Standard and non-standard Lagrangians for dynamical systems

17 Pages·2008·0.21 MB·English

by  Z E Musielak

IOPPUBLISHING JOURNALOFPHYSICSA:MATHEMATICALANDTHEORETICAL
J.Phys.A:Math.Theor.41(2008)055205(17pp) doi:10.1088/1751-8113/41/5/055205
Standard and non-standard Lagrangians for
dissipative dynamical systems with variable
coefficients
ZEMusielak
DepartmentofPhysics,TheUniversityofTexasatArlington,Arlington,TX76019,USA
E-mail:[email protected]
Received10October2007,infinalform6December2007
Published23January2008
Onlineatstacks.iop.org/JPhysA/41/055205
Abstract
Dynamicalsystemsdescribedbyequationsofmotionwiththefirst-ordertime
derivative(dissipative)termsofevenandoddpowers,andcoefficientsvarying
eitherintimeorinspace,areconsidered. Methodstoobtainstandardandnon-
standard Lagrangians are presented and used to identify classes of equations
ofmotionthatadmitaLagrangiandescription. Itisshownthattherearetwo
general classes of equations that have standard Lagrangians and one special
class of equations that can only be derived from non-standard Lagrangians.
In addition, each general class has a subset of equations with non-standard
Lagrangians. Conditions required for the existence of standard and non-
standard Lagrangians are derived and a relationship between these two types
ofLagrangiansisintroduced. ByobtainingLagrangiansforseveraldynamical
systemsandsomebasicequationsofmathematicalphysics,itisdemonstrated
thatthepresentedmethodscanbeappliedtoabroadrangeofphysicalproblems.
PACSnumbers: 45.20.−d,45.30.+s,05.45.−a
1. Introduction
Thefactthatallfundamentalequationsofmodernphysicscanbederivedfromcorresponding
Lagrangians is well-known and strongly emphasized in many textbooks and monographs
[1–7]. A lesser-known fact is that practically all of those Lagrangians were not part of an
aprioriprocessthatoriginallyledtotheequations,andthattheirexplicitformswereusually
obtainedinadhocfashionsinsteadofbeingstrictlyderivedfromfirstprinciples[4–7]. The
mainpurposeofthispaperistopresentgeneralmethodstoderivestandardandnon-standard
Lagrangians, and use these methods to identify classes of equations of motion that can be
derivedfromtheLagrangians.
WeconsiderLagrangianstobestandardiftheycanbeexpressedasdifferencesbetween
‘kinetic energy terms’ and ‘potential energy terms’; these Lagrangians may, or may not,
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J.Phys.A:Math.Theor.41(2008)055205 ZEMusielak
explicitlydepend ontimethrougheitheranexponential factororanotherfunction, andthey
arealsoreferredtoasS-equivalentLagrangians[8]. AllotherformsofLagrangiansdiscussed
inthispaperarecallednon-standardLagrangians[6]; notethatArnold[9]referstothemas
non-naturalLagrangians.
Dynamical systems discussed in this paper are described by second-order ordinary
differentialequationswiththefirst-ordertimederivative(dissipative)termsofevenandodd
powers, and coefficients varying either in time or in space. We refer to these systems as
‘dissipative’ because of the first-order time derivative (velocity) terms in their equations of
motion[10, 11]. Onemustkeepinmindthatthisgeneraldefinitionencompassesbothnon-
conservativesystemswiththelinear(orhigheroddpowersof)velocitytermsandconservative
systemswiththequadratic(orhigherevenpowersof)velocityterms;thelatterarealsocalled
‘damping-like’terms[10]becausetheresultingequationsofmotionareinvariantwithrespect
totimereversal,whichmeansthatthesystemsareconservative.
Itiswell-knownthatNewtonianmechanicscanbeappliedtobothconservativeandnon-
conservative systems; however, the Lagrangian and Hamiltonian formulations of mechanics
arelimitedtoconservativesystems[1–5]. Thevalidityofthelatterisguaranteedbyacorollary
resultingfromwell-knownmathematicaltheoremsfirstprovedbyBauer[12]. Thecorollary
shows that it is impossible to apply the Lagrangian formulation and Hamilton’s variational
principletoalineardissipativesystemdescribedbyasingleequationofmotionwithconstant
coefficients;thisisaproblemrelatedtothetimereversibilityoftheactionprinciple,whichdoes
notallowselectingaspecialdirectionintime[6]. However,Bateman[13]foundloopholesin
Bauer’sresultsandconstructedLagrangiansfordissipativesystems.
