metric properties of differential equations lewis

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Metric Properties of Differential EquationsAuthor(s): D. C. LewisReviewed work(s):Source: American Journal of Mathematics, Vol. 71, No. 2 (Apr., 1949), pp. 294-312Published by: The Johns Hopkins University PressStable URL: http://www.jstor.org/stable/2372245 .

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METRIC PROPERTIES OF DIFFERENTIAL EQUATIONS.*

By D. C. LEWIS.

1. Introduction. The purposeof this paper is to discussthe depen-dence of solutionsof a system f differentialquations,

(1. 1) dx,/dt Xi (x1, * * * , xn, t) = Xi [x,t , i = 1,2, *,-n,

on the initial conditions. The classical existenceproofsyield inequalitieswhich how thatthe solutionsdependcontinuouslyn the initialconditions,

at least, if the righthand members atisfy Lipschitzcondition. In this

paper we shall refine nd supplement hese classical inequalities, ssumingthattheXi are ofclass C'. In so doingwe call attentiono thefundamentalimportance f a certainquadraticformQ to be introducedn the sequel.

Still further,ut less useful, efinements aybe obtained n theassumptiolithat theXi are ofclass C", in which ase a certainbiquadratic ormB plays

the decisiverole. It is also indicated, y considerationf simpleexamples,that our fundamentalheorems re the bestpossibleones of theirtype. A

simple applicationto the theory f qualitative ntegrations given.

Our theoremsre specifiedn termsof a metricwhichwe assignto thespace of the (x1, .* , x,,). For mostpurposes t is sufficiento assume a

Riemannian pace. Much of the analysis,however,s not moredifficult,f

we assumea generalFinsler space.Let f[x,x] f(x1, * , xn, p, .* ,n) be of class C'" and positively

homogeneousfdegree 1 in x, ,. Furthermore,upposef[x,x] > O,

unlessall the x's are zero, nd that i.j [x,fl]AAi > 0 forall setsAX, A

not proportional o 1 ,* *n. Iere, as in the sequel, the summation

conventionf tensor nalysis s used,with ll repeatedndices ummed rom

ton. The engthfan arc, i-Xi(T), i = l * n, 0< 1, is defined

as the valueofthe ntegral,f [x(r), x'(r) ]dT, whichexists if the arc is

sufficientlyegularand is, moreover,ndependent f the parametrizationn

* Received May 11, 1948.1 The purpose of this condition is to insure the " regularity" of the variational

problem 8ftx, ,]d-r = 0 in parametric form. Cf., for instance, Marston Morse, "The

calculus of variations in the large," American Mathematical Society Colloquium Pub-lications, vol. 18 (1934), p. 121.

294



METRIC PROPERTIES OF DIFFERENTIAL EQUATIONS. 295

accountof the assumedhomogeneityf f[x, . The distancebetween wopoints is the length of the shortest rc connecting he two points. The

parametric quations of such a geodesicalarc are well known to satisfvEuler's equations,

(1. 2) f [x(T), x'(T)]- (d/dT)fxi[x(T), x'(7)] O, i=1, * , n.

2. Fundamental resultson thehypothesis hat the Xi are of class C'.

THEOREM 1. Let us consider wo integral arcs of the system (1. 1).

CO: xi,x,(t,O) and C,: x,=x,(t,i), toC tCto + h.

We let D(t) denote the distance between x(t,O)] and [x(t, 1)] measuredalong the geodesicalarc gt. We furthermnoressume that all thesegt areincludedwithin n open regionR, and that no completegeodesicobtainedby extending gt containsentirelywithinR an arc whoseend pointsareconjugate o eachother in the senseof the calculusof variations).

If

(2. 1) a (t) ?{f$:[x, X]XA[ix ] + fPI[x, A]X [x, t]XAi :(t)

for every et An, , X such thatf[x,A] 1= and at everypoint [x] on

gt, then

(2.2) D (to) exp a (u) du < D(t) < D (to) exp /8(u) du,to t

to t < to+ h.

Beforeproceedingwith the proofof this theoremwe note that in theimportant iemannian ase,when

(2. 3) f[x, ] = (gij[x] xisii) ,

themiddlemember f (2. 1) can be replacedbya very implequadraticformin At, . *,An,namely,

(2. 4) Q[A] Xi 1VXV (iXkagij/axk gikXkx3AXAi.

Here the A's are componentsf a unit vector, nasmuch s f x, A] cannow be written n the formg1jAiAi 1. In formula (2. 4), X,j is thecovariant erivative fthe covariant ectorXi - gk Xk

To prove hetheorem, e let

(2.5) xixi(t,T)) i 1,- * ,n, 0<T<1,



296 D. C. LEWIS.

be thecoordinatesf thepointon gt whichdividesgt into two arcsofshorter

length n the ratio r: (1 -r) from x(t, 0)] to [x(t, 1)]. The functions

xi t,T) may be shownto be of class C". It is here that the hypothesisconcerninghe non-existencef certain airsof conjugatepoints n R is used.

