Stability of Motion, (Mathematics in Science and Engineering, Volume 30)

Stability of Motion A. M. L iap unov with a contribution by V. A. Pliss and an introduction by V. P. Basov

Translated by FLAVlA N A B RAMOV ICI

and MICHAEL SHlMSHONl

Department of Applied Mathematics Weizmann Institute of Science, Rehovoth, Israel

Academic Press

New York and London 1966

COPYRIGHT 0 1966,

BY

ACADEMIC PRESSINC.

ALL RIGHTS RESERVED. NO PART OF THIS BOOK MAY BE REPRODUCED I N ANY FORM, BY PHOTOSTAT, MICROFILM, OR ANY OTHER MEANS, WITHOUT WRITTEN PERMISSION FROM THE PUBLISHERS.

ACADEMIC PRESS INC. 111 Fifth Avenue, New York, New York 10003

United Kingdom Edition published by A C A D E M I C PRESS INC. (LONDON) LTD. Berkeley Square House, London W. 1

ORIGINAL RUSSIAN PUBLICATIONS Liapunov, A. M., “Issledovanie odnogo iz assobenych sluEaev zadsci ob estoijrivasti doyenia izadel’stve,” Leningrad University, 1963. K vopruso o b ustoyEioesti dvijenia, “Collected Works of Liapunov,” Vol. 2, pp. 267-271, 1956. Issledovanie odnogo iz assobenych slutaev zadsci ob estoijEivasti doyenia izadel’stve, “Collected Works of Liapunov,” Vol. 2, pp. 272-331, 1956. (This paper appeared earlier in Matem. Sbornik 17,253-333 (1893). Pliss, V. A., Issledovanie odnogo transcendentnogo slucaya teorii ustoirivosti diviieniya, Isvestia Akademia Nauk SSSR Mat. 28, 911-924 (1964).

LIBRARY OF

CONGRESS CATALOG C A R D

NUMBER: 66-26264

PRINTED I N THE UNITED STATES OF AMERICA

Preface In a celebrated MCmoire dated 1892 A. M. Liapunov founded modern stability theory and provided a powerful technique to test for it. It is not generally known, however, that about the same time he wrote important complements. They have now been translated for the first time into a western language, and form the topic of the present volume. Academic Press deserves considerable credit for having undertaken this noteworthy scientific task. We recall that a French translation of the MCmoire appeared in 1907, in the Annales de Toulouse and was photoreproduced in 1949 by Princeton University Press as No. 17 of the Annals of Mathematics Studies. Thus the full contribution of Liapunov to the theory of stability is now available to the West. It was natural for Liapunov to test his theory in his Mkmoire on the following first difficult question: To study the stability behavior of the origin for a real n-vector system X

=AX

+ X(X),

X ( 0 ) = 0,

( * ) = d/dt,

where A is a constant matrix and X is a vector whose components are convergent power series in the components of x beginning with powers at least two. Let Al, A, = A,, be the eigenvalues of A . The first difficult cases treated in full by Liapunov in the MCmoire are: (a) A, = 0; Re Ah < 0 for h > 1. (b) A = iw, A2 = -iw, o > 0; and Re A,, < 0 for h > 2. The next step was to attack the far more arduous case of A, =A, = 0 and A of the form

+

A=[ B O

1, .-[“‘I.

o c

0 0

and the eigenvalues of C all with negative real parts. Liapunov was able to dispose of A = B, n = 2, and this is the topic dealt with in Section 4. However, a last subcase of the general case vii

viii

Preface

eluded him. The related MCmoire was not published, and was only discovered in his Archives about a decade ago. It was published as a special monograph by the Soviet Academy of Sciences with an excellent summary by V. P. Basov (Sections 1 and 2). The “ missing link ” was recently solved by V. A. Pliss (Section 5). I have not yet said anything about Section 3. It contains a beautiful solution of the following problem: To prove that if A has eigenvalues with zero real parts, where Xmay be so chosen that one obtains stability or instability at will. In other words, the eigenvalues of A could serve. to distinguish between stability and instability only when none of the real parts of the eigenvalues were zero. To sum up, this volume offers a welcome addition to the growing literature centering around Liapunov’s classical stability (theory), and available to the non-Soviet world. S. LEFSCHETZ

September, 1966

Translators Note

Three of A. M. Liapunov’s works on stability problems, and a preface to one of them by V. P. Basov, are collected in the present translation. A paper by V. A. Pliss on a related subject is included as well. In the translation we tried to give the authors’ meaning as faithfully as possible. We did not change their way of presentation, but we did not hesitate to break up some long and involved sentences into shorter ones wherever we felt that clarity demanded it. As to the mathematical terms, we used the ones generally accepted in stability problems. However, for some terms used frequently by Liapunov we could not find English equivalents, and we literally translated the Russian terms into English. In all cases these translations seem to be self-explanatory, except for the term “ sign-definite.” A sign-definite function is one which is either positive-definite or negative-definite. A few footnotes were added by us pointing out some misprints in the “ Collected Works ” by Liapunov. Flavian Abramovici Michael Shimshoni

July, 1966

ix

Contents

Prefac:~

1fanslawrs· Note

vii

ix

Lntroductioo-A Resume--v. P. BAsov

References

12

An Invest igation of One of the Singular Cases of the Theory o f Stability of Motion, 1-A. M. LtAPUN
13

On the Problem of the Stability o f Motion-A. M. LtAPUNov

123

An Investigation of One of the Singular Cases of the Theory of Stability o f Motion, 11-A. M. LIAPUNov

128

Chapter I: Systems of Se<:ond Order

131

An Investigat ion of a Tr-..nscendental Case of the Theory of Stability of Motion- ¥ . A. Pctss '

185

References

202

l!
203

Introduction In 1893 A. M. Liapunov published a long paper dealing with the problem of stability of motion [l]. This paper is evidently unfinished. At the beginning, he mentioned that he would investigate the problem of stability of the unperturbed motion in the following case: the system of differential equations for the perturbed motion is autonomic and the characteristic equation of the linear system which gives the first approximation, has a double zero root. When the order of the system is larger than two (n > 2), all the remaining roots have negative real parts. The double root corresponds to a multiple elementary divisor of the characteristic matrix. He wrote further: " Before treating the general case, we consider .. . when the system of equations under investigation is of order two. " In this case the problem allows many simplifications which, in general, are impossible for systems of higher order. " ... On the contrary, when dealing with systems of higher order, we meet some new situations causing in the known cases, very peculiar difficulties." After this follows Chapter I, entitled " Systems of second order." In this chapter he considered the system dx

dt = y

+ X(x, y ) ,

2 dt = Y ( x , y )

when the functions X and Y are holomorphic with respect to x and y , and their series expansions do not contain terms of order less than two. The problem of stability of the unperturbed motion for such systems was completely investigated by Liapunov; this is the end of the paper. Liapunov intended to publish a second chapter, in which he would consider the general system of n equations, with n > 2. This work, however, was never published. 1

2

Stability of Motion

On the centenary of Liapunov’s birthday, the Soviet Academy of Sciences undertook the task of publishing the complete collection of his works. At the time of the preparation of this publication in 1954, Academician V. I. Smirnov discovered in Liapunov’s archives a long manuscript with the same title as the abovementioned work [ 13. According to its contents, this manuscript corresponds to the second chapter that Liapunov intended to publish. In the following sections we shall briefly describe the contents of that manuscript, emphasizing the methods of investigation that are different from those used by Liapunov in his published works on stability.* In the course of the description we shall mention the problems that were left unsolved. In the manuscript the system dx -= y dt

+ X(x, y, XI, ...

+ ... + psflxn+ Xs(x, y , XI, ... , xfl) (s = 1, 2, ... , n )

_ -- p s l x l dx, dt

is considered, assuming the real constants pS&, c = 1, that all the roots of the characteristic equation

... , n) such

-x

... ... Pnl ... P n n - X have negative real parts. X, Y, X,(s = 1, ... n) are holomorphic functions with respect to x, y , xl, ... ,x,,, and we suppose that their expansions in powers of these quantities do not contain terms of order less than two. Let P11

Y(x, y , 0, ... ;o> = YdX) + Y1(x)y

... 0) = Xl0’(X)

XS(X, y, 0,

7

+ W(x,y ) y 2 ,

+ xp(X>y + Ws(x,y)y2

(s = 1, ... , n),

* The notation adopted here does not always correspond to that used in the manuscript.

3

Introduction

where Y,(X),

Y,(X), X p ( X ) , Xj”(X)

( s = 1 , ... > n)

(2)

are holomorphic functions of x, while W, W, are holomorphic functions of x and y. If the functions (2) are not identically equal to zero, suppose that their expansions in powers of x are of the form

+ glxm+’ + ... ( m 2 2), Y1(X)= axa + a 1 x z + l + ... (CI 2 l), xp = a(s)xas+ ... Xjo’(x) = gsxms + Y,(X)

= gxm

1 . .

>

(s = 1,

... , n),

where 9, a, g s , and a ( ” )are constants different from zero. With the aid of transformations which do not change the form of system (11, one can always manage to satisfy the following conditions : (i) If Y,(x)= Y,(x) = 0,then X:’)(x) = X:’)(x) = 0 (s= 1, ... , n ) ; (ii) If Y,(x)=O, Y,(x)$O, thencr,>a(s= 1, ..., n); (iii) If Y,(x) $ 0, then m, > m; in this case a, > a if a < rn, or a , > m , i f a > , m ( s = l , ..., n). Moreover, we can always suppose, that X ( x , 0, 0, ... , 0) = 0.

As in reference [ 11, Liapunov distinguishes between the following possible cases : I. Y,(x)-O; II. meven; III. m odd, g > 0; IV. nz odd, g < 0. Cases I, 11, 111, and also IV, when CI t m and CI is even, are studied using the second method, and all the obtained results are analogous to the corresponding results for the systems of two equations [l]. We remark that for establishing instability, Liapunov sometimes constructs functions that are not within the frame of the conditions

4

Stability of Motion

of his well-known Theorem 11of the second method, and sometimes he uses additional considerations. For instance, in the third paragraph, he constructs a function V(x, y , xl, ... , x,,), variable in sign, which does not contain terms which depend only on x. The derivative of this function with respect to t is by virtue of system (1) of the form

= Y 2 + X I 2 + ... + X,2 + vy2 + y f v,xs + dt s= 1 s,u= dV

1

v,,x,x,,

(3)

where u, u s , 0, are some holomorphic functions of x, y , xl, ..., x,, vanishing for x = y = x , = - . . = x , = O . We see that function (3) is positive but not positive-definite, as it vanishes for y = x, = .-.= x, = 0 and arbitrary x. The function V, constructed in this way, does not satisfy the conditions of Theorem 11 of the second method. The instability is proved by Liapunov by making the following considerations : “ From this we must conclude that the unperturbed motion is unstable. For, if it was stable with respect to x,then considering x as a given function of t, the numerical values of which never exceed some sufficiently small bound, we would find that the derivative of V is a sign-definite function of the variables y, xl, ... , x, and according to the principles exposed in the above-mentioned work [2], we would conclude that the unperturbed motion is unstable with respect to y, x,, ... , x,. We remark that this function V satisfies the conditions of cetaev’s theorem [3] (published for the first time in 1934), from which the instability follows directly. We must say that it was not Liapunov’s intention to improve his second method. This becomes very clear if we remember that in reference [2], after proving the theorems of the second method, he wrote: “By varying the conditions satisfied by the unknown functions, one could, of course, propose many other theorems, similar to the demonstrated ones. For the applications which we have in view, however, these theorems are perfectly sufficient and therefore we limit ourselves to them.” 99

Introduction

5

The unstudied Case IV, with the assumption ci 3 m or a odd, is divided by Liapunov into three subcases: IV,. a>-

m-1. 2 '

IV,. a<-

m-1. 2 '

m-1 IV,. a = -. 2

Let us consider in more detail the first of these subcases. Here occur for the first time the " peculiar difficulties '' in comparison with the systems of two equations, mentioned in reference [l]. By letting m = 2q - 1, q being an integer larger than 2, the subcase considered is characterized by the inequality ci > q - 1. After some preliminary transformations of system (l), Liapunov introduces the new variables r and 9 by the formulas x

=

rC(9),

y = -rqS(9),

(4)

where C(9) and S(9) are functions* defined by the differential equations

with the initial conditions C(0) = 1, S(0) = 0. As it is known, these functions are periodic, with period o. Making the substitution (4), the system (1) becomes

* In reference [l], Liapunov denoted these functions by the symbols Cn8 and Sn9,respectively.

Stability of Motion

6

where R1,0, are holomorphic functions of Y with o-periodic coefficients with respect to 9; R, , 0 , , X , (s = 1, ... , n) are holomorphic functions of r, x l , ... , x, with o-periodic coefficients with respect to 9 such that R,(9, Y, 0, ... , O)-X,(9, r, 0, ... , O)=O, and the expansions of the functions Xs(9, Y, 0, ... , 0) in powers of Y have no terms of order lower than 2q. Eliminating dt, we get the system r 4 + l R 1+ rR, d3 - r 4 - l + rqO1+ 0, '

dr _

In general, the right-hand sides of Eqs. (6) are not holomorphic with respect to r, x,, ... , x,, and this makes the essential distinction between the cases n > 2 and n = 2. Liapunov shows that system (6) can always be satisfied formally by series of the form

us('),which are indewhere c is an arbitrary constant, while d'), pendent of c, are either o-periodic functions of 9, or finite sums of periodic and secular terms. If the series (7) are periodic, and if the supplementary conditions = 0 (1 = 2, 3, ...) are satisfied, the coefficients of these series are determined uniquely. If not all the coefficients of (7) are periodic, the stability problem is solved by methods similar to those used in [2] for the case where the characteristic equation has two imaginary roots. If, however, all the coefficients of series (7) are periodic, solving the problem of stability encounters very serious difficulties. The point is that since the right-hand sides of system (6) are not holomorphic, the series (7) are, in general, divergent and therefore one cannot make use of considerations similar to those applied in the case of two imaginary roots. Liapunov overcame these difficulties partially, by using a new

7

Introduction

and very interesting method of investigation. This method consists of the following. One introduces in system (6) the arbitrary constant c by the substitution

r = c(l + p), x s = c2q-1ts (s = 1, ... , n), (8) where p and 5, are new variables. As a consequence, the system is brought to the form

where P and Zs (s = 1, ... , n) are some holomorphic functions of t l , ... ,5, and c, with coefficients of period w with respect to 9. Further, instead of (9), one considers the system

p,

dP 5 = CP(9, P , d 4 S

A-=p d9

si

41,

..., 5 , , 4 + q(c, 4, (10)

51 + ... + Psn4n + cEs(9, P, 4 1 , ... 4,)

(s = 1,

9

... , n),

in which A is some real parameter and p(c, A) is some function of c and A, which is unknown for the moment. Liapunov proved that p(c, A) can always be chosen so that system (10) has a periodic solution which can be represented by the series

c v"'(9, m

p

=

i=l

c vpy9, A)ci a,

i)c',

(5", =

i= 1

(i

=

1, ... , n).

(11)

The coefficients of these series, u ( ~ )us'), , depend on 9 and A but not on c, and they are of period o with respect to 9. With the supplementary conditions di)(0,A) = 0 (i = 1,2, ...) series (I 1) are determined uniquely, and are uniformly convergent for all real 9 and A, and for all sufficiently small IcI. In these conditions, the function q(c, A) is also determined uniquely in the form of a series m

~ ( c A) , =

C qi(A)c', i= 1

(12)

8

Stability of Motion

which is uniformly convergent for all real 1 and sufficiently small ICI.

Liapunov stated that the coefficients d i ) ,ZI;~),cpi of series (11) and (12) are continuous and bounded, together with their partial derivatives, with respect to 9 and A up to an arbitrary order, for all real 9, A. These coefficients can be represented asymptotically by series in nonnegative integer powers of 1, with coefficients continuous and periodic with respect to 9. The above-mentioned asymptotic series are infinitely differentiable term by term with respect to 9 and 1. He also showed that series (11) and (12) are differentiable term by term with respect to A. Let m

cp(c) = cp(c,

cq- 1)

=

1 i=

cpi(cq- 1)Ci 1

According to (8), it will be sufficient to suppose that c 3 0. From the form of systems (9) and (10) it follows that if ~ ( c= ) 0 for all sufficiently small c, then the series obtained from (I 1) by putting 1 = cq-', define a periodic solution of (9). Taking into account substitution (8), we see that to this solution corresponds a periodic solution of (6), defined by the series r

=c

+ ci C ~"'(9, cq-')ci, = 03

1

1UP'($, 00

x,

= c2q - 1

cq- ' ) c i

i= 1

(14) (s = 1,

... , n).

This solution is, in general, not holomorphic with respect to c, but can be expanded asymptotically in positive powers of c. As it is easy to see, the above constructed formal series (7) are just the asymptotic expansion of this solution. Suppose now that q(c) 0. By rearranging series (13) in increasing powers of c, we get the asymptotic expansion of this function in powers of c. It follows that only two cases are possible: either one can find a positive integer k, so that

+

lim cp(4 c++O

Ck

Introduction

9

exists and is different from zero, or for each given k this limit is zero, i.e., all the coefficients in the asymptotic expansion of q(c) are equal to zero. The first case occurs when not all the coefficients of the series (7) are periodic. For this case it was shown above how the stability problem is solved. As to the second case, a problem, appears at once, which is still open. Namely, Liapunov put into doubt even the possibility of such a case occurring. He writes: " One may ask the question, whether in this case the function q(c) is identically zero. " Since we cannot prove this, let us consider the hypothetically possible case when q (c) is such a function of c that it is not identically zero, although expression (15) vanishes for any k . " In this case, all the terms are zero in the asymptotic expansion representing the function q(c), similarly to what happens to function exp( - I / c 2 ) . " The study of the properties of q(c) .is, of course, a very difficult problem. But nevertheless we suppose it to be investigated and known." After this there arise two possibilities: (1) There exists c' > 0 such that the function q (c) is not zero on the open interval (0, c'). (2) For each arbitrary small c' > 0, there exists an infinite set of zeros of q(c) on the interval (0, c'). If the second possibility is true, then to each root of the equation q(c)=O corresponds a periodic solution of system (6), so that each neighborhood of the unperturbed motion contains an infinite set of periodic solutions. Further, Liapunov used series (14)in order to transform system (6). Thus, he introduced into (6) new variables z, zl, ... , z, by putting r

=

z

+ z Cdi)(9,zq-')zi, 10

i= 1

10

Stability of Motion

For the system obtained in this way, he used the second method, and solved the problem of stability of the unperturbed motion, when in a sufficiently small interval (0, c’) one of the following conditions : (2)

=0; d c >z 0;

(3)

dC>2

(1)

q(c)

0

is valid. Moreover, he solved the problem of orbital stability of the periodic solutions neighboring to the unperturbed motion (when such solutions exist). From these considerations it follows that, in order to solve completely the problem of stability for u > q - 1 , one has to prove that from the vanishing of all the coefficients of the asymptotic expansion of q(c) in powers of c, it follows that q(c) = 0 (for all c sufficiently near to zero) or, if that is not true, then one has to study the stability problem when q(c) vanishes an infinite number of times on the arbitrary small interval (0, c’), so that it is not constant in sign or q(c) S 0. We want to mention that Liapunov was, seemingly, the first who used asymptotic series in the study of the stability problem. Let us turn now to subcase IV2 . It is characterized by the inequality a < (rn - 1)/2 = q - 1 (u = odd). One can consider also that a > 0, as this can always be obtained by a transformation of the system. In order to investigate the stability problem, Liapunov eliminates from system (1) the quantity dt, and shows that the system obtained in this way has the solution a

u+l x, = .“+2f,(X)

(s = 1,

... , n),

in which f ( x ) , fs(x) (s = 1, ... , n ) are some functions, defined and continuously differentiable on the interval [0, A ] , A being a sufficiently small positive constant.

Introduction

11

Substituting the quantities (16) into the right-hand side of the first part of Eqs. (l), we convince ourselves easily that in the considered subcase IV, the unperturbed motion is unstable. Finally, considering the last subcase IV, , characterized by the equality CI = (m- 1)/2 = q - 1 (a= odd), Liapunov divides the investigation into two parts corresponding to the following inequalities :* a 2 2 4q,

a 2 < 4q.

(17)

If the first inequality is valid, the instability of the unperturbed motion can be stated by considerations completely similar to those of subcase IV, . If, however, the second inequality of (17) is valid, then after substitution (4), some transformations of the obtained system, and the elimination of dt, we get a system perfectly similar to (6). Here occur the same difficulties as in subcase IV,. Correspondingly, the similar problems remain unsolved as well. In 1939, before the discovery of Liapunov’s paper, Kamenkov [4]studied the problem of the stability of the unperturbed motion of system (1). He gave the solution of the stability problem of all the cases except of those in which all the coefficients of series (7), which formally satisfy system (6) (or a similar one in subcase IV3), are periodic. These last cases are simply omitted by Kamenkov. In conclusion we want to mention that one can state with a large degree of certitude that this manuscript was written by Liapunov in 1892. Actually, at the beginning of the manuscript, there are two notes on reference [2] in which page references are given. Thus, the manuscript could not have been written before 1892. On the other hand, the manuscript finishes with a summary of the obtained results, referring to second-order systems. The same summary, in an improved form, is at the end of reference [l]. From here it follows that the manuscript was written before the publication of reference [ 11 and that this work is an extract of

* Here it is assumed that g = - 1. This can always b e achieved by some transformation of system (1).

12

References

its results concerning second-order systems. In reference [5] there is a communication that on December 4,1892, at the session of the Char’kov Mathematical Society, Liapunov reported a work with the same title as this manuscript and reference [ 11. There are in the manuscript so’me nonessential mistakes which are corrected in the text. The character of the corrections is mentioned in footnotes. Words, written shortly in the manuscript, are reproduced fully in the text; the added parts of words are included in square brackets.* V. P. BASOV

* These brackets are omitted in our translation into English (Translators’ note).

References 1 . Liapunov, A. M., Issledovanie odnogo iz osobenych sluEaev zadaEi ob ustoiEivosti dviieniya, MatematiEeskii sbornik 17, No. 2, 253-333 (1893); See also A. M. Liapunov, “Collected Works”, (Edited by the Academy of Sciences, USSR), Vol. 11, Moscow-Leningrad, 1956. 2. Liapunov, A. M., ObshEaya zadaEa ob ustoiEivosti dviieniya, “Collected Works,” (Edited by the Academy of Sciences, USSR), Vol. 11, MoscowLeningrad, 1956. 3. Cetaev, N. G., UstoiEivosti dviieniya, Gesudarstvenoe Izdatelstvo TechnoteoretiEcskoi Literatury (Governmental Press for Technical and Theoretical Literature), Moscow, 1953. 4. Kamenkov, G. V., Ob UstoiEivosti dviieniya, Tr. KAI, No. 9, Kazan’, 1939. 5 . SoobshEeniya Char’k. Matem. obshEestva (Communications of the Charkov Mathematical Society), Second series, Vol. IV, No. 5 and 6, Charkov, 1895.

An Investigation of One of the Singular Cases of the Theory of Stability of Motion, I

1. We suppose that, for the unperturbed motion, the quantities with respect to which one investigates the stability are identically zero. For the perturbed motions, these quantities satisfy a system of differential equations having a normal form. We call these equations the differential equations of the perturbed motion. When the above-mentioned quantities are small, the right-hand sides of the equations can be expanded into positive integer powers of these quantities. We call singular all the cases when in solving the stability problem, one has to take into account the high-order terms in the indicated differential equations. Let us limit ourselves to the case when all the coefficients of our differential equations are constant quantities (suppose them to be real), i.e., the case of a motion which is stable in some known sense. Singular cases of the stability problem occur when the algebraic equation, on the solution of which the integration of the linear differential equations of the first approximation depends, has no roots with positive real parts, but has roots with zero real parts. We shall call this algebraic equation the characteristic equation. Two of the simplest singular cases were investigated by me in the work : “ The General Problem of Stability of Motion ”.* One of these cases occurs when the characteristic equation has a zero root, all the remaining roots having negative real parts. The second

* Here and in the following the author is referring to the work: “The General Problem of Stability of Motion. Considerations by A. Liapunov ” (Char’kov, 1892) (Editor’s note). 13

14

Stability of Motion

case occurs when the characteristic equation has two pure imaginary roots, the other roots having negative real parts. I intend to consider here the more complicated case, when the characteristic equation has two zero roots, assuming that all the remaining roots (if its degree is larger than two) have negative real parts. However, I shall not investigate this case in all its aspects. I limit myself to the assumption that the zero root does not cause at least one of the Jirst nzinors of the fundamental determinant to vanish (i.e., the determinant the vanishing of which constitutes the characteristic equation). From the principles expounded in my above-mentioned work, it follows that in this case, by a linear substitution with real coefficients, the differential equations of the perturbed motion can be brought to the form

dx

d= y -

-=y+x, dt

dt

P11-

Pz 1 P nt

x

Y,

P12 P22

-

...

Pn2

x

I . .

..*

...