OneofBateman’stechniquesistoaddtoasystemunderconsiderationanotheronethat
isreversed intimeandhas negative friction. Themethodleads totwoequations ofmotion,
andtheresultingHamiltoniangivesextraneoussolutionsthatmustbesuppressed[6,7]. An
interesting modification of this method was done by Dekker [14] who introduced two first-
orderequationsthatwerecomplexconjugateofeachotherandshowedhowtocombinethem
toobtainonereal,second-orderequationofmotion. Bateman’sothertechniqueistoconsider
aLagrangianthatdependsexplicitlyontimethroughanexponentialfactorandtoobtainthe
desiredequationofmotionbyignoringthetime-dependentterm[6,7]. Theproblemwiththis
approachisthattheresultingmomentumandHamiltonianmaynotbephysicallymeaningful.
A generalized method to deal with non-conservative systems was developed by Riewe
[15, 16], who formulated Lagrangian and Hamiltonian mechanics by using fractional
derivatives. Hismainresultisthatnon-conservativeforcescanbecalculatedfrompotentials
thatcontainfractionalderivatives. Afterthemethodwasappliedtoseveralsystems[17,18],
itbecameclearthatitsbroadapplicationsarelimitedbythecomplexityoffractionalcalculus.
Another problem with the method is that its equations are acasual and that the procedure to
change these equations into casual ones is not well defined [19]. In addition, the method
cannotdirectlybeusedtoquantizelineardissipativesystems.
IntheliteraturetheinverseproblemoffindingaLagrangianforagivenequationofmotion
is called the Helmholtz problem, which requires solving Helmholtz equations [20, 21]. An
alternativeapproachwasdevelopedbyMusielaketal[22],whointroducedageneralmethod
toderiveastandardLagrangianforanonlineardynamicalsystemwithaquadraticfirst-order
timederivativetermandcoefficientsvariableinthespacecoordinates.
In this paper, the method [22] is used to determine forms of equations of motion that
have standard Lagrangians. Another method is developed to identify equations of motion
that can be derived from non-standard Lagrangians. It is shown that there are two general
classesofequationsofmotionwithstandardLagrangiansandonespecialclassofequations
ofmotionwithnon-standardLagrangians. Aninterestingresultisthateachgeneralclasshas
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also a subset of equations of motion with non-standard Lagrangians. Conditions required
for the existence of standard and non-standard Lagrangians are derived and a relationship
betweenthesetwotypesofLagrangiansisintroduced. ByobtainingLagrangiansforseveral
dynamicalsystemsandsomebasicequationsofmathematicalphysics,itisdemonstratedthat
thepresentedmethodscanbeappliedtoabroadrangeofphysicalproblems.
Theoutlineofthepaperisasfollows: amethodtoderivestandardLagrangiansandbasic
equations of motion are presented in section 2, equations of motion with time and space-
dependentcoefficientsthatcanbederivedfromeitherstandardornon-standardLagrangians
aredeterminedinsections3and4,physicalimplicationsoftheobtainedresultsarediscussed
insection5andconclusionsaregiveninsection6.
2. Methodandbasicequations
TointroducethemethodthatisusedinthispapertoderivestandardLagrangiansfordynamical
systems with variable coefficients, we consider two second-order ordinary differential
equationswithdifferentvariablecoefficients:
x¨ +C(t)G(x)=0 (1)
and
x¨ +c(x)g(x)=0, (2)
where x¨ = d2x/dt2,C(t) is an arbitrary function of t only, and G(x),g(x) and c(x) are
arbitraryfunctionsofxandtheydonotdependexplicitlyont.