By definitionfD (t) we have

(2. 6) D(t)- f[x(t, T), xT(t, T)]dT,

whichwe differentiatenderthe integral ign,thusobtaining

41D'(t) = (fx [x, xr] &x,/aT) + f$s[x, r] aOx/t/ataT) )dr.

The secondtermin the integrand an be integrated y parts, and, in a

mannerfamiliarto all devotees f the calculus of variations,we findthat

^1

D'(t) { (fxj x,Xr] - (0a/aT ) f; [x, XT] axi/a} dr

+ {f x t, ),p Xlr(t, ) ] xi(tp )/aot

-f$i[x(t,

O), ( t, 0) ] Oxi(tp 0) lot}.

In virtueof (1. 2), which the xi t, r) considered s functionsof r are

known o satisfy,heintegral n the aboverepresentationfD'(t) disappearsat once. RememberinghatthecurvesCO nd C, are integral urves f (1. 1),

the restof the expression an also be simplified, iththe resultthat

(2. 7) D"'(t) {f,;,[x (t,,1) ,x (t~,1) ]X Ex t,,1), t]

f$i [Z(t, O), Xr(t, O) ]XI[x(t, O), t] }

J(t, 1) -J(t,0 ),

wherewe define as follows:

J t,,r) fl$,Z(t,'r) XTr(t5 T)X [x (t, r), t].

Differentiatinghis last equalityand again using (1. 2), we obtain

(2. 8) Jr (tN 7-) fxiEx(t, 7-) x ( tN 7-) X'[x (t,, -), t]

+ f 1[x(t, 7), Xr(t, 7)]Xxj [x(t, 7), t]aXj/8T.

We now introduce he newvariables defined y

rT




whichhas the obviousproperties:

(2. 10) aS/aT = f[x(t, r), X (t, T)],

(2. 11) s t, 1) D (t)(2. 12) f[x, x/as] = 1.

The last of theserelationsfollowsfrom the fact that, on accountof theassumedhomogeneityf f[x,x], the parametern (2. 9) can be changedfromT to s', where ' s(t,T), with theresult,deducedfrom 2. 9), that

s = f[x(t,s'), x8(t, ')]ds'.

Again using the homogeneitypropertyof f x, x] and the relationsx= =XS(&0r]= x8f[x, 7], we obtainfrom 2. 8) the resultthat

Jr(t, T)= [fU[x(t, s), xj]XXE[x(t,), t]

+ f$ x, x8] (aX'/a1Xj) (OX/as) ]f[X, XT];

Fromthis we findwiththe aid of (2. 1) and (2.12) that

a(t) f[xI xr] _ J7-(t, 7-) C J8(t) f[X, Xr].

Integrating rom = 0 to T = 1 and referringo (2. 6) and (2. 7), we findthat(2. 13) a(t)D(t) ? D'(t) ? 3(t)D(t).

The proofof the theorem s completedby integration f these last twoinequalities.

The theoremust provedyields t once a resultmorereminiscentftheclassicalinequalities,fwe take fora (t) and p t) constants and : whicharerespectivelyower nd upperbounds n R for he middlemember f (2. 1)

or (in the case of a Riemannianmetric)for the quadraticformQ given by(2. 4). We thus obtainthe result

(2. 14) D(to)ea(tto) < D(t) < D(to)ee(tto), to t,

whichmaybe comparedwithsuch a classical formula as

(2. 15) D (t) ? D (to) eM(t-to).

2Cf., for instance, formula (2), p. 43, of Bieberbach's textbook,Differentialgleich-

ungen,where a different otation is used in connectionwith a first order system andthe usual Euclidean metric.

4



298 D. C. LEWIS.

HereM is thepositive onstantppearingn the Lipschitz onditionor somepositive onstantmultiple hereofn the more complicated ases). In other

words,ftheX's are of class C' and if weneglect possiblepositive onstantfactor,M is an upperboundfor the IOX10xj. As we shall show in 6,thereare cases when Q is negativedefinite nd hence 8 may sometimes etaken to be zero or even negative,whereas except forthe trivialcase whentheX's are independentf x1, * *, xn), M must be positive. In this sense,at least, (2. 14) is an essentialrefinementf (2. 15).

THEOREM 2. Let C, and C1 be defined s in the preceding heorem.Let rto be an arbitrary ufficientlyegularcurve,wvith arametric quations,x= 4(r), i =

1,,n,

0 < <r

1,whoseend points re at

[x(to,0)] and

[x(t0, 1)] respectively.Let C, be the curve whoseparametric quations,x= xi (t,T), satisfy hesystem 1. 1) foreach fixedvalue of r on the unitinterval nd take on the initial valuesxi (t0,-) 4 1 Tr). Let rt be the arcwhoseparametric quations, oreach fixed , (to ? t < to+ h), are likewisex= xi (t,-r), <r _- 1. Let L(t) denote he lengthof rt.