P1. P2n Pnn

-X

=0

(2)

Investigation of One of th e Singular Cases of Theory of Stability of Motion

15

the inequalities 1x1 < E , lyl < 6 , Ix,I < E (s = 1, 2, ... , n) are valid for all the positive values o f t . Whenever such a possibility is established, we call the unperturbed motion (x = y = x,= 0) stable with respect to the quantities x,y , x,. If, on the other hand, the impossibility of choosing such a number 1 is proved, in other words, if one shows that there exists a positive number E such that for any arbitrary small positive 1, among the real values of the quantities a, b, a, satisfying (3), one can find such ones that for some positive value t, at least one of the following (n 2) equalities holds: 1x1 = E, lyl = E, lxll = E, ... , Ix,I = E, then the unperturbed motion is unstable with respect to the quantities x,y , x,. The solution of our problem will be based on the principles presented in the work “The General Problem of Stability of Motion,” and will follow the method given there.

+

2. y

Consider the system of equations

+ x = 0,

ps*xl

+

ps2x2

+ ... + psnx, + x,= 0

(s = 1, 2,

... , n),

that defines the quantities y , x, as functions of x. Under the assumptions adopted, these equations can be satisfied in a unique manner, by holomorphic functions of x,vanishing for x = 0. Obviously, these functions do not contain terms lower than of second order. Let y

= v,

X I = u1,

x2

=

u2,

... , x, = u,

(4)

be such a solution. By making the substitution (4)in the expression of the function Y, one can meet the following four cases : (1) the result of the substitution is identically zero ; (2) the expansion in increasing powers of x begins with an even power of x;( 3 ) the expansion begins with an odd power of x which has a positive coefficient; and (4) the expansion begins with an odd power of x having a negative coefficient. Each one of these four cases represents a kind of singularity and requires a special investigation. This will be done successively

Stability of Motion

16

by transforming system (1) with the aid of the substitution + y , x, = us +i?, (s = 1, 2, ... ,n). If the transformed equations are

y =u

dy - -- Y

d x-y+z,

dt

dt

then, obviously,*

x,=

pslul

+

ps2u2

du, + ..- + psnu, + x,- ( y + z) dx

(s = 1, 2,

... , n).

We see from here that 1,F, XS,as functions of x, j,&, have the same properties as X , Y, X , as functions of x,y , x,, and are such that for j = Z1 = 2, =:. = 2, = 0 the following equalities hold: X = 8, = T2= = X , = 0, P = ( Y ) , where ( Y ) denotes the result of performing substitution (4)in Y. In the following we shall consider that this transformation was already performed for the system (1). In other words, we shall suppose that for y = x1 = x 2 = ... = x, = 0 the functions X , XI, X , , ... , X , are identically zero.

3. Consider system (1). If it satisfies the conditions of the first case, i.e., not only X , X , vanish for y = x1 = xz= = x, = 0, but so does Y. First of all let us perform the following transformation. By the system pslxl+ps2x2 ... +psnxn X , = 0 (s = 1, 2, ... , n) we define the quantities x,, x2, ... , x, as holomorphic functions of x and y , vanishing for x = y = 0. Obviously, these functions are of the form

+

x1

= Yfl(X2

Y ) , x2

+

= Y f 2 ( X , Y)7

... > x, = YfrI(X, Y ) ,

* In the following two lines, 8 was written instead of script (Editor’s note).

jj

(5)

+ 8 in the manu-

Investigation o f One of the Singular Cases of Theory of Stability of Motion

17

where f,, f 2 , ... , f n are holomorphic functions of x,y , vanishing for x = y = 0. Instead of the variables x,, we introduce 2, by the substitution x,= yfs(x, y ) 2, (s = 1, 2, ... , n). The transformed equations are of the same form as above dx dY = y - -- y + X , dt dt '

+

but the functions

x

= Y(Pslf1

+ P s 2 f 2 + ... + P s n f n )

+X,-y(y+X)--

afs

ax

(

Y j,+y-

( s = 1 , 2 , ..., n )

have the following property. For 2, = Z2 = ... = 2, =O, either there are no first-order terms in y present in their expansions, or, if such terms are present, then they are also present in the expansion of Y (after setting XI = X 2 = ... = 3, = 0), and the lowest power of x in these terms in X , is greater than the lowest power of x in the corresponding terms in Y. In other words, if by substitution ( 5 ) , all first-order terms in y disappear for Y, then surely such terms are not present in X , for Z1= 2 , = = 2, = 0. If, however, such terms remain after performing this substitution and if gyx" (where g is a constant) is the term of lowest order among them, then the lowest order of the similar terms in the functions X,,if there are such terms, surely will contain x to a power higher than m. Suppose that the substitution was already performed and therefore the functions Y, Xs have, with respect to variables xu, the properties of the functions Y,1,with respect to variables Zu. First consider the case when, in the terms independent of the quantities xu , Y, and therefore all X,,do not contain y to a power lower than the second. Let Y

= ay2

+

n

s= 1

x,cp,(x)

f

Py

+ Q + R,

Stability of Motion

18

where a is a constant. q,(x) are holomorphic functions of x, vanishing for x = 0. P is a linear form, Q is a quadratic form in the quantities x, with constant coefficients, and R is a holomorphic function of x, y , xs not containing terms lower than of the third order in x, y , x, and lower than of the second order in y , x,. Further, let

c n

xs

=

u=

1

xu(Pso(x)

+ R,,

where q,,(x) are holomorphic functions of x, vanishing for x = 0 and R,are holomorphic functions of x, y , x, , not containing terms lower than of the second order in y , xu. Define the holomorphic functions $,(x) by the equations s= 1

Since the determinant of these equations is equal to a constant different from zero [Eq. (2) has no zero roots], for x = 0, the functions $,(x) are perfectly determined. Obviously, these functions vanish for x = 0. Set now

+ (1 - a)x)y + u y + w + s = 1xs$Jx), n

I/ = (1

where U is a linear form and W a quadratic form in the quantities x,, defined by the equations

From the assumption about the roots of Eq. (2), it follows that such forms are always perfectly determined (see “ The General Problem of Stability of Motion 7. By (I), we have dV dt

- = y2

+ x12 + x22 + ... + xn2 + s,

Investigation of One of the Singular Cases of Theory of Stability o f Motion

where S = ( 1 - a )x (a y2

19

+ P y + Q ) + [l + (1 - u ) x J R+ (1 - a ) y X

i

il

+ xs = 1xs+s'(x> + Y s = 1xsC+S'(x> - $,'(O>l. Obviously S can be written in the form "Y2 +

such that

ZI,u s ,

c n

c n

"SYX, s= 1

+ s=l

n

u=l

"Suxsxu

us, are holomorphic functions of the quantities

x,y , xi,vanishing for

x=y=x

= x2 =

... = x, = 0.

We found therefore a function V , vanishing for y = x1= x2 = = x, = 0, and capable to take an arbitrary sign, the derivative of which is positive for sufficiently small 1x1,Iyl, Ix,I and vanishes only for y = x1 = x2 = ... = x, = 0. From this we must conclude that the unperturbed motion is unstable. For, if it was stable with respect to x, then considering x as a given function of t, the numerical values of which never exceed some sufficiently small bound, we would find that the derivative of V is a sign-definite function of the variables y , xl,... , x,, and according to the principles presented in the above-mentioned work, we would conclude that the unperturbed motion is unstable with respect to y , xl, x2 , ... , x,. Consider now the case when for x1 = x2 = = x, = 0, firstorder terms in y are present in the expansion of Y. Suppose that gxmyis among them the term of lowest order. If such terms do also appear in the expansion of X,, then the lowest order of them contains x to powers higher than m. e f .

4. Let us first prove the following auxiliary assertion. Suppose that y is defined as a function of the independent variables x, xl,x 2 , ... , x,, satisfying the equation

20

Stability of Motion

It can be proved that in general, if X , Y, X , vanish for y = x1 = xz= ... - x, = 0, then the equation can be satisfied in a unique manner, by taking for y a holomorphic function which vanishes for x1= x 2 = = x, = 0. Moreover, this function will not contain, in its expansion, terms lower than of second order with respect to x, XI, xz, ... , x,. In order to convince ourselves that our assertion is true, we remark that according to one of the theorems, proved in “The General Problem of Stability of Motion” (p. 97), there always exists one pair of holomorphic functions x and y of the variables x, , that satisfy the equations

c n

s= 1

(PSlX,

ax + PSZXZ + ... + PsnX, + X,)-= y + x, ax,

n

dY C (~slx1+ ~szxz+ ... + PsnXn + Xs)=Y s= 1 ax,

(7)

and vanish for x1= xz= = x, = 0. These functions do not contain in their expansions terms lower than of second order with respect to the quantities x, . The last property is true whatever X , Y, X , are, with the condition that they do not contain terms lower than of second order. When, moreover, as in our case, they vanish for y = x1= xz= = x, = 0, Eqs. (7) can be satisfied by two holomorphic functions x, y so that y vanishes, as previously, for x - xz= ... = x, = 0, and x takes an arbitrary numerical value c, sufficiently small. In this case the functions x and y are holomorphic not only with respect to the quantities x,, but also with respect to x,, c. Indeed, suppose that by making the substitution x=c+x

we get

x,= CS1X1 + c,zxz + + c,,x, + c,y + x, x = a1x1 + azx2 + + a,x, + ay + x, Y = b , x , + b,x, + ... + b,x, + by + **.

*.a

(s = 1, 2, ... , n),

Investigation of One of the Singular Cases of Theory of Stability of Motion

21

where c,, ,c, ,a,, b,, a, b are some holomorphic functions of the constant c, vanishing for c = 0. X, X , F, are holomorphic functions of the quantities 2,y , x,,c, which do not contain terms lower than of second order with respect to the quantities 2,y , x,. Set

x = A , x l + A 2 x 2 + ... + Anxn+ 5, Y = B1x1

+ BZx, + ..* + B,x, + T / .

We can always choose the constants A , , B, (defining them as some functions of c) so that if we perform the substitution x = c + C A x , + 5,

Y

=

C BSY, + T/

in Eqs. (7), 5 and T/ satisfy new equations of the form (7), the righthand sides of which do not contain first-order terms in the quantities x,. A , and B, are to be taken as solutions of

c c n

(Psu s= 1 n

s=l

(PS,

+ c,, + c,B,)A,

= a,

+ c,, + c,B,)B,

=

b,

+ ( a + 1)BU + bB,

9

(0

=

1, 2, ... , n),

*

According to the assumption about the coefficients p,,, one can determine from here A , and B, as holomorphic functions of c, vanishing for c = 0. For 5 and q we therefore get a system of equations of the form

where*

* In the manuscript, the quantity a was missing in the expression for p (Editor’s note).

22

Stability of Motion

For sufficiently small Icl, these equations satisfy the conditions of the theorem shown in a previous work (p. 100). We can assert therefore, that t and q are aIways holomorphic functions of xl, x 2 , ... , x, that do not contain terms lower than second order. The coefficients of these functions are perfectly determined and are holomorphic functions of c. We can prove in the same manner as for the similar case considered in the same work, that 5 and q are holomorphic with respect to x, , c. Hence, we found the following solution of system (7):

Here cp and $ are holomorphic functions of the quantities x,, c, vanishing for x1= x2= ... = x , = O and not containing terms lower than second order with respect to x, , c. By eliminating c from Eqs. (9), we get Y

f

= (XI,

x2 > ... xn > x),

(10)

3

wheref is a holomorphic function of the quantities x, , x,which does not contain terms lower than second order and vanishes for x1= x2 = ... = x, = 0. Moreover, we have

a$ af +-af aP ax, ax, ax ax,

( s = 1 , 2 ,...) n).

-=-

These equalities become identities by substituting c taken from the first equation of (9). As a consequence, Eqs. (7) easily lead up to an identity which shows thatfis the solution of (6). Our statement was therefore proved. Before getting on, we call the attention to the following fact. Consider the solution of system (I), that satisfies Eq. (10). For x, xl, ... , x, as functions of t, we get a system of equations of the form dx -= X', dt

dx, - = p,lx, dt

+ ps2x2 + + psp, + X,' .'*

where X ' , X,' vanish for x1 = x2 =

= x, = 0.

(s = 1, 2,

... , a),

Investigation of One of t h e Singular Cases of Theory of Stability of M o t i o n

23

According to the properties proved in my work, we can affirm in this case that for perturbations satisfying condition (lo), the unperturbed motion is stable. 5. Returning now to our problem, let us perform in (1) the substitution y =f(xl, x 2 , ... , x,, x) + J . The transformed equations are of the form dx

dy -d t= y ,

-dt= y + x , dx,

- = pslxl

dt

+~

+ + psnx,+ X,

~ 2 x 2

(s = 1, 2,

... , n).

The function

vanishes for jj = 0. We remark that the functions

R=

x +f, Y,x,

contain J in the terms independent of the quantities x,, in the same manner as y is contained in the similar terms in the functions X , y,

x,.

As a consequence, we can suppose that the function Y appearing in (I), vanishes for y = 0, and therefore y is either identically zero, or it cpnserves the sign of its initial value, at least when the quantities 1x1,( y l , (xsI are sufficiently small. Let Y = uy2 gyx" yR, where, in the terms independent of the quantities y , x, ,R does not contain x to powers less than m 1. Moreover, the first-order terms in R represent some linear form in x1, x2 - * + , x, * Suppose first that the number rn is odd; then take the function V = yx + W , where W is a quadratic form in the quantities x,, satisfying the equation

+

9

+

+

24

Stability of Motion

We have

i:

dV

= Y 2 + gyxm+’ f x,2 + s, dt s=l

where S

=yX

+ x ( a y 2 + y R ) + Cn X , aw ax,

s=l

is always of the form yxm+lu

+ y2u + y c u,xs + c c u,,xsx,, S

s

o

in which u, u, u s , vso are holomorphic functions of the quantities x, y, x,,vanishing for x = y = x1 = x2 = ... = x, = 0. We see from here that if one imposes the condition that the variable y has the same sign as the constant g, then the function dV/dt is always positive for sufficiently small 1x1,lyl, IxJ, vanishing only for y = x1 = x2 = - - .= x,= 0. But whatever the sign of y is, the function V can obviously be made to take any sign. This function vanishes for y = x1 = x 2 = .-.= x,= 0, and if we consider x in it as a given function of t, the numerical values of which never exceed some sufficiently small bound, then, when the quantities Iyl, Jxs( do not exceed some sufficiently small bound E , the function 1 Vl has an upper limit which tends to zero together with E. In other words, according to the terminology of the work, “ The General Problem of Stability of Motion,’’ the function V has aninfinitesimal upper limit. From here it follows that when g y > 0, if the unperturbed motion is stable with respect to x,it is surely unstable with respect to the quantities y , x,. Hence, in the case considered, the unperturbed motion is unstable. Although it is not stable unconditionally, it has, as we saw, a conditional stability, namely, it is stable for y=O. Let us show that it is also stable in the more general case, when g y < 0. Let R = x”f(x)

x

= YV(X)

+ P + R,

+

c x,cps(x> + x, S

Investigation of One of the Singular Cases of Theory of Stability of Motion

25

where f ( x ) , cp(x), cp,(x) are holomorphic functions of x,vanishing for x=O. P is a linear form with constant coefficients of the quantities x,; 17 is a holomorphic function of the quantities x,y , xs, vanishing for y = x1= x 2 = = x, = 0 and not containing terms lower than of second order with respect to x,y , x,;and B is a similar holomorphic function not containing terms lower than of second order with respect to the quantities y , xs. Put

v=

--

m+l

xm+l[l

+ $(x)] + y[1 + (9 - a)x + U ]

S

where $(x), $,(x) are some holomorphic functions of x,vanishing for x = 0. U is a linear form in the quantities x,and W a quadratic form of the same quantities, both with constant coefficients. They are defined by the equations

c (PSlXl + PSZXZ + ... + S

PsnX,)

au

- + EJ = 0,

ax,

(11)

The functions $(x), $s(x) are chosen so that in the expression of the derivative dV/dt, formed by virtue of equations (I), no zeroorder terms with respect to the quantities y , x, appear. For this purpose we determined the above-mentioned functions from the equations :

c

{Psu

+

cp,u>*,

- g[l

*

+f)x" = 0, (12) X

+ + m + 1 3 u= 0,

(a= 1,2, ...) n),

(13)

S t a b i l i t y of M o t i o n

26

where qsu(x),fs(x) are holomorphic functions of x, vanishing for x = 0. These functions appear in the expressions X,

=

C x,(P,,(x) + xrnyfs(x)+ R,

(S

= 1, 2,

... , n),

U

where R, does not contain first-order terms with respect to the quantities y , xu. System (1 3) gives $,

=

(1 +

l+b

X + r3)F , , n+ldx

where F, are known holomorphic functions of x, vanishing for

x = 0. By using these relations, Eq. (12) becomes d$ xdx

+ (rn + l)$ = F ( x ) ,

where F(x) is some known holomorphic function of x, vanishing for x = 0. Hence, we find

$(x)

5

Yrn-l

From the definitions of the functions $, $,, U, W we have dV - = g(y2 + i 1 2 dt

+ x22 + ... + X“2) + s,

where S is of the form

s = uy2 + y 2 u,x, + 2 S

s

a

u~ax,x,.

Here u, u s , usuare some holomorphic functions of x,y , x,, vanishing when these quantities are simultaneously equal to zero. We see from here that for sufficiently small y , x,x, , the derivative dVjdt can not take values which are opposite in sign to g. If gy < 0, then V is a sign-definite function of x,y , x,,opposite in

Investigation of One of the Singular Cases of Theory o f Stability o f Motion

27

sign to g. For, W satisfying Eq. (1 l), is a sign-definite form of the quantities x, , opposite in sign to g, as was shown in " The General Problem of Stability of Motion." According to what was said in this work, we must conclude that for gy < 0 the unperturbed motion is stable. Suppose now that rn is even. Take the function

v = [I + (1 - a)x + U-Jy + w,

where the linear form U and the quadratic form W satisfy the equations :

where s=[(l-a)x+

U]Y+(l-a)yX+yCX,-

,

au ax,

2W

+ C x,+ y(R - P ) axs can always be represented in the form

if u, u, u s , v,, are holomorphic functions of x, y , x, , vanishing when these quantities are simultaneously equal to zero. Since V vanishes for y = x1 = x2 = ... = x, = 0, and can be made positive for y > 0, we conclude as before, that if g > 0, the unperturbed motion is unstable. It has only conditional stability for y = 0. Let us study, finally, the caseg < 0 (rn odd). Take as Vthe abovedefined function for y < 0, and V , = [l - ( 1 a)x U ] y- W for y > 0, U and W being the same as before. Denote by t,bs(x),

+ +

28

Stability of Motion

holomorphic functions of x chosen so that the derivative with respect to t of the expression

for y=O does not contain first-order terms with respect to the quantities x,. We find that ~-xm+z(~+~x,tj,)

if y < o

and V , +xm+z(1

+~x,+~)

if y 2 0

is a sign-definite function of the variables x,11, x,, the derivative of which has also a constant sign, opposite to that of the function. We must conclude therefore that the unperturbed motion is stable. Moreover, we can affirm that in this case the quantities y , x, tend to zero for t tending to infinitiy, if their initial values are sufficiently small. We express this by saying that each perturbed motion, for which the perturbations are sufficiently small, tends asymptotically to a motion belonging to the continuous set of motions defined for different values of the constant c, by x = c, y = XI = xz= ... == X" r 0. Indeed, if we consider x as a given function of t, the numerical values of which do not exceed some sufficiently small bound, then V and V , are functions with infinitesimal upper limits, and moreover, V is negative definite if y < 0 and V, is positive definite, if y 3 0. In the same conditions, their derivatives are sign-definite, with signs opposite to those of the functions themselves. According to the principles established in the work already mentioned, the validity of our claim follows.

6. In order to finish the investigation of the case when for y = x1 = ... = x, = 0 the functions X , Y, Y, are all equal to zero, what remains to be studied is the stability of the motions x = c, y = x, = xz= ... = x, = 0, corresponding to sufficiently small values of the constant c.

Investigation of One of the Singular Cases of Theory of Stability of Motion

29

It is easy to see that the stability of these motions depends on the quantity I , defined by formula (8). For I 3 0 the motions considered are unstable and for I < 0 they are stable. If for x1 = x, = = x, = 0 the function Y does not contain first-order terms in y , then the equalities b = c, = c2 = = c, = 0, hold, and therefore" I = 0, p = 1 a. Hence, in this case, the perturbed motion is unstable. If, however, for x1 = x, = = x, = 0, the function Y contains first-order terms in y and gyx" is the lowest degree term of them, then the expansion of b, and therefore that of I , in increasing powers of c, will begin with the term gc". Hence, if m is odd, the motions considered are unstable for gc > 0 and stable for gc < 0. If, however, m is even, the motions are unstable for g > 0 and stable for g < 0.

+

1 . .

7. For what follows, it is useful to show one property of Eqs. (1) in the case when not only X , X , but also Y vanish for y = x, = x, = ... = x, = 0. Suppose that Uis the derivative of an entire function of the quantities x,y , x,, vanishing for y = x1= x 2 = = x, = 0. It is easy to see that there always exists one entire function Y of the same quantities, vanishing for x = x1 = x2 = ... = x, = 0, of an order not higher than m so that in the expression dV/dt - U, formed by virtue of Eqs. (l), no terms lower than of the (rn 1)th order with respect to x, y , x, occur. This function does not contain terms of an order lower than of the lowest term of the function U. Indeed, if

+

u = u(1)+ u'2' + I/ = 1/(1) + p) + ... 1 . .

where in general U ( j ) ,Y c j )are forms of thejth order with respect to the quantities x,y , x, , then for the successive determination of

* In the manuscript, the quantity a is omitted from the expression for p as well as from formula (8). This, however, has no effect on the further considerations, for, by means of a linear transformation which conserves the form of the system, one can make p = 1 (Editor's note).

30

Stability of Motion

the forms V ( ' ) , V('), etc., one gets a system of equations of the form :

Here W'j) is a form of the jth order with respect to x, y , x,, determined in a known way from the forms V" ), V(' ), ... , V ( j - l ) . Moreover, W ( j )vanishes for y = x1= x, = = xn= 0. Suppose that all the forms V ( ' ) ,V ( 2 ) ,... , V ( j - ' ) have already been found and that, therefore, W'j) is known. Let W"'

= yQ0

+ + + + Q1

Q2

.**

Qj,

where Qlis a form of the fth order in the quantities x, with coefficients depending on x and y if f <j , and with constant coefficients if f =j. Qo is a form of the ( j - 1)th order of x and y . Put V j ) =

Po + P ,

+ P , + ... + P i ,

where P, is a form of the Zth order of the quantities x,. We have

The forms P o , P , , P j are separately obtained from these equations and they are perfectly determined. As P o must vanish for x = 0, we have, among other things:

investigation of One of the Singular Cases of Theory of Stability of Motion

31

where in the integration y is considered as a constant. One determines the form Pjwithout difficulties. Each one of the forms P, is also easily obtained. Let

Ql = Boxk B,Xk-'y Y , = A0xk + A , x k - ' y

+ Bk- lXyk-' f Bkyk, + ... + A k - ' ~ y k - '+ Akyk, ***

where k =j - I, B, and A, are forms of the Ith order of the quantities x,with constant coefficients. For the determination of the forms Ai we have the equations

A. C(P,~X, + ~ ~ 2 +x 2... + P ~ , X , , ) a2 + ( k - i + 1)Ai-1 = Bi ax,

(i

=

1, 2, ... , k),

which give successively A o , A,, A , , ... , A k . All these forms are perfectly determined. From this one sees that if U ( ' )is the collection of lowest order terms in U,then V ( ' )is the collection of lowest order terms in V and that

+

does not contain terms lower than of the (m 1)th order. From here it follows that it is always possible (and moreover in a unique manner) to find an entire function of the quantities x, y , x,,of an order not higher than the mth order, equal to y for x = x1 = x2 = ..-= xn= 0, and the derivative of which with respect to t , formed by virtue of Eqs. (l), does not contain terms lower than of the (m 1)th order. Indeed, if the function to be found is y V, where V vanishes for x = x, = x, = ... = x, = 0 and does not contain terms higher than of the mth order, then Vmust be chosen so that in the expression dV/dt - U, all the terms lower than of the (m 1)th order disappear. According to what we proved, this is always possible and moreover, in a unique manner.