Depending on the physical meaning of the independent variable t and the dependent
variable x, and on the form of the coefficients C(t) and c(x) and the functions G(x) and
g(x), equations (1) and (2) may describe many different physical systems. The best known
examples of such systems are the simple linear oscillator, with C(t) = c(x) = ω2 = const
o
and G(x) = g(x) = x, and the nonlinear pendulum, with C(t) = c(x) = ω2 = const and
o
G(x)=g(x)=sinx,whereω isthenaturalfrequencyoftheoscillatorysystem.
o
Other examples of different linear and nonlinear oscillators that are represented by
equations (1) and (2) are given in two monographs [23, 24]. Moreover, equation (1) can
be cast into the form of either the Airy, Mathieu or Hill equation [10, 11], or the time-
independent Klein–Gordon equation with constant [1] and variable [25] coefficients or the
time-independentSchro¨dingerequation[1];notethatfortheKlein–GordonandSchro¨dinger
equations,thevariablestandxhavedifferentphysicalmeanings.
WritingLagrangiansL (x˙,x,t)andL (x˙,x)forequations(1)and(2),respectively,isa
C c
straightforwardprocedure[22]aslongasG(x),g(x)andc(x)arecontinuousandintegrable
functions. Assumingthatthisisthecase,wehave
(cid:1)
1 x
L (x˙,x,t)= x˙2−C(t) G(x˜)dx˜, (3)
C
2
(cid:1)
1 x
L (x˙,x)= x˙2− c(x˜)g(x˜)dx˜. (4)
c
2
AccordingtoCourant[26],theaboveintegralsareindefinitebecauseanyvalueofthelower
limitoftheseintegralscanbechosen. Sincedifferentindefiniteintegralsofthesamefunction
differonlybyanadditiveconstant,thelowerlimitoftheintegralscanbechoseninsuchaway
thattheresultingconstantisalwayszero. Becauseofthisproperty, theintegralsarewritten
withouttheirlowerlimits.
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ForG(x),g(x)andc(x)beingcontinuousfunctions,thederivativesoftheaboveintegrals
withrespecttoxaregivenbytheirintegrands[26]. Thispropertyisessentialtodemonstrate
that equations (1) and (2) are obtained when the above Lagrangians are substituted into the
Euler–Lagrangeequation:
(cid:2) (cid:3)
d ∂L ∂L
C,c − C,c =0. (5)
dt ∂x˙ ∂x
SinceL (x˙,x,t)andL (x˙,x)areexpressedasthedifferencebetweenthe‘kinetic’and
C c
‘potential’energyterms(seeequations(3)and(4)),werefertothemasstandardLagrangians
(see section 1). For the Airy, Mathieu and Hill equations, and the time-independent Klein–
Gordon and Schro¨dinger equations, the integral in equation (3) can be evaluated and the
explicit form of the Lagrangian L (x˙,x,t) can be obtained. However, this may not be the
C
case for physical systems described by the Lagrangian L (x˙,x); depending on the form of
c
c(x),theintegralinequation(4)mayleadtoanon-localLagrangian[22].
Letusnowadd‘dissipativeterms’toequations(1)and(2),andwritethemdowninthe
followinggeneralform:
x¨ +B(t)F(x˙)+C(t)G(x)=0, (6)
x¨ +b(x)f(x˙)+c(x)g(x)=0, (7)
whereF(x˙)andf(x˙)arearbitraryfunctionsthatsolelydependonx˙,andthecoefficientsB(t)
andb(x)dependonlyontandx,respectively.
Thebasicideaofthismethodistofindtransformationsthatremovethedissipativeterms
from equations (6) and (7), and allow casting these equations in the form of equations (1)
and (2); obviously, the existence of such transformations depends on specific forms of the
functionsF(x˙)andf(x˙). Oncethetransformationsarefound,thecorrespondingLagrangians
for the transformed variables can easily be derived from equations (3) and (4). Then, the
standardLagrangiansfortheoriginalvariablesarealsoobtained.
WeassumethatthefunctionsF(x˙)andf(x˙)aregivenaspowerlaws,whichmeansthat
equations(6)and(7)canbewrittenas
x¨ +B(t)x˙m+C(t)G(x)=0 (8)
and
x¨ +b(x)x˙n+c(x)g(x)=0, (9)
wheremandnareintegers1,2,3,....ThechoiceofthepowerlawsforthefunctionsF(x˙)
andf(x˙)isjustifiedbynumerousphysicalsystemsdescribedbytheaboveequations.