If

(2. 16) cc*t) ---Y [x,Ak]i[x, t] + f;i x,k]X [x,x t]Aj C /3 (t),

forevery et of theA's,such that f[x,A] 1 and at every oint [x] of Ft,then

(2.17) L(t0) expJ *(u)d)du?l>L(t) L(t0) expf3*(u)du.to . to

To prove this,we observe hatby our definition f length

(2. 18) L(t) = (f [x(t, T) xT(t, T)]dT.

Differentiatingnderthe ntegral ign,wehave (aftermakinguse of (1. 1))1

(2. 19) L (t) = {fi[x(t, T), XT(t Ti)]X [x(t, T), t]

+ffM[x(t,), xT(t, T)]X%x?[X(t, -), t]axj/ai-}dT-.

As in thepreceding roof,we introducehearc length as a newparameter.

T

s ff[xEt, T i) , X (t, I-) ] dIT, fX, x/as] 1.

Again usingthehomogeneityf f x,x] and hence the samehomogeneityf

the integrandn (2. 19), we find hat




>L(t)(2. 20) L' (t) Jaf [X,Ox/as]Xt x, t] + f.[x,,X/OS]X$,JI [x, t]axjl/s}ds.

It nowfollows t once from 2. 16) that

(2. 21) * t)L(t) L'(t) /38*t)L(t),

and hence theproof s completed y integrationf (2. 21).

The relationship etweenTheorem and Theorem is made sufficientlyobviousbythefollowing emark:

Althoughrt is not, in general, geodesicfor all values of t (or even

foranyvalue

of t),it is possible o choose rt0 n such a waythat forsome

particular , Pt is a geodesic. For sucha rt it is clearthat

D(t) L(t)whereas

D (t + t) vL(t + t).Hence

{D(t+At)-D(t)}/\t-?{L(t+ At)-L(t)}/At, if At>0,

while

{D(t + At)-D(t) }/zt>{L(t + At)-L(t) }/At, if At< 0.

Hence, allowingAt to approachzero first hrough ositivevalues and then

through egativevalues,we find hatD' (t) =L' (t). Thus the inequalities(2.13) used n theproof f Theorem canbe deducedfrom hecorrespondinginequalities 2. 21) used in theproofof Theorem . The. ndicated lterna-

tive in theproofTheorem1 was not adoptedbecause of the essential mpor-tance offormula 2. 7) in 3.

THEOREM 3. Let CJo,,, D(t), a (t) be defined s in Theorem1. Letthe rt, introducedn Theorem2 be chosen s a geodesic, nd then et L(t)and ,3*(t) be defineds in Theorem . Then

0 ? L (t)-D (t) < D(to) [exp 8* (u) du -exp c(u) du],

to? t < to+ h.

This theorem,which s a trivialconsequence f L (to) =- D (to) and the

two previous heorems, ives some informationbout the manner n which

geodesics re deformed nderthe transformationsefined y the differentialsystem 1. 1).



300 D. C. LEWIS

3. Estimates orDI'(t) and L"(t) on thehypothesishat he Xi areof class C". So farwehavenot made use of the particularmanner n which

theparameter was chosen n the proof f Theorem . Actually t was chosenin such a manner hat

s(t,yr) TD(t) f[X(t,T'), Xr(t, 7) ]dT',

whence, ifferentiatingith respect o r, we have

(3.1) D(t) =f[x(t,Tr),x7(t,T)]

(3. 2) 0=-w @z/) ?s(,2xt/8t2).

On accountof the assumptionsmade in 1 on fx,x], it may be shown hat,although he n equations (1. 2) are not independent, hese equations takelntogether ith 3. 2) can be solveduniquely for 2x,/Or2 in terms 1, *, x.and ax,/1, - *, Ox/r. We accordinglywrite

(3. 3) =2Xt0r2 F= [x,Ox/ah].

It is easy to verify hat the functions P[x,:x], here introduced,mustbehomogeneous f degree 2 in x1, . ., x". It is perhapsalso worthwhile orecord he fact that these F's are determinedy the following ystem f nlinearequations:

(3. 4) (fffj + f1fj;,)F ffi -(ffjix. + f$fjk) , n.

We introduce he abbreviations,

(3. 5) P[t, x, ] = f$jfj{XiXIjxFk -XiXkFJxJk (XiXk)xli,zFi$k}

? fwJ (X'XjXk) ih4k + ft$j (X'Xi)$k-4 + ? x x

+ f;,{ XiXi>,X) Xk X t$k}k

+ f$i{X$Xi + X4t,

3These are essentially well known facts in the Calculus of Variations. It is alsoeasily shown that a set of n independentequations equivalent to the n + 1 equations(1. 2) and (3. 2) may be written

[Od(f)/ Ox] - [8/OTJ] Oa(f)/0j] = 0,

where 4 is an arbitraryfunction f class C"whose first nd second derivativesare positivefor positive values of its argument. Formula (3. 4) was obtained by taking 1' f) = f2.

