5.J.J ltahinoIVilch IIIIUIIII/;/JIIIII )01".,11I1of '/1.t'or('lic"II'hpics. \1.1. .16. No.2. 1997

(( is inten:sling 10nOleIhalthe oblained conSlant (71) orlhe high-cnergyinleraction or protons is in at:cord with its e:<perilllem:tIvaluc. whit:h isapproxill1alcly cqual to 15 (Acosta ('I III.. IlJ7J).

Hence the obtained resulls arc in accord wilh the c:<perilllelllal valucsor the binding cncrgies or nuclei. their radii. and the constant or slronginteraction.

Example of Indeterminacy in Classical DynamicsI~EFEnENCES

S:lIIjay P. Bh:lt I and Dennis S. Bernstein I

,'<:usla. V.. CDwan. C. L.. allll Gmh:nn. B. J. (I '.J73). E,u,'III;lIls of Moduli Physics. Harper &Row. New YOlk.

Ericson. '1'.. ami Weise. W. (I '.JX!!). I';,m." IIIItI Nllc!.';. Chlrellllon Press. O.~ford.

Feynlllan. R. I'. (1961). rh.' n",.I1I)' of F,,"'llIo/l.IllIIII'm '.s...\.. Uenj:nnin. New Yurko

Naulllo\'. A. I. (I '.JS.J). fi~;k" 1IItJ/l/IIO.~II.I'lIIlm;d"/I/t'lIIlIrtI.I'kh dIllSI;I.f. I'rosveschenie. Moscow.

Rahinowilch. A. S. (I I)I).J). 11111'/'111I/;/1/11I1)/II/I'IllIllIf 'f1,,'u''<'I;Cl/IPhy.\';('s. JJ. 2O.J1). .

IlI.a;' 1Mil)' IJ. 1996~.._---

The case or :\ parlicle lIIovinG :alonG :\ nonsnlllolh constr:ainl ullIl.:r Ihe :u:lion ufuniforlll Gmvily is presenled as :11Iexmnple of illllel.:nnin:u:y in :\ d:assicalsillmlion. The indelennill:ley :arises frolll cen:lin inili:11 culllliliuns havinG

nonunique sulutions :lIId is due 10 Ihe f:lilore uf Ihe Lipschill. cOlllliliun :11IhecorrespondinG poinls in the ph:lse sp:lce of Ihe e1lu:llion of nlolion.

1. INTRODUCTION

An oflen unstated assumption or classical Il1echanics is that the laws urdynamics yield deterministic models. This assumption is rormally capturedin Newton's principle or determinacy (Arnold, l!Jlltl, p. 4):

'/11t:inililll po.\'ilitJI,.I'lIlIl/w:loc:itie.I'of lI/I tht'lmrticle,l' oflllllt'dltllliclll

.I'Y_I-wmuniquely tletermine till of il.l-motioll,

The developmentsin physicssince theearly decadesof thiscenturyhaveshown that our physical world is not completely empirically deterministic. thatis, the motion of a mechanical system cannot be fully determined rrolllphysical measuremelHs or the initial positions and velocities or ils points_Inparticular, chaos theory has shown that inlinilc precision is requircd in Ihemeasuremcnts of initial conditions ror the motion 10 be fully predicted evenqualitatively. On the other hand, Heisenberg's unccrlainty principle holds thatsimultaneous measurements of positions and velocities can be made onlywith limited precision. The prcsence or noise further limits the accuracy ormeasurcment. In spite or these rundamentallimitations on our ability to makepredictionsfrom empirical observations, it is gcnerally believed that modelsobtainedfrom classicalmechanicsarc complelelydeterministicand,if obser-

I DCI':lrl/llenl of Aerusp:\ce EnGineerinG. University or Michigan, Ann Arhur. MichiGan 4K I(~).211 K; Ihh:al.llsh:\eru)@enGin.lllllich.edu.

5.15tJ"~U.11"KI'J1l1t~UU.US.U' I !.)IIIU t) I'JIJ'II'IIo'IIUI" 1'-.hli\hina: ('UllkM:lli.."

