We show by some counterexamples that Lagrangian sysytems with nonlocality of
finite extent are not necessarily unstable.Comment: 8 pages, 1 figure Submitted to Phys. Rev. 

Josep Llosa

M. Ostrogradski

R. P. Woodard

R. S. Hamilton

X. Jaén

English

arXiv

The paper by Woodward [Phys. Rev. A 62, 052105 (2000)] claimed to have proved that Lagrangian theories with a nonlocality of finite extent are necessarily unstable. In this Comment we propose that this conclusion is false

Llosa, Josep

Diposit Digital de la Universitat de Barcelona

PHYSICAL REVIEW A 67, 016101 ~2003!Comment on ‘‘Canonical formalism for Lagrangians with nonlocality of finite extent’’Josep Llosa*Departament de Fı´sica Fonamental, Universitat de Barcelona, Diagonal 647, E-08028 Barcelona, Spain~Received 11 January 2002; revised manuscript received 10 July 2002; published 15 January 2003!The paper by Woodward @Phys. Rev. A 62, 052105 ~2000!# claimed to have proved that Lagrangian theorieswith a nonlocality of finite extent are necessarily unstable. In this Comment we propose that this conclusion isfalse.DOI: 10.1103/PhysRevA.67.016101 PACS number~s!: 03.65.Ta, 11.10.Lm, 45.20.JjI. INTRODUCTIONIn Ref. @1# a canonical formalism for nonlocal Lagrang-ians with a nonlocality of finite extent is established. It iscompared with the Ostrogradski formalism @2# for localLagrangians that depend on a finite number of derivatives ofcoordinates. One of its central conclusions is that Lagrangiansystems with a nonlocality of finite extent have no ‘‘ . . .possible phenomenological role @{{{#. They have inherited thefull Ostrogradskian instability . . . .’’The aim of the present Comment is ~i! to point out somedefects in Ref. @1# ~Sec. II! concerning the application of thevariational principle that underlies the derivation of the non-local Eq. ~2! from a Lagrangian; ~ii! to stress the importanceof the functional space where the variational problem is de-veloped; this is also the functional space where the solutionsmust be searched ~Sec. III!; and ~iii! to illustrate by twosimple counterexamples ~Sec. IV! that ~a! Lagrangian sys-tems containing derivatives of a higher order than the firstare not necessarily unstable and ~b! nonlocality of finite ex-tent does not inevitably lead to instability.II. THE NONLOCAL ACTION PRINCIPLEAlthough the canonical formalism set up in Ref. @1# isderived on a general ground, it is basically illustrated by thesimple nonlocal Lagrangian systemL@q#~ t !512 mq˙2S t1 D2 D2 12 mv2q~ t !q~ t1D! ~1!and the equation of motion for this Lagrangian is written asE0DdL@q#~ t2r !dq~ t !dr52mH q¨ ~ t !1 12 v2q~ t1D!112 v2q~ t2D!J 50. ~2!It must be noticed that the latter equation as it reads doesnot properly correspond to the standard action principle ofmechanics. Indeed, the latter states that @3# ‘‘The motion ofan arbitrary mechanical system occurs in such a way that theaction integral S becomes stationary for arbitrary possible*Electronic address: pitu@ffn.ub.es1050-2947/2003/67~1!/016101~6!