If we set gij (x, x) = ff + fjfi Gi = - iFi, etc., the relation (3.4) is seen tohave obvious connectionswith certain equations given by E. Cartan, "Les espaces deFinsler," Actualit6s scientiflqueset industrielles, 79, Exposes de Geometrie, I, pp. 16and 17.



METRIC PROPERTIESOF DIFFERENTIALEQUATIONS. 301

and

(3. 6) P* [t,x,x] fwi,Xi AX'j skik + 2fxjtjXXjxkxk + fJiCXtXj

+ fPItXiXkXk)+ XiXjyi;j f${{X'x,Xi+ X%t},

so that the following heoremsmay be stated in reasonably ompactform.

THEOREM 4. With the same assumptions as those of Theorem 1regardingCO,CO,D(t), gt, we assume in addition that

(3. 7) C(t) -<P [ t, A~] _ (t)

xt every oint [x] on gt and for every et Al,*,An such that f[x,A] = 1,then

(3. 8) b(t)D (t) < D"(t) _q (t)D (t).

THEOREM 5. With the same assumptionsas those of Theorem 2regarding O,C1,rt, L(t), we assume n addition that

(3. 9) +8 (t) -<P*Et, x, Ak] <+*(t)

at every oint [x] on rt and for every etAX' *n such thatf x,A]-1then

(3. 10) 0* (t) L(t) :!EL"'(t) +(t) L(t).

In theRiemannian ase we may write,

(3. 11) 5 B [A] = [f x,A]]]3P[t, x,A] - Y,jkl(x, t)xtxijkkV( B* A]=- [f x,A] 3P* [t,x,.A] Y*i.kl (x, t)Xi'Xjkk .

These biquadraticforms, n their relation to the appraisal of the secondderivatives f D (t) and L (t) are analogous to the quadraticformQ A]givenby (2. 4) in its relation o theappraisalof the first erivatives.Thereseems,however,o be no particularly imple nterpretationorthecoefficientsY jkl or Y*ijkl such as we had for the X-j in (2. 4) as the covariantderivative f Xi.

To proveTheorem , differentiate2. 7) withrespect o t and rememberthat xi = xi(t, T), i =1, * n, for r= 0 and 1 satisfy (1. 1). We thus

find hat

(3. 12) Jr t) K(t, 1) -K(t, 0) J- X (t, T)dT,



302 D. C. LEWIS.

where

K (t, T) f$i$j,[x (t, T), xr t, T) ]XjxSk[x (t, T), t]X[Ex (t, T), t]oxkl/

+ f$jxjXiXi + f XiXiXj + f$Xit.

Now differentiatinghis ast equalitywithrespect o T and usingboth (1. 2)

and (3. 3) as well as the easilyprovedrelations,

(81/9T)ff,j fx,-j -f$,kF-

f$x 1 [f$i$k FklJ + jj.k'P$i]

(0/0'r)Ai$j f$i$j faq!kFwxp

we obtain

Yr (t, T) ;P [t, x t, T), x, (t, T) ]

Now, sinceP[t, x,x] is homogeneousf degree+ 1, we may introduce he

parameter as in the proofof Theorem1, thusobtaining

Kr(t,r) ==P[t,x(t,s),xs(t,s)]f[x(t, r),x7(t,s)], f x(t, s), xs(t, s)] = 1.

Fromthis,we findwiththe aid of (3. 8) and (2. 12), that

+O(t)fEX, T] C< K,(t, 7) C< +(t) fEx,x,].

Integrating rom = 0 to Tr 1 and referringo (2. 6) and (3. 12), we find

that (3. 8) has beenproved, s desired.

To proveTheorem 5, we differentiate2. 19) underthe integral ign

and makeuse of (1. 1). We thusobtain

X (t)Et, x(t, r), xT(t, r)]dr.

Since P* [t,x, t] is homogeneous f degree 1 in the x's, we have, uponintroducinghe arc length s a newparameter,he following ormula:

fL(t)L'(t) - J P*t, x(t, s), xs t,s) ] ds.

It nowfollowsmmediatelyrom 3. 9) that (3. 10) is true, huscompletingtheproof.

It is interestingonotice hedifferenceetweenhe twoforms andP*:

P p* = fj{X'X xkF' - XiXkFiwk - 2 (XXk) xFj$kkt + XfXXjxhxk;hk}

+ fi~j {XxwkXj - X Xjxk-




Upon rearranginghe termsand writing

(3. 13) Si - fj,;f XixkFl - - 2x $,Fijj91 +, ~h;4}

lf;jXkx $Fj$t't + (fxj$Xiwk - fcxj$jXk) Xk,

we thusobtain

(3. 14) P-P*-=XiSi.

4. Special case of a Riemannmetricwithconstantcoefficients.Formost purposes it is sufficiento take a Riemann metricwith constant

coefficients.f this is done, practically ll formulas re enormouslyim-

plified.n this ectionwethereforessume hatf2=

gjX 'j, where heg'sare constants. From (2. 4), we have

(4. 1) Q[A] = gikXajXWA;.