5.1(, IIh:11 and IIt:rnslcin Examplc or IlIllclcrmimlcy ill Classic:11Dynamics 547

v:ltions coullJ he made wilh inlinite precision, then predictions could be madewith unlimited accuracy. In this paper, we present a counterexample to thiswidely held notion.

The countt:l"example,given in Section 2, consists of a panicle movingalong a 110nsmooth(C1 hut not twice dilTerentiahle) constraint in a uniformgravitatinnal field. II is shown Ihat, for cenain inilial conditions, the equationof motion possesses multiple Solulions. The mOlion of the panicle slaningfrom Ihe~e initial conditions cannot, therefore, be uniquely dclermined basedon physil.:al laws. Thus Ihis example provilles an instance of indeterminacyin classical dynamics as a lIirecl counterexample to the principle of detenni.nacy statcd above.

In Seclion 3, we presenl a modilicalion of Ihis counlerexample. Themodilication consiSISin replacing the original conslrainl by a spatially peri-ollic nonsmoolh constrai'nllhal divides the conliguration space of the panicleinlo "potential wells." The equation of motion in Ihis case possesses multiplesolutions for initial conllilions thai corresponu 10zero tOlalmechanical energy.For a smooth (C") constraint, the parlicle is forever conlined to remain inthe potential well in which il is inilially located if Ihe IOlalmechanical energyis zero (or less). In the case we consider, if Ihe panicle is inilially locatedin one of these potenlial wells with zero lotal mechanical energy, Ihen thereexist solutions of Ihe equation of motion which correspond 10 the panicleleaving the potential well after a linile amount of lime. At any given instanl,the only prediction that can be made about the panicle is Ihal il is localedsomewhe re in anyone of a cenain number of potential wells and, funhermorc,this number increases with the passage of time.

Both of the examples menlioned above possess equilibria that arc linite-time repe lIers: solutiuns starling inlinitesimally close 10 such poinls escapeevery given neighborhood in linite time. Mechanical systems can exhibilsimilar bt=havior in the presence of non-Lipschilzian dissipalion (Zak, 1993)or controls (Bhat and Bernstein, 1996). However, the examples presentedhere are completely classical anu involve neilher dissipalion nor controls.

The Lagrangian yields Ihe equalion of motion

.\'11 -I-"'(.Ifl -I-:i'~"'(.\')""(x) -I-"'(.\') := II (])

Now, consiller

"(.\') ::: -Ixl", x (0 R (.\ )

where U E (312, 2). Figure I shows a plot of this constrainl for (I := 1)f5.We claim Ihal wilh 11(.)given by (4), equal ion (3) aLlmitslIonuniquesolutionsfor the inilial conuitions

.1'(0) = 0, .W» = 0 (5)

To show Ihis, consiuer Ihe uifferenlial equalion

(j(t) = (2 - tt)/)21l1 -I- u~({I(t».I(..-I)'I2'''11-1I2 (6)

Nole thai a E (312, 2) implies that 4(u - 1)/(2 - u) > 4. Hence Ihe right-hanu siue or (6) is C' in q anu bounueu on R. It Ihus follows thaI there existsa unique function T(') on 10, (0) Ihal salisfies (6) and Ihe inilial conditionT(O) = O. Moreover, T(') is Iwice continuously uilTerenliahle.

It follows by direct SUbSlilulionthat Ihe function IT(')12'r~-"I salisfies(3) and (5). In facl, this same funclion uclayeu in lime by an arbitrary posiliveconstanl T also satisfies (3) anu (5). To make Ihis prccise, delinc

.\'1'(t) = 0, I :5 T (7)

= (T(r - nf'(2-..), I > T (H)

y

2, AN EXAMPLE OF INDETERMINACY

Con~ider a particlc of unil mass cOllstraineu 10 move without friclionin a vcrtical plane along the curve y = "(.\') unuer the aclion of uniform

gravily. For convenience, assllme the gravitalional acceleralion 10 be unily.The total mechanical energy or the particle is given by

E(.\',.\') :::{.i.211-I-/r'(.lfl -I-"(.\') (I)

while the Lagrangian for lhe particle is given by

I.(.\', .i') = t.i'!11 -I-/r'(.lrl - /r(.I') (2) Fi/:. I. CUlls(r:lillcd particle ill IIlIirunll cravily.