Examplesofforce-freedynamicalsystemswithlineardamping(m = n = 1)includea
damped pendulum whose length increases in time [27], a damped Duffing oscillator [28], a
dampedDuffing–vanderPolsystem[29]andLie´nardandothertypesofoscillators[8,29–31];
specifically,theLie´nardequationisobtainedwhenn=1andc(x)=1. Numerousdynamical
systemswithnonlineardamping(m = n = 2)havebeenconsideredintheliterature[8,24,
32–38]. Dynamical systems with higher powers of m and n have also been discussed [39],
and an important example of such systems is the nonlinearly damped double-well Duffing
oscillator[40].
Havingdescribedthemethodandspecifiedthebasicequationsofmotionusedinthispaper,
wenowdeterminewhichofthoseequationscanbederivedfromstandardandnon-standard
Lagrangians. Letusbeginwiththeequationsofmotionwithtime-dependentcoefficients.
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3. Equationswithtime-dependentcoefficients
3.1. ExistenceofstandardLagrangians
The general form of the equation of motion with time-dependent coefficients is given by
equation (8). To determine the value of m for which the first-order derivative term can be
removedfromtheequation,weusethefollowingintegraltransformation:
x(t)=x (t)eI1(t), (10)
1
where
(cid:1)
t
I (t)= φ (t˜)dt˜, (11)
1 1
andφ isanarbitraryfunctiontobedetermined;therestrictiononthisfunctionisthatitmust
1
beintegrableanddifferentiable. Inaddition,ifφ isacontinuousfunction,thentheindefinite
1
integralI (t)alwayspossessesaderivativesuchthatdI (t)/dt = φ (t)[26]. Alsonotethat
1 1 1
theabovetransformationguaranteesthatthevariablesx(t)andx (t)havethesamezeros.
1
Thetransformedequationofmotioncanbewrittenas
(cid:4) (cid:5)
x¨ +2x˙ φ + φ˙ +φ2 x +(x˙ +φ x )mB(t)e(m−1)I1(t)+C(t)G(x eI1(t))e−I1(t) =0. (12)
1 1 1 1 1 1 1 1 1 1
Sincethetermswithx˙ havedifferentpowers,theycannotberemovedbythefunctionφ (t)
1 1
unlessm=1;notethatonlyform=1thepowerofthetwotermswithx˙ isthesame,which
1
means that one can find such φ (t) that allows removing these terms. This is an important
1
result as it shows that the integral transformation used here is limited to the equations of
motionwiththelineardampingterm.
Takingintoaccountthefactthatm=1,equation(12)becomes
(cid:6) (cid:7)
x¨ +[B(t)+2φ (t)]x˙ + φ˙ +φ2+B(t)φ x +C(t)G(x eI1(t))e−I1(t) =0. (13)
1 1 1 1 1 1 1 1
Thisshowsthattheconditionrequiredtoremovethetermwithx˙isφ (t)=−B(t)/2;notethat
1
substitutionofthisconditionintoequations(10)and(11)givesawell-knowntransformation
[28].
Usingtheabovecondition,wewriteequation(13)as
(cid:4) (cid:5)
x¨ − 1 B˙ + 1B2 x +C(t)G(x e−IB(t))eIB(t) =0, (14)
1(cid:8) 2 2 1 1
where I (t) = tB(t˜)dt˜/2. Since the form of equation (14) is similar to that given by
B
equation(1),weuseequation(3)towritetheLagrangianL (x˙ ,x ,t)inthefollowingform:
(cid:2) (cid:3) (cid:1) 1 1 1
L (x˙ ,x ,t)= 1x˙2+ 1 B˙ + 1B2 x2−C(t)eIB(t) x1G(x˜ e−IB(t))dx˜ . (15)
1 1 1 2 1 4 2 1 1 1
It is easy to show that substitution of L (x˙ ,x ,t) into the Euler–Lagrange equation (see
1 1 1
equation(5))leadstotheequationofmotiongivenbyequation(14). Alsonotethatbasedon
ourdefinitiongiveninsection1,L (x˙ ,x ,t)isastandardLagrangian.
1 1 1
HavingobtainedtheLagrangianforthetransformedvariablesx˙ andx ,wenowexpress
1 1
L (x˙ ,x ,t) in terms of the original variables x˙ and x and derive the standard Lagrangian
1 1 1
L(x˙,x,t). Itsexplicitformisgivenbythefollowingproposition.