Since f is independentf the undottedx's, we have fx-6==fI fxi =0o,so that (3. 4) yieldsFi =-0. From (3. 5), we now obtain

(4. 2) f3P = B[x] = (9ag9j - gaigfij) (XXxk) xat04hXk

+i g9a8gh (X'Xji ) a4Xaxfl4hJk

From (3. 13), we have

(4. 3) f3S, = (gaPgij - g9agflj)XiBj)X7 wxhaphkw.

Still furtherpecializing o the Euclidean case we obtain

(4. 1E) Q A.]= Z$$AA

(4. 2E) f3P (XXXk) Xhi~a5iackz ( XXjk) Xh^itj4;7

+ (X $jXj) x7aMiXk + X txXaXaXtik

(4. 3E) f3St Xx$xaialhxk

- Xj$a,XtXjXhW

5. Second orderlinear systems. The simplestnontrivial systemtowhichwe mayapplythe abovetheorys the secondorder inearsystem,

(5. 1) dx/dt A(t)x + B(t)y, dy/dt C(t)x + D(t)y,

whichwe shall consider n connectionwiththemetric,(5. 2) ds2 Edx2+ 2Fdxdy Gdy2, EG-F2 >0, E > O.



304 D. 0. LEWIS.

Here E, F, G are constants.Thus, takinggl,, E, g12=-F, g22 = G,A'=A,

A2 ,u,x1= x, x2= y, thefundamental uadraticformQ may be computed

from 4. 1).

(5. 3) Q= (AE + CF)A2 + (BE + DF + AF + CG)Al + (BF + DG)A2.

The extreme alues of Q under the condition, A2+ 2FAM G12_ 1, thatA and jube the componentsf a unitvector, an be foundbythe method fLagrange'smultipliers. The maximum, B(t), and the minimum,x(t), ofQ A] are thusseento be the rootsof thefollowing uadraticequation n s:

(5. 4) S2_ (A + D)s + AD-BC

-I(BE -AF+ DF- CG)2/(EG-F2) =0.

It is interestingo notice that the sum of the rootsof this quadratic s thesame as the sum of the rootsof the equation,

(5.5) S2 -(A +D)s+AD -BC=0,

which, n the case of constant oefficients,, B, C, D, is commonlyalledthe characteristicquation. We shall call it the "characteristic quation,"

evenwhenA, B, C, D are not constants. A mostfavorablemetric forsomefixed alue of t in thenon-constantase) is defineds a metricwhichmakes

,/ a a minimum. Since a +,B = A + D is independent f the metric,it is clear that a most favorable hoicealso makes/3 minimum nd a amaximum. We summarizeour results on most favorablemetrics n thefollowinghreetheorems:

THEOREM 6. If

(5. 6) (A +D)2 -4(AD-BC) = (A-D)2 + 4BC > O,a mostfavorablemetric ivesto a and/8 he valuesofthe rootsofthe charac-teristicequation (5. 5). There exist infinitelymany essentiallydistinctmostfavorablemetrics.

Proof. Since (,8 - a) 2 is to be a minimum, emustchooseE, F, and Gso as to minimize he discriminant f (5. 4). But the discriminantf (5. 4)

4Two metrics, EdX2 + 2Fdxdy + Gdy2 and E'dX2+ 2F'dxdy + G'dy2, are for thepurposes of Theorems 6 and 7 regarded as essentially the same, if, and only if,

E9/EJ' FI/' =GG'.




is the same as the discriminant f (5. 5) augmentedby the non-negative

quantity,

(BE -AF + D1F-CG) 2/ (EG .F2).

Hence the requiredminimizations obtained y choosing , F, and G, so that

(5 )BE -AF+ DF-CG = OEG F2 > 0, E > 0.

If BC > 0, we satisfy (5. 7) by taking E=l C' F 0, G= B.If BC < 0, we take E= + (A-D)C, F i 2BC, G + (D-A)B,where ? is chosen so that + (A - D)C > 0. This makes EG - F2

-- BC[(A-

)2 + 4BC], which nview f 5. 6) mustbe positive npresentcircumstances.f B = Oand C 0, takeE =1, F =+ C, G=-+ (D -A)

> 0. Similarly,fB =- 0 andC 0. Finally, fB=C ==0, takeE == G 1and F 0. From considerations f continuity,t is clear that in each ofthese cases there re infinitely anyotherchoices.

THEOREM 7. If

(5. 8) (A +D)2-4(AD-BC) = (A-D)2 + 4BC < O,

there existsan essentially nique mostfavorablemetric,making ac /3real partof one of theconjugate omplex ootsofthecharacteristicquation.