+

+

+

Stability of Motion

32

Obviously, V does not contain terms lower than of the second order. From this it follows that the partial differential equation determining the first integrals of system (1) can always be satisfied, at least formally, by a series expansion in the positive powers of the quantities x, y , x, . Moreover, there always exists one series of this type, in which the unique first-order term is y and for x = x1 = x2 = ..- = x, = 0 all the other terms vanish. When, for sufficiently small x, y , x,, this series is convergent, it represents one of the first integrals of system (1). However, this series is not always convergent, and therefore system (1) has not always a holomorphic first integral. Thus, for instance, the system d_x -- y , dt

d-=y zx dt 1-x’

dz _ dt

- =,

which is a particular case of system (l), has no holomorphic first integral, independent of t. Suppose, however, that system (1) has some holomorphic first integral U, independent of t. For x = x l = x 2 = . . - = x , = O it becomes F(y), where F is a holomorphic function* of y . Let y V be a series, satisfying formally the condition for the first integrals of system (1) and which is equal toy forx = x1 = x2 = ... = x, = 0. From this series we deduce a new one F(y V ) , which also formally satisfies the condition for the first integrals of (1) and so will be the series U - F(y V ) .In this series, however, all the terms vanish for x = x1 = x2 = ... = x, = 0. It follows that all the terms of this series are identically zero, for if W is the collection of lowest order terms, then W satisfies the equation

+

+

+

c

(P,lXI

+ PS2X2 + ... + PsnX,)

aw

aw

-+ y - = 0.

ax, ax From this it follows that if for x = x1 = x2 = ... = x,=o, then W = 0 identically. S

* I n the manuscript

the superfluous phrase

... = x,,= 0” is written after y (Editor’s note).

“

vanishing for x

w=o

= xI= x 2 =

Investigation of O ne of t h e Singular Cases of T h e o r y of Stability of M o t i o n

33

Suppose that the expansion of F(y) in increasing powers of y is aoym+ a l y m + ' ... , where a,, a,, etc., are constant, a, being different from zero. In this case, the expression U - F(y) does not contain terms lower than of the (m 1)th order and, therefore, by putting

+

+

x = yz,

x1 = y z , ,

... ,

x2 = y z , ,

x, = Y Z , ,

(14)

the function U is represented in the form

u = y"{a, + F(Y, 2 , z1, 2 2

9

**.

7

z,>>,

where F is a holomorphic function of y , z, z l , z z , ... , z, , vanishing when these variables are equal to zero. We see from here, that if we seek the function W from the equation F( W )= U, we get for it m, and only m,different definitions in the form of series expansions in the positive powers of y , z, zl, z 2 , ... , z,. For sufficiently small values of Iyl, IzI, lzll, 1z21, ... , IzJ, all these series are absolutely convergent and one cannot find divergent series of the same type which, if the operations specified by F a r e performed upon them, will result in the given function U . We saw before that the series y V has the desired property, and therefore by making substitution (14), it is convergent for sufficiently small values of Iyl, Iz(, 1zJ. It follows that this series was also convergent before making the substitution. Thus, if system (1) has a holomorphic first integral independent of t, then it has a first integral of the form y V and every other holomorphic first integral independent of t is a holomorphic function of the last one.

+

+

8. Let us go now to the case, when Y does not vanish for y = x,= x, = = x, = 0. Suppose that it takes the value Y o , the series expansion of which in powers of x is Yo = gxm g1xm+ ..., where g, g1, etc., are constants and g is different from zero. Suppose further that Y = Yo Y l , Y , vanishing for y = x1 - x, = ... = x,= 0. Suppose first that m is even. According to what was said above, we can always find an entire

+

+

+

34

Stability o f Motion

function V of the quantities x,y , xs, with the property that in the expression

+

no terms appear lower than of the (m 1)th order. We can suppose, moreover, that the function V becomes equal to y for x = x1= x 2 = = x, = 0. By making this assumption, we obviously find that dV - = g(xrn dt

+ + y2

XI2

+ x22 + ..' + x,2) + s,

where the derivative dV/dt is taken in virtue of Eqs. (1) and S is a holomorphic function which does not contain terms of an order lower than m 1. Thus, the derivative dVjdt is a sign-definite function of the variables x, y , x,. As V can be made to take an arbitrary sign, according to the principles exposed in the above-mentioned work, we must conclude that the unperturbed motion is unstable. Suppose now that m is odd. Define an entire function Vsuch that in expression (15) all the terms of an order lower than m 2 disappear and V vanishes for x = x1= x, = ... = x, = 0. Under these assumptions, the function V does not contain terms of order lower than two and if V , is the collection of its second-order terms,

+

+

= g(y2

+ x12 + x 2 2 + ... + x,,).

If V , is equal to Vi0)for x1 = x, = ... = x, = 0, we find from here Via) = gxy. Moreover, it is obvious that V 2- Via) does not depend on y . In consequence, we find that in virtue of system (l), we have

where S does not contain terms of order lower than m

+ 2.

Investigation of One o f the Singular Cases of Theory of Stability of Motion

35

We see from here that if g is a positive number, then dV/dt is sign-definite. As V can be made to take an arbitrary sign, we must conclude that the unperturbed motion is unstable. What remains to consider is only the last case: when for m odd, g is negative. 9. Considering system (1) with the assumption that it satisfies the condition just shown, substitute in the expression of Y for the quantities x , the holomorphic functions of x and y defined by the equations (s = 1 , 2, *.., n ) . ps1x1 + ps2xz + ... psnxn+ x,= 0 Since these functions vanish for y = 0 (page 16), the terms of Y which depend only on x remain unchanged by this substitution. From the other terms, let us consider those in which y appears in the first power. Suppose that by making the above substitution, such terms remain and let ayx" be the one in which x appears in the lowest power. Here a is a constant supposed to be different from zero, and CI a positive integer. We shall suppose that c( < m. Next we make the transformation of (1) by the substitution from page 17. Let us suppose that system (1) is already transformed. In this case Y is of the form: Y = gx" + g l x r n + l + ... + U ~ X "+ u , ~ x " + + ' ..* + R , where R contains terms either depending on the quantities x,, or containing y 2 . As above, X vanishes for y = x , = x 2 = = x, = 0. The functions X , do not necessarily vanish but each time one puts these zero values into their expression, terms containing x to powers lower than m 1 do not appear. Moreover, these functions are such that in the terms independent of the quantities x , and containing y to the first power, x does not appear to a power lower than CI 1. We can further suppose that the function X is identically zero and that the functions X , are such that they do not contain any terms independent of x, lower than an arbitrary large order. For such an assumption is equivalent to the use of a transformation that does not change the type of (l), as it was defined above. Such a transformation is always possible.

+

+

+

36

Stability of Motion

As a matter of fact, the first assumption is reduced to the change of the variable y into j by the substitution y X = j . The second assumption is connected with the possibility of finding series expansions in integer positive powers of x, y which do not contain terms of an order lower than two and which satisfy formally the system of equations

+

(s = 1, 2,

... , n).

As it is easy to see, it is always possible to find such series. They are unique. Let x1 =fi(x, y ) , x 2 =f2(x, y ) , ... , x, =fn(x, y ) be the collection of terms of order not higher than k . In this case, if instead of the variables x, we put 3, defined by x, =fs(x, y ) 2, (s = 1, 2, ... , n), then in the transformed equations, which are of the same type, the functions playing the role of the functions X,, obviously do not contain terms independent of the quantities R,, which are of order lower than k 1. Moreover, in accordance with the assumption about X , , the functionsfs(x, y ) do not contain terms independent of y , which are of an order lower than m 1 in x and first order terms in y , in which x appears to a power lower than a 1. As a consequence, the transformation considered does not change the type of equations (I), because it is determined by the numbers g, a, m, and a. We shall suppose therefore, that X = 0 identically, and that in the expressions of the functions X , there are no terms, independent of the quantities x,, of order lower than m 1. Thus, we can assume that g = - 1, for we can achieve this by replacing x and y by x[(-"'"~-')]-' and y[(-g)'""-')]-', respectively. Under these conditions, let us assume that

+

+

+

+

+

Y

= -x"

+ g1xm+1 + ... + yY1 + Y , ,

where Yl does not depend on the quantities x, and vanishes for x = y = O . Y,vanishesforx,=x,=...= x, = 0. Suppose now, that y = c q(x, c) is the general integral of the equation dy/dx = Y,. Here c is an arbitrary constant and ~ ( x c) ,

+

Investigation of One of t h e Singular Cases of Theory of Stability of Motion

37

a holomorphic function of x and c, vanishing for x = 0. Replace now in Eqs. (l), x and y by the variables t; and y, defined by Y

= rl

+dx,

$7

From these equations we can find x and y in m different ways as holomorphic functions of 5 and y, vanishing for t; = y = 0. Of the m definitions obtained in this way, only one is real. This is the one we will use. It gives for x and y expressions of the form x = 5c1 + $(<,dl,

Y = rl

+ qt;,y),

where $ and 8 are holomorphic functions of t; and y, vanishing for t; = v] = 0. Let us see what form the equations dx _ -Y, dt

dY dt

-=

Y

will take after the transformation. We have: Y , = acp(x, v])/ax and, therefore

whence dv]- -tm+ dt

y2

1

+ acp/ay.

Thus the transformed form of Eqs. (16) is

38

Stability of Motion

where the right-hand sides are supposed to be expressed in terms o f t and y. We remark now that, according to the assumption about Y,, the function q(x, 0) is represented by a series, in which the lowest order term is [a/(. l ) ] x a + Hence l. e(t, 0) is represented by a series in which the lowest order term is [a/(a l ) ] t " + 'As . a consequence, the right-hand side of the first equation of (17) is of the form

+

+

a

y+cc+l

+ r,

ta+l

where T does not contain terms of an order lower than two, and for y = x1= x2 = ... = x, = 0 it becomes a holomorphic function of 5, the expansion of which does not contain terms of an order lower than c( 2. For the sake of simplification we change a by a(a I). Denoting the variables 5 and y~ by the previous letters x and y , we get the following transformation of our equations :

+

+

where X , Y, X , do not contain terms of order lower than two. -Moreover, the function Y vanishes for x, = x2 = ... = x, = 0 and the functions X , become with this substitution holomorphic functions of y and x, not containing terms of order lower than m + 1. Finally, the function Xis such that for y = x, = x2 = ... = x,= 0 the terms of order lower than a + 2 with respect to x disappear. All our transformations are such that the problem of stability with respect to the previous variables is completely equivalent to the problem of stability with respect to the variables x, y , x,, satisfying Eqs. (18). 10. From what was shown in page 31 it follows that one can always find an entire function U , which does not contain terms

Investigation of One of the Singular Cases of Theory of Stability of Motion

39

of an order lower than three or higher than k, and vanishing for x = x1= x2= ... = xn= 0.U has the property that in the expression of the derivative

d

(Y'

+ U),

formed in virtue of Eqs. (18), there are no terms, which vanish for y = x1 -- x 2 = = x, = 0, of an order lower than k 1. The number k can be arbitrarily large. We shall determine the function U under the assumption k = m +a 1. Let us form, in this case, the lowest order term of (19) which does not depend on y , x,. Put

+

+

where X ( j ) , Y c j>) X 'sj )> U ( j )are forms of o r d e r j in the variables x, Y , xs * According to our assumptions, all the forms Y(j)'and those of the forms X6j) for which j < m 1, vanish for x1= x2= ... = Xn = 0. The forms X"' for which j < a 2 vanish for y = x1= x 2 = ... = x, = 0. From our assumptions it also follows that the forms U ( j )must vanish for x = x1 = x2= = x, = 0. For the determination of the U") we have equations of the form*

+

+

where P ( j ) is a form of order j , vanishing for y = x1= x2= ... = x, = 0. This is deduced in a well-known manner from the forms U ifor ) which i c j . These equations are always possible and perfectly determined. As we assume that a < m, for j < a 2 we obviously have

+

* The original manuscript has ( j > 3) instead of ( j 2 3) (Editor's note).

Stability of Motion

40

In this case P 3 )= -2yY"). We see from here that V 3 is ) of the form U ( 3 )= y(L,x Lzy Q), where L1 and Lz are linear forms, and Q a quadratic form of the quantities x, . According to this it is easy to prove, that for j < u 3 all the U ( j )are of the form

+

+

+

u(i) = v(i) + w(i),

(21)

where Y ( j )and W ( j )vanish for x1 = x2 = = x, = 0, and W ( j ) does not contain terms of an order lower than two with respect to the x, . As a matter of fact, if what was said above takes place for all the values of j smaller than some integer I < u 3, then, as one sees from (20), the form PI), and therefore V'), is of the same type. Considering the values o f j larger than a + 1, in order to obtain the expressions of P ( j ) ,we have to replace, in Eq. (20), X ( " ) by + $ a + l ) , Y('")by -xm Y ( m )and , to throw away from the result all the terms which are independent of the quantities y , x s . According to this, it is easy to deduce that all the forms U") for which j < m 1, are also of the type (21). Hence, for P('"+')we have the expression

+

+

+

+

p(m+l)

= 2y(x"

- Y(m))-

aU(m-*+l)

ax

axa+'

From here we conclude that U('"+')is of the form

This type of the form U ' j ) for higher orders is of no interest to us. For this reason we will not deal with its determination. Assuming that j > m I, consider now the expression

+

Investigation of One of the Singular Cases of Theory of Stability of Motion

41

+

It is easy to see that for j < m a + 1 it does not contain terms independent of the quantities y, x,, for obviously, there are no such terms in the expression in the first line. The expression in the second line may contain such terms only in the case when there are values of i, satisfying the inequalities: i > m,j - i 1 > a 1, from which it follows that j > CL m 1. Thus, we find

+

+ +

if j < m + a

Q'" + a + 1 '

+

+

+ 1.

Moreover, it is easy to see that P ( m + a +l) 2ax" a . As a consequence we find +

+

au

au

+y + c x,a+ (uxa+l+ ax x S

au

au

X ) - - (Xm - Y ) ax dY

Hence, the required term is 2axm+a+1 . Moreover, the function U is of the form:

u=-

m + l

xm+'

+y

u,x,

+ c u,,x,x, + R ,

where u s ,us, are entire functions of x,y , x, , vanishing for x = y = x 1 -- x 2 = = x, = 0, and R does not contain terms of order lower than m 2. Defining U in this way, let us construct the function V = y 2 U 2aym+' x W, where Wis a quadratic form of the quantities x, , satisfying the equation

+

+ +

+

In this case, according to Eqs. (18) we have dV - 2u(xm+a+ 1 + ym+a+ 1 -_ dr

+ X I 2 + X Z 2 + '.. + XnZ) + s,

42

Stability of Motion

where

+ 2a(m + a)ym+a-lx(Y - xm)+

aw c x,ax, -

s=l

contains only terms of the following three kinds: (a) of an order higher than m a+ 1 (a > 0 and m > 1); (b) linear with respect to the quantities x, with coefficients of an order higher than m with respect to x and y , or (c) of an order not lower than two with respect to the quantities x, with coefficients vanishing for x = y = 0. Since by assumption CY < m and, therefore, 2(m 1) > m a + 1, we see from here that if m a 1 is an even number, then dV/dt is sign-definite. Its sign is identical with that of a. This fact occurs, thus, each time when for c( < m,a is even, for m is supposed to be odd. Since W is a sign-definite form, opposite in sign to that of a, the function V is positive definite when a < 0. But the function V can also take positive values for a > 0. Hence, according to the principles presented in the work previously mentioned, we must conclude that for a > 0 the unperturbed motion is unstable, and for a < O it is stable. This conclusion was drawn from two assumptions : (1) a < rn and (2) a is even. If at least one of these conditions is not valid, the conclusion is no longer true. Moreover, in the cases when a is an odd number, one cannot use the same method. Hence, in order to investigate the remaining possible cases, we must use methods of a different sort. Our analysis requires another classification of all the possible cases, which is different from the classification that follows from the previous considerations. Namely, we have to divide all the cases into the following three categories: (1) a > (m- 1)/2, including the case when after performing the substitution shown in page 36, all the first-order terms in y disappear from the

+

+ +

+

+

Investigation of One of the Singular Cases of Theory of Stability of Motion

43

expression of Y ; (2) a < (m - 1)/2 (this case is possible only for m 2 5 ) ; and (3) a = (m- 1)/2. We shall consider these three cases successively. 11. As in the previous paragraphs, we shall suppose g = - 1. Let us begin with the case when a > (m - 1)/2, or when after performing the substitution considered in page 36, the first-order terms in y disappear from the expression of Y. We shall consider first the case when for y = 0, Y becomes of function of x only, -xm +glxm+' g2xm+ ..., and for x = y = 0, X equals zero. According to our assumptions, X vanishes also for y = x1 = x2 = ... = x, = 0. Consider the transformation of system (1) by the substitution from page 17. Suppose therefore, that

+

Y

= -xm

+

+ g1xm+1 + g2xm+2 + ... + y(ax" + u1xa+1 + ... + Y'),

where Y , vanishes for y = x1 = x 2 = ... = x,=O and that for x1 -- x2 = .-.= x, = 0, the functions X , contain x to powers not lower than (m I) in the terms independent of y. In the first-order terms in y they do not contain x to powers lower than a 1, if a < m ,and lower than m,if a 3 m . Since m is odd, we let m = 29 - 1 where g is an integer larger than 1. Introduce thereafter in our equations instead of x and y the variables r and 9, by putting

+

+

x = rC(9),

y = -r4S(9)

(22)

where C(9) and S(9) are functions of 9 defined by Cy9)

+ qS2(9) = 1,

with the conditions C(0)= 1, S(0) = 0. From these equations we deduce dC(3) -- - C'(9) = -S($), d9

dS(9) - S'(9) = C Z 4 - ' ( 9 )

d9

(24)

44

Stability of Motion

and

By these equations, the functions C(9) and S(9) are perfectly determined for all the real values of 9, as well as for all the complex ones in which the coefficient of v'? does not exceed some sufficiently small bound. These functions are periodic with the period

If we like, we can replace equalities (25) by more definite relations. Namely, if we introduce the angle cp by

supposing that the square root is positive for real values of cp, we have sin cp

C(9) = cos cp,

(1

+ cos2 cp + *.. + cos2q-2 (p)1/2.

To the values of cp: 0, 4 2 , n, 3x12, 2n, 5x12, ... correspond the values of 9: 0, w/4, 012, 3014, o,5014, .... By these formulas, the functions C(9) and S(9) are uniquely determined for the real values of cp, and therefore, of 9. However, by these formulas they are uniquely determined also for all the complex values of 9 in which the coefficient of v ' r l does not exceed some bound L. From these formulas follows @(cp

+ kn) = O(cp) + 2 k 0

for each integer k. Further we have

@(;

+ .) + @(;

-

6)

.)2 0 ( 3 =

Investigation of One of the Singular Cases of Theory of Stability of Motion

46

Since O(n/2)= 4 4 , we get from here =(-l)kC(9),

S

Further we have C(-9) = C(9),

C(8

+);

=

-C(;

S(-9) = -S($), -9 ),

s(8 +);

=

s(;

-9 ).

These conditions satisfied by C(9) and S(9) remind one of the functions cos9 and sing, into which they get transformed for q = 1. The smallest of the values of q, however, which we must consider, is 2. In this case our functions become elliptic functions. They are also elliptic functions for q = 3. Let us consider in more detail these two cases. In the case q = 2 we have

whence cp = am 9(of modulus 1/45). Thus, for q = 2 our functions are transformed into elliptic functions C(9) = cn 9,

of modulus 1/2/2. In the case q = 3 we have:

Putting

S(9) = sn 9d n 9

Stability of Motion

46

where k = (2 - $)/4. (2 - J 3 ) " 2 / 2 ) ] . We have further cos2q

=

From here $ = am 29/

*

+ cos + (1 - JS)cos*

1

1

+J3

I

+ C O S ~ + COS"

9=

, sin2cp =

$ [of modulus

J3(1 - cos *)

1

+ JS+

(1 - J3)cos$'

12(1 - k 2 sin2 $)

[l

+ $ + (1 - J3)cos *y

Thus, our functions are defined by the following formulae:

(S9) =

23(1- k 2 sin2 t,b)'/2 sin($/2)

(c0s2(*/2> + J3 29

'

2-J3 k 2 = -, 4

C(9) and S(9) are therefore elliptic functions when q = 2 and q = 3. For q = 2 they are single-valued for all the values of 9.

Investigation of One of the Singular Cases of Theory of Stability of Motion

47

For q = 3 however, they have branch points, in which they become infinite. These points are determined from the equation 29 1+J3 cn= -___ I-JS' These are the only points at which our functions become infinite. In general, for real 9, C(9) varies between the limits - 1 and 1. The function S(9) varies between the limits -l/dq and +l/dq

73

+

12. By using substitution (22), we deduce from the equations

_-

dx -y+X, dt

!iLY, dt

the following relations

from which, according to the expression for Y written above, we get dr - = -rqS(9)[glrC2q(9) + dt

+ rS2(9)[araCa($)+ ... + Y,] + c2q-'(9)x,

+ C(9)S(9>[araC"(9)+ ... + Y,] - qS(9)X-r . From the assumptions about Y , and X , the right-hand sides of these equations are holomorphic functions of Y, xl, x2, ... , x, with coefficients periodic in 9. Moreover, as a consequence of CY > q - 1, these equations have the form dr dt

- = rq+'R1 + rR,, (28)

48

Stability of Motion

where R1 and O1 are holomorphic functions independent of the quantities x,. R2 and 0 , are holomorphic functions of r, xl, x 2 , ..., x,, v a n i s h i n g f o r x , = x 2 = . . . = x, = 0. Performing the same substitution in the equations

we find that all X,are transformed into holomorphic functions of the quantities r , x, with coefficients, periodic in 9.In these functions, the terms independent of the quantities x,, do not contain r at powers lower than 2q. Our problem is reduced to the investigation of the stability with respect of the quantities r, x,. According to the first equation of (28), r is either identically zero, or it conserves the sign of its initial value, as long as the quantities (rI, Ix,I are sufficiently small. Without loss of generality, we can limit ourselves to the assumption r > 0. Consider now the system of equations that is obtained from (28) and (29) by eliminating dt. In this new system we consider the quantities r, x, as functions of 9. We shall endeavor to satisfy this system by series of the form

where c is an arbitrary constant and d'), u:') are functions of 9 independent of c. They are either periodic with the period o,or are represented in the form of finite sums of periodic and secular terms. We call secular, the terms of the form gkq(9),where k is a positive integer and 4 3 ) is a periodic function of 9 with period o,defined and continuous for all the real values of 3. It is easy to see that our equations can always be satisfied (at least formally) by similar series and that, moreover, if we impose the condition that all u ( ' ) vanish for 9 = 0, these series are perfectly determined.

Investigation of One of t h e Singular Cases of Theory of Stability of Motion

49

Indeed, performing substitution (30) we have r4+1R1 fr-1

=

+

+ r4@1 + @ 2

u(4+1)c4+1

...

+ u(4+2)c4+2 +

- c4-1 + J m C 4 + v ( 4 + 1 ) c 4 + 1

-

+

x,= u:2q)C2q + U:2q+1)C2q+1 +

... (S

=

1, 2,

...,!I),

where U ( ' ) ,V ' ) ,U,") are some known rational entire functions of the quantities d i ) ,us.') with coefficients periodic with respect to 3. These functions are independent of c. As a consequence, we get for the determination of the functions d i ) ,u : ~ )the , following equations:

( s = 1 , 2 ,...) n).

Obviously, the functions U q + ' U ) ,: 2 q )do not depend on the quantities d i ) ,usi), and therefore, they are completely known. The functions U ( 4 - l ), V 4 ' - ) depend only on those d')for which i < 1. Moreover, for I < q 2, these functions and the function V Z q - l ) do not depend on the quantities ~ 5 ' ) . For I 2 q + 2 they can depend only on those u;') for which i < q 1- I, while V ( q + ' - 3certainly ) does not depend on the quantities u : . ~ + ' - ~ ) . Finally, the functions Ui') (for which always I2 2q) can only depend on those u ( ' ) for which i < 1 - 2q 2 and on those ~ 1 . ~ ) for which i < 1- 1. +

+

+

+

+

Stability o f Motion

50

We see from here (1) that the functions u ( ' ) ,uj') can be calculated in the following order: u ( ~ )d, 3 )... , , u ( q + l ) the , group of functions u j Z q ) , u ( q + ' ) ,the group of functions ~ ( q + ~ the ) , group of functions u:'q+'), etc. (2) that after finding all u : Z q + jfor ) which j < i and all u ( j )for which j < q + i + 2, the functions u $ ' ~ + 'are ) completely determined, and moreover they are periodic if all their predecessors are periodic; ( 3 ) that function u ( ' ) ,when all its predecessors are found, can contain only one additional arbitrary constant, and therefore is completely determined whenever one imposes the conditions that this function vanishes for 9 = 0; (4) that if among u ( ' ) , u$')there are nonperiodic functions then there are such functions already in the set u ( 2 ) u ( 3 ) u(4) >..., (31) and that if the first nonperiodic function among them is d k )then , it is of the form 3

U(k)

2

=

v

+ h9,

(32)

where v is a periodic function with the period o and h is a constant different from zero. If one does not impose the condition that all u ( ' ) vanish for 9 = 0, it is easy to see that the arbitrary constants that can enter into the expressions of the functions d'), u:') cannot influence the numbers k and h. Suppose that in the set of functions (31) there are indeed nonperiodic functions and that the first of them u(")is given by Eq. (32). In this case, if by u6" for i < 2q, we mean zero, then we can affirm that all u:') for which i < q k - 2, are periodic. Supposing that all d'), u:') are defined so that for real values of 9 they are real, put r = z + U ( 2 ) Z Z + u ( 3 ) 2 3 + ... + u ( k - l ) Z k - l + v Z k ,

+

and x,= z,, if k < q

./,= u ( 2 d z 2 4 S

if k

q +2.