Proposition1. Theequationofmotion
x¨ +B(t)x˙ +C(t)G(x)=0, (16)
withB(t),C(t)andG(x)beingcontinuous,differentiableandintegrablefunctions,admitsa
LagrangiandescriptionandtheresultingstandardLagrangianis
(cid:9) (cid:10) (cid:1)
1 1 x
L(x˙,x,t)= x˙2+Bx˙x+ (B˙ +B2)x2 e2IB(t)−C(t)e2IB(t) G(x˜)dx˜, (17)
2 2
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where
(cid:1)
1 t
I (t)= B(t˜)dt˜. (18)
B
2
Proof. Substitution of the Lagrangian L(x˙,x,t) into the Euler–Lagrange equation (see
equation(5))givesequation(16),whichvalidatesproposition1. (cid:1)
An important result obtained here is that the presented method of writing a standard
Lagrangianislimitedtotheequationofmotionwiththetime-dependentcoefficientsandthe
linear (m = 1) damping term. The explicit dependence of L(x˙,x,t) on t will be discussed
in section 3.2, where the results described in proposition 1 are applied to selected physical
systemsandtosomeequationsofmathematicalphysics.
3.2. ApplicationsofstandardLagrangians
To apply the results of proposition 1 to physical systems, we begin with a pendulum whose
length increases with time like in Poe’s thrilling tale ‘The Pit and the Pendulum’. Actually,
weassumethatthependulumisnonlinearanddampedanditslengthl(t)increaseswithtime
t at the constant rate a so that l(t) = l +at, where l is the initial length of the pendulum.
0 0
Ifθ(t)isadisplacementandg isgravity,thentheequationofmotionforthissystem[27]is
givenby
2a g
θ¨ + θ˙ + sinθ =0. (19)
l +at l +at
0 0
By comparing this equation to equation (16) and using equation (17), we may immediately
writetheLagrangianL(θ˙,θ,t). Theresultis
(cid:2) (cid:3)
1 2a 1
L(θ˙,θ,t)= θ˙2+ θ˙θ (l +at)2+ aθ2+g(l +at)cosθ. (20)
0 0
2 l +at 2
0
OursecondexampleistheLane–Emdenequation,whichdescribesaself-gravitatingand
sphericallysymmetricobjectcomposedofafluidwiththepolytropicindexκ. Thisequation
isusedinstellarastrophysicstostudyinternalstructuresofstars[41],anditsgeneralformis
d2ψ 2dψ
+ +ψκ =0, (21)
dξ2 ξ dξ
where ψ is dimensionless stellar density and ξ is also dimensionless and related to stellar
radius.
The Lagrangian L(ψ(cid:1),ψ,t), with ψ(cid:1) = dψ/dξ, can be written by using equation (17),
whichgives
(cid:9) (cid:10)
1 2 1 2
L(ψ(cid:1),ψ,t)= (ψ(cid:1))2+ ψ(cid:1)ψ + ψ2− ψκ+1 ξ2. (22)
2 ξ ξ2 κ +1
To discuss other applications, we take G(x) = x and writethe equation of motion (see
equation(16))as
x¨ +B(t)x˙ +C(t)x =0. (23)
For this equation, the integral in the standard Lagrangians L(x˙,x,t) can be evaluated (see
equation(17)),andtheLagrangianbecomes
(cid:6) (cid:7)
L(x˙,x,t)= 1 x˙2+Bx˙x− 1(2C−B˙ −B2)x2 e2IB(t). (24)
2 2
Depending on the form of B(t) and C(t), equation (23) can be cast in the form
of the Sturm–Liouville equation if B(t) = p˙(t)/p(t) and C(t) = q(t) + λr(t) [41],
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which means that many important equations of mathematical physics can be represented by
equation(23). ExamplesincludetheBessel,Legendre,Laguerre,Hermite,Chebyshev,Jacobi,
hypergeometricandconfluenthypergeometricequations[1, 42]. Animportantresultofthis
analysis is that for each of these equations, the standard Lagrangian L(x˙,x,t) can formally
beobtainedfromequation(24). Todemonstratethis, weapplytheresultstotheBesseland
Laguerreequations.
WewritetheBesselequationinthefollowingform:
(cid:2) (cid:3)
d2u 1du ν2
+ + 1+ u=0, (25)
dy ydy y2
whereν isaparameter. Notethatthephysicalmeaningoftheindependentvariabley andthe
dependentvariableudependsonaspecificproblemthatisconsidered. Usingequation(17),
weobtainthestandardLagrangianL(u(cid:1),u,y),withu(cid:1) =du/dy,fromwhichtheaboveBessel
equationcanbederived. TheLagrangianisgivenby
(cid:9) (cid:2) (cid:3) (cid:10)
1 1 ν2
L(u(cid:1),u,y)= (u(cid:1))2+ u(cid:1)u− 1+ u2 y. (26)
2 y y2
Similarly,wemayobtainthestandardLagrangianfortheLaguerreequation,whichcan
bewrittenas
(cid:2) (cid:3)
d2v 1 dv k
+ −1 + v =0, (27)
dz z dz z
wherekisaparameter. ThestandardLagrangianL(v(cid:1),v,z),withv(cid:1) =du/dz,becomes
(cid:9) (cid:2) (cid:3) (cid:10) (cid:9) (cid:2) (cid:3)(cid:10)
1 1 1 k 1 2
L(v(cid:1),v,z)= (v(cid:1))2+ −1 v(cid:1)v ze−z− + −1 v2ze−z. (28)
2 z 2 z 2 z
ItmustbenotedthatamongthefourdifferentLagrangiansderivedabove,onlyL(v(cid:1),v,z)
depends explicitly on the independent variable z through an exponential factor; this type of
dependence is characteristic for Lagrangians first discussed by Bateman [13]. Clearly, the
otherthreeLagrangiansareofdifferenttypesastheirdependenceontheindependentvariable
isnotexponential.
The above examples demonstrate that the method presented in this paper can be used
to obtain standard Lagrangians for many equations of mathematical physics as well as for
equationsofmotiondescribingoscillatoryandothermechanicalsystemswithlineardamping
terms.
3.3. Existenceofnon-standardLagrangians
It was demonstrated that the nonlinear second-order Riccati equation admits a Lagrangian
formulation[31]andthatitsnon-standardLagrangianisgivenby
1
L(x˙,x,t)= , (29)
x˙ +αu(x,t)
where α = const and u(x,t) = v (t)+v (t)x +v (t)x2, with the functions v (t),v (t) and
0 1 2 0 1
v (t)tobedetermined.
2
Bygeneralizingthisresult,thefollowingformofnon-standardLagrangianisconsidered:
1
L(x˙,x,t)= , (30)
P(t)x˙m+Q(t)H(x)+R(t)
wherethefunctionsP(t),Q(t),R(t)andH(x)aretobedetermined;notethatitisrequired
thatthesefunctionsaredifferentiable.
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WeusetheEuler–Lagrangeequation(equation(5))toderiveageneralequationofmotion
resultingfromtheabovenon-standardLagrangian. Theresultis
A (x˙,x,t)x¨ +A (x˙,x,t)x˙m+A (x,t)=0, (31)
1 2 3
where
A (x˙,x,t)=m(m+1)P2x˙2(m−1)+(QH +R)Px˙m−2, (32)
1
A (x˙,x,t)=(1+2m)PQH(cid:1)+2m(Q˙H +R˙)Px˙−1+mPP˙x˙(m−1)−m(QH +R)P˙x˙−1
2
(33)
and
A (x,t)=(QH +R)QH(cid:1). (34)
3
In addition, H(cid:1)(x) = dH(x)/dx and P˙(t),Q˙(t) and R˙(t) are time-derivatives of these
functions.
In order to determine the functions P(t),Q(t),R(t) and H(x), the coefficients
A (x˙,x,t),A (x˙,x,t) and A (x,t) must be compared to the corresponding coefficients in
1 2 3
equation(8). ThecomparisonshowsthatA hastobe1,oranyotherconstant,andA hasto
1 2
beafunctionoftonly. Tosatisfythesetwoconditions,wemusttakem=1andalsoassume
thatP(t)=const;withoutanylossofgenerality,wesimplytakeP(t)=1.