Proof. In view of the knownrelationship etween he rootsof (5. 5)and (5. 4), it is sufficiento showthatE, F, and G can be chosen n essen-tially ust one way in suchwise that (5. 4) have a doubleroot. Equatingto zero the discriminant f (5. 4) and makingsome algebraicmanipula-tions,we find hatE, F, and G mustsatisfy he relation,

(5. 9) (BE + CG)2-(2BF + DG -AG) (2CF + AE -DE) = O,if a and ,l are to be equal. While thereare obviously n infinite umberof waysof satisfyinghis relation, t turns out that there s (neglectingcommon actor) onlyone real set ofvaluesforE, F, G, namely

(5.10) E=+C>O, F=zt(D-A), G +TB,

which atisfies 5. 9).In fact, if we set p=-23F + DG-AG and q-2CF + AE - DE,

we find t oncethat(5.11) (BE + CG) (DI-A)=Cp - Bq.



306 D. C. LEWIS.

It followsfrom 5. 9) that p and q must satisfy he relation

(5.12) (Cp-Bq)2 (D -A)2pq=-O.

The discriminant f this quadratic form in p and q is found to be(A -D)2[4BC+ (A-D)2], which,because of (5. 8), is surelynegativein case A #A. Since the left hand member f (5. 12) is therefore ositivedefinite, he only real values for p and q which can satisfy (5. 12) arep q 0. The statedresult follows t once, n case A # D. In the con-trary ase,A = D, we are justified, n view of (5. 11) and the fact deducedfrom (5. 8) that BC 0, in writingp=lkB and q kcC. Thus (5. 9)becomes BE + CG)2 2BC. The left hand side of this relation s non-

negative,while, n viewof (5. 8), the righthand side is non-positive.Hencek 0, and we are led to p- q 0, as before, s well as to BE + CG= 0,fromwhichthe proof s readilycompleted.

THEOREM 8. If

(5.13) (A+D)2- 4(AD- BC) (A -D)2+4BC 0,

there s no mostfavorablemetric, xcept whenB =C= 0. But, for anynumber > 0, a metriccan alwaysbe foundsuchl hat 2(A + D) -e < a

2(A + D) <3 < (A + D) +e

Proof. We provethe secondstatement irst. If A #ZD, thenBC < 0.Then, if we take E= -+(A-D)C>0, > F= 2B0 G ?(D -A)B,0 <0 <1, we find hat

EG-F 2 BC (A --D) 2- 4B2C202

BC[(A-D)2 + 4BC] - BC[4BC(02-1)],

which, n accountof (5. 13) reduces o 4B2C2(1- 02) > 0, so thatEG -'> 0, as required. We also find hatequation (5. 4) becomes

s2- (A+D)s+1(A+D)2- [(A-D)2(- 0)]/( +0) ==0.

Since 0 < 0 < 1, the roots of this equation are real; and, by taking 0sufficientlylose to 1, we can make the rootsdiffer rom1 A + D) by lessthanE. If, however, D, eitherB or C is zero or bothare zero. We nowtakeE CjII+, F O, G B >+, >0. ItfollowsthatEG-_2

> 0, while (5. 4) becomese2_ 9.Ael A lrt([7-C2,q1/{ R I 1 I C7 1




whichyieldsreal rootsdifferingromA by less than E, if v is sufficiently

small.

To provethe first tatement, e note that the established econdstate-ment implies that for any most favorablemetric,ac 3 , 1 A + D), a

doublerootof (5. 5). Thus, equations (5. 4) and (5. 5) mustbe identical

equations, nd thuswe are led to the relation,BE - AF + DL - CG - O,to be satisfied ythecoefficientsf anymostfavorablemetric. This relationcan also be writtenn the form

(5. 14) (A -D)2F2= (BE - CG)2.

From (5. 13), we have (on multiplication y EG)

(5. 15) - (A-D)2EG 4BCEG.

Addingcorrespondingmembers f (5.14) and (5.15), we obtain

- (A-D)2(EG- F2) = (BE + CG)2 ? 0.

Since EG - F2 is to be positive, his relation s impossible nless A = D

and BE = - CG. From these two relationstaken with (5. 13), we find

at once that we would have to have B = C = 0 as stated.

We shall not dwell on the obviousconnections etweenTheorems6, 7,and 8 and theexplicit olutions f (5. 1) whichare availablewhen A, B, C,

and D are constants. We merelyremarkthat such considerationshow

thatTheorems and 2 are the bestpossibletheorems f theirtype.We nowconsider hebiquadraticformsB [A] and B* [x] B - f3XiSi,

whosesignificances explained n 3. We see at once from 4. 3) that,for

any linear system not merely econd ordersystem)and forany Riemann

metricwithconstant oefficients,ll the St mustvanish. Hence, our first

result s to the effect hatB [A] B* [A].

In orderto writedownthe expression or B [x] in reasonably ompactand yet lluminating orm,we introduceA*, B*, C*, D*, and H, defined s

follows:A*B* AEF AB 2 (AtBt

_C*D(= FG (CD CtDt

H=EG F2.

We thenfindfrom 4. 2) that

(5.16) B[x] H[-C'2+ (A -D)X +By2]2+ [Ex2 + 2F5!j+ Gy2] A*;.2+ (B* + C*)>z + D*y2].



308 D. C. LEWIS.

Hence, lowerand upperbounds forB, for sets (x,y) satisfying 2 Ei_2

+ Fxy+ Gy2 1, can be writtenn the formHo2 + p or p,wherep being

an extreme value of the quadratic form, A*x2+ (B* + C*)xy+ D*2,satisfies he quadraticequation,

Hp2 [A*G+D*E B* - C*F]p + (A*D* B*C*) -(B*-C*)2

0,

whilea, being an extreme alue of the form, C Q2 + (A - D) + By2,satisfies he quadratic

HU2_[BE-AF +DF-CG]a + (AD-BC)

-'(A +D

)2 O.

~~~~~~~~~~~~~~~~In view of Theorem 7 and (5. 10), the occurrenceof the form,

? C*.2 + (A - D)x + By2, in (5. 16) is rather interesting. It seems

worthwhile thereforeo investigatewhathappenswhen,fora certainvalue

oft, (5. 10) maybe assumed o hold. The resultdependsuponthefollowingmatric dentity,whichthe reader will have no trouble n verifying:

symm (1(D-A) B C)( D)2]

C ~ (D-A(A2 + D2 + 2BC)(l(D-A) -BA)

Here the symbol ymmM is used for the symmetric art of the matrix

i. e. symmM =I (M + M). Using the definition f A*, B*, C*, D*, and

HRwe find rom 5. 16) that,when

Ex2 + 2Fxy + Gy2 ?.?2 i (D - A) + By2 1,

we necessarily ave

B=w(A +D)2 {[CAt + 2(D- A)Ct]X2

[CBt - BCt + 2 (D-A) (Dt + At)P]x

+ [1(D- A)Bt--BDt]2} P[t,x,x].

This result s fully o be expectedn case the coefficients, B, C and D

of (5. 1) are constants. Thus, if (5. 10) holdsfor a particularvalue of 1,it holdsforall t. Our result howsthat theestimates iven n Theorems

and 5 are the best of theirtypes. For we have exhibited n exampleinwhichthe estimates ive the exactvalues.




6. Applicationto a simple non-linear ystem. Consider the secondorder ystem,

dx/dt - k(X2 + y2)'Xdy/dt q-l(x2 + y2) y,

wherek and g are positive onstants,with the Euclidean metric,ds2 dx'

+ dy2. Thus taking g1= g22-1, g12= O,xl-x, X2 Y, Al= A,X2 U,the fundamentaluadraticformQ may be computed rom 4. lE). We thusfind

(1/kl)Q[A] [X2/(X2 + y2) + (X2+ y2) ]X2+ (2xy/(X2 y2)i)XIA

+ [y2/(x2 +y2)+ (x2 +y2) ]2.

If we set V = (X2 + y2)-, cos 0 x/v, sin 0 - y/v,we obtain

- (1/kv)Q[X] (1 + cos2 0)X2 (2 cos0 sin0)Xkt (1 + sin 0),/2

A2 + /12+ (ACOS0 +?sin 0)2

1 + (Acos0 + /u in9) 2 forunit vectors A,p),

forwhichX2 ,U2 1. Evidently, or uchunit vectors, chwarz's nequalityyields

0? (X cos 04+-u sin 0)2? .

Therefore1 ? - / lV) Q X] ? 2,

-2kv Q ] < -kv.

We thushave an exampleof a systemn whichQ is negativedefinitetall

pointsotherthan the origin. The biquadraticformB, however, eemstoo

complicated o botherwith.

7. Applications o thequalitative ntegration f differentialquations.We close our paper with a few indications f the way in which our ideas

may proveto be of value in the study of functions efined y systems f

ordinary ifferentialquations.

THEOREM 9. Suppose that theregionR of Theorem is such thattheXZ x, t] are defined or x] in R and forto t < + oo. Suppose also that



310 D. C. LEWIS.

nosolution urvewhich ort to is withinR everpassesthrough boundary

pointofR fort > to. Let A be theleastupperboundof

(7. 1) f [x, ]X [x,t] + f-i x,AIX,$j [x, t]Ai

for[x] in R forevery et ofX'ssuchthatf x,A] 1 and fort ? to.

Then,ifA < 0, any twosolutions,whichfort - t1 to are withinR,mustapproacheach other symptotically.f R is finite, hisphenomenons

uniform.

Proof. In fact,from itherTheorem or Theorem , it is obvious hat

thedistancebetween orrespondingointson thetwotrajectoriess less than

or equal to the distance t t t1,multiplied y exp [A(t -t1)]; (A < 0).Hence it tendsto zeroas t --> 0o. The uniformityesultsfrom hefactthat

the distanceat t t1 for any two trajectories s bounded, f R is finite.

Furtherdetailsare omitted.

THEOREM 10. Supposethat, n addition o thehypothesisfTheorem ,

it is assumedthattheX's dependperiodically n t withperiodr and thatR

is finite. Then thereexists n R a unique periodicsolutionof period .,

towardwhich ll other olutions ntering mustapproach symptotically.

Proof. Let x,= x? (t), i=1< ,n, be an arbitraryolutionwhich

fort? to is withinR. Then xi x,O$(t) x,(t+ MrT), i=1,2 *

foreach positive ntegralvalue of m, is also a solution. By the previous

theorem,t is knownthat any two of these solutions pproacheach other

uniformly.Hence, givena number > 0 one can alwaysfind numberT,

independentfp, suchthat

Ix iO(t) -xP(t)I < e for t > T, p1,2,*

In particularIx (mr) - xiP(mTr) < E for integralm> T/r. From the

definitionfxim(t) this inequality an be written

Ixim0) - xm+P(0) < E form > T/r.

Hence the points[xm0)] form Cauchy equence nd we can write

(T. 2) ~~lim im0) tn- 1, n.m-ooo

The solution, i = xi t), such thatxi 0) = at thenturnsoutto be periodic.To establish his it is sufficienton accountof the,uniqueness heorem or




solutions f (1. 1) takentogetherwiththe periodicity f the X's) to show

that xi T) = a,. By the known continuity f solutions n respect o their

dependence n the initial conditions,we knowthat (7. 2) impliesx,(t) = lim xe, (t), i 1,2, ,

in->oo

uniformlyverthefinitenterval < t < T. In particular,

xi (T) = lim Xim (T) = lim xm+1 (O) - lim xim O) =ai,m->oo m->oo m-ooo

as wewished o prove. Finally, heperiodic olution x(t)] mustbe unique.

For, if thereweretwoperiodic olutions x(t)] and [x(t)] it is clear,from

the fact that 2 Ix,(t) x, t) 1 iS continuouson 0 ? t < T, that this

quantity as (on 0 ? t? T) a minimum alue,which lso serves on account

of periodicity)as the minimumvalue for all t. From Theorem9, this

minimumvalue must be 0. By the uniquenesstheoremfor differential

equations,xi t) and , (t) wouldthenhave to coincide, nd the theorems

established.The readerwill noticethat in the special case in whichtheX's do not

dependexplicitly n t,theperiodic olution,mentionedn Theorem10, must

be an equilibriumpoint (i. e. a point whereall the X's vanish), towardwhichall othersolutionstend asymptotically. n fact,we can provethe

followingheorem or thenon-existencefperiodic olutions,which s some-

whatreminiscentf a resultof Bendixson n the case n 2.5

THEOREMI 11. If the X's do notdependexplicitly n t it is impossible

to have a periodic olution otherthanan equilibrium oint) in any point

set in which the expression 7. 1) is eitherpositivedefinite r negative

definite.

Proof. We consider nlythecase when (7. 1) is negativedefinite.Theother ase reduces o this, f t is replacedby - t'. Supposewe had a given

periodic olution x t)] withperiodT onwhich 7. 1) werenegative efinite.

Then,referringo Theorem , we takeCOas thecurvexi = xi t), C, as the

curvexi = xi(t +? ), and rt0as the curvewith one end point at [x(to)]

drawnaroundCOto the point [x(to + T) ] Since xi (to+ T) xi (to), we

see thatrt0, nd hencert, is closed. The lengthL (t) is nothing ther han

the ength f thegivenclosed rajectory, hichof course s a constant. Since

5var Bendixson, " Sur les courbes definiespar des equations diff4rentielles," cta

Mathematica, vol. 24 (1901), pp. 1-88, especially p. 78.



312 D. C. LEWIS.

(7. 1) is negativedefinite, he ,8*of (2. 17) may be taken ess than a fixednegativenumber. Here we makeuse of the fact that the closed trajectorys

compact, but we omit details. Hence, fromTheorem 2, L (t) -> 0 as t-> oo.Hence L ( t) - 0. Thus the given periodic solution would reduce to anequilibrium oint, and the theorem s proved.

In conclusion,we point out certain obvious connections etweentheresultsof this paper and a paper by Wintner.6 We refer, orexample, ohis theorem o the effect hat, if A t) is a matrix of n2 real continuousfunctions atisfying

t(7. 3) limsup max (yA(s)y)ds < oo

m-*ao O 11=1

then everysolution vector x == x t) of dx/dt A t) x is bounded as t -> 00.

Here thequadraticformyAyplaysthe sameroleas ourQ, and, in fact,this theorem f Wintnerresults mmediately,f we apply our Theorem1or 2 to a linearsystem nd a Euclidean metric.

UNIVERSITY OF MARYLAND, COLLEGE PARK, MD.

NAVAL ORDNANCE LABORATORY,HITE OAK, MD.

6 Aurel Wintner," Asymptotic ntegrationconstants," American Journal of Mathe-

matics, vol. 68 (1946), pp. 553-559,especially p. 558.

metric properties of differential equations lewis

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