/$20.00 67 0161variations of the configuration of the system, provided thatthe initial and final configurations of the system are pre-scribed.’’On the other hand, the action integral whose variationwould be the left-hand side of Eq. ~2! isS~@q# ,t !5E0DdrL@q#~ t2r !5Et2Dtdt L@q#~t! ~3!and Eq. ~2! is equivalent todS~@q# ,t !dq~ t !50, ~4!where t is the same both in the numerator and the denomi-nator. However, the Euler-Lagrange equation that followsfrom the action principle dS50 isdS~@q# ,t !dq~ t8!50, ; t8,which is much more restrictive than Eq. ~4!.Moreover, an equation like2mH q¨ ~ t !1 12 v2q~ t1D!1 12 v2q~ t2D!J 50, ~5!valid for 2‘,t,‘ , cannot be derived from an action inte-gral like Eq. ~3!, extending over a finite interval. Indeed, thevariation of the action ~3! isdS~@q# ,t !5Fmq˙ S t1 D2 D dqS t1 D2 D Gt2Dt2mEt2DtdtFq¨ S t1 D2 D dqS t1 D2 D1v22 q~t1D!dq~t!1v22 q~t!dq~t1D!G .~6!The extremal condition dS50 then leads to the boundaryconditions dq@ t1(D/2)#5dq@ t2(D/2)#50 and to theequations of motion:©2003 The American Physical Society01-1COMMENTS PHYSICAL REVIEW A 67, 016101 ~2003!~a! t2D,t,t2D2 , q~t1D!50,~b! t2D2 ,t,t , q¨ ~t!1v2/2q~t1D!50,~7!~c! t,t,t1D2 , q¨ ~t!1v2/2q~t2D!50,~d! t1D2 ,t,t1D , q~t2D!50,which has the only solution q(t)50 for t2D,t,t1D , asit ineluctably follows from sequentially exploiting ~d!, ~b!,~a!, and ~c!.Furthermore, if we alternatively try with an action ex-tended over a larger interval,S5E0Tdt L@q#~t!,the Euler-Lagrange equations are~ i! q~t1D!50,~ ii! q¨ ~t!1v22 q~t1D!50,~ iii! q¨ ~t!1v22 @q~t1D!1q~t2D!#50, ~8!~ iv! q¨ ~t!1v22 q~t2D!50,~v! q~t2D!50,where the domains ~i! to ~v!, respectively, correspond to 0,t,D/2; D/2,t,D; D,t,T; T,t,T1D/2, and T1D/2,t,T1D . Equation ~8! only looks like Eq. ~5! in theinterval D,t,T .Conditions ~i! and ~v! in Eq. ~8! then yieldq~t!50, D,t,3D2 , or T2D2 ,t,Tthat act as constraints on the possible solutions of ~ii!, ~iii!,and ~iv! in Eq. ~8!. As a consequence, Eqs. ~8! can be re-duced to an ordinary differential equation, whose order de-pends on the number of times that the elementary length Dfits into @0,T# .We have thus illustrated the important role played by theintegration bounds in the nonlocal action ~3! as far as theEuler-Lagrange equations are concerned. The integrationbounds in the action and the problems associated with themare commonly overlooked in theoretical physics literaturebecause, in standard local cases no trouble is usually entailedby proceeding in this manner. Nonlocal cases are, however, anew ground where nothing can be taken for granted.01610For a local action, the bounds of the integral also deter-mine the functional Banach space where the variational cal-culus is meaningful @4#, e.g., the space C 2(@a ,b#) for anaction integral extending over @a ,b# . This is also the spacewhere the solutions to the Euler-Lagrange equations have tobe sought.A way to derive Eq. ~2!, for t extending from 2‘ to ‘ ,from an action principle could consist in taking the integralover the whole R:S5E2‘‘dt L@q#~t!, ~9!but then two additional difficulties arise: on the one hand, theaction S does not converge anymore for all qPC 2(R) and,on the other, C 2(R) is not a Banach space. ~The variationalcalculus should be then approached in terms of Fre´chetspaces @5,6#.! To my knowledge, it remains an open problemto establish the appropriate mathematical framework where anonlocal equation like Eq. ~5! can be derived from an actionintegral like Eq. ~9!. This results in a lack of preciseness inthe definition of the functional space where the nonlocalequation has to be solved.On possible nonstandard statements of the action principleIt could be thought1 that the derivation of Eq. ~2! from theaction integral ~1! does rather rest on a nonstandard versionof the action principle than on the standard statement quotedright before Eq. ~3!, namely, ‘‘the action S5* t1t2dsL@q#(s) isstationary with respect to variations dq(s) that vanish for s<t11Dt and t22Dt<s .’’ Notice that it is similar to thestandard statement of the action principle, where the usual‘‘boundary point conditions’’ —initial and final conditions—are replaced by a sort of ‘‘boundary layer condition.’’The latter nonstandard action principle leads both to thenonlocal equationq¨ ~ t !1v22 @q~ t1Dt !1q~ t2Dt !#50 fort11Dt,t,t22Dt ~10!and to the ‘‘boundary layer conditions’’q~ t !5q1~ t ! for t<t11Dt and~11!q~ t !5q2~ t ! for t22Dt<t ,where q1(t) and q2(t) are given functions defined in theirrespective half-lines.Equations ~10! and conditions ~11! have to be solved to-gether and, in most cases, given any set of layer data,$q1(t),q2(s); t<t11Dt ,s<t22Dt%, Eq. ~10! has no solu-tion. Indeed, as is pointed out in Eq. ~3! in Ref. @1#, Eq. ~10!can also be written as1Suggested by R. P. Woodard ~private communication!.1-2COMMENTS PHYSICAL REVIEW A 67, 016101 ~2003!q~t!522v2q¨ ~t2Dt !2q~t22Dt ! for t112Dt,t,t2,~12!which, after n iterations permits to obtain q(t) in terms ofq(t2nDt) and q(t2@n11#Dt), whenever t ,t2(n21)DtP@ t112Dt ,t2# .Now, taking n the least integer such that (n11)Dt>t22t1 , q(t)5q2(t), for t22Dt,t,t2 can be expressed interms of q(t2nDt) and q(t2@n11#Dt), which only de-pend on the ‘‘lower layer datum’’ q1(t) because t2nDt<t11Dt . This leads to a consistence condition between theupper and lower layer data, whence arbitrarily chosen datacould be incompatible.III. THE STABILITY PROBLEMLeaving aside the difficulties just mentioned, supposethat, for some physical reasons whatsoever, we are only in-terested on the solutions of Eq. ~5! in the Banach space,B5$qPC 2~R!; uq~ t !u, uq˙ ~ t !u and uq¨ ~ t !u are bounded%.The general solution of Eq. ~8! is thusq~ t !5(l~Aleiklt1Al*e2iklt!, ~13!where 6kl are the real solutions2 ofh~k ![k22v2cos~kD!50 ~14!and Al* is the complex conjugate of Al , to ensure that q(t)PR.Notice that the number of real roots of Eq. ~14! is finite. Alook at Fig. 1 is enough to get convinced that they can beindexed so thatk j,ki if j,i and l51,2, . . . ,N .N can be either odd and then all roots are simple, or even, inwhich case 6kN are both double. It should also be remarkedthat the greater is D , the denser is the wiggling in the graph-ics ~Fig. 1!. Therefore, N increases with D .2A complex value of kl would result in an exponential growtheither at 1‘ or 2‘ and then q„B.FIG. 1. h(k)50 has a finite number of real roots, and the signof the derivative h8(kl) at each root is alternating.01610The space of solutions of Eq. ~5! in B can be hence coor-dinated by 2N,‘ real parameters, namely, the real andimaginary parts of Al that can be put in correspondence withthe initial data: q0 , q˙ 0, . . . q0(2N21).The solutions of Eq. ~5! in B are stable because a smallchange in the initial data dq0(a) results in a small change inthe complex parameters: dAl . Indeed, from the linearity ofEq. ~5! and from the general solution ~13! it follows that(l51N@dAl~ ikl!a1dAl*~2ikl!a#5dq0(a),a50,1, . . . ,2N21 , which can be