Withtheseassumptions,theresultingcoefficientsare: A = 2,A (x,t) = 3Q(t)H(cid:1)(x)
1 2
andA (x,t)=2(Q˙H +R˙)+QH(cid:1)(QH +R). SinceA (x,t)mustbeafunctionoftonly,it
3 2
isrequiredthatH(cid:1) =1,whichmeansthatH(x)=x. Thus,equation(31)becomes
(cid:6) (cid:7)
x¨ + 3Q(t)x˙ + Q˙(t)+ 1Q2(t) x+ 1Q(t)R(t)+R˙(t)=0. (35)
2 2 2
A comparison of this equation to the original equation of motion given by equation (8)
withm=1showsthatQ(t)=2B(t)/3and
(cid:6) (cid:7)
Q˙(t)+ 1Q2(t) x+ 1Q(t)R(t)+R˙(t)=C(t)G(x). (36)
2 2
Tosatisfyequation(36),wetakeG(x)=x andR(t)=0. Then,wehave
C(t)=Q˙(t)+ 1Q2(t). (37)
2
The main result is that there is only one special form of the equation of motion, with
m=1,G(x)=x andC(t)givenbyequation(37),forwhichaLagrangianformulationwith
anon-standardLagrangianispossible. Thefollowingpropositionsummarizesthisimportant
result.
Proposition2. Theequationofmotion
x¨ +B(t)x˙ +C(t)x =0, (38)
with B(t) and C(t) being continuous and differentiable functions of t, admits a Lagrangian
descriptionwiththefollowingnon-standardLagrangian:
1
L(x˙,x,t)= , (39)
x˙ +Q(t)x
if,andonlyif,
C(t)=Q˙(t)+ 1Q2(t) (40)
2
and
Q(t)= 2B(t). (41)
3
Proof. Substitution of the Lagrangian L(x˙,x,t) into the Euler–Lagrange equation (see
equation (5)) yields equation (38) with C(t) and Q(t) given by equations (40) and (41),
respectively;thus,proposition2isproved. (cid:1)
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J.Phys.A:Math.Theor.41(2008)055205 ZEMusielak
3.4. Equationwithstandardandnon-standardLagrangians
Usingequations(40)and(41),wewritetheexplicitformofequation(38)as
(cid:6) (cid:7)
x¨ +B(t)x˙ + 2 B˙(t)+ 1B2(t) x =0. (42)
3 3
Accordingtoproposition2, thisequationofmotioncanbederivedfromthefollowingnon-
standardLagrangian:
1
L(x˙,x,t)= . (43)
x˙ + 2B(t)x
3
However, if C(t) = 2(B˙ +B2/3)/3 and G(x) = x, then the form of equation (42) is
the same as that of equation (16). This means that the above equation of motion has also a
standardLagrangian. ToderivethisLagrangian,weuseequation(17)andobtain
(cid:6) (cid:4) (cid:5) (cid:7)
L(x˙,x,t)= 1 x˙2+Bx˙x+ 1 5B2−B˙ x2 e2IB(t). (44)
2 6 3
Aninterestingresultobtainedhereisthatamongallequationsofmotionwiththetime-
dependentcoefficientsconsideredinthispaper(seeequation(8)),onlyequation(42)canbe
derivedfromeitherthestandardorthenon-standardLagrangian.
4. Equationswithspace-dependentcoefficients
4.1. ExistenceofstandardLagrangians
Wenowconsiderequation(9)anddeterminevaluesofnforwhichthedampingtermcanbe
removedbythefollowingintegraltransformation:
x(t)=x (t)eI2(x2), (45)
2
where
(cid:1)
x2
I (x )= φ (x˜ )dx˜ , (46)
2 2 2 2 2
and φ is an arbitrary function to be determined; other restrictions on φ and the rules for
2 2
takingderivativesarethesameasthosealreadydiscussedforφ belowequation(11).