+

+ 2,

u(24+1)z24+' S

+

... + u s( q + k -

2)zq+k- 2

+ z,,

51

Investigation of One of the Singular Cases of Theory o f Stability of Motion

Introduce afterwards in our equations instead of the variables r, x,,the variables z , z, .Thence, if we put the transformed equations in the form

(s = 1, 2, ... , n), (33) d9 - = 0, dt

(34)

then 2, 2, , 0 are holomorphic functions of the quantities z, z,, vanishing for z = zl = z 2 = ... = z, = 0, and they have periodic coefficients with respect to 9.Moreover, according to the properties of our transformation, function 2 is such that for z1 = z 2 = ..+= z, = 0 the lowest power of z appearing in it is k f q - 2 and the corresponding coefficient h is constant. The functions 2,, in which in general no terms lower than second order appear, for z1 = z 2 = ... = z, = 0 do not contain z to powers lower than q k - 1. Finally, the functionais transformed for z1 = z 2 = ... = z, = 0 into a series expansions in increasing powers of z , in which the lowest power of z is (q - 1). In addition to these, the functions 2, Z , , 0 are such that their expansions in powers of the quantitiesz, z, , when they are different from zero but still sufficiently small, are uniformly convergent for all real values of 9. We shall say that such functions are uniformly holomorphic for all real values of 9. Our problem is obviously reduced now to the problem of stability with respect to the quantities z, z, . The first of these quantities can be made to satisfy the condition z 3 0.

+

13. Let* = h z k + 4 - 2 + p ( 1 )+ ~ ( 2 +) ~

+

... + ~ ( k + q - Z ) ~ k + q - J + R ,

+

where R does not contain z to powers lower than k q - 1 in the terms independent of z,. In the terms linear with respect to

* This equation in the original manuscript reads Pckf q P k+*- (Editor’s note).

3,

instead of

Stability of Motion

52

+

q - 2. P iare ) linear forms in the quantities z, with coefficients periodic with respect to 9. Further, let 2, = P:”z +P:’)z2 ... R, , where f ‘ : i ) are linear forms in the quantities z, with coefficients periodic with respect to 9. R, does not contain terms linear in z, and the terms independent of these quantities do not contain z to powers lower than k+q-l. Finally, let 0 = z 4 - ’ + 0 , z 4 +Ozz4+’ ... T, where O,, O2 , ... , are some periodic functions of 9. T vanishes for z1 = z 2 = ... = z, = 0. Consider a quadratic form W of the quantities z,, that satisfies the equation

z, it does not contain z to powers lower than k

+ +

+ +

c

aw

n

(PSlZ1

s= 1

+ PSZZZ + ... + PsnZn) az, - = h(Z12 +:z + ... + Z,Z),

and is definite with a sign opposite to that of h (see “The General Problem of Stablity of Motion ”). We shall also consider the linear forms of the quantities zs, U“ ), U 2,) Vk+¶-’) with coefficients periodic with respect to 9, determined successively from the equations 7

and in general for j


Investigation of One of t h e Singular Cases of Theory of Stability of Motion

63

and for j 3 q

Put now I/ =

... + ~ ( k + q - z ) ~ k + q - + 2 W

+ ~ ( 1 +) ~ ~( 2 ) + ~ 2

and take the total derivative of this function with respect to t in virtue of Eqs. (33) and (34). This derivative is obviously of the form

where S is some expression of the type

in which ZI, us, are holomorphic functions of the quantities z , z,, vanishing for z = z1 = z2 = ... = z, = 0. They are uniformly holomorphic for all the real values of 9. We see from here that whatever the real function expressing 9 in terms of t is, for z 3 0 the derivative of the function V is signdefinite having the same sign as h. Considering in V the variable 9 as a given real function of t, we can affirm that V is a function having an infinitesimal upper bound, and that for z 3 0 it is positive definite when h is negative. We thus conclude that for positive h the unperturbed motion is unstable and for negative h it is stable. In the last case any perturbed motion sufficiently near to the unperturbed one will tend to it asymptotically. 14. Let us make some remarks about k and h. We have imposed on z the condition z 3 0. However, with the same right we

54

Stability of Motion

might impose the condition z GO as well, and we would get the same result. If, however, we consider the function

v1 --

+ ~ ( 1 +) ~

+

... + ~ ( k f q - 2 )Z k + ¶ - 2

+(-l)k+q-lW,

where U"), Ware the previously used functions, we obviously find

where S , is an expression of the same type as S. The derivative d VJdt is obviously sign-definite having the same if z satisfies the condition z GO. But under sign as (-1)"'"-'h, the same condition, the function V , is negative definite, if (-l)k + ¶ - ' h> 0. Thus, when this last inequality holds, the unperturbed motion is stable. And because this is true for h < 0 we must conclude that (-l)k < 0. Thus, we arrive at the conclusion that k q is even. Moreover, we can obtain some inequalities, giving a lower bound for k. Return to Eqs. (27) and (28). By assumption, a is larger than q - 1. Suppose that a 3 2q - 1. Then, if we put

+

+ Ri1)r + R i Z ) r 2+ ... , @ 1 -- @p + @ p r+ @ y r 2 + ... , R,

= RjO)

where R(",OF) are functions of 3, independent of r, we deduce easily from the expressions in the right-hand sides of Eqs. (27) that all Ry)for which s < q - 1 are of the form S(3)P(3), P(3) denoting a rational entire function of C(3). At the same time, all OC,.)for which s < q - 1 are of the form P(3). From here it follows that all U ( ¶ + ' - - lfor ) which I < q 1 are entire functions of the quantities u ( ' ) with coefficients of the form S(3)P(3).Also the V(q+'-3)forwhich I < q 2 are entire functions of the quantities di), with coefficients of the form P(3). Thus, taking into account that S(3) = - dC(3)/d9, we must , 3 ) ,... , u ( ¶ ' are necessarily rational entire conclude that d 2 ) d functions of C(9), and therefore periodic. For this reason, k cannot be smaller than q 1. And because k q is even, k is necessarily larger than q 1.

+

+

+

+

+

Investigation of One of the Singular Cases of Theory of Stability of Motion

55

+

Thus, if a 3 2q - 1, then necessarily k > q 1. Suppose now that a < 2q - 1. Then, similar to what was shown previously, we conclude that d 2 )d, 3 )... , , u ( ~ - ~ +are ’ ) entire rational functions of C(9), and therefore k is not smaller than CI - q 2. Thus, if. < 2q - 1, k > M - q 1. Moreover, if a is odd, then necessarily k > CI - q 2. Suppose, on the contrary, that a is even. Since RP- q , will necessarily be of the form

+

+

+

+

Rja-q) = aS2(9)Ca(9) S(W(9),

+

if we replace the quantities zdi) for which i < M - q 2, in the expression of U(“ I, this function also becomes an expression of the form aSz(9)Ca(9)+S(9)P(9). Therefore, u ( ‘ - ~ + ’ is .) obtained from an equation of the form +

du(a-q + 2 )

d9

+ s(GjP(9).

= as2(9)Ca(9)

Multiplying the right-hand side of this equatian by d9 and integrating from 0 to o,we get a

s:

S2(9)Ca(9)d9,

Since the integral is not zero, we conclude that

The constant h obtained here has obviously the same sign as the constant a. We thus conclude that if a < 2q - 1 and a is even, then the unperturbed motion is stable for negative a, and unstable for positive a. This result, which was obtained here from the assumption that M > q - 1, confirms the result of page 42, which was arrived at without this assumption. 15. Consider now the case in the series (31), where all the functions appearing are periodic no matter how far we go. If our system of differential equations was of second order and thus could be reduced to the form (28), where R, and 0,would be identicallyzero, thentheseriesr = c + u(’)c(’)+ d 3 ) c 3 + ... ,formed

56

Stability of Motion

on the assumption that all u ( ' )vanish for 9 = 0 would be absolutely convergent for sufficiently small (c( and would represent the general integral of the equation dr r2Rt -=d9 1 + r O , '

Thus when all the functions d i )are periodic, the stability is clear. If, however, our system is higher than of second order, the case considered presents very serious difficulties, which can not always be overcome. We want to point out what these difficulties are and the problem to which they lead in the case considered. When all u ( ' )are periodic functions, so are also all u!). Thus, the series (30) are periodic in the present case. However, these series are in general not convergent, no matter how small the modulus of the constant c is and whatever the arbitrary constants are upon which the functions di), uai) depend. We can indicate only a few particular cases for which these series are surely convergent. Such is the case, for instance, when all X , vanish for xt = x 2 = = x, =. 0. Such is also the more general case when the system of equations 1..

(y

+ X ) & + Y aY ax

= pslxl

+ ps2x2 + *.. + P,"X, + x, (s = 1 , 2,

... , n )

can be satisfied by holomorphic functions of the variables x and y , vanishing for x = y = 0. If for 9 = 0 we put all d iin ) series (30) equal to zero, these series define for sufficiently small CI some periodic solution of the system of differential equations derived from (28) and (29) by the elimination of dt. As we shall see further, the unperturbed motion is in this case stable. Without considering other similar particular cases, we want to show how one has to treat our equations in the general case. Denoting by c a positive constant sufficiently small in magnitude, put Y = c( 1 p), x , = cBts(s = 1, 2, ... , n), where p is an arbitrary

+

57

Investigation of One of the Singular Cases of Theory of Stability of Motion

integer satisfying the condition g < p < 29. Instead of Y, x, we introduce in our equations the variables p , 5, and then eliminate dt. The equations obtained in this way will obviously be of the form dP - = CP, d9 d5 d9 - P s 1 5 1

c4- 1 s-

+ ~ s 2 5 2+ ... + P s n t n + zs

(s = 1,2, ... 9

(35) ?I),

where P, Es are some holomorphic functions of the quantities p , tl, t 2 , ... , 5, and c with coefficients periodic with respect to 9. All Z, vanish for c = 0. Consider, instead of (35), the following system: dP d9

- = CP

+ cp(c, A), (36)

where A is some real parameter and cp(c, A) some function of c and A, still unknown. Let us show that cp can always be chosen so that the system (36) has a periodic solution, representable by the following series expansions in positive integer powers of c: p =d1)C

+ d 2 ) C Z + d 3 ) C 3 + ... ,

5, = Uj2q-fl)czq-P + u s( ' q - f l + ' ) c ' q - f l + '

+ ...

(s = 1 , 2,

(37)

... , n ) ,

Here di), u j ' ) are periodic functions of 9 with the period w. The dependence of these functions on the parameter A is known. They do not depend on c. For convenience we shall denote them by ~ ( " ( 9 ,A), v $ ~ ) ( SA). , Suppose that for each i, ~("(0,A) = 0. We show that cp(c, A) can be represented as a series expansion in positive integer powers of c d c , A) = cp,(4c

+ cpz(A)c2 + ...

9

(38)

and that for I CI sufficiently small, this series is uniformly convergent for all real values of A.

58

Stability of M o t i o n

The coefficients q 7 , ( A ) of this series are functions of A which in general cannot be expanded in series of positive integer powers of A. These functions can be represented by such series asymptotically for values of A tending to zero. The derivative ~~'(1) of these functions with respect to A has the same property. Suppose that from (37) one gets the following expansions : p == I/@) + I/""C + 1 / ( 2 ) c 2 + ... ,

zs= c 2 q - p ( v p + V:"c + v / ! 2 ) c 2+ ...} (s = 1, 2, ... , n), where Vi),V/!')are known rational entire functions of the quantities ZI('), u:') with coefficients periodic with respect to 9. Moreover, the functions V O )V:') , do not depend at all on the quantities u ( ~ )u, : ~ ) ,whereas the functions Vi), V : i )depend only on those u ( ' ) 00( 2 q - p + 1 - 1 ) for which 16 i. Trying to satisfy the system (36) by series (37) and (38), we obtain systems of equations of the form 7

( s = 1 , 2 , ... , n ) ,

where we put for brevity ,u = 2q - p - 1. These systems of equations give us the possibility of finding all our functions successively for i = 1, 2, 3 , .... In this way, because of the condition ~("(0,2 ) = O all our functions are completely determined. If we admit that all ZI('), for which 1 < i are already found, the functions di),'piare obtained from 9

u ( i ) = J0v(i-1)

d9

+ qi(A)$,

(39)

We can remark that ql(l)is always identically zero. The functions u ~ P + ~are ) found also in a completely determined fashion.

Investigation of One of the Singular Cases of Theory of Stability of Motion

59

If we assume that all p s c are zero with the exception of the following P 2 2 = XZ ... , P n n = X n P 1 1 = XI, 9

p21 =

p 3 2 = c2 >

3

...

7

pnn-1

(41)

= On-19

then these functions are found by the formulas

( j = 2, 3, ... , n). These formulas are always meaningful, as J. and co are real numbers, whereas x,,as roots of Eq. (2), have negative real parts which are different from zero. The general case can always be reduced to the one shown, by introducing as unknown functions, instead of the quantities t,, some linear functions of them which have constant coefficients. As a consequence, the quantities v6”’ i , are in the general case some linear combinations of the right-hand sides of the equalities (42). The formulas in (42) are of use for all real values of A which are different from zero. However, these formulas can be replaced by simpler ones, which depend on the knowledge of the sign of A. For if A < 0 we have in general:

and for A > 0

We can therefore replace (42) by the following:

Stability of Motion

60

+

where in the lower limits of integration " " corresponds to a negative A and " - " to a positive A. For A = 0 these formulas must be replaced by =

@+i) 1

_ _1

p4i-1) 1

9

x1

q + i )

=

1 __

{Oj-

p;.Y+;)

+V

(45) y ) }

( j = 2, 3, ... , n ) .

x j

By defining functions d i ) , in this way, it is easy to prove the convergence of the series (37) and (38) for sufficiently small values of JCI. Let -Al, - A 2 , ..., -An be the real parts of the numbers xl, x 2 , ... , xn. Suppose further that d i ) , are some upper bounds for the moduli of the quantities u f ) , u:~),suitable for all the real values of 9 and A. W j i )are supposed to be the expressions, obtained from V i ) ,V j i ) by replacing the quantities u ( * ) , ui') by MI('), wj') and replacing the coefficients of the expansions of the functions P,Zsin powers of the quantities p, 5, , c, by some upper bounds of their moduli, suitable for all real 9. These last bounds we will suppose to be such that the series representing the functions P,zSremain convergent for sufficiently small ( p ( , /&,I, ( c ( after replacing the coefficients by the upper bounds shown before. After this substitution they become holomorphic functions f(p,

5 1 9 t29

...

9

tn,

c), f X p ,

51, 527

*..

9

t n , C)

(46)

of the quantities p, 5,, c. Of these functions, the last n vanishes for c = 0. Suppose that all p,,, not appearing in series (41) are equal to zero. Using formulas (39), (40),(44), and (43, we obviously have Iqi(n)l <

(47)

V i - I )

for all real A, and, therefore, we can take w(i) A.W(?+i) J

J

= 20w(i-1)

=

lgj-llEJy+;)

A

1

w(p+i)

1

+ w(!-1) ,

= w(i-1) 1

9

( j = 2 , 3 ) . . . )n).

These formulas can be used for all values of i, beginning from

i = 1.

Investigation of One of the Singular Cases of Theory of Stability of Motion

61

By virtue of the mode of generation of the expressions FVi), WLi),the series

w(')c+ W ( Z ) C 2 + ... , c p + l { w : o ) + Wj'k + w p c 2 + .*.} w(0)+

(s = 1, 2,

... , n )

are the expansions in powers of c of the functions (46), after replacing in their expressions the quantities p, 5, by the series

Therefore the series (48) are the expansions in powers of c of the quantities p, 5, which satisfy the equations p = 2 ~ C f ( p , 5 1 , 5 2 , ... Y 5.,c>, 4 5 1

=f,(p,

tl, 52,

Aj5j=luj-115j-l

*..>

5fl,

c>,

+fj(P,51,52,...,5n,c>

( j = 2 , 3 , ...,n>

and vanish for c = O . It follows that these series are absolutely convergent for sufficiently small IcI . We conclude from this, that the series (37) are uniformly convergent for sufficiently small for all real values of A. Moreover, we deduce from (47) that series (38) is also uniformly convergent for all real values of A. Thus, if q(c, A) is defined by the series (38), then for sufficiently small (cl, the series (37) represent some periodic solution of the system (36). Let us conclude that it follows from our analysis that, assuming p to vanish for 9 = 0, the series (37) and (38) are the only ones to satisfy the conditions of our problem. In order to prove the convergence of our series we have considered some linear transformation of the system (36). Returning to the initial system (36) in which the coefficients of the expansions of the right-hand sides are real functions of 9, we see that for this system, all the coefficients in the series (37) and (38) are real functions of 9 and A.

(cI

Stability of Motion

62

16. Consider some properties of the functions Cp,(A),

V(’)(9,

A), u(/’+i) s 4.

(49)

($3

We want to prove (I) that these functions together with their derivatives with respect to 9 and ;1 of an arbitrary order are defined and continuous for all real values of 9 and A*. These functions are periodic with respect to 9 and bounded with respect to 1; (2) that these functions together with their derivatives with respect to 9 and 1 of an arbitrary order can be “ represented asymptotically ” (in the sense of PoincarC) by series expansions in integer positive powers of i with coefficients which are defined, continuous and periodic with respect to 9;(3) that if $,(9) A2$2(9) ... is a series which represents asymptotically one of these functions, then

+

+

+

is a series representing asymptotically its mth order derivative with respect to A, and nth order derivative with respect to 9;and (4) that for sufficiently small IcI, the series

+ ‘pi( h ) C 2 + ‘p3’(h)C3 + . * .,

‘pl’(h)c

are uniformly convergent for all real values of /z and, therefore, represent the derivatives with respect to A of the functions Cp(G

4,

V(’)C

+ V(2)CZ + V ( 3 ) C 3 + ... ,

Ve+l)cIc+l

+ vja+2)ca+2 +

...

* In the cases we deal with we suppose that the values of 8 and A are always determined, and therefore finite.

Investigation of One of the Singular Cases of Theory of Stability of Motion

63

We will begin with the first three points, in which essentially the properties of our functions are expressed. These properties obviously hold for the functions pl(A), ~(”(9, A), which do not depend on A (the first is identically zero) and can be proved easily for the functions v : ’ + ~ ) ( $A). , Therefore, if we assume that they hold for all the functions (49), for which i is smaller than some integer 1, and if we prove that in this case they hold also for i = 1, then these properties are proved in general. However, by the formulas given above for the successive calculation of the functions (49), the proof of what we just said is reduced to the proof of the fact that iff($, A) is a function having the three enumerated properties, then the functions

also have these properties, x being an arbitrary constant with a negative real part. The sign of the lower limit of the last integral must be taken as opposite to that of A. From these formulas follows directly the validity of the first statement for the functions F(A)and F(9, A). What remains therefore is to investigate only the function @($, A). Integrating by parts we easily find

This formula is valid for every A, different from zero, whatever the nonnegative integer m is.

64

Stability of Motion

Denoting by O(9, 0) the limit to which O(9, A) tends when A tends to zero, we get from the last formula 1 w,0) = --f(9, x

O),

which corresponds to the formulas in (45),as the expression

tends to zero when A tends to zero. We convince ourselves that this is true by remarking that this expression does not exceed the quantity

where x' is the real part of x and M is an upper bound of Iam+'f(9, A)/~9m'11for all real values of 9 and A. We can affirm therefore, that @($, A) is defined and continuous for all real values of 9 and A. Moreover, it is bounded and periodic with respect to 9. Consider now its derivative. By putting in (51) rn = 0 we get

whence

Remarking that the function @ is a periodic solution of the equation

and that, therefore, aO/aA is a periodic solution of the equation

Investigation of One of the Singular Cases of Theory of Stability of Motion

65

we find

and thus

This formula can also be obtained directly by taking the derivative of @(S, A). The expressions (52) and (53) are of the same type as @(9, A), and therefore the functions

w s , 2) as

and

a@($, 4 an

~

are defined, bounded and continuous. The continuity of these functions for A = O may be doubtful, as formulas (52) and (53) were deduced supposing A to be different from zero. In order to prove the continuity for this case it is sufficient to show that*

and that lim{8@/8A}a=

n=o.

= the

derivative of the function @ for

This can be proved in the following manner. We find from (52) l j m ( g )a = o =

1 w s , 0) x as '

and from 1

a($,0) = --f($, 0 )

x

* The symbol lim{a@/a6},= lim(a@/&) (Editor's note). A-0

corresponds in modern notations to

Stability of Motion

66

we find

whence follows equality (54). Further we find from (53)

However, equality (51) for m

=

1 gives

whence

By taking here the limit for l - = O , we get the expression of the derivative a@/aI corresponding to I = 0, which coincides with (56) [in virtue of (55)]. Thus the continuity of the first derivatives of Q, was proved. Using now formulae (52) and (53), we get expressions for the second derivatives of this function, similar to the expressions (52) and (53). From these expressions it follows that the derivatives are defined, continuous and bounded. In this way we prove successively that all the derivatives of a) have these three properties. Thus, point (1) can be considered proved. Consider now point (2). If the series a,

+ a,x + a2x* + 4 3 x 3 + ..'

(57)

(convergent or not) is such that for each positive integer m, the equality f ( x ) = a,

+ a1x + ... + amxm+

Xm+lqm(X)

Investigation of One of the Singular Cases of Theory of Stability of Motion

67

is valid, wheref(x) and (P,(x)are some determined functions of x, finite for x = 0, then following the terminology of PoincarC, the series (57) represents the function f(x) asymptotically f o r x = 0. Suppose that the function f ( 9 , A) is represented asymptotically by a similar series with coefficients which are defined, continuous and periodic with respect to 9. We suppose therefore that for each positive integer rn

f ( g , A > =fo(s>+ fi(s>A + ... + . f m ( s > A m

+ A m + l ~ m ( 9 ,A>>

wherefo(9), fl(9), etc., are functions of 9 which are defined, continuous and periodic. (~"(9,0) is finite. From here we deduce directly that F(A) and F(9, A) can also be represented by asymptotic series expanded in powers of 1,. If we suppose further that the derivatives of the function can be represented by similar asymptotic series, we deduce that such a representation is also possible for the derivatives of F(A) and F(9, A). From (51) follows then the possibility of such a representation for the function (D(9, A), as well. But the derivatives of Q, are determined successively by formulas which are similar to the one which gives 0.Hence the same is also true for these derivatives. Thus, point (2) can be considered as proved. Finally, the validity of point (3) for the functions F(A) and F(9, A) follows directly from their expressions, if we admit that it is valid forf(9, A). For the function @(9, A) however, its validity follows from formula (51) which by (52) can be written in the form

Let us now prove point (4). Consider in Eqs. (36) the quantities 4, as functions of A. Take the derivative of these equations with respect to A and put

p,

Stability of Motion

68

We find

+

ass c -5.’-n

j=l a t j

at,

as

( s = 1 , 2 ) . . . )n).

Replace in these equations the quantities p, 5, by their expressions (37) and consider p’, 5,’ as unknown functions of 9, whereas $ is an unknown constant. Then, they become linear equations. However, these equations are of the same type as equations (36) if we do not consider the fact that the coefficients of the expansions of the right-hand sides in powers of the quantities p’, C,,’ c depend not only on 9, but also on 1. From the assumptions on the series (37) it follows that these coefficients are such that the right-hand sides of our equations are holomorphic functions of c uniform with respect to all real values of 9 and 1. Therefore, as for Eqs. (36), we can prove that by a suitable choice of the function $, our equations admit a periodic solution represented by a series expansion in positive integer powers of c. Moreover, there is a unique way of choosing such a series for the function $, for which in the afore-mentioned solution p’ vanishes. In this solution all our series are uniformly convergent for all real values of 9 and 1, if c is sufficiently small. However, from the way our equations were generated, they are obviously formally satisfied by replacing $, p’, 5,’ respectively by series (50). Since, by assumption, in the series for p’ all the coefficients of the powers of c vanish for 9 = 0, our equations can be formally satisfied in a unique manner by periodic series expansions in powers of c. Thus it follows that the above-mentioned periodic solution is defined by series (50). Thus, these last series are uniformly convergent for all real values of 9 and 1. Therefore, series (50) indeed represent the partial derivatives with respect to 1 of the function q(c, 1)and of (37). In the same

Investigation of One of the Singular Cases of Theory of Stability o f Motion

69

way we can prove that the series (37) and (38) are indefinitely differentiable with respect to A, 9 and c. The series obtained after differentiation are uniformly convergent for all real values of 9 and A, as long as IcI is sufficiently small. 17. We formed series (37) assuming that all u ( ' ) vanish for 9 = 0. We could make a more general assumption, by putting

~("(0,A) =ai,* whereaIi, a, , ... , arearbitraryconstants, satisfying only the condition that the series a,c + a 2 c 2 + a 3 c 3 ... are convergent for sufficiently small 1 , different from zero. Let us see how all our functions vary as a consequence of this assumption. From the way Eqs. (35) were generated it follows that if we put into these equations

+

CI

+ a2cz + ... + (1 + a,c + u2c2 + ...)p, = (1 + a,c + a2c2 + ...)Pts (s = 1, 2, ... , n), c + u1c2 + a2c3 + ..- = c, -

p = a,c

5,

(58)

the result is simply as if the quantities p, t,, c are replaced by p , [,, C in the same equations. Taking this into account, replace in system (36) the function q(c, A) (which corresponds to the periodic solution in which p vanishes for 9 = 0) by the function $(c, A), which corresponds to the periodic solution for which, for 9 = 0, p = a,c a2c2 a3c3+ ... . Afterwards, we perform substitution (58) and at the same time put A(1 + a,c + a2c2 + = 2. (59) System (36) is thus reduced to

+

+

..a)"-'

dp d9

- = cP+ (1 + a,c

x

-

--

dts - pslgl d9

+ u2c2+

...)$(c, A), -

+ p s , t , + ... + psflgfl+ 3,

(s = 1, 2,

... , n),

where P , Z, are obtained from P, 3, replacing the quantities t,, c by P , 5,, c. *Here u' was written instead of ai (Translators' note).

p,

Stability of Motion

70

By our assumption, this system must have a periodic solution, representable by a series expansion in positive integer powers of c, with coefficients depending on A, in which p vanishes for 9 = 0. On the other hand, the same system has a periodic solution

m

satisfying the last condition, if (1

+ u,c + u2c2 + ...)$(c, 2) = p(C,

1).