inverted to obtain dAl asa linear function of dq0(b). Therefore, there exists K.0 suchthat udAlu<Kuudq0uu , where uudq0uu[sup$udq0(a)u; a50,1, . . . ,2N21% . The deviation from q(t) evolves withtime asudq~ t !u5U(l51NRe~dAleiklt!U<(l51N2udAlu<2NKuudq0uu,which proves the stability3 of the solutions of Eq. ~5! in thespace B.Notwithstanding, if we now have a look at the Hamil-tonian @Eq. ~48! in Ref. @1##,H~ t !512 mq˙2~ t !112 mv2q~ t !q~ t1D!212 mv2E0Ddsq˙ ~ t1s !q~ t1s2D!,on substituting the general solution ~13!, we obtain thatH~ t !52m (l51Nklg~kl!AlAl* ~15!withg~k !5k1v22 D sin~kD!. ~16!As expected, H(t) is an integral of motion, but it has not adefinite sign. Indeed, notice that g(k)5h8(k)/2 alternatessign at each root kl @see Eq. ~14! and Fig. 1#. Therefore,g(kl) is positive or negative depending on whether l is evenor odd, respectively @moreover, g(kl)50 if kl is a doubleroot#.IV. TWO SIMPLE COUNTEREXAMPLESA. The so-called Ostrogradskian instabilityIn Ref. @1# it is proved that the Hamiltonian formalism fora nonlocal Lagrangian can be obtained as a limit case forN→‘ of the Ostrogradski formalism @2# for a Lagrangian3In the sense of Liapounov, see @7#.1-3COMMENTS PHYSICAL REVIEW A 67, 016101 ~2003!that depends on the derivatives of the coordinates up to orderN.4 For N.1, the Ostrogradski Hamiltonian is linear on allthe canonical momenta but one, namely, P1, . . . , PN21,therefore it has not a definite sign. The fact that the energy isnot bounded from below is then argued to conclude that thesolutions of the equations of motion are ineludibly unstable.This is what is called the Ostrogradskian instability. It is alsoshown in @1# that this drawback also holds in the limit N→‘ .Actually what has been proved there is only that the en-ergy cannot be taken as a Liapunov function @8# to concludethe stability of the equations of motion derived from an N thorder Lagrangian (N.1). However, the fact that a sufficientcondition of stability is not met does not imply instability.Let us consider the following simple counterexample:L~q ,q˙ ,q¨ !512q¨2112 Bq˙2112 Cq2, ~17!where B and C are two parameters which we shall later tunein order to get stability.According to Ostrogradski theory, the canonical coordi-nates and momenta are @in the notation of Ref. @1#, Eqs. ~6!and ~7!#Q15q , Q25q˙ , P15Bq˙ 2q (iii), P25q¨and the Hamiltonian is @Eq. ~9! in @1##H512 P221P1Q2212 BQ22212 CQ12. ~18!IntroducingXW 5S Q1Q2P1P2D and M5S 0 1 0 00 0 0 1C 0 0 00 B 21 0D ,the Hamilton equations for Eq. ~18! can be then written asthe linear system:ddt XW 5MXW ,the stability of whose solutions depends on the real part ofthe roots of the characteristic polynomial pM(l)5det(M2lI4), that is,l56AB6AB224C2.If the parameters are tuned so that B,0 and 0,C,B2/4,then all roots are imaginary and the system is stable @8#. Thelatter is not an obstacle to the fact that the Hamiltonian doesnot have a definite sign.4A similar result was also obtained by Ref. @9#.01610It could be argued that, although the Lagrangian system~17! is stable, it is physically irrelevant unless it can becoupled to anything else. It seems that, since the energy hasnot a lower bound, an unending flow of energy leaving thesystem cannot be prevented. To show that this is not the case,we shall now see how Eq. ~17! can be stablily coupled to aharmonic