifu=(^u1,...,uⁿ)∈Rⁿ≈Tq_eM Q˜

(

^x

)(

^u

)

⁼

n

k=1

Q_,k

(

^qe

)

⁺ n

=1

∂

^Q,k

(

^qe

)

∂

^{q x}

u^k,

andtheexteriorderivativeofQ˜(^x)^reads:

dQ˜

(

^x

)

=dQ˜

(

⁰

)

= n

k=1

<k

∂

^Q,k

(

^qe

)

∂

^{q −}

∂

^Q,

(

^qe

)

∂

^{qk dx∧dxk.}

Becausethematrix(^∂Q_∂^,k_q^(qe))k,isthestiffnessmatrixK=K(^qe),thendQ˜(^x)=dQ˜(⁰)^{isthelinearform}

φ

^oftheprevi-

oussection.

Toconcludethisparagraph,itmaybenotedthattheissueoftheextension tothenon-linearframeworkofthedecom- positionofthestiffnessmatrixintoitssymmetricandskewsymmetricpartisinvestigatedin[21].Theproposedsolution doesnotusethelanguageofdifferentialformseventhoughthetrickofPoincaré’sLemmaisstronglyusedinthepaper.The proposeddecompositionaimstoextendtothenon-linearcasetheremarkable(butobvious)followingpropertyoftheskew symmetricmatrixKa:x^TKax=n

k,=1K_a,kx_kx=0.Inourcontext,thisinterestingextensionisnotusable.

3. Solutionofthenon-linearproblem 3.1. Thesolution

IfthereisasuchpdimensionalsubmanifoldNofM,thentherestrictionQ_,NofQtoNmustbeclosed.Locallyboth conditionsareequivalentandthelocalconditioninthelocalcoordinatesystem(q,U)ofMreads:

∀

^q∈

φ (

^U∩N

) ∀

X,Y∈TN dQ

(

^q

)(

^X

(

^q

)

,Y

(

^q

))

=0, ⁽¹³⁾ wherethedifferential2-formdQ∈²(^M)^.Straightforwardcalculationsgive:

dQ

(

^q

)

=

<k

∂

^Q,k

(

^q

)

∂

^{q −}

∂

^Q,

(

^q

)

∂

^{qk dq∧dqk,}

andthevectorfieldsX,YbelongtoTNifandonlyif(locallyandwithabovenotations) dfⁱ

(

^X

)

=dfⁱ

(

^Y

)

=0,

foralli=1,...,n−p(dependencyofqisomitted).

WesupposenowthattheformdQisregularonMmeaningthatitsclassrisconstantonM.Thenhere,sincetheform dQisitselfa closedform(d²=0),its classisalsoequaltoits rankandiseven: r=2s.sistheuniquenumbersuchthat (^dQ)^s=0and(^dQ)^s+1=0.Wethendeducethat2s≤n.

Darboux’stheoremgivesthelocalmodelingofdQonanopensetUofMandreads:

dQ= s

k=1

dy^k∧dy^k+s, (14)

wherey¹,...,y^2sare2sindependentfunctionsonU.Choosenowthesfunctions f^k=y^konUforallk=1,...,s.ThenifX andY∈TNthenX(^fi)=dfⁱ(^X)=0=dyⁱ(^X)^andY(^fi)=dfⁱ(^Y)=0=dyⁱ(^Y)^foralli=1,...,s.Thus,

dQ

(

^X,Y

)

= s

k=1

dy^k∧dy^k+s

(

^X,Y

)

= s

k=1

dy^k

(

^X

)

^dyk+s

(

^Y

)

−dy^k

(

^Y

)

^dyk+s

(

^X

)

=0.

Itistheproofofthefollowing

Proposition1. Suppose thattheclassof dQ is constant(namely maximal). The(non-linear) degree ofnon-conservativity of ^{isthenthehalfsoftheclass2sofdQ}andthus p=n−s.ThelocaldefinitionofthesubmanifoldNis givenbythefamily f¹=0,...,f^s=0ofequationsonMwherefⁱisanylinearcombination(inthevectorspaceonRandnotinthemodulusonthe ringonthefunctionsonR)ofyⁱandy^i+sforalli=1,...,s.

Proof.Theproofhasbeengivenfor f^k=y^konUforallk=1,...,s.Supposenowthatfⁱisanylinearcombinationofyⁱand y^i+sforall i=1,...,smeaningthat fⁱ=

α

iyⁱ+

β

iy^i+sfori=1,...,swith(

α

i,

β

i)=(0,0).Then,choosinggⁱ=−_α2^βi

i+β_i²yⁱ+ αi

α²_i+β_i²y^i+swededucedy^k∧dy^k+s=df^k∧dg^kwhichallowtoconcludebyasameargumentasabove.