The series (60) however can be transformed into periodic series expansions in positive integer powers of c, for it is easy to prove that each one of the functions ~ ( " ( 9 ,X), v:"(9, X) can be expanded in such series if (cl is sufficiently small. Thus, we deduce from series (60) the periodic series expansions in powers of c, which satisfy formally the equations obtained from (36) replacing q(c, A) by

Such series are completely determined on the assumption that for 9 = 0, the series for p becomes u,c

+ a2c2 + ... .

(62)

They coincide therefore with those representing the periodic solution giving for p the quantity (62) when 9 = 0. The convergence of this last series for sufficiently small IcI can be proved in a manner similar to that of the convergence of the series (37). Let n

p =

CS =

1 w'"(9,

i= 1

A)ci,

,cws"+i)(9, A)cp+' n

(s = 1, 2,

... , n )

L = l

be the series representing this solution and let $(c,

A) = $l(A)C

+ l j Z ( A ) C 2 + ... .

Investigation

of One of the Singular Cases of Theory of Stability of Motion

The functions di), w:""), with respect to c:*

c w'"(3, m

i=l

A)c'

71

are determined by the identities

$i

= a,c

+ u2c2 + .'.

+ (1 + a,c + a2c2 + ...) c u'"(9, A)?', m

c w6'+')(9, A)C"+i c $i(A)c'

i=l

a)

i=l

= (1

+ a,c + - . ) B

=

Cup(9, A)?:,

i=l

di'2),

m

i=l

m

1 + a,c

+ u2c2 +

Here it is convenient to consider one property of the function d c , 4. Let ql(c, A) be a function such that if it replaces q(c, A), system (36) has a periodic solution representable by series expansions in powers of c, and p vanishes for 9 = 4 2 . Suppose that for 9 = 0 we have in this solution p = a,c

+ u2c2 +

*..

.

Note now that the functions C(9) and S(9) have the property C(9),

Therefore, if we put

s

(y -

1

(- 1)49 = (- 1)?3(9).

w

9=--(-1)49,, 2

r = -rl,

equalities (22) give x = rIC(9,),

y = -rlqS(9,).

From here it follows that the result of substituting (62) in Eqs. (28) and (29)is simply reduced to the replacing of r by r1 and 9 by gl, and therefore, if in system (35) we put c = -cl,

w

9 = - - (-1)491, 2

* In the original manuscript

ts= (-- 1YS, ,

c was omitted (Editor's note).

(63)

72

Stability of Motion

the result is simply the replacing of c by c, 9 by 9, and 5, by f,. Finally, we deduce from here that if we replace in Eqs. (36) the function cp(c, A) by cpl(c, A), and afterwards we perform the substitution (63), putting at the same time

A, = (-

l)q-IA,

the result is equivalent to the replacing of c by cl, 9 by 9,, 5, by by (-l)q-lcpl(-cl, (-l)q-lAl). By assumption, this new system has a periodic solution in which p must vanish for 9, = 0. Thus, by comparison with system (36) we deduce that

5, and cpl(c, A)

(-l)q-lcp,(-c,,

(-1)q-14)

=dc,,

Ad,

whence cp,(C,

A) = (-

1)q-

'p( - c, (-

1 ) q - 1A).

On the other hand, according to what was proved above, we have

From this follows cp(-C,(-l)q-9)

=(-1)

q-l

cp{c

+ a1c2 + ..*, (1 + a,c + -.->"-'A> 1 + a,c + a2c2 + ... (64)

This property of cp was just what we intended to show. The constants a, appearing here are in general functions of A, representable asymptotically by series expansions in positive integer powers of A.

18. Let ~ ( cc,q - - ' ) = cp(c). If for any sufficient small c, cp(c) = 0, then the series (37) with A = cq-' represent some periodic solution of the system (35). To this solution corresponds some periodic solution of the system (28), (29), in which Y, x, are periodic functions of 9, and r becomes equal to c for 9 = 0. In this solution the quantities Y, x, are represented asymptotically by the series expansions in powers

Investigation of One of the Singular Cases of Theory of Stability of Motion

73

of c (30). Let us find the relation between 9 and t corresponding to this solution. Put in the right-hand side of the second equation (28) the expressions of Y, x, corresponding to the periodic solution. As we saw, these expressions are of the form r

=c

+ C’$($,

c),

(s

x, = c2q$,($, c)

=

1 , 2 , ... , n),

where Ic/, $s are known periodic functions of 9 with period w , defined and continuous for all real values of 9,and for all sufficiently small real values of c. We find obviously rq-l+

rqOl

+ O,

=

1

1

+ cO(9, c) ’

where O(9, c) is a function of the same type as $(9, c), $,(9, c). All these functions can be represented asymptotically by series expansions in positive integer powers of c. We find therefore [l

whence 9

+ cO(9, c)] d9 -= dt

+ c ”!

cq-1,

9 0

O(9, c) d9 = c q - ’ ( t - to),

where to is an arbitrary constant. Put c

o

0

0

1 + - J” O(9, C) d9 = h.

Then Eq. (65) takes obviously the form: h 9 + cO(9, c) = c4- ( t - to),

where O(9, c) is a periodic function of 9 of the same type as O(9, c). According to this equation, for sufficiently small IcI, t varies in the same sense when 9 grows, and to a variation of 9 equal to w corresponds a variation o f t of ho/(cq-l).It follows that this equation defines 9 as a function of t, varying in the same sense when t grows and increasing by o each time t increases by hw/(cq-’). Hence by expressing everything by t and returning to the variables x and y , we find that in the case under consideration, system

74

Stability of Motion

(1) will have a periodic solution depending on two arbitrary constants c and t o and having a period with respect to t expressed by the formula

Since q > 1, this period can be made arbitrarily large by taking c sufficiently small.

If necessary we can represent T asymptotically by a series expansion in increasing integer powers of c. If the expression q(c) is not identically zero, it represents some function of c having an asymptotic expansion in positive integer powers of c. Since q(0) = 0, only two cases can occur: either there exists a positive integer k so that the limit lim{

F}

c=o

exists and is different from zero, or, for any given k, this limit is zero. In the first case the series (30) are obviously not periodic and we have shown here how in this case the stability problem is solved. What remains to consider is the second case. One may ask the question whether in this case the function q (c) is identically zero ? Since we cannot prove this, let us consider the hypothetically possible case, when ~ ( c is) a function of c which is not identically zero, although expression (66) vanishes for any k. In this case, all the terms are zero in the asymptotic series representing the function q(c), similarly to what happens to the functionexp[ - l/c2]. The study oftheproperties of q ( c ) is, of course, a very difficult problem. But nonetheless, we suppose it to be investigated and known. We shall meet with one of the following two cases: (1) there exists a sufficiently small positive number c' such that for 0 < c < c' q(c) vanishes nowhere, and (2) for any positive E , arbitrarily small,

Investigation of One

of the Singular Cases of Theory of Stability of Motion

75

the function q(c) vanishes an infinite number of times in the interval between 0 and €. In the first case one finds another positive number cl, such that for -cl < c < 0 the function q ( c ) is nowhere zero. In the second case, for any positive 6, when c increases from - 6 to 0, the function q(c) passes through zero an infinite number of times. This follows from equality (64), which for q(c) takes the following form: q ( - c ) = (-l)

q-l

+

+

+

alc2 u2c3 -..) ' 1 + a l C+ U Z C 2 + ...

q(c

where in the expressions of a,, 1 must be replaced by c 4 - ' . In the first case, among the perturbed motions which are sufficiently close to the unperturbed motion, there are no periodic ones. On the contrary, in the second case there exist periodic motions in any vicinity of the unperturbed motion and similar to those considered above. But they differ from these last ones, for which c might take any small value, in that c must coincide with the roots of the equation q(c) = 0, which are sufficiently close to zero. In this case we have, therefore, some discrete subset of the infinite set of periodic motions, containing also the unperturbed motion. It will be shown that when q(c) vanishes identically and, therefore, there exists a continuous set of periodic motions, the problem of stability is solved in the affirmative. In the case when, for q(c) not being identically zero, the equation q(c) = 0 has in an arbitrarily small neighborhood of zero an infinite set of roots, the problem is so involved that we give up any hope of solving it. 19. Supposing p = 2q- p - 1 = 0, introduce in Eqs. (28) and (29), instead of variables r, x, , variables z, z, , putting I' =

z

m

+ 2 2 1o(')($,

zq-1)zi-l

= @(z,$),

i=l

m

I

x, = z 2 q x o1"(9, i=l

zq-1)zi-l

+ z, = (Ds(z,9) + z ,

(s = 1, 2,

... , n )

76

Stability of Motion

If we denote for the moment the right-hand sides of Eq. (28) and (29) by

$1,

N r , xl, x2, ... , x,, Xs(r, ~

xz

1 ,

@ ( r , xl, x z , ... , x,, 91, 7

9

xn

3

91,

we have from the properties of the functions @, 0,

a@ - - R(@,a,, @z, a3 - @(@,

...,a,, 9)

a,,QZ, ... ,a,,

9)

+ zrp(z),

But from the properties of the function 0 we have @(@, @,, @z

, ... , a,, 3) = 2 4 - 1

+ z4q(z, $),

where q(z, 9) is a function representable for sufficiently small IzI by a series similar to (67). Hence, by our substitution one gets the following equation: d9 - = z'-' dt

+ zqrp(z, 9) + T ,

where T vanishes for z , = z2 = .-.= z, = 0. By the same substitution, we have R(r, x,,xz,... , x,, 9) = R(@, QZ, ... , @, , 9) a function vanishing for z = 0 and also for z1 = z 2 = ... - z, = 0; < I Xs(r, xi, xz,... , x,, 3) = A',(@, @,, QZ, ... , @, ,9) a function which vanishes for z1 = z2 = ... = z, = 0. Since

+ +

dr dt

a@dz az dt

-=--

, @ i , @ z ,..., @ , , 3 ) + R ( @z4-I + z4q(z, 9) [z"-' + z 4 q ( z , 3) + TI

dx, _ _ -d z , +-a@,,dz + X,(@, dt

dt

az dt

z4-I

@*,

..., @,,,$)

+ z4rp(z, 9)

CZ"-'

+ z4q(z, rp) + T I ,

Investigation of One of the Singular Cases of Theory of Stability of Motion

77

or

dz, dx, - -- dt dt

aCD, + d@,ddz -- + - T + X,(a,, 3.z dt a$

@I, 0 2 ,

..., an,9),

we get, therefore, the following transformation of our equations :

dzs _ -P dt

(69)

+

+

S I ~ ~~ s 2 ~ *2* .

+ PsnZn + Zs +f(z>fs(z, 9) (s = 1, 2,

... , n).

Heref(z, 9),fs(z, 9) are functions of z and 9 of the same type as @(z, 9), QS(z, 9). Moreover, these functions have the following property : f(0,9) = 1,

fS(O, 9) = f,'(O, 9) = ..*

=

fyy o , 9) = 0,

where we wrote in general

For constant z and 9, 2 and 2, are holomorphic functions of zl, z 2 , ... , z,, vanishing for z1 = z 2 = ... = z, =O. Moreover, these functions are uniformly holomorphic for all real values of 9 and for all sufficiently small real values of z. The coefficients in these expansions are functions with the same properties asf(z, 9), fs(z, 9). Finally, functions 2, are such that in their first-order terms

78

Stability of Motion

with respect to the quantities zl, z , , ... , z, , the coefficients vanish for z = 0. For our functions it is possible to have an " asymptotic expansion " in integer positive powers of z with coefficients periodic with respect to 9. We must remark that by Eqs. (67) for sufficiently small IzI, r is an increasing function of z, vanishing together with z. Hence, for any sufficiently small r and for any real 9, Eq. (67) gives one perfectly determined value for z, sufficiently small in magnitude, which has the same sign as r. It follows that if one imposes on r and z the condition that they do not exceed some bounds, sufficiently small but different from zero, then z is completely determined from Eq. (67) as a continuous, increasing function of r, vanishing together with r , and periodic with respect to 9 with the period w . From here it follows that the problem of stability with respect to the variables r, x, is completely equivalent to the stability problem with respect to the variables z, z,. In solving this last problem we can suppose that z 3 0. 20.

Let

z = P , z , + P,z, + ... + P,z, + Z', z,= Z ( P , , Z , + Ps,Z, + ... + PsnZ,) + Z,'

( s = 1, 2, ... , n ) ,

where P,, P,, do not depend on the quantities zi,whereas Z', 2,' do not contain terms lower than second order with respect to the last ones. Denoting by A l , A , , ... , A, the functions of z and 9, periodic with respect to 9, determined from the equations zq-"1

+ z q ( z , 9>]dA. + c (psi + ZP,i)A, + ZPi = 0 a3 &-

s=l

(i

and by W, a quadratic form of zl, equation

c n

s=l

(PslZ1

+ P s 2 Z 2 + ...

2,

=

1, 2,

... , n ) ,

, ... , z, , determined from the

Investigation of One of the Singular Cases of Theory of Stability of Motion

79

in which h is some constant, let us put I/ = z

+ A , z , + A2z, + ... + A"2, + w.

Dividing the equations which determine the functions Ai by

1

+ zcp(z, 9),we bring them to the form z4-l

dAi

-+

as

C (psi + zP;i)A, = zPir

s=l

( i = 1,2, ... , n),

(70)

where, by the properties of the functions P i , P s i , cp(z, 9), the functions Pi', Pii can be represented for each real 9 and sufficiently small real z by series of the form Uo(9, Z"-')

+ Z U l ( 3 , Z"-') + Z2U2($,

Z"-")

+ ... .

(71)

In these series, 9,($,zq--')are some functions of 9, z q - l , defined and continuous for all real values of 9 and z 4 - l , periodic with the period o with respect to 9, and representable by'asymptotic series expansions in integer positive powers of the quantities zq-'. Moreover, these functions are such, that for sufficiently small IzI, the series UO(9,

A) + zu,(9, A) + Z 2 U 2 ( 9 , A )

+ .*.

(72)

converges uniformly for all real values of 9 and A. Replace for each of the functions Pir,P i i , the series of the form (71) by series of the form (72). Denote by P:, P;i the transformed functions. In this case, if instead of (70) we consider the system

in which the coefficients are holomorphic functions of z , which have coefficients which are periodic with respect to 9, then, by the properties of the constants p s i , we can always find one and only one periodic solution of the last system in which all A i are uniformly holomorphic functions of z for all real values of 9 and A. Replacing in this solution 1 by z q - l , we find the desired periodic

80

Stability of Motion

solution of system (70). In this solution, the quantities A i are represented by series of the form (71), in which ~ ~ (zq--') 9 , is zero. Determining in this way the functions A i and the form W , we form the derivative of V with respect to t by Eqs. (68) and (69). This derivative will be

+ h ( Z l Z + Z Z 2 + ... + zn2) + s, where n

S = Z Z ' + CAsZ,'+ s= 1

JA 1 LaA -z,+ asz , T + z Z C -aZ s=l

s=i

c -awZ s , n

s=l aZs

can obviously be represented by the form:

cc n

n

s =s=l a=l

~ S U Z S ~ O

such that all us, vanish for z = z1 = z2 = = z, = 0 and for any real 9 all us, can be made arbitrarily small by choosing for the quantities z, z, real values, not exceeding some sufficiently small bound independent of 9. For positive and sufficiently small values of z, let the function f(z) take only values of a fixed arbitrary sign and let the constant h which is dependent on our choice be opposite in sign tof(z). In this case it is clear that for z 3 0 the sign of dV/dt is constant and opposite to that off(z), 9 being a completely arbitrary real function of t. Suppose that for sufficiently small positive z, the function f(z) cannot be negative and take h < 0. Then dV/dt is a negative function of the quantities z, z, for z 3 0. In the same conditions, however, V is positive-definite, as is easy to see. Therefore, we must conclude that if for a positive z which is sufficiently small, f(z) >O, the unperturbed motion is stable. Remarking that by the assumption on cp(z), the quantitiesf(z) andf(-z) in the case considered are always opposite in sign for + . +

Investigation of One of the Singular Cases of Theory of Stability of Motion

81

sufficiently small IzI, the condition just shown can be expressed in the following manner: for sufficiently small IzI, the function zf(z) must take only positive or null values. With this condition any perturbed motion sufficiently close to the unperturbed one, tends asymptotically to one of the motions, for which z1= z2 = = z, = 0, f(z) = 0. If the functionf(z) is such, that for sufficiently small IzI it can vanish only for z = 0, then all these motions are reduced to the unperturbed one. If f(z) = 0 identically, then these motions are periodic, and belong to the continuous set of such motions which exist in this case. Finally, if forf(z) not identically zero, the equation f(z) = 0 has an infinite set of solutions, arbitrarily close to zero, these motions belong to a discrete set of periodic solutions. Suppose now that for positive z which are sufficiently small, the functionf(z) is different from zero and always negative, or, in other words, for 1zI sufficiently small, zf(z) vanishes only for z = 0, whereas for other values of z it is negative. Then, if we take h > 0, forz 3 0 the derivative dV/dt is a positive-definite function of z, z, . Since under the same conditions one can assign to the function V itself positive values, we must conclude that the unperturbed motion is unstable. Thus, our analysis includes the following cases : (1) f ( z ) = 0 identically, in which case the unperturbed motion is stable; (2) for IzI sufficiently small but different from zero, zf(z) conserves its sign being also different from zero. In this case for zf(z) > 0 one has stability and for zf(z) < 0 instability; (3) equation f(z) = 0 has an infinite set of roots arbitrarily close to zero, whereas for sufficiently small IzI, zf(z) is nonnegative. In this case one has stability. There remain only the cases when the equationf(z) = 0 has an infinite set of roots and either when passing through them the function zf(z) may change sign, or for sufficiently small IzI it is never positive. In such cases the problem remains open and we shall not deal with its solution. Let us remark that although the problem of stability of the

a2

Stability of Motion

motion z = z1 = z 2 = = z , = 0 in the cases belonging to the last category is very difficult, the stability of the periodic motions for which z1 = z 2 = ... = z, = 0,f ( z ) = 0, z being different from zero, does not present any difficulties.* Indeed, Eqs. (68) and (69) are such that, if we put in them z = c(l + i), where c is a real constant sufficiently small, then their right-hand sides become functions of the quantities i,z,, uniformly holomorphic for all real values of 9 with coefficients periodic with respect to 9. Moreover, eliminating dt from these equations, we get, for the determination of the quantities i,z,, as functions of 9, equations from which the derivatives di/dt, dz,/dt are also found as holomorphic functions of the quantities i,z,. This follows from the fact that the right-hand side of Eq. (68) for i= z1 = z 2 = ... = z, = 0 becomes a function of 9, c4-’ +c%p(c, 9), which for sufficiently small IcI can never be zero. Thus, we get a system of equations with coefficients periodic with respect to 9. Since in order to investigate the stability of these motions one has to assign to the constant c values which makef(z) vanish, the right-hand sides of these equations vanish for i= z1 = z2 = ... = z, = 0. Moreover, for IcI sufficiently small, the characteristic equation of this system, obviously, cannot have more than one root of modulus 1. We come therefore to one of the cases considered in the work, “ The general problem of stability of motion.” Let us remark that in the case c > 0 the variable 9 plays the same role in the solution of our problem as t ; in the case c < 0 however this role is performed by the variable (If, in the first case, we consider the characteristic equation corresponding to the period w (w > 0), and in the second case, we take the one corresponding to the period (- 1 ) 4 - 1 w , this equation, for sufficiently small IcI, has n roots with moduli less than 1. In the case when f ( z )= 0, and when therefore c can take any *The stability of these motions with respect to z , z, is not equivalent, however, to that with respect to x , y , x , .

Investigation of One of the Singular Cases of Theory of Stability of Motion

83

value that is sufficiently small, we prove in this way that all the corresponding periodic motions are stable. Consider now the case when forf(z), which is not identically zero, the equationf(z) = 0 has roots arbitrarily close to zero, and suppose that c is one of its positive sufficiently small roots. We shall not consider negative roots since they do not give new motions. Suppose that the considered root c is of multiplicity k ; in other words, f ( c ) = f ' ( c ) = = f ( k - l ) ( c ) = 0, but f ( k ' ( ~ ) is different from zero. We shall suppose here c to be sufficiently small in order that the function 1 ccp(c, 9) is different from zero for any reai value of 9. Putting z = c( 1 i)we will have

+

+

d9 dt

- = cq-"1

+ ccp(c, 9)] + 0,

where 0 is a holomorphic function of the quantities i,z, ,vanishing for i= z1 = z2 = ... = z, = 0. Thus if we put az

we get for the determination of i,z, as functions of 9, a system of equations of the form

(s = 1, 2, ,,. , n),

where 2, Z, are holomorphic functions of i,z, not containing terms lower than of second order and for z1 = z 2 = ... = z, = 0 they become holomorphic functions of inot containing powers of ilower than k 1. p , , p,, are periodic functions of 9, the last ones being such that the difference ps6 -p,, can be made arbitrarily small for each real 9, by taking c sufficiently small.

+

84

Stability of Motion

Thus we can suppose that c is so small that all the roots of the characteristic equation corresponding to the period w of the system _ dzs - P s l z l + P . 5 2 ~ 2+ ... + P s n Z n d9 cq-"1 ccp(c, 9)]

+

(s =

1, 2, ... , n)

(74)

have moduli smaller than 1. As to the roots of the characteristic equation of system (73), for k > 1, n of these roots coincide with the roots of the characteristic equation of system (74) and the (n 1)th root is equal to 1. Let us consider first this case (k > l), when c is a multiple root of the equation f ( z ) = 0, or q(z) = 0. In order to apply to this case the method given in the abovementioned work, we have first to perform some linear transformation of system (73), namely, we have to introduce instead of C as the unknown function the quantity Cl = 5 Alz, ... A,z, , where A l , A , , ..., A , , are periodic functions of 9 determined from the condition that in the expression of the derivative dC,/d9 do not appear terms lower than the second order. This problem is always possible, since it is reduced to the search of periodic solutions of the system

+

+

+

Determining the functions As in this way, we get for dC1/d9an expression of the form dC,/d9 = - ~ p ( ~ ) ( (c, c ) $9) Clk Z', where 2' is a function of the quantities C,, z, of the same type as 2, and

+

c A,F,(c, 9). n

$(c,

9)

= F(c, 9) -

s=l

Let us remark now, that from their mode of construction the quantities p s are functions of c such that the products cq-lpS are finite for c = 0. Thus, the periodic functions as determined from system (75) are also finite for c = 0. From here, taking into account that F(O,9) = 1/(1.2 ... k), Fs(O,9) = 0, we conclude, that $(O, 9) = 1/(1.2 ... k), and that c can be considered so small that the function $(c, 9) is not zero for any real value of 9.