oscillator. Since classical electromagnetic field isactually an infinite set of harmonic oscillators, each one char-acterized by its polarization and its wave vector, the nextexample will also serve as an indication that the stable cou-pling of the system ~17! can be extended to a Maxwell field.Consider the second-order Lagrangian,L512 ~q¨21Bq˙ 21Cq2!1gqx1m2 ~x˙22v2x2!. ~19!We shall see that the parameters B and C can be tuned so thatthe system is stable.The equations of motion areqiv5Bq¨ 2Cq2gx ,~20!x¨ 52v2x1gmq ,and can be written in matrix form asddt X5GX, ~21!whereX5S q. . .q¨q˙qx˙xD and G5S 0 B 0 2C 0 2g1 0 0 0 0 00 1 0 0 0 00 0 1 0 0 00 0 0 g/m 0 2v20 0 0 0 1 0D .The ordinary differential system is stable if the character-istic roots of G are all imaginary and simple. This amounts tosaying thatP~y !5~y1v2!~y22By1C !1g2m~22!has three simple negative roots. A little bit of algebra showsthat the latter is achieved if, and only if, the parameters B andC are chosen so that ~1! B is a real root of the cubic equationb~B2v2!31av2~B2v2!21Bv41g2m50,~2! C is1-4COMMENTS PHYSICAL REVIEW A 67, 016101 ~2003!C5b~B2v2!3v22g2mv2,~3! and a and b are two real parameters in the nonemptyregion of R2 defined by the inequalities,max$0, 9a22222~12a!3/2%<27b<9a22212~12a!3/2.Now, assuming that the parameters meet these conditions,the system ~21! is stable in spite of the fact that the Hamil-tonian does not have a definite sign. Notwithstanding, a posi-tive definite integral of motion ~a Lyapunov function @10#!can be found such that it forbids the existence of runaways.Indeed, if all characteristic roots of Eq. ~21! are imagi-nary, the matrix G can be diagonalized and all its eigenvaluesare imaginary, namely, 6ila , a51,2,3. Therefore, a realregular matrix F exists such thatG5F21KF withK5S 0 2l1 0 0 0 0l1 0 0 0 0 00 0 0 2l2 0 00 0 l2 0 0 00 0 0 0 0 2l30 0 0 0 l3 0D . ~23!Then, the quadratic formf ~ q. . .,q¨ ,q˙ ,q ,x˙ ,x !5XTFTFX ~24!~a! is positive in the whole phase space and ~b! is an integralof motion. Indeed, taking Eqs. ~21! and ~23! into account, wereadily have thatd fdt 5X˙TFTFX1XTFTFX˙5XT@GTFTF1FTFG#X5XTFT@KT1K#FX50.Thus, any orbit of the dynamical system ~21! remains on theellipsoid,f ~ q. . .,q¨ ,q˙ ,q ,x˙ ,x !5 f 05const,which is bounded byf 0<k2 uuX0uu2,where k5uuFuu is the norm of the linear map F and X05( q. . .0 ,q¨ 0 ,q˙ 0 ,q0 ,x˙ 0 ,x0).01610B. A case of finite extent nonlocalityIn the next example, the boundaries of the actionintegral are finite. Consider the nonlocal action S@q#5*0TdtL@q#(t), withL@q#~ t !512q˙2~ t !212 v2q2~ t !1v42 q~ t ! E0Tdt8 G~ t ,t8!q~ t8!, ~25!where, for (t ,t8)P@0,T#2,G~ t ,t8!521v sin vT @sin v~T2t8! sinvt u~ t82t !1sin v~T2t ! sin vt8 u~ t2t8!# ~26!and is the solution of] t2G~ t ,t8!1v2G~ t ,t8!5d~ t2t8! ~27!for the boundary conditions: G(0,t8)5G(t ,T)50.The variation dS50 with the boundary conditionsdq(0)5dq(T)50 leads to the equations of motionq¨ ~ t !1v2q~ t !2v4E0Tdt8 G~ t ,t8!q~ t8!50. ~28!The solutions q(t) must be sought in the Banach spaceC 2(@0,T#).Differentiating twice ~28! and taking ~27! into account,we arrive atq (iv)12v2 q¨ 50. ~29!Hence, the solutions of Eq. ~28! must be among the generalsolution of Eq. ~29!,q~ t !5Aeiat1A*e2iat1Dt1E ~30!with a5vA2. The parameters A , A*, D , and E must fulfillthe following constraints:D50 