5.UIIIhal and IIcrnslcln I~xillnplc or IlIdclcl'mirmcy III Clnssic:lt Dynamics

5-11)

Then it follows hy direct suhstitution Ihat, for every T 2: 0, Ihe functions

:!.:.I',.(.)salis fy (J) and (:)). Thc funclions x,. and -x,. correspond 10 Ihe particlercmaining at resl at .I' = 0 for timc r ami then moving off 10 the righl andleft. rcspectively.

Figurc 2 shows the phase portrait for (3) with n = lJ/5. The origin is asaddle-poil\! cquilibrium and the sets ~' = I(x, .\"):£(x, .\') = 0, X.\' SOl andCl1[= I(x. .(-): £(x. .\") = 0, X.t 2: 0 I. which arc shown in Fig. 2, arc thecorresponding stable and unstable manifolds, respectively. Solutions 10inilialconditions con wined in ~ converge to the origin in finite time, while solutionsto initialconditionscontainedinCl1Lconvergeto the origin in backwardtime.For the Solulions x}"tlescribed above, (.\At), .\'.,.(1»lies in Cl1Lfor aliI 2: O.IIis easy to s~e thm for every initial condition in ~, (3) possesses mulliplesolutions. For such initial conditions, the motion of the particle cannot beuniquely dclcrminetl. This phenomenon represents indeterminacy in a classi-cal situation and is a counterexample to Newton's principle of determinacystated above.

.J~1

1 3111

7/'01

~Fl\:. 3. P:lrliclc umlcr pcriudic ClIlISIWill1.

hex) = -I cos(x) I", xeR (9)

where u e (3/2, 2) as before. As Fig. 3. which corresponds 10 n " 1)15.

shows, the constraint divides the configuration space into the potential wells1V"= Ix:~(211 - 1)1T S X S ~(211 -I- 1)1T). II = . . . , -I, 0, I, . . . .

The p~)ints (t(211 -I- 1)1T,(), II = ... , - I. 0, I, . . . , arc saddle-poinlequilibria that nre connected by heteroctinic orbits. The set I(x, .i'): I~'{.I',,i')= O} is the union of these heteroclinic orbits. Because of the non-Lipschit;>,iannature of the equation (3). solutions starting in the swble manifold of anyone of these points converge to that poil\! in finite time, while solutionsstnrting in the unstable manifold leave every neighborhood of that point ina linite time.

Suppose the particle is initially located in 'Wn with i'.ero10lalmechanicalenergy. Then, depending on the direction of its initial velocity, the heteroclinicstructure will bring the purticle to rest ut one of the crests :!.:1T/2in a linitenmount of time. As in the previous exnmple, there exist solutions whichcorrespond to the purticle stuying at rest at x = :!.:1T/2for an urbitraryamount of time before sliding off to the right or the left. Every solution thatcorresponds to the pnrticle moving off brings the particle to rest at someother crest in u linite time. This argument C~1I1be used repe.lledly to show

that given un initial condition in 1Vuwith i'.ero total mechanical energy, forevery integer /I there exists a solution x(.) with x{t) e 'W" for all t 2: '/' forsome 7: In other words, it is not possible to predict in which potential wellthe particle will bc found after a certain linite :unount of time fWIIIthe initiajinstant. The only prediction that C.IIIbe made ut any instant is that the particleis located somewhere in anyone of a certain numbcr of potential wells.Moreover, this numbcr incrcases with time. This means that we can prcdictIcss and less about the particle as time passes.

J. A FURTHER EXAMPLE OF INDETERMINACY

The indetcnl1inacy seen above can be made even more striking byreplacing (4) by

.i

4, CONCLUSIONS1.5 2

The examplcs given in the previous scctiuns show that classicalmechani-cat situations can exhibit a lack of detcnllinacy even in thc absence ordisturbances and noise. This lack of determinacy is distinct from thc enlpirical

xFi\:. 2. Phase Jlonrail for t3).