Investigation of One of the Singular Cases of Theory of Stability of Motion

85

By making these assumptions we find that the integral J:$(c, 9) d9 is a positive quantity. Since in the expressions dz,/d9 the terms which are independent of the quantities z, do not contain il in powers lower than k, the problem of stability depends on the number k and the sign of the expression V ( ~ ) ( C )J;$(c, . 9) d9, as was shown in my previously mentioned work. For even k, instability always holds if only this expression is different from zero, whereas for odd k, when this expression is positive, the motion under consideration is stable. When the expression is negative, the motion is unstable. Thus, we come to the conclusion that if c is a sufficiently small positive root of the equationf(z) = 0 (or its equivalent, q(z) = 0) of even multiplicity, the corresponding periodic motion is unstable. It is very remarkable that when we are dealing with the case of a discrete set of periodic motions, corresponding to the roots of even multiplicity in the intervals between them, the function f ( z ) being positive, the motions of this set which are sufficiently close to the unperturbed motion are all unstable though, as we saw, the unperturbed motion itself is stable. We obtain, therefore, an infinite set of unstable motions, among which one can find motions arbitrarily close to some stable one. When c is a root of odd multiplicity, the stability of the corresponding motion depends on the sign of v ( ~ ) ( coff‘k)(c). ) Namely, when v ( ~ ) ( c or ) f(”‘(c) is positive, it is stable, otherwise it is unstable. Thus, if the functionf(2) changes its sign from - to when z passes through the multiple root c in the increasing direction, stability holds, in the opposite case instability holds. Let us show that the same is true even for a simple root c. In this case the problem is reduced to the investigation of the characteristic equation of the system

+,

86

Stability of Motion

Let us show that for sufficiently small c one can pick up periodic functions A l , A , , ... , A , of the variable 9 so that in the expression of the derivative with respect to 9 of = [ Alzl A,z, ... Anz,, expressed in terms of il,z, , there are no terms depending on the quantities z, and that for any c which is sufficiently small, the functions A , do not exceed some unknown bound. Indeed, the functions A , must be determined from the equations :

+

+

-q’(c)A,

c

Fj(C, $ ) A j = 0

j = 1

(s = 1 , 2,

+

+

... , n),

which we will represent in the form

(s = 1, 2,

... , n).

In this case the functions P j , , Q j , P , will be conveniently represented by series of the type (71) ( z being replaced by c). In consequence, reasoning as in the case of Eqs. (70), we prove that the quantities A , which satisfy the equations* considered here, are also found as series of the type (71). Determining the functions A , in this way, we find

+ cp’(CF,(C,

9)il

(s = 1, 2, .. . 2 n ) ,

where $(c, 9) is an expression constructed from the quantities A , in the same manner as the one considered above.

* Equations (70) were linear. The equations obtained now, are nonlinear. However, this fact does not make the proof more difficult.

Investigation of O n e of the Singular Cases of Theory of Stability of Motion

87

We see from here that one of the roots of the characteristic equation of our system is

and that the remaining n roots coincide with the roots of the characteristic equation of the system, formed with the n remaining equations for il = 0. For sufficiently small c, however, these n roots have obviously moduli which are smaller than 1. Relative to the root (76) we see that for sufficiently small c, the integral f:$(c, 9) d9 is positive, being less than 1 for cp'(c) > 0 and more than 1 for cp'(c) < 0. Thus, when the functions q(z) orf(z) change sign from - to + for z passing through c in increasing direction, the motion considered is stable. In the opposite case it is unstable. One sees from here, that if all the positive roots of the equation f ( z ) = 0 are simple, then by passing successively through the periodic motions belonging to the considered set in the direction of the unperturbed motion, we alternatively meet stable and unstable motions. 21. Let us return now to page 43. There we did not consider the case a > (rn - 1)/2 for the most general assumptions about system (l), since we assumed that X vanishes for x = y = 0, whereas for y = 0, Y becomes dependent on x only. We want to show now that the general case can be reduced to the particular one by some transformation. Let us consider system (1) under the general assumption that for y = x, = xz = ... = x, = 0, the functions X , X, are zero, whereas Y is a series of the form - x" +glxm+' -... Suppose, moreover, that Y = - X" +glxm+' Y,, where Y , vanishes for y = x1 = x2 = ... = x, = 0,and does not contain terms lower than second order with respect to the quantities x, x, , y. In page 22 it was shown that if in equations (6) we replace Y by Y l , one finds one and only one solution of the form y = f(x, xl,x2, ... , XJ, where f is a holomorphic function of the

+ +

+

88

Stability of Motion

quantities x, x,, not containing terms lower than second order and vanishing for x1 = x 2 = ... = x, = 0. If, however, the functions X and Y , vanish for x = y = 0, it is not difficult to see that the functionfvanishes for x = 0, no matter what the quantities x, are. With this remark in mind, consider system (7) under the general assumptions stated above. We can always find a solution of this system, of the form x = cp(xl, x z , ... , x,,),y = $ ( x l , x2 ... x,,), where cp and $ are holomorphic functions of the quantities x,, not containing terms lower than of second order. Transform now system (1) by means of the substitution

x = cp(x,,

x2,

... , x,)

+ x,y =

$(XI,,

x2,

... , x,)

+j.

The transformed equations are of the form:

dx -- j + X _ dt

where R, P, X , vanish for j = x1= x2 = ... = x, = 0, whereas R and P vanish also for X = jj = 0. As a consequence of the above remark we can find for the equation

a holomorphic solution of the form jj =f(.f, xl, x 2 , ... , x,,), in which the function f has the property f(Z, 0, 0, ... , 0) = f(0,XI, XZ , = 0. Let us now transform the system (77) by means of the substitution j = f ( X , xl, xz, ... , x,,)+ y l . . a * ,

xfl)

Investigation of One of the Singular Cases of Theory of Stability of Motion

89

It is thus transformed to the form: dx -= y , dt

+ X',

where 8', Y , , X , vanish for y , = x1 = x2 = ... = x, = 0; the function 8' vanishes also for 2 = y l = 0 and the function F1 vanishes for y , =O. We arrive therefore at the case considered above. The transformation of system (l), which ought to be performed in this connection, consists of replacing the quantities x and y by new variables iand y , , related by the equations x =

x + (o(x,,

y = yl +f(x,

x2,

... , XJ,

x2

7

...

9

xn>

+

x2 >

... >

xn>>

(78)

which, when solved with respect to X and y , give

x = x - cp(x1, x2, ... , x,), y,

=Y-f(X-(P,Xl,xz,.

(79) ..,Xn)--(X,,x2,..',X,).

Suppose that by expressing everything in terms of x and y , solving the equations

+

+ ... + psnx, + X,= 0

~ ~ 1 x ~1~ 2 x 2

(S =

1, 2,

... , H ) ,

(80)

we find x, = yf,(x, Y >

(s = 192,

". > a>,

(81)

wheref,(x, y ) are holomorphic functions of x and y , vanishing for x=y=o.

Let us suppose that by making the substitution- (81) we get Yl = y(ux" ulxa+l+ ...) ... , where the omitted terms are proportional to the higher powers of y.

+

+

Stability of Motion

90

Suppose now that by expressing everything in terms of R and yl, one gets, solving Eqs. (80): (s = 192, ... n>,

x, = YlCp,(X, Y l )

7

(82)

where q , ( X , y , ) are holomorphic functions, vanishing for R = y1= 0. Suppose that by performing substitution (82) we get y, = y,(a'x"' + .,rxa'+L + ...) + ... where the omitted terms are proportional to higher powers of yl. With Eqs. (80), we find from (78) dx - = - dF dt dt '

d y -- -d + y ,- dt

dt

df dx dx dt

Thus, because of the relation between the quantities x, y and 2, y , which follows from (78) or (79) and from (81) or (82), we find

+ 8' = y + x, df( y + X ) , + glim+l + ... + 7, = - x " + g , x ' " + l + .'. + Y, - xox y,

-2"

or -xm

+ g1xm+1+

... + y(axU + a,xa+1 + ...) + ... = - x m + g , x- m +

1

The above-mentioned relation is obviously expressed by equations of the form x =x Y = y,

+ Y I 2 F ( X , y,),

+ xY,@(x, Y1) + Y 1 2 W ,

Yl>,

Investigation of One o f the Singular Cases of Theory o f Stability of Motion

91

where F, @, Y are holomorphic functions of X and y,, vanishing for X = y , = 0, whereas F and Y do not contain terms lower than of second order. Therefore, takingonly the terms which are not higher than the first order in y,, we have -Xm+glXm+l

+

... + y ( a x a + a , x " + '

+ ... + y,(a'p'+ a,'x"'+' + ...) +

+ ...)+ ... =

-:m+gl:m+l

... .

We note that by substitution (82) X ' vanishes for y , = 0. With the same substitution af/ax, which already equals zero for x, = x2 = ... - x, =0, will also vanish for y , = 0. As a consequence of this and of the last equation we find from (83) that c i = a, a' = a. Thus the numbers a and CI do not change by our transformation. ~

22. All the cases that one can meet when solving any problem of stability can be divided into two categories: in one of them, it is sufficient to consider in the differential equations of the perturbed motion t e r m that are not higher than of some order, as one gets the same result whatever the high-order terms are. In the second category it is necessary to consider all the terms in these equations. We call the cases belonging to the first category algebraic and those belonging to the second, transcendental. So the case considered in 13, where the series (30) are not periodic, is algebraic. On the contrary, the cases we dealt with in the following paragraphs are transcendental. Supposing that m is odd, g=--1 and a >(m - 1)/2, let us assume that we are dealing with an algebraic case. As we saw, the problem depends in this case on the sign of some constant h. Let us show that for its determination it is not necessary to perform all the preliminary transformations shown in the previous paragraph. Supposing that we are dealing with the transformed system (1) and, therefore, the functions X , Y , X , satisfy the conditions from

92

Stability of Motion

page 43, we form the following system of partial differential equations

+ P s n X n + xs (S =

1 , 2, ... , n).

(84)

This system can be always satisfied formally, and in a unique manner, replacing xsby series expansions in positive integer powers of x and y , not containing terms lower than of second order. Replace in these series x and y by their expressions (22) and then replace Y by the first of the series (30). By expanding the result in increasing powers of c they are obviously transformed into the old series (30). Replace the quantities x, in the equations dx - -- y + x , dt

dY - y dt

__

by the above considered series taken up to an arbitrary order N and then transform these equations by substitution (22). We obtain the equation dr _ - R, d9

defining Y as a function of 9. We look for the general solution of this equation in the form of a series expansion in positive integer powers of the arbitrary constant c, so that for 9 = 0 the coefficients of this series take the corresponding values of the functions di).As it is clear from the above, in this series all the terms containing c to powers not higher than N are identical with the corresponding terms of the first series (30). Thus, if N 3 k, the first nonperiodic term of our series is identical with the first nonperiodic term of the first series (30), d k ) c k .The constant g is found, therefore, by considering this term, in which d k is ) of the form u(")= g9 v, where v is a periodic function of 9. Let us suppose now that we are dealing with the initial system (1)-

+

Investigation of One of the Singular Cases of Theory of Stability of Motion

93

In order to go over to the case considered in 11, we perform transformation (78). Suppose now that this transformation was already performed and that one forms series expansions in powers of 2 and y , up to the terms of order k, satisfying formally the equations which determine the quantities x, as functions of .? and y1 . Now put these series into the right-hand sides of Eqs. (78) instead of the quantities x,. Since in these series all the terms vanish for y , = 0, one gets for x and y expressions of the form: x =2

+ YI2F(x-,

Yl),

Y = YlC1

+ @KY 1 ) L

(87)

where F and @ are holomorphic functions of .? and y , , containing no terms lower than of second order. Suppose after this, that in the expressions of X and Y, the quantities x, are replaced by series satisfying system (84) up to terms of order k, and consider with these assumptions, Eqs. (85). In order to go over from these equations to Eq. (86), in the righthand side of which it is sufficient to retain only the terms up to order k , we have to perform transformation (87), and then put X = rC(9), y1 = - r4S(9),and then, eliminating from the result dt, retain in the expression of dr/d9 only the terms up to the kth order. It is not difficult to see that in order to determine the constant g, we could [instead of Eq. (86)] treat in a similar manner the equation dp--

dcp

P,

obtained from system (85) by eliminating dt and replacing x and y by p and cp with the aid of the equations x = pC(cp),y = - p4S(cp). Indeed, by (87), the variables Y, 9, cp, and p are related by the equations

+

p~(cp= ) ~ ( 9 )r 2 4 s 2 ( 9 ) ~ [ r C ( 9-) ,r4S(9)], p4S(cp) = r4S(9){1

+ @[rC(9),- r 4 S ( 9 ) ] } .

+

(89)

We deduce from these equations p = Y[ 1 Y ~ ( Y , 9)], where f is some holomorphic function of Y and 9with coefficients periodic with respect to 9.

Stability of Motion

94

Putting this expression for p in the previous equations, we deduce the following ones: C(cp> = C(3) + rf1(r,

$1,

S(cp) = S(9) + rf2(r, $1,

(90)

wheref, and f 2 are holomorphic functions of r and 9 with coefficients periodic with respect to 9. It is not difficult to see that the functions f , fl, andf, vanish for 9 = 0. We remark now that by the assumption about C and S the functions C(cp z ) , S(cp + z ) are holomorphic with respect to z for any real cp. Thus, assuming that cp vanishes simultaneously with 9, we deduce from Eqs. (90) for sufficiently small ~ Y J

+

cp = 9

+ r$(r, $1,

(91)

where $ is a holomorphic function of Y with coefficients periodic with respect to 9, vanishing for 9 = 0. Thus, we expressed p and cp as functions of Y and 9. One can find similar expressions also for Y and 9 as functions of p and cp. Put in the right-hand sides of Eqs. (89) and (91) Y = c d 2 ) c 2 ... d k ) c k As . a consequence, the equation takes the form

+

+

p =c

+ u 2 c 2 + u3c3 + ... + UkCk + ... ,

q = 9 + vlc+v2c2+-~,

+

(92)

where U i , Viare functions of 9, independent of c, of which ...) & - 1 are indeed periodic, whereas Uk is of the form g9 + a periodic function of 9. It is not difficult to see that all the functions U, I/ are such that if we put in their expressions 9 = cp z , where cp is any real number, they are holomorphic with respect to z. Thus, supposing cp to be real and /cI sufficiently small, we deduce from Eq. (92): 9 = cp Wlc W,c2 ..., where W1, W , , etc., are functions of cp independent of c, of which W,, W , , ... , Wk-l, are indeed periodic. As a consequence we get for p an expression in the form of a series u2,

u3,

...?

uk-1,

vl> vZ,

+

+

+

+

p=c+v2c2+v3c3+

...)

(93)

Investigation of One o f the Singular Cases of Theory o f Stability of Motion

95

in powers of c, with coefficients us which are functions of cp. Among these u 2 , v 3 , ... , v k - are indeed periodic, whereas uk is of the form v k = gcp, a periodic function of cp. By the manner of its generation, series (93) is such that up to the term V k C k inclusive it coincides with the series which represent the general solution of equation (88) expanded in powers of the arbitrary constant c, as long as in the last series one assigns convenient values to the arbitrary constants introduced in the forming of its terms. Hence, the above affirmation was proved. Thus, if we replace the quantities x, in the right-hand sides of Eqs. (85) by series which satisfy Eqs. (84), taken up to terms of sufficiently high order, and afterwards treat Eqs. (85) as the proposed differential equations of the perturbed motion by the method shown above, we arrive at the solution of the problem of stability whenever we are dealing with an algebraic case. 23. Let us consider now the second of the three cases mentioned above. We suppose therefore that system (1) is such that if we replace in Y the quantities x, by series satisfying the equations pslxl + p s 2 x Z ... +psnx, X , = 0 (s = 1, 2, ... , n), then if the expression -xm +glxm+' ..- +y(axa + a , x a + ' ...) represents, in the result of the substitution, the set of all the terms not higher than of first order with respect to y , the number a satisfies the inequality a < (m - 1)/2. Since a can not be zero and m is odd, this inequality holds only if nz 3 5. By putting, as in the first case, m = 2q - I, we have therefore a < q - 1, q 3 3 . We shall suppose moreover that a is odd, since the case of an even ci was analyzed in 10. Assume first that we are dealing with the case when the series which formally satisfy system (84) represent holomorphic functions of x and y. These functions then represent a solution of system

+

+ +

+

(84).

Let XI = YlClI(X,

be this solution.

Y),

x2

= YlClZ(X2

Y ) , ..' x, = YlCl,(X, Y > 3

(94)

Stability of Motion

96

Here all $s are holomorphic functions of x and y , vanishing for x=y=o.* Let us show that even for perturbations satisfying the conditions (94), the unperturbed motion is unstable. For such perturbations, the perturbed motion is determined from Eqs. (94) together with

-d _x - y + x , dt

_ dy - Y. dt

(95)

Supposing that in the expressions of X , Y, the quantities xl, x2,... , x, are replaced by their expressions (94), we show that the equation dY _ -- - y dx y + X

can always be satisfied under the conditions assumed replacing y by some holomorphic function of x, vanishing for x =O. Indeed, the right-hand side of Eq. (96) has the form dY - -X"cpo(X) dx

+ ax"cpl(x>Y + c p Z ( X > Y + ... YC1 + d x , Y ) l

where cp,(x), cp(x, y ) are holomorphic functions of their arguments of which cpo(x),cpl(x),cp(x, y ) satisfy the conditions cp,(O) = cp,(O) = 1, cp(0, 0) = 0. Put y = x P ( b z), where is some positive integer and b is a constant different from zero. To determine z as a function of x we get the equation

+

dz x - = -p(b dx

+ z)

Within the framework of our assumptions the numbers b and

* Here there is an inaccuracy. As it follows from the considerations of page 35, the right-hand sides of expressions (94) cannot contain y as a factor. This, however, is not essential for the following, since the system ( 1 ) can always be transformed so that X vanishes identically for y = 0 (Editor's note).

Investigation o f One o f the Singular Cases o f Theory o f Stability of Motion

97

p can always be chosen so that the right-hand side of this equality is a holomorphic function of x and z , vanishing for z = x = 0. Indeed, in order that this holds, the numbers b and p must be taken so that the expression

vanishesfor x = 0. But this is only possible in two cases: (1) when the numbers m - 28 1 and CY - p 1 are equal to one another and (2) when oneof these numbers is zero and the other is positive. The first case is impossible, since it implies B=m-a and, therefore, the number

+

+

m - 2p + 1 = c( - f l + 1 = 2.

+ 1 - rn

would be negative. Hence, it is possible only in the case when %(X>

- Cpo(x>

x m - ~ a - ~

m-

=-

a'

+dx),

where ~ ( xis) a holomorphic function of x, vanishing for x = 0. In this case we should take b = l/a. Thus, this case is only possible when there exists some known relations between the coefficients of the expansions of q0(x)and cp,(x). We shall not insist on them. As to the second case, it is not possible to make m - 2p 1 = 0, since then we should have c1- p 1 = CY - (m - 1)/2 < 0. We must put therefore c( - p 1 = 0. With this assumption, which is always possible, B = c( 1, and therefore m - 2p 1 = YYI - 1 - 2. > 0. If we put b = a/(. I), the required condition is satisfied. Thus, Eq. (96), transformed with the aid of the substitution, y = x a + [a/(. 1) z ] , gives for x(dz/dx) an expression, holomorphic with respect to x and z, vanishing for z = x = 0. If, moreover, we put x=O, it becomes -p(b + z ) + a = -(a 1)z. Hence, the transformed equation is of the form

+

+

+

+

+ + +

+

x

dZ dx

- = -(.

+

+ l ) z + zx,

where 2 is some holomorphic function of x and z.

98

Stability of Motion

As in the right-hand side of this equality the coefficient of z is negative, we always find by a well-known theorem one perfectly determined holomorphic function of x, satisfying this equation and vanishing for x = 0. Let z = xf(x) be this function. Then, putting in the expression of X (which vanishes for y = 0)

we transform it into a holomorphic function of x, the expansion of which does not contain powers of x lower than the (a 2)th. Hence, the first of .these equations is in this case transformed into the following :

+

dx

-=-

dt

a cc+l

XU+l

+ XU+%p(X),

where q(x) is a holomorphic function of x, vanishing for x = 0. Thus, if one solves the problem of conditional stability with the assumption that the quantities x,y , x, are related by Eqs. (94) and (97), the problem is reduced to the investigation of Eq. (98). However, since a + I is even, this equation leads to a negative result. We must conclude therefore that the unperturbed motion is unstable. Let us consider the general case, when system (84) does not have a holomorphic solution. In the case considered in the preceding paragraph, system (1) has a solution in which the quantities xl, x 2 , ... , x, , y are expressed as functions of x by equations of the form: 24.

a

a+l x, = X " + * f S ( X )

(s = 1, 2,

... , n),

(99)

where f ( x ) ,fs(x) are holomorphic functions of x. In the case which we intend to consider now, such a solution does not generally exist. We will prove, however, that there always

Investigation of One of the Singular Cases of Theory of Stability of Motion

99

exists a solution, which corresponds to equations of the form (99). where f ( x ) ,f,(x) are not holomorphic functions. They are some real functions of the real variable x, defined and continuous for all values of x sufficiently small in magnitude and represented by asymptotic series expansions in positive integer powers of x. If such a solution is possible and if the initial value of x is supposed to be different from zero, then x in this solution conserves the sign of its initial definition at least as long as it does not exceed some bound. Hence, considering such a solution, we can limit ourselves to the assumption that x takes values with a fixed arbitrary sign. For our goal it will be sufficient to consider only the values of x which are of the same sign as a. First of all, in order to simplify our calculations, we conveniently arrange the value of the constant a. By our assumption that c( is odd, this is always possible, since by replacing x and y , respectively, by Cx and Cy, we replace a by P a , and by conveniently choosing the real constant C, this quantity can be given any numerical value different from zero, with an arbitrary sign. We shall suppose that a = (ct l)/ct, and correspondingly x will be considered positive. Let us remark that by doing this, we can no longer suppose that g = - 1 . We shall assume therefore that g is some negative constant, its actual value being unimportant in our calculations. In order to simplify our analysis we assume that the transformation of page 17 has been performed on system (1). We suppose therefore that

+

where Y , does not contain y to powers lower than the second one in the terms independent of the quantities x,. X vanishes for y=x,=x,=...=x,=O and for x , = x , = . . . = x , = O , the quantities ,'A are of the form qo(x)+ y q 1 ( x ) + y 2 q 2 ( x ) ..., where cpo (x), q l ( x ) do not contain powers of x lower than the (m 1)th and (ct l)th, respectively.

+

+

+

100

Stability of Motion

With these assumptions, we transform system (1) using the substitution : 1

y=-

(1

a

+ z), x, = X 2 @ + 1 2 ,

(s = 1,2, ... , n).

(100)

As it is not difficult to see, the transformed system is of the form dx xa+l _ --- a (1 dt

+ z)(l + XX),

dz dt

+ z){ -(a + 1)z + XZ'},

x-=-

d- z=, dt

xa+'

(1

a

pslzl

+ ps2zz + '.. f psnz, + x z :

(s = I, 2, ... , n ) ,

where A',, Z', 2; are some holomorphic functions of the quantities x, z , z l , z z , ... , 2 , . For this we have only to take into account that r n > 2 a + l anda>O. By eliminating dt from these equations, we get a system of the form x

dz dx

-= -(a

(102)

xa+ 1

---d z , - P S l Z l o!

+ l)z + x z ,

dx

+ P S 2 Z 2 + '.. + P s n G + z,

(s =

1, 2, ... , n ) ,

where 2, 2, are holomorphic functions of the quantities x, z , z, . The quantities 2,are such that for x = 0 2, = [ -z/( 1 z ) ] ( p , , z 1 ps2zz+ ... +psnz,,).Let us show that for x 3 0 one can always find a real solution for system (102). In this solution, the functions z , z, vanishing for x = 0, are, together with their derivatives, defined and continuous for all sufficiently small values of x.They are represented by asymptotic series expansions in positive integer powers of x. Let

+

xz

= F(x, z , z 1 , z 2 ,

2, = F,(x, z ,

21,

... , Z"),

z z , ... , Z")

(s =

1,2,

... , n ) .

+

Investigation of One of the Singular Cases of Theory of Stability of Motion

101

Consider instead of (102) the system x

dz dx

-=

-(a

+ l)z + F(cx, z, z1, z2, ... , Z"),

1

- dz, -= PSlZl + P S Z Z 2 + ... + PsnZn + FLEX, z , z1, z 2 , ... , Z") a dx

(103)

(s = 1, 2, ... , n).

We look for a solution of this system in which z, z, are represented by series expansions in positive integer powers of the parameter E with coefficients vanishing for x = 0. Let z

z,

= Z ( l ) € + z ( 2 ) E 2 + z(3)63 + =Z p E

...

+ Z p E 2 + Z j 3 ) € 3 + ...

(s = 1, 2,

... , n )

(104)

be these series. Replacing them in the expressions of F(cx, ...), F,(tx,...) and expanding the results in increasing powers of E , we find F = Cxe + F ( 2 ) t 2+ P 3 ' E 3 + ..*, (s = 1, 2, ... , n), Fs = C,X€ + F y c 2 + F53'€3 + ... where C, C, are some constants, whereas F ( k ) FJk) , are some entire z ( k - ' ), z;l ), )';. , ... functions of the quantities x, z(l), z ( ~ ... ) , zp-'). The last ones are such that if x is considered a first-order quantity and z ( ~ )z:,) quantities of ith order, then F ( k ) FJk' , are quantities of kth order. For the successive calculation of the functions z ( ~ )zy) , we get, therefore, the following system of equations : 3

3

(s = 1, 2, ... , n ) ,

01

dx

(s = 1,2, ... , n).