and E5A1A*,which result from substituting Eq. ~30! into Eq. ~28!.The general solution of Eq. ~28! is thereforeq~ t !5A~eiat11 !1A*~e2iat11 !. ~31!The phase space for our system is thus two dimensional, andevery solution is determined by the initial values q0 and q˙ 0,q052~A1A*! and q˙ 05ia~A2A*!.By direct inspection of Eq. ~31!, we see that the solutionsof Eq. ~28! are stable, although the latter is derived from aLagrangian with a nonlocality of finite extent. That is, for1-5COMMENTS PHYSICAL REVIEW A 67, 016101 ~2003!any e.0 there exists r.0 such that udq0u1udq˙ 0u,r im-plies that udq(t)u1udq˙ (t)u,e , for all t, which proves thestability.V. CONCLUSIONWe have intended to stress the crucial importance ofclearly precising the Banach space where the variationalprinciple for a nonlocal Lagrangian is formulated. This de-gree of precision is usually obviated in theoretical physics~i.e., for local Lagrangians! without any major problem.However, such a nonrigourous way of proceeding cannot beextrapolated to systems with a new complexity. The rel-evance of the above-mentioned Banach space is twofold: ~i!it is where the solutions of the equations of motion must be01610sought and ~ii! it is the function space where path integralsare to be calculated in an eventual quantization of the sys-tem.We have also analyzed the stability of the equations ofmotion for a Lagrangian system presenting a nonlocality offinite extent. We have shown that the choice of the Banachspace where the variational principle is meaningfully formu-lated is crucial to decide the stability or unstability of thesystem. Furthermore, we have seen that a system can bestable in spite of the fact that the Hamiltonian does not havea minimum.Finally, we have shown by a counterexample that higherorder Lagrangian systems are not necessarily unstable. Thefact that a sufficient condition for stability is not fulfilleddoes not imply instability.@1# R. P. Woodard, Phys. Rev. A 62, 052105 ~2000!.@2# M. Ostrogradski, Mem. Acad. St. Petersburg VI 4, 385 ~1850!.@3# C. Lanczos, The Variational Principles of Mechanics ~Dover,New York, 1986!, p. 113.@4# I. M. Gelfand and S. V. Fomin, Calculus of Variations~Prentice-Hall, Englewood Cliffs, NJ, 1963!.@5# R. S. Hamilton, Bull. Am. Math. Soc. 7, 65 ~1982!.@6# Y. Choquet-Bruhat, C. Dewitt-Morette, and M. Dillard-Bleick,Analysis, Manifolds and Physics, revised ed. ~North-Holland,Amsterdam, 1987!, Chap. VI.A.@7# L. Pontriagine, E´ quations Diffe´rentielles Ordinaires ~MIR,Moscow, 1975!, p. 282.@8# W. E. Boyce and R. C. DiPrima, Elementary DifferentialEquations and Boundary Value Problems ~Limusa, MexicoCity, 1998!.@9# X. Jae´n, R. Ja´uregui, J. Llosa, and A. Molina, Phys. Rev. D 36,2385 ~1987!.@10# C. Hayashi, Nonlinear Oscillations in Physical Systems~Princeton University Press, Princeton, 1964!, p. 74.1-6

Physical Review A

Comment on "Canonical formalism for Lagrangians with nonlocality of finite extent"

Crossref

Comment on “Canonical formalism for Lagrangians with nonlocality of finite extent”

Comment on "Canonical formalism for Lagrangians with nonlocality of
  finite extent"

Comment on "Canonical formalism for Lagrangians with nonlocality of finite extent"

Abstract

Similar works

Full text

Available Versions

Diposit Digital de la Universitat de Barcelona

Crossref