55U11111I1nlld lIerll~tcin

indclcrminacy Ihal ariscs from scnsitivc depcndcncc on inilialcondilions,lhcIIIH.:crtainl)' principlc, and random noisc. Our cxamplcs providc SilU:lliollswhosc Olllt.'omc call not bc predicted thcoretically. Finally, Ihis clTeel woulll

bc diflicult 10 dClllOnstrme cmpirically duc 10 noisc and dissipation.

ACKNOWLEDGMENT

This researchwassupportedin partby thc Air ForccOfficeof ScienlificRescarchundcr grant F49620-95-I-OO19.

REFERENCES

Arnuld, Y.I. ( 19H9).Malhl!lIImiml M"llwtlso/ClllJ"siclIl M"c/"/IIic.r, Sprillger-Yerhlg, New Yurk.Bh:ll, S. 1'., n...1 Bernslein, D. S. (1996). CUlllinuuus, buunded, linile-lime slnbiJizntinn uf Ihe

tr:lnshllionnl :lIId rut:lliullIIl duuble inlegr:lturs, l'men'tliIIJl.\' of II,,: 11)1)6IEEh' Cmifl!rl!/II'I!01/ CO/llral """lielllitm.,', Dearburn, MI, pp. 185-190.

Zak, M. (1993). Telll1in:llllludcl uf Newluniall dyn:lIl1ics, 1/III!rIImimwl JmlTlwl of 11/t'orl!liL'llI"hysics. J2( I), 159.

,.-

~.

'f:I.

.,

r,""1I"rIIlllill/lII/ JmmwllJf TI'l!o/"t:riml 1'/I)'siL'.r, Vol. J6, No.2. 11)1)7

Lie Symmetries of Quadratic 'l\vo-DimcnsionalDifference Equations

M. A. Almciun,l F. C. Snnllls,l nnd I. C. Mureira I

Ill!c";I,,',1 JIIIU' -I, 11)1)(,

We lind nil syslems uf Iirsl-nnler IIlI:lllr:llic :IUlllnllmllllSIWII.,limensinnnldifference e1llmliuns which h:lye (wnline:" Lie symllldries. Knuwkllge IIflhesesYlllmelries permils Ihe syslems In be illlegr:lled hy n retlucliun p ccllure.

The idcntification of integrable systems for continuous or discrcte cqua'tions is all important problem in applied mathcmatics. Discretc dynamicalsystems have been studied in many contexts in the reccnt years. They appcarin discretization procedurcs of continuous systems, or, more naturally. inmodels described within a discrete space, for instance, in many hiologicalsystems. Two-dimensional continuous systems of lirst-onler aulonomousordinary differential equations have no chaotic behavior: however, there arcch:lOlictwo-dimensional autonomous difference equations, Hcnon's map, forexample (I-lenon, 1976), In many cases, discretization of completely integrablecontinuoussystcms also can exhibit chnotichehavior (Dnle 1.'1al., IIJH2).

Although integrnble discrete systems have been known for decades(MacMillan, 1971). few systematic studies were undertakcn in Ihis direction(Hirotn, 1979:Mneda. 1987:Granllllaticosel al.. IIJIJI: Quispcland Sahade-van, 1993).We study here two-dimensionalsystems of lirst-orderquadraticaulOnomous difference equations. These equatiolls arc discrcle counlerpartsof Lotka- Volterra continuous systems, which arc important in populalion

dynamics (Gardini 1.'1al.. 1987). We annlyze the invariance of these discreteequations under a continuous group of symmetries for detcrmining intcgrablecases.Maeda (1987) cxtended Lie's algorithm for finding symmetriesofdilTerellce equations and constructed :I procedurc for making a reduction of

'InSliIUIU de Fisica, lJFIU, 211).15.1)711,Itio tic J:lllcim, 11 1.il; c-mail: :lIIlunieI6!.if.ufrj.hr.

551

.ltt1U.11,Uv'n/U2un.uH1\ 11..\U/U ~ .'I'J'/11""um I'uhh\hin. CUII illinn