102

Stability of Motion

+

The first of Eqs. (105) gives z(') = [C/(a 2)]x. The required solutfon of the last n equations (105) is obtained in the following way. Let xl, x 2 , ... , x,,be the roots of Eq. (2). We know that we can always find constants aso (real or imaginary) such that a s a result of the substitution Z, = a s l ~ + ~ l~ ~ 2 +. ... ~ 2 + asfly,, ( S = 1, 2, ... , n ) , (107) the system of equations dz,/dt =pslzl +ps2z2 2,... , n) is transformed into the form

+ ... +psnz, (s

=

1,

dY1 -= X l Y l , dt

where ojW1are some constants which can always be assumed to be real and nonnegative. By performing a similar transformation on the system considered, we bring it to the form

xa+' dyj') CI

dx

- xjyj

+ oj-lyjt), + B j x

( j = 2, 3,

... , n ) ,

where B,, B 2 , ... , B,, are some constants, depending in a known manner on the constants C,. From here we find successively a dx

( j = 2, 3, ... , n ) .

For x 3 0, the integrals appearing here are defined, since all X, have negative real parts. It is not difficult to convince ourselves (see the following paragraph), that the functions yi' ), defined by formulas (108), indeed vanish for x = 0.

Investigation of One of the Singular Cases of Theory of Stabilityof Motion

103

Let us suppose, that all z ( ' ) ,2:') for which 1 < i were found and that they vanish for x = 0. Then the first of Eqs. (106) gives

Next, denoting by y$i)the quantities corresponding [because of (107)] to the quantities z : ~ ) and , by @,:i) the quantities which are obtained from F6') in the same way as the constants B, are obtained from the constants C,,we find

( j = 2, '3, .. . , n).

By such formulas we determine successively all y ( i )and thus also all z : ~ ) z, ( ~ ) One . can state that all the functions z ( ~ )z, : ~ ) , defined by our formulas, are real. Indeed, if we assume that those for which i < k are real, the functions P k )will also be real and then, if Z ( k ) = U ( k ) + 4 T u ( k ) z ; k ) - U, (k) V'-_ - Iulk), where all u, u , will satisfy the equations are real functions, d k ) ukk)

+

y + l

dujk)

a

~

dx

= p sl u (1k ) + p s2 U 2( k ) +

1..

+ psnup

(s

=

1, 2, ... , n).

These equations however have for x 3 0 no other solution in which all the unknown functions vanish for x = 0, except d k )= u;k) - 0 2( k ) ... = uik)=O. For this reason all the functions z ( ~ ) , zy) are real. Since also the functions z(' ), zl' are real, it follows that all the remaining ones are real as well. ~

25. Let us now consider more closely the properties of the functions z ( ~ )zjk). , Supposing that f ( x ) is defined and is continuous together with its derivatives for any x satisfying the condition

104

Stability of Motion

0 < x < A , where A is some positive number, let us then prove that the function

will also have the same property, where 2. is an arbitrary number with a positive real part. By q(0) and $(0) we mean the limits of q(x) and $(x) for x = 0. For this purpose we note that by integrating by parts, we get

f (XI q(x)= i " x a + ' f ' ( x )d x , a + 1 (a+ l)xa+l 0

(111)

From here we find

and therefore, in general

Further, we find from (1 12)

By putting for any function F(x), xa+'F'(x)= Fl(x), we can write this as

For this reason, if we put x"+'F;(x)= F,(x),x"+'F;(x)= F3(x), etc., we have in general a dx 1

.

(114)

Thus, the functions cp'")(x)and t,b,(x) are represented by formulas completely analogous to those giving q(x) and $h(x). All these

Investigation of

One of the Singular Cases of Theory of Stability of Motion

105

formulas are of course valid unconditionally, only as long as 0 < x < A . On the other hand, the definiteness and continuity of our functions for x f0 is obvious. What remains to be investigated is the case x = 0. We agreed already to denote by q(0) and $(O) the limiting values of the functions q(x) and $(x). Let us prove that these values exist. Suppose that x does not lie outside the limits 0 and A . Denote by M' the upper bound of the function If'(x)l in this interval and by A' the real part of 1. We have

One sees from here that the limiting values for x = O of the right-hand sides of (1 11) and (1 12) are f(O)/(a 1) and f(O)/ll. The existence of the two limits is thus proved and therefore d o ) =f ( O ) / ( @ $(O) = f ( O ) / l l . In order to prove now the continuity and definiteness of the derivatives of our functions, it is sufficient to show that

+

+

1>7

$(")(O) = lim $(")(x). x=o

Integrating (1 11) by parts we find

f (XI

q ( x ) = -@

whence

+1

xf'(x)

(a

+ l)(E

-t 2)

+

(a

+

1 l)(@+ 2)xa+'

106

Stability of Motion

Going to the limit for x = 0, we get equality (1 15) for n = 1. Proving equality (1 15) for any given n and using formula ( I 13), we prove it also for n 1. Thus it is valid in general. Let us now prove equality (1 16). By (1 14), one can write formula (112) in the form $(x) = [f(x)/1] - [i,bl(x)/la], whence

+

...... We find from here

However, in general,

where all A$) are some positive integers. For this reason, inside the interval from 0 to A , the modulus of the fraction [fm(x)]/ [xma+l] has some upper bound. Denoting it by M,, , we find from (114) x ( " + l ) a + le(L'/X")

I$fl+l(.X)l< M,+1

J:

e-(L'/x")

a dx

y +1 -

From here l*;+l(x)l

=

Ii,b.+z(x)I xa+l


M n + 2 x(n+l)a 3

M,+l~("+l)a+l

1'

Investigation of One of the Singular Cases of Theory of Stability of Motion

107

In general we have

$I?:

(k)

l(x)I < M n + 1x

(n+l)a-k+l 2

where M,'?, is some positive number. Differentiate now equality (1 17) k times and put x = 0 into the result. For k < (n 1)a, we find

+

From here we find, among other things lim $'(x) = f ' ( O ) / R . Forx=o

mula (1 17) gives, however, for rz = 0 $(XI

-

X

$(O) --f ( x ) - f ( O > -$I@) AX Iax '

whence (as lim [$l(x)]/x= 0) $'(O) =fr(0)/A= lim I+V(X). x=o

x=o

Suppose now that it has been proved that $("(O)

= lirn x=o

$("(x).

We find from (117) by (118):

whence, supposing k

< (n + 1)a,

As a consequence, we can consider equality (116) as proved. At the same time, we have proved also the definiteness and the continuity of our functions together with all their derivatives for values of x and between 0 and A . Let us consider the particular case where f ( x ) = xm for integer and positive values of m. In this case q(x) = xm/(a m 1). Further, fn(x) = m(m + a)(m + 2a) ... (rn + n - la)xm+"".

+ +

Stability of Motion

108

Thus, formula (I 17) takes the form Xb

+ m(m + a)A2a2 X2z

x3a

- m(m + a)(m + 2M) -+ ... + ( - l ) " m ( m -ta) A3a3

x (m

1

+ 2a) ... ( m + n - l a ) A"a" - - (Xna

1)"2* +n +l a n1(x) +l '

where

and, therefore,

I$"+

l(4l

< m(m +

... ( m + na)

Xm+(n+l)a

1'

*

We conclude from here that for f(x) = x'", $(x) is represented asymptotically by some series expansion in integer positive powers of x. Returning to the general hypothesis about the function f ( x ) , we conclude from here also, that iff(x) can be represented asymtotically by such a series, then the functions q ( z ) and $(x) can also be represented asymptotically by such series. Suppose that if one differentiates k times the asymptotic series forf(x), one gets the asymptotic series for the function f'"(x). Then, by differentiating k times the asymptotic series of q(x) and $(x), one gets the asymptotic series for the functions qck)(x)and $ ( k ) ( x ) .For q(x) this results directly from formula (I 13). For *(x) this follows from equality (117), which if one differentiates it k times, gives for $(k)(x)an expression in terms of the derivatives of the function f(x) with the remainder -(- I)"[~f:+l(x)]/ (~'l a"+l), the modulus of which, for 0 < x < A , does not exceed, as we saw, the quantity (2+ lan+ )-l&fLylx(n+ 1 ) a - k + l Returning to the formulas of the previous paragraph, we deduce easily from what was proven here, that all z ( ~ )z,: ~ )for , x 3 0, are

Investigation of One of the Singular Cases of Theory of Stability of Motion

109

functions of x defined and continuous together with all their derivatives. These functions, and all their derivatives, can be represented asymptotically by some series expansions in positive integer powers of x,the asymptotic series of the derivatives being the series of the derivatives of the terms in the asymptotic series of the functions.

26. Consider now the problem of the convergence of the series (104). We can consider instead of series (104) the similar series for the functions z , y , where y , are quantities related to z, by Eqs. (107). For the quantities z, y s we have a system of equations similar to (103) in which the functions corresponding to F and F, will be denoted by 4, 4,. Let ( ( i ) , be some upper bounds of the moduli of the quantities z ( ~ )y,i ' ) for x varying from 0 to some arbitrary positive number 5. We take

?ai)

Further, denoting the real parts of the numbers xl, x2, ... , xn by -I,, - I z , ... , - A n , respectively, and supposing all oj to be positive or zero, we can take by (108)

Suppose now that all 5 ( k ) , v f ) , for which k < i, were already calculated. F(i),d : i )are some polynomials depending on the quantities x, d k )y:k) , for which k < i. The coefficients of these polynomials are linear forms of the coefficients in the expansions of 4(x, z, Y l , Y 2 > .*. Y n ) , 4s(x, 2, Yl, Y z > * - . > Y E ) in powers of x,Y , y, , with positive numerical coefficients. Denoting by Y ('), '3':') the quantities obtained from Fi), 4:i) replacing the quantities x, z ( ~ )y:k) , by the quantities 5, (("I,qik), and the coefficients in the 7

110

Stability of Motion

expansions of the functions 4, q5s by their moduli, we deduce from (109) and (110)

We can use these formulas for all the values of i, starting with 2. Defining i ( k )qSk' , in this way, we find that the series

are the expansions in powers o f t of the quantities i,qs vanishing for E = 0 and satisfying the equations:

+ 1)i =

i,q1, q 2 ... ?A 2 1 ~ 1 = Y'l(tt, i,~ 1 ~ ,l >2 ... > q n > , i j q j = y j ( c 5 , t, ~ 1 ~2, ... > q n > + o j - 1 Y j - 1 ( j = 2, 3, ... a), where WX,z , Y,,y 2 , I . . , vn),y s ( x ,z , y l , y 2 , ... , y,) are obtained > Yn) by replacing from @(x,Z, ~ 1 Y,Z > yn), @ s ( ~ , Z, ~ 1 ~2, > (a

W E 5 9

2

2

5

9

3

the coefficients by their moduli. For this reason, the right-hand sides of our equations are holomorphic functions of the quantities tt, i,q1 , q 2 , ... , qn , vanishing when these quantities are zero and not containing i,q l , q 2 , ... , q,, in the first-order terms. From here it follows that for lt51 sufficiently small, our equations have a solution in which iand all qs can be represented by absolutely convergent series expansions in positive integer powers of €5, vanishing for c t = O . These series, considered as expansions in powers of E , coincide with (1 19). We can suppose 5 to be sufficiently small, so that series (119) remain convergent for E = 1 as well. In this case series (104) also remain absolutely convergent for

Investigation o f One of the Singular Cases of Theory of Stability of Motion

111

1 . Moreover, these series are uniformly convergent for all values of x between 0 and inclusive. We can therefore affirm that for sufficiently small x,the series 2 = dl) + 2 Q ) + z ( 3 ) + ... , Z, = ~ 6 +~ zb2) ) + ~ 6 +~ ...) ( S = 1, 2, ... , n ) represent some solution of the system (102), in which the functions z , z, vanish for x = 0. Let us remark about the character of these series, that if we put z ( j )= ~ i ( ~ )z x: ~~= ) , u6i)xi (s = 1, 2, ... , n), then u C i ) ,usi) are functions of x defined and continuous for x30. As one sees from the proof of their convergence, these series are such that we can take Z ( k + l ) + p + 2 ) + ... = X k + l y ( k f l ) , E =

<

$+I)

+ Z ; k + 2 ) + ... =

Xk+l

v,

(k+l) 3%

where V k + ' )Vs(''+') , are functions of x, defined and continuous for all sufficiently small positive values of x and also for x = 0. By what was shownin the previous paragraph we can conclude that the functions z, z, in our solution can be represented asymptotically by some series expansions in integer positive powers of x. It is not difficult to see that for the derivatives of these functions a similar representation is possible, and that the asymptotic series for the derivatives are obtained by differentiating the asymptotic series of the functions term by term. One sees from here that if we want to form these asymptotic series, we can determine their coefficients starting from Eqs. (102) and using the ordinary method of unknown coefficients. Return now to the problem of stability. Replace z , z, in the righthand side of Eq. (101) by the expressions found (supposing x > 0). This equation then takes the form

wheref(x) is a function, defined and continuous for any sufficiently small positive values of x and also for x = 0. The right-hand side of this equation is, therefore, such that for

112

Stability of Motion

nonnegative values of x not exceeding some sufficiently small bound, it remains positive, vanishing only for x = 0. From here we must conclude that however small the initial value of x will be, as long as it is positive, there always comes a moment when x reaches some fixed bound. For this reason, we conclude, as we did in the case considered in page 98, that the unperturbed motion is unstable.

27. We have left for consideration only the case when m-1 -4-1. 2

u=--

If in this case q is odd, we have a case which was already considered in page 42. For this reason with assumption (120), it is sufficient to limit ourselves to the case when q is even. We suppose, as in page 99, that the system (1) is considered after performing on it the transformation of page 17. With this, leaving a arbitrary, we t a k e g = -1. Therefore, the function Y will be of the form Y = - x Z 4 - ' g1x2¶ ... +y(ax4-' + a l x Y ...) Y,, where Yl has the same properties as in page 99. For x1 = x2 = .-.= x, = 0, the functions X , become expressions of the form q o ( x )+ycpl(x) y z q z ( x ) ..., in which cpo(x) and cpl(x)do not contain x to powers lower than 2q, and q, respectively. By making these assumptions and denoting by b some real constant different from zero, let us transform system (1) by the substitution y = bxy1 + z), x, = x2q-12, (s = 1,2, ... , n ) . (121) Then, if the number b can be chosen so that one can find, from the transformed system, an expression for x(dz/dx),holomorphic with respect to x, z, z,, and vanishing for x = z = z1 = z 2 = ... =z,=O, we are dealing with a case similar to that considered in 24. Let us see if such a value of b can always be found, and if not, under what conditions this happens. By substituting (121) one obviously gets for dxldt an expression of the form: dx - = bxy1 + z)(l + X X , ) , ( 122)

+

+ +

+

+

dt

+

Investigation of One of the Singular Cases of Theory of Stability of Motion

113

where X , is some holomorphic function of the quantities x,z, z, . We also find

where 2,‘ are holomorphic functions of the quantities x, z , z,. Further, we have : dY

dz

dt

dt

- = bX4 -

- p l - 1

+ qb2X’“-’(l + z)’(1 + X X , ) + g1xZ4+ ... + b x Z 4 - ’ ( 1+ z)(a + a l x + ...) + x 2 * Y z ,

where Y , is related to Yl by the equality: Yl = x z 4 Y 2 ,and therefore it represents, by (121), a holomorphic function of the quantities x, z , z,. From here we deduce dz bx - = - x 4 { 1 - ab(1 dt

+ z ) + qb’(1 + z)’) + x q f l Z ’ ,

where 2’ is some holomorphic function of the quantities x,z , 2,. We see, therefore, that the number b must be determined from the quadratic equation

ab + 1 = 0 (124) (which, if we would leave g arbitrary, would take the form qb’

-

q b 2 - ab = g). For this reason, our problem is only possible under the assumption a2 B 4q,

(125)

and admits one solution if this condition is satisfied with the equality sign and two solutions when it is satisfied with the inequality sign. Considering condition (125) fulfilled and taking as b one of the roots of (124), we find dz = -xq{(2qb - a ) + ~ qbz’} dt

X -

+ -b1 x 4 + 1 Z ’

Let us suppose first that the condition (125) is fulfilled with the inequality sign and of the two roots of (124) we take the larger one if a > 0 and the smaller one if a < 0. We have in this case: 2qb-a=a

, (1---, 3)”’

if the square root is taken positive.

Stability o f Motion

114

With this, we deduce from our equations dz

where C is some constant and Z a holomorphic function of the quantities x, z, z,, not containing terms lower than of second order. As the numbers a and b are necessarily of the same sign, the coefficient of z in the right-hand side of (126) is negative. For this reason Eq. (126) does not differ essentially from the first Eq. (102). Moreover, replacing x by Ax in Eq. (122) we get an equation of the same form, in which instead of b the number Aq--'bappears, which, for even q, can be made equal to any real number different from zero, by conveniently choosing the real number 2. By taking A4-'b = l / ( q - l), we reduce the equation considered to the form (101). Hence, the system of equations obtained by eliminating dt from (122) and (123) does not differ essentially from the system of the remaining n equations (102). As a consequence, we can affirm that when for even q, condition (125) is fulfilled with the inequality sign, the unperturbed motion is unstable. Suppose now that for even q the condition u2 = 4q is satisfied. Then Eq. (126) takes the form dz x -= c x dx

+ z.

Performing the above-indicated transformation, Eq. (122) becomes d x - x4 (1 dt q - 1

+ z)(l + X X , ) .

Finally, in order to determine z, as functions of x we have a system of the form x4 dz, q - 1 dx

- - = pslzl

+ ps2zp + ..* + psnzn+ c , x + z, (s = 1, 2,

... , n),

Investigation of One of the Singular Cases of Theory of Stability of Motion

115

where the C, are some constants, and the Z, are holomorphic functions of the quantities x, z , z, , not containing terms lower than of second order. Supposing q to be even and x 3 0, one can show that the system (127)-(128) has a solution similar to that found for system (102). T o prove this one can perform a completely analogous analysis. However, in order to perform the analysis from 24 and 26 for the present case, we have to make some modifications. With some changes in the notations, we can affirm, that for nonnegative x, which is sufficiently small, the system (102) has a solution which can be represented by series

in which z ( ; ) ,z;'), for x 2 0, are functions of x,defined and continuous together with all their derivatives. They can be represented asymptotically by series expansions in positive integer powers of x. We could prove also, that these functions are bounded. We prove now that the system (127)-(129) has a solution of the , Suppose form (130) of the same character as the functions z ( ~ )z,:~). that by performing the substitution (130) we get Z

=

Z(2)x' + z(3)x3+ ... ,

Z,

=

zpX2+ z;3lX3+ ...

(s = 1, 2,

... , n ) ,

where Z ( k ' ,Z i k )are known entire functions of those z ( ~ )z,: ~ for ) , from the which i < k . We determine now the quantities z ( ~ )zji) equations

x-

d(z(')x) = cx, dx

(s = 1,

2, ... , n )

116

Stability of Motion

and for i > 1

+ psnz$)xi+ 2S')xi

(s

=

1, 2, ... , n).

By putting now instead of .jib the functions ys", using substitution (107), and denoting afterwards the expressions corresponding to 2(i), Z J i )by , Y ( ' ) ,Y:i), we find from (131), (132), and (133) successively:

( j = 2, 3,

( j = 2, 3,

..

... , n ) .

Suppose now that i(i), u ] : ' ) are some upper bounds of the moduli of the quantities ,z(~),y:i) for x 3 0 and Y ( i ) "',6') the expressions, deduced from Y ( i ) ,Y,") by replacing the quantities z ( ~ )z, : ~ )by 5 ( k ) , qjk), and the coefficients of the expansions of 4, 4s(introduced after substitution (107) instead of 2, 2,) by their moduli.

Investigation of One of the Singular Cases of Theory of Stability of Motion

117

According to (134) and (135) we can take = $1)

ICI,

=t IB I

4

1 (1) q;') = : (oj.-lyj-l

,

'.i

+ Bj)

( j = 2, 3, ... , n),

((i) = yCi)

1 qy)=-yy)

1 ilj

11(.9 = - (a. J - l q (i) j-l+Yy))

'

1 1

( j = 2 , 3 ,..., n).

If we denote by Y, Y sthe functions deduced from 4, 4 s , by replacing the coefficients with their moduli, we deduce from here that the series 1; = px +p qs = q61'x

+ ... ,

x 2

+ 7 p x 2 + ...

( s = 1, 2, ... , n )

are the expansions in powers of x of the functions' L and qs satisfying the equations 1; = l C b + V

X ,

5, 111, 112, ... 11A 9

+ Y , ( X , i,vll?q 2 ... V n ) , A j y j = oj-lvj-1 + IBjlx + Y j ( x 3 5, 111, 112 ...

A,?,=

P l l X

9

9

3

9

qn)

( j = 2, 3,

... , n)

and vanishing for x=O. For this reason, these series are surely convergent for sufficiently small 1x1.We conclude from here, that series (130) are equally convergent for sufficiently small positive x. Thus, we can consider the existence of the solution as having been proved. Using this solution we prove as in page 112, that the unperturbed solution is stable. 28. Let us investigate now the case, when Eq. (124) has no real roots, i.e., when a2 < 49.

(136)

Assume that system (1) has been brought to the form, which we

Stability of Motion

118

used in page 43. Transforming it with the aid of substitution (22) in the present case (a = q - l), we bring it to the form: dr dt

- = aS2(9)Cq-'(9)rq+ rq+'R1 + r R 2 ,

dxs

= ps,xl dt

+ p s z x 2 + ... + psnxn+ x,

(s = 1, 2,

... , n ) ,

where R1,R, , O,, 0, , X, are functions of the same type as in system (28)-(29). With the present assumptions, one can treat system (137) in a manner similar to that in which system (28)-(29) was treated. One has only to make a few modifications of the methods because of some differences between systems (137) and (28). However, by a known transformation, system (137) can always be brought to form (28). In this connection, let us remark that with condition (136), the integral

s

S2(9)Cq-'(9) d 9 0 1 aS($)C4(9)

+

is a function definite and continuous for all real values of 9. Indeed, by (24) the maximum and the minimum of the function S(9)C4(9)are determined from the equation C Z 4 ( 9-qS2(9) ) = 0, which gives by (23) Czq(9)= qS'(9) = 5, and therefore, S(9)Cq(9)= &+2/q. For this reason the minimum of the function 1 +as($) xCq(9) is equal to 1 - lal/(22/q), and therefore by (136) it is positive. If, however, q is even, which, of course, we suppose here, then the integral (138) is, moreover, a periodic function of 9 of period o.Indeed, by assumption, the functions S and C are such that

Investigation of One of t h e Singular Cases of Theory of Stability of Motion

119

For this reason, if q is even

j wS2(9)Cq-'(9) dS-JI12 1 + aS(9)Cy9) 0

+s,.

S2(9)C4-'(9) d9 iw =-I 1 nS($)Cy9) Ol2

0/2

0

-

s

I'

=

012

0

S2(9)Cq-'(9)d9 = 0, 1 aS(9)C4(9)

*

0

S2(9)Cq-'(9) d9 = 0. 0 1 aS(9)C4(9)

J5!'

+

I',,.

+

Noting this, introduce instead of of the equation r=pexp[aJ 0

Y

the new variable p by means

I

S2(9)Cy-'(9) d9 1 + aS($)C4(9) '

(139)

and instead of 9, the new variable cp through the equation

The variable cp defined by this equation is an increasing function of 9, varying by some determined quantity CT each time when 9 varies by o.This quantity CT is given by * S2(9)C4-'(9) d$] d9 0 1 + aS(9)Cy9) 1 + aS(9)C4(9)'

w

(141)

Conversely, by (140), 9 will be defined as an increasing function of cp, varying by the quantity o each time cp varies by CT. For this reason, any periodic function of 9 of period o is transformed by (140) into a periodic function of cp with period 0. Introducing now the variables p and cp into Eqs. (137), we bring them to the form dP

dt = pqC1P1+ p P , ,

* In the manuscript, as the upper limit of the external integral in formula (141), 8 is written instead of w (Editor's note).

Stability of Motion

120

where PI and Q1 are holomorphic functions of p , independent of the quantities x,,in which the coefficients are real periodic functions of the real variable cp with the period 0,defined and continuous, whereas P, and Q 2 are holomorphic functions of the quantities p , x,, vanishing for x1 = x 2 = ... = x, = 0, and having similar coefficients. The system (142) has thus all the characteristic properties of system (28). Therefore, in the present case the stability does not depend just upon the quantity a, but in order to solve the problem, we have to make use of the method presented in page 46, and in the following pages. Let us remark that the integral (138) is expressed in terms of the functions S and C in the following manner: J'

0

S2(3)Cq-'(9)d9 1 1 aS(9)C4(3) q(4q - a )

,

+

1

--

2qa

log(1

arctan

(4q - a2)1'2S(9) 2cy3)

+

+ aS($)C4(9)).

Here arctan is taken so that it vanishes for 9 = k(w/2) for any integer 4. For 9 = (4k l)(w/4) it is equal to an angle between 0 and 4 2 , the tangent of which is (4q - ~ ~ ) " (under ~ / a the assumption a > 0) and for 9 = (4k 3)(0/4) it is an angle situated between -(7r/2) and -71, having the same tangent. These angles are the maximum and the minimum of this arctan.