1
Thetransformedequationofmotioncanbewrittenas
(cid:9) (cid:2) (cid:3)(cid:10)
1 dφ
x¨ + 2φ +x φ2+x 2 x˙2+(1+x φ )n−1b(x eI2(x2))e(n−1)I2(x2)x˙n
2 1+x φ 2 2 2 2 dx 2 2 2 2 2
2 2 2
e−I2(x2)
+ c(x eI2(x2))g(x eI2(x2))=0. (47)
2 2
1+x φ
2 2
Theconditionthatφ mustsatisfyis
(cid:9) (cid:2) (cid:3)2(cid:10)
(cid:6) (cid:7)
dφ
2φ +x φ2+x 2 x˙2+ (1+x φ )nb(x eI2(x2))e(n−1)I2(x2) x˙n =0. (48)
2 2 2 2 dx 2 2 2 2 2
2
Noting that (1+x φ ) (cid:2)= 0 (see equation (47)), one sees that there is no φ that satisfies
2 2 2
equation(48)anddependssolelyonx ,unlessn = 2. Thelattercaseisexceptionalbecause
2
suchφ canindeedbefound[22]. Toshowthis,wetaken=2andwritetheconditiongiven
2
byequation(48)as
dφ
2φ +x φ2+x 2 +(1+x φ )2b(x eI2(x2))eI2(x2) =0, (49)
2 2 2 2dx 2 2 2
2
and φ becomes a function of x only as required by the integral transformation (see
2 2
equation(45)).
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J.Phys.A:Math.Theor.41(2008)055205 ZEMusielak
Sincethereisφ (x )thatsatisfiesequation(49),wewriteequation(47)as
2 2
x¨ +(cid:11)2(x )=0, (50)
2 2 2
with
e−I2(x2)
(cid:11)2(x )= c(x eI2(x2))g(x eI2(x2)). (51)
2 2 1+x φ 2 2
2 2
Using the results obtained in section 2, the standard Lagrangian L (x˙ ,x ) for
2 2 2
equation(50)is
(cid:1)
1 x2
L (x¨ ,x )= x˙2− (cid:11)2(x˜ )dx˜ . (52)
2 2 2 2 2 2 2 2
Afterevaluatingφ (x )andreplacingthetransformedvariablesbytheoriginalvariables[22],
2 2
onemayobtainthestandardLagrangianL(x˙,x). TheexplicitformofL(x˙,x)isgivenbythe
followingproposition.
Proposition3. Theequationofmotion
x¨ +b(x)x˙2+c(x)g(x)=0, (53)
withb(x),c(x)andg(x)beingcontinuous,differentiableandintegrablefunctionsofx,admits
aLagrangiandescriptionanditsstandardLagrangianis
(cid:1)
1 x
L(x˙,x)= x˙2e2Ib(x)− c(x˜)g(x˜)e2Ib(x˜)dx˜, (54)
2
where (cid:1)
x
I (x)= b(x˜)dx˜. (55)
b
Proof. TheLagrangianL(x˙,x)yieldstheequationofmotion(seeequation(53))afterL(x˙,x)
issubstitutedintotheEuler–Lagrange equation (seeequation (5)); thisshowsthattheproof
ofproposition3isstraightforward. (cid:1)
An important property of the standard Lagrangian L(x˙,x) is that it does not diverge as
thesystemevolvesintime. Alsonotethatinthespecialcaseofg(x)=x,L(x˙,x)reducesto
theformoriginallyderivedbyMusielaketal[22].
4.2. ApplicationsofstandardLagrangians
We now apply the result of proposition 3 to some physical systems. Our first example is a
well-knownprobleminclassicalmechanics[9,21]. Considerabeadslidingwithoutfriction
along a wire that is bent in the shape of a parabola z = ar2, where a = const. The wire
rotatesaboutitsverticalsymmetryaxiswiththeangularvelocityω. Theequationofmotion
for this problem is typically derived by calculating the kinetic and potential energy, writing
theLagrangianandusingtheEuler–Lagrangeequation(seeequation(5)). Theresultis
4a2r 2ga−ω2
r¨+ r˙2+ r =0, (56)
1+4a2r2 1+4a2r2
wheregisgravity.
Letusreversethisprocessbyassumingthattheaboveequationofmotionisgiven,sowe
canuseproposition3toderivethestandardLagrangianL(r˙,r). Bycomparingequation(56)
toequation(53),calculatingI (r)andevaluatingtheintegralsinequations(54)and(55),we
b
obtain
L(r˙,r)= 1(r˙2+4a2r2r˙2+ω2r2)−gar2, (57)
2
10