Build a family of non-linear constraints converting the system into a conservative one is then brought back to find (thankstoDarboux’stheorem) thedecomposition(14).Thisdecompositioncomes itself(byexterior derivative)fromDar- boux’sdecomposition forthe differential 1-form Q. There are several proofsof the existence of such a decomposition.

Beforeexamininganexample,wefirstasktheeffectivenessofthesolution.Inthefollowingparagraph,theissueoftheset ofallnon-linearsolutionsisonlyformalizedinthelanguageoffiberbundles.

3.2.Effectivenessofthesolution

The issueof the effectivenessof the solutionis a significant problem forthe physicist. Here, theeffectiveness of the calculation ofthe degree ofnonconservativity isclear atleast underthe assumption ofmaximal rank (or class)of dQ. CalculatingthesuccessivepowersofdQleadstothevalueofs.TheexampleinSection3.4illustratesthispoint.

Onthe contrary,the effectivenessof thecalculation ofthe constraints defining an appropriate submanifoldN of Mis broughtbacktotheoneofthecanonicalexpressionofdQthankstoDarboux’stheorem.InSection1,weinvestigatedthe effectivenessofbothsolutionsofthesameissuewithinthelinearframework:theoneproposedin[12]andtheoneofthis paper.Theformerisconstructivebutnotreallyeffectivewhereasthelatteriseffective.

We tackle now the corresponding non-linear issueof the effective calculations of family of constraints f_i,i=1,...,s whichisequivalenttoDarboux’s theoremfordQ.There areseveralproofsofDarboux’s theorem.Roughlyspeaking,one mayfindtwo differentkinds ofproof of Darboux’stheorem. Withournotations, thefirst one isdone by inductionon s andwemayfindsuchatypeofproofsinseveralbooksofAnalyticMechanicslikein[18]orin[22].Thenon-effectivestep thenliesinthecalculationoftheflowofano time-dependingvector fieldateachstep oftheinductionnamelyherethe calculationsofthe flows of sno time-depending vector fields(these vectorsfields haveto be calculated ateach step of theinductionreasoning).The secondkindofproofsofDarboux’s theoremusesaMoser’slemma whichisareasoningby homotopywithoutinductiononsbutinvolvingthecalculationoftheflowofatime-dependingvectorfield(seeforexample [23]–[25]).Bothkindsofproofshaveitsownadvantagesanddisadvantagesbutbothneedtocalculatetheflowofnon-linear vectorfields oratleasttointegrate thesevector fieldswhichcannot be done generallyanalyticallybutonly numerically.

Thatisthemainobstacletoan analyticsolutionofthisissue.ThefollowingexampleofthefourdegreeoffreedomBigoni systemleadstosuchasituation.

3.3.Setofsolutions

Theyare asmanysolutionsasfunctionsy_k,k=1,...,sinvolvedin(14).Butthispointofviewisnot geometric.Itisa similarissueastheonemetinthelinearcase:asolutionisafamilyoflinearformsbutthesetofsolutionsisdescribedby (themanifoldof)Lagrangianspaces.Heretogetan analogousobjectforthedifferentiablenon-linearcase, onehastouse thelanguageofvectorbundle.The2-differentialformdQ_π maybeviewedasa sectionofthevector bundle^2(M)ofthe 2-differentialformsonMwhichisitselfa vectorbundleassociatedtothe tangentbundle

τ

(^M)=(^T(^M),pM,M)^thefiber beingRⁿ. Supposingthat the class2s ofdQ_π is constant onMthen thefield q→kerdQ_π(^q) ^{definesa vectorsubbundle}

τ

0(M) over M whose the fiber is R^k withk=n−2s. The total space of this vector bundle is

q∈M

{

^q

}

×kerdQ_π(^q)^{. On} thequotientvector bundle

τ

˜(^M) ^{overM of}

τ

^{(M) by}

τ

0(M), the 2-differentialformdQ˜_π induced bydQ_π(q) on eachfiber TqM/kerdQ_π(^q)^{definesastructureofsymplectic vectorbundleover M.Thesetofsolutionsisthen builtbythesetofall} LagrangianmanifoldsLofthissymplecticvectorbundle.Theverydifficultissuetobuiltitislettofurtherinvestigations.