+

+

29. Let us summarize the results found, limiting ourselves, for simplicity, only to the case of second-order systems. Let the system

dx _ -y+x, dt

dY

-=

di

y,

(143)

be given, in which X and Yare holomorphic functions of x and y , not containing terms lower than of second order. From the equation y X = 0 we deduce

+

Y = F(x),

(144)

where F(x) is a holomorphic function of x, not containing x to powers lower than the second one.

Investigation of One of the Singular Cases of Theory of Stability of Motion

121

Suppose that replacing y by this function in the expressions Y and (8Yjdy) - F'(x)[l (axjay)]we get

+

y ay _._

6Y

=

f(4,

3

F'(x) 1 + -

= q(x).

Expanding in increasing powers of x we can meet the following cases : I. f(x)=q(x) = 0. In this case the unperturbed motion is unstable. f 11. ( x ) = 0, q(x) = gx" +glxn*+' ... . In this case, for nz odd, the unperturbed motion is unstable, whereas for m even it is stable if g < 0 and unstable if g > 0. 111. f ( x ) = gx" + g l x m f l ... , m even or odd and g > 0. The unperturbed motion is unstable. IV. f ( x ) = gx" g l x m+' ... , m odd and g < 0. q(x) = axa alxa+l ... , or q(x) = 0. There are the following possible cases : (1) CI even, less than m. In this case for a < 0 the unperturbed motion is stable and for a > 0 it is unstable. (2) CI odd, less than (m - 1)/2. The unperturbed motion is unstable. ( 3 ) CI odd = (m - 1)/2 and a2 2(m 1)s 3 0. The unperturbed motion is unstable. (4) c( > m , or q ( x ) = O , or CI is an odd number larger than (m - 1)/2, or for CI odd, CI = (m - 1)/2 and a2 2(m 1)s< 0. In this case, denoting by Fm(x) the group of terms of the function (144) that contain x to powers not higher than (m 1)/2, let us transform Eqs. (143) by using the substitution x = ( - g ) - li(m-1)a$),

+

+ +

+ + +

+

+

+

+

+

= F , ( ~ )- ( _ g ) - l / ( m - l ) r ( m + l ) / 2 ~

($13

where S(9) and C(9) are periodic functions of 9, determined from the equations C'(9) = - S(9), S'(9) = C"(9) with the conditions S(0) = 0, C(0)= 1.

122

Stability of Motion

Eliminating dt from the transformed equations, we obtain from them the following one: drld9

+ R,r2 + ... ,

= R,r

(145)

where R,,R, , etc., are some periodic functions of 9, of which R, is identically zero except in the case when a = (m - 1)/2. On the other hand, however, R,is such that the integral JR,dS is also a periodic function. Denoting by c the value of r for 9 = 0, we integrate Eq. (145) by using the series r = ulc u2c2 u3c2 ..., where u,, u, , u s , etc., are functions of 9 independent of c, of which the first one u, = exp[J;R,dS] is always periodic. It may happen that so are also the remaining ones (which really is the case, for instance, when Eq. (145) has a holomorphic first integral, independent oft). In this case the unperturbed motion is stable. If not all these functions are periodic, the first nonperiodic one in the sequence u,, u, , u3 , ... , which we denote by uk (the number k is always such that k (m 1)/2 is even), can be represented in the form exp[J~R1d9](hS u), where h is a constant different from zero and u is a periodic function. In this case, for h > 0, the unperturbed motion is unstable, and for h < 0 it is stable. We remark that the functions us are, of course, all periodic, if the system (143) is such that by replacing x by -x and t by -t, it remains unchanged. The same holds also in the case when the system remains unchanged by replacing y by -y and t by -t. In both cases, Eq. (145) possesses a first integral, holomorphic with respect to r , and periodic with respect to 9. In these cases it is possible to have for system (143) a holomorphic first integral independent of t. However, for the existence of this first integral it is necessary to fulfill some supplementary conditions. In general, from the periodicity of the functions us the existence of a holomorphic first integral independent of t does not follow as a necessary consequence.

+

+

+

+ + +

On the Problem of the Stability of Motion* The present note represents a small supplement to the work “The General Problem of Stability of Motion” (Charkov, 1892; Izdanie Char’kovskogo Matematiceskogo Obshcestva-edited by the Charkov Mathematical Society.) We supposed there that the right-hand sides of the differential equations of the perturbed motion, reduced to canonical form, are represented by series expansions in positive integer powers of the unknown functions. Adding some general assumptions, which will be described in the sequel, we exhibited a condition, under which the solution of the stability problem does not depend upon the high-order terms of the series mentioned, and we proved its sufficiency. The purpose of the present note is to show how the necessity of the same condition can be proved. Let x l , x 2 , ... , x, be the quantities with respect to which stability is investigated. In the differential equations of the perturbed motion, they are the unknown functions of the time t. These quantities are some given functions of positions and velocities of the considered material system. Their expressions can depend explicitly upon the time. These functions are supposed to vanish for the motions, the stability of which is investigated. We call the latter unperturbed motions. As for the perturbed motions, they satisfy differential equations of the form: dxs -= pslxl di

+ ps2x2 + ... + psnx, + x,

(s = 1, 2,

... , 0).

(1)

* Zapiski Char’kovskogo Universiteta, 1, 99-104, (1893); SoobshEeniya Char’kovskogo MatematiEeskogo ObshEestva, Ser. 11, 3, No. 6, 265-272, (1893). 123

Stability of Motion

124

Herep,&, 0 = 1,2, ... , n) are real constants, whereas X,,X,, ... , X,, are known functions of the quantities x,, x 2 , ... , x,,and t. For sufficiently small Ix,I, these functions are represented by series

x,=C p , ( m i . m z , . . . , m , )

x1 m ix 2 mz

...p (ml

+ m , + ... + m, > l),

which are expansions in positive integer powers of the quantities x,, not containing terms lower than second-order with respect to them. The coefficients of these series are either real constants, or continuous real functions of time. We suppose them to be so that one can find positive constants M and A such that inequalities of the form

P11

--x

P21

Pn1

P12

PZZ

-x

Pnz

..*

... ...

Pln P2n Pnn

-X

= 0.

125

On the Problem of the Stability of Motion

condition reads as follows: the smallest of the numbers ( 2 ) is diferent from zero. The sufficiency of this condition can be shown in the following way. For the cases when the smallest of the numbers (2) is positive, one proves that the unperturbed motion is stable and for the cases when it is negative, one proves that it is unstable. Here only the general assumptions about the functions X,,formulated at the beginning, are taken into account.* In order to prove the necessity of our condition, one has to prove the following. Whatever the constants pso are, the functions X , can always be chosen so that stability or instability holds as desired, provided that the smallest of the numbers ( 2 ) is zero. The fact that under this assumption the named functions can be chosen so that instability holds, follows already from results given in the above-mentioned work but, of course, it is also very easy to prove this directly. Thus it only remains for us to show that if the smallest of the numbers ( 2 )is zero, we can always find functions X , for which the unperturbed motion is stable. We consider first two particular cases in which the numbers (2) are all zero. Let system (1) be of the form : dx1 _-

dt

- x-1, (3)

dxi _ - X , - ~ = X ~ ( i = 2 , 3 ,..., a ) dt

and cpl,

...,

the functions determined successively (for cpo,”+ provided that (P,+~ = O . It is easy to see that if X,= - 2 ~ , + ~ q 1(s= ~ +1,~ 2, ... , n), the function b1 is a first integral of system (3). This function is continuous and single-valued and can vanish for real values of x , , only when x 1 = x 2 = ... = x, = 0. Thus for the chosen X,, the unperturbed motion is surely stable. cpz,

(P,

s = n, n - 1, ... , 2 , 1) from the equations cps = x,’

* “The General Problem of Stability of Motion”

+

(see 26).

126

Stability of Motion

Suppose now that system (1) is of even order, n = 2m,and of the form :

dxi - = -pyi dt

+ xi-1 + x i ,

dYi - P X i + yi-1

dt

+6

(i = 2, 3,

(4)

... , m ) ,

where y,, Y, are new notations for x, +, , X,,, +, . Let q l ,c p 2 , ..., cpn be the functions determined successively from the equations cp, = xS2 yS2 q,"+ provided that cp, + = 0. Then, if

+ +

x,=

-%+I

( P S+ l ,

,

y, = -2Y,+lqS+l

(s = 1 9 2 , ... , m>,

the function q, is a first integral of (4), as it is easy to see. Since this function, for real x, , y , , can vanish only for x1 = x 2 = ... = x , = Y l = Y 2 = ... - y , = 0, we must conclude as before that for this choice of the functions X,, Y,, the unperturbed motion is stable. Let us now consider the general case. We remark that whatever the constants psc are, one can always find a linear substitution with real constant coefficients, which transforms system (1) into another one, which is separated into groups of equations, pertaining to one of the following two types: d_ Y, - -lY, dt

+ Yl,

or

dzi _ -- pyi - l Z i + z i - I + zi dt

( i = 2, 3,

... , k ) ,

On the Problem o f the Stability of Motion

127

where Y , , 2, are the collections of high-order terms with respect to the unknown functions. Here the case k = 1 is not excluded, where the groups of form (5) are reduced to the first equation whereas the groups of type (6) are reduced to the two equations of the first line. In these equations 1 is one of the numbers (2). Thus, if among the latter there are no negative ones, in order to make the unperturbed motion stable one has only to put Y,= Z , = 0 in all the groups with 1 > 0, and also in those for which k = 1 . In the groups with 1= 0, k > 1, one has to take the highorder terms as was just shown in the two particular cases. The necessity of the above-mentioned condition can therefore be considered as proved. As a matter of fact, this condition is necessary only as long as all the systems of the form (1) are considered. If it,is required to consider only systems of some determined type, then, this condition remaining sufficient,may be no longer necessary. Thus, for instance, if one considers only canonical systems with constant coefficients, this condition is surely not necessary.

An Investigation of One of the Singular Cases of the Theory of Stability of Motion, IF 1. The problem we are talking about here consists of the following. Given a system of differential equations of the form dx _1 -- x1, dt

dx,

dt =

x,,... ,

-dx, dt_

-Xn,

where XI, X,, ... , X,are known real functions of the real variables xl, x,, ... , x,, t, vanishing for x1= x, = ... = x,= 0. The quantities x, are the unknown functions of the variable t. It is required to decide whether for every positive number 1 one can choose another positive number E such that, whenever the initial values of the functions xn, corresponding, say, to t = 0, are submitted to the conditions lxll < E, Ix2i < E , ... ,Ix,I < 6 , the following inequalities hold for all positive values off: 1x11 < 1,

< 1, ... Ixnl < 1. (2) Suppose that f is the time and x l , x, , ... , x,, are given functions of positions and velocities of a material system under the action of some forces. These functions may depend explicitly upon the time. Let us assume that some motion of the material system corresponds to every real solution of system (1). In this case, the formulated problem is the problem of the stability of the motion x1= x, = ... = x, = 0 with respect to the quantities xl, x2, ... , x,.t If the answer to the stability problem is in the affirmative the motion is called stable. If the answer is negative, it is called unstable. 1x21

9

* MathematiEeskiy Sbornik, 17,No. 2, 253-333 (1893). t One can consider a more general problem: the stability of the same motion not with respect to all the quantities x l , xz , ... , x,,, but only with respect to x,, x 2 ,... , x, (rn < n). One obtains this problem from the previous one replacing inequalities (2) by lxll

< Z,lxzl < 1, ... , Ix,I < 1. 128

Investigation of One of the Singular Cases of Theory of Stability of Motion

P11

-x

P21

P n1

P12 P22

-x

...

Pn2

... ... ..’

129

Pln P2n

Pnn

= 0,

(3)

-X

* See “On the Problem of the Stability of Motion” (Soobsheeniya Char’kovskogo matematiEeskogo obshEestva series 11, vol. 111, 1893). This paper is included in the present book (Translators’ note).

130

Stability of Motion

Two of the simplest singular cases were considered in “The general problem of stability of motion ” (Charkov, 1892; edited by the Charkov Mathematical Society). In one of these cases equation (3) has one zero root, the remaining roots having negative real parts. In the second case the equation has two purely imaginary roots, the remaining ones being as in the first case. Here we intend to treat some more general case, when equation (3) has two zero roots and if the order of the system is higher than two, all the remaining roots have negative real parts. We do not consider this case in all its details, but limit ourselves to the assumption that at least one of the,first minors of the determinant appearing in Eq. (3) does not vanish for x = 0. Under this assumption it is always possible to find a linear substitution with constant real coefficients, which transforms Eqs. (1) into

where X , Y, 2, are holomorphic functions of the new variables x,y , zl, z2, ... , z k (k = n - 2), not containing in their series expanssion termslower than second order. Here qso are real constants such that the equation 411-x

-.. I . .

qkl

‘**

qlk 4kk-X

has only roots with negative real parts. Equations (4)represent the starting point of our investigation. The solution of the problem will be based upon the principles established in the above-mentioned work and we will be guided by considerations made there for similar problems, though less complicated ones. Before treating the general case, we consider the case when k = 0, i.e., when the system of equations under investigation is of order two. In this case the problem allows many simplifications, which, in general, are impossible for systems of higher order.

Investigation of One of the Singular Cases of Theory of Stability of Motion

131

We shall see that in this special case the problem is not essentially different from that for the cases treated in the above-mentioned work. On the contrary, when dealing with systems of higher order, we meet some new situations causing in the known cases very peculiar difficulties.

Chapter I SYSTEMS OF SECOND ORDER 2. Suppose that the stability problem under consideration is reduced to the investigation of the equations

-dY= K

-dx =y++, dt

dt

where X and Y are holomorphic functions of x and y , the series expansions of which do not contain terms lower than of second order. These functions are independent of t. It is easy to see that replacing y by some new variable y , , this problem can be reduced to the investigation of equations of the same form - = y , + X I , -dY1 dx - = y,, dt

dt

where X I is an arbitrary given holomorphic function of x and y,, not containing terms lower than second order. Indeed, to this end it is sufficient to impose the following relation between x, y , and y , : y X = y , X,. In view of this relation, for sufficiently small 1x1,ly( and lyll, y , is a holomorphic function of x and y , vanishing for x = y = 0, whereas y is a holomorphic function of x and y,, vanishing for x = y1 = 0. With such relations between the variables, the problem of stability of the motion x = y = 0 with respect to x and y is equivalent to the problem of stability of the motion x = y1 = 0, with respect to x and y,. Suppose that our transformation is taken so that X , vanishes for y 1 = 0. Under this assumption, expand the function Y, in increasing powers of y , : Yl = f ( x ) cp(x)yl ..-. Here f ( x ) , cp(x),

+

+

+

+

Stability of Motion

132

etc., are holomorphic functions of x. Iff(x) is not identical zero, then expanding it in increasing powers of x and assuming a to be a constant different from zero, we writef(x) = axa alxa+' .... Here the number tc is not less than 2. In the same manner, if q(x) is not identical zero, then, expanding it in increasing powers of x, and assuming b to be a constant different from zero, we have q(x) = bxp blxp+' .... Here the number /?is not less than 1. The numbers a, a, b and p play a very important role in our investigation. From the infinite set of transformations submitted to the condition that XI vanishes for y1 = 0, we show here one yielding the simplest mode of constructing the functions f ( x ) and q(x). This transformation is given by the substitution y = F(x) y,, where F(x) is the holomorphic function of x vanishing for x = 0, obtained by solving the equation y X = 0 with respect to y. It is obvious that using this transformation one finds f ( x ) replacing y by F(x) ir. the expression for Y.* In order to find q(x) one has to perform the same substitution in (axjax) (8 Yjay). If we use other transformations, we obtain, of course, other expressions forf(x) and cp(x). However, as it is not difficult to see, whatever our transformation will be, these expressions come out in the form:

+

+

+

+

+

+

+

f(x)=f1(x>II1+O(x>I, q(x) = qdx) + f I ( X ) W .

Herefl(x) and q l ( x ) are the functionsf(x) and q(x) corresponding to some particular transformation, whereas O(x) and $(x) are holomorphic functions of x, depending on the choice of our transformation. $(x) may be any function, but O(x) is always such that it vanishes for x = 0. From these expressions it follows that if, for a particular transformation, the functionf(x) is identically zero, it will be so for all transformations. In this case the function q(x) will not depend upon the choice of the transformation. If f ( x ) is not identically

* In Liapunov's collected works, Y , is mistakingly written. In the original paper, it is written correctly as Y (Translators' note).

Investigation o f One of t h e Singular Cases o f Theory of Stability of Motion

133

zero, then, though it can vary from one transformation to another, the numbers a and u remain always the same. If p < u, the numbers b and p are also invariant. However, if p > a, b and p can be made equal to any values, provided that the last condition is satisfied. It follows that b and fi are of interest to us only when p < u or when f(x) = 0. We start our investigation from this last case.

3. Here, and in the sequel, unless we state otherwise, we assume that the function X vanishes for y = 0. As we saw, this assumption may require the transformation of system (1) by some substitution which is always possible. Put, therefore, Y =f (x) cp(x)y ..., wheref(x) and q(x) are the functions considered in the previous paragraph. Let us begin with the case when f ( x ) = 0, identically. In this case, the right-hand side of the equation

+

+

obtained from system (l), is holomorphic in x and y . Thus, considering y as a function of x, and denoting its values for x = 0 by c, we find by a known theorem y =c

+ $(x,

c),

(3)

where Icl(x, c) is some holomorphic function of x and c, vanishing for x = 0. Using this expression for y , we bring the first equation (1) to the form

-dx_ - c + 6(x, dt

c).

(4)

The function O(x, c) is also holomorphic and moreover, like $(x, c), it does not contain terms lower than second order with respect to x and c. If not onlyf(x), but also cp(x) is identically zero, the right-hand side of Eq. (2) vanishes for y = 0, and therefore the function $(x, c ) vanishes for c=O. In this situation the function O(x, c) has the same property, and therefore the left-hand side of the equation

i

0

cdx

c

+ 6(x, c) = ct + c',

134

Stability of Motion

deduced from (4), is holomorphic in x and c. Here c' is an arbitrary constant. From this equation we deduce therefore that in the case q(x) = 0, the unperturbed motion is unstable. Suppose now that the function q(x) is not identically zero. For y=O, the right-hand side of Eq. (2) becomes a holomorphic function of x with the lowest order term bxP.Therefore, the lowest order term of the function $(x, 0) is [b/@ l)]xP++'. This also holds for the lowest order term of the function Q(x,0). As a consequence, if we consider the particular solution of system (1) corresponding to the assumption c = 0, Eq. (4) takes on the form:

+

where the unwritten terms contain higher powers of x. One deduces easily from this equation that when p is odd, or when p is even and b is positive, the unperturbed motion is unstable. Therefore, what remains to be considered is only the case when is even and b, is negative. Let us first remark that whatever the red constant c may be, when p is even we can always write c - [b/(p l)]yP+', where y is also a real constant. Replacing c by this constant in the right-hand side of Eq. ( 3 ) , this becomes a holomorphic function of xand y , of which [b/(p I)] (xfl+l- y P + ' ) is the collection of lowest order terms, with respect to x and y. Thus, putting c $(x, c) = 0, we get p 1 definitions for x as a holomorphic function of y. Denote the only one of them which is real, by o(y). The lowest order term of o(y) is y. Therefore, if instead of x we introduce the variable 5 by putting x = o(y) + & the right-hand side of Eq. (3) becomes a holomorphic function of 4 and y, vanishing for 4 = 0. The collection of lowest order terms, with respect to 5 and y, of this function is [b/@ I)][(< y ) B + l yfl+l]. The right-hand side of Eq. (4) will be of the same form. The latter is transformed into

+

+

+

+

+

dC/dt

=

bH5,

+

(5)

where H is a holomorphic function of 4 and y. The collection of

Investigation of One of the Singular Cases of Theory of Stability of Motion

135

+

lowest order terms of this function is expressed by [(4 Y ) ~ + ' Y P "IMP 1MI. For p even this form is positive definite, and therefore, it is possible to find an upper bound Z, different from zero, for the absolute values of y and c, such that for any real y and 5 satisfying the conditions j y ] < I , 151 < f , the inequality H > 0 holds. Let us assume that y and to(the value of the function 4 for t = 0) are taken to be less than f in magnitude. In this case, writing Eq. (5) in the form

+

it follows that if b is negative the condition (i( < (to(holds for all positive values o f t . Taking into account the existing relation between x,y , 5, and y , it is not difficult to deduce from here that the unperturbed motion is stable. 4. As we saw, when p is odd the unperturbed motion is unstable. We can show, however, that in this case it is somehow conditionally stable. Namely, we can show that it is stable with respect to the perturbations satisfying the condition by

< 0.

(6)

To this end consider the equation y

+ Y(X,

Y> = c,

(7)

obtained from (3) by solving it with respect to c. Here Y(x, y ) is a holomorphic function of x and y able to be represented in the form Y(x,y ) = yv - [b/(P l)]xs+'(l u). Here u and v are holomorphic functions, vanishing for x = y = 0. Thus, with fi odd, the left-hand side of Eq. (7), as a function of x and y , has permanently the opposite sign of b when condition (6) is satisfied and as long as the absohte values of x and y are sufficiently small. Furthermore, it vanishes only for x = y =O. In other words, in conformity to the terminology used in our abovementioned work (and used also here), this is a sign-definite function when condition (6) holds.

+

+

136

Stability of Motion

As the function Y vanishesfor y = 0, this condition being valid at the initial moment of time, it will necessarily hold,at least all the time the quantities x and y remain sufficiently small in magnitude, to ensure that X and Y can be represented by series expansions in powers of x and y. Thus, reasoning, as is customary in such cases, we deduce from Eq. (7) that no matter how small the given positive number 1 is, one can find a positive number E such that when the conditions by < 0, 1x1 < 6 , lyl < t are satisfied at the initial moment of time, the functions x and y always remain smaller in magnitude than 1. One can notice that in the case considered (i.e., for B odd), the unperturbed motion is also stable with respect to the perturbations satisfying the conditions y = $(x, O),

b x < 0.

(8)

However, the stability corresponding to condition (6) differs essentially from that corresponding to the last conditions: in the case of condition (6), when the initial values of x and y are sufficiently small in magnitude, they vary within arbitrary small limits not only when t increases beyond all bounds but also when it decreases without limit, whereas in the case of conditions (8) the situation is such only when t increases beyond all bounds. We see here, therefore, examples of two kinds of stability: one which is conserved when the time t is replaced by -t and the other which is not conserved by such a transformation. The stability of the first kind is called conservative. We have shown in the previous paragraph that when B is even and b is negative, stability holds. This stability is, obviously, not conservative. Cases of unconditional conservative stability that can be met in the problem under discussion, will be shown in their proper place. 5 . It is easy to see that in all the cases of conditional or unconditional stability considered so far, x and y tend to some limits when t increases beyond all bounds, if their initial values are sufficiently small in magnitude. The limit for y is always zero.

investigation of One of the Singular Cases of Theory o f Stability of Motion

137

We express this fact by saying that the perturbed motions tend asymptotically to one of the motions defined by the equations x = C, y = 0, C being an arbitrary constant. In the case when f ( x ) = O* identically, such motions always exist. Supposing I C( to be sufficiently small, let us consider the stability problem for these motions with respect to the quantities x - C and y. We suppose C to be so small in magnitude that by performing the substitution x = C 5 the right-hand sides of Eqs. (1) become holomorphic with respect to 5 and y . Under this assumption, the equations mentioned are brought into the form

+

- -- (1

dt

dY

+ C > y+ ... ,

- = q(C)y

dt

(9)

+ *..

Only the first-order terms with respect to 4 and y are written here. The constant C' is a holomorphic function of the constant C, vanishing for C = 0. Since the characteristic equation? of system (9) has the roots: q ( C ) and 0, whenever q(C) > 0, instability holds. When q(C) = 0 we have the same situation if 1 C' is not zero, since this case can be reduced to that considered in page 133 when q(x) vanishes identically. If, finally, q(C) < 0, then, as the right-hand sides of Eqs. (9) vanish for y = O , according to results proved in the abovementioned work, we must conclude that stability holds. In conclusion, assume that ICI is so small that if the func+:on q(C) does not vanish identically, it has the sign of its lowest-order term bCBand that 1 C' is different from zero. Then we can affirm that if q(x) = 0 identically, all the motions considered are unstable. In the opposite case, if ,8 is odd, of the motions considered those are

+

+

* f i s missing in Liapunov's collected works but present in the original text (Translators' note). -f We refer thus to Eq. (3) corresponding to system (1) (page 128).

138

Stability of Motion

unstable for which bC > 0 and stable are those for which bC <0. If, however, is even, all the motions considered are unstable for b > 0 and stable for b < 0. 6 . Let us consider now the case whenf(x) is not identically zero. In most of the cases pertaining to this category, the solution of the problem under discussion follows immediately from a general statement, which is easy to prove. This statement plays a very important role in my work, mentioned above more than once. This statement consists of the following. Suppose that it is requested to find a real holomorphic function V of the variables x and y , the derivative of which with respect to t, V’= ( y X)(dV/dx) Y(aV/ay),formed by virtue of Eqs. (I), is a sign-definite function of the same variables. If V , which is supposed to vanish for x = y = 0, can be made to take the same sign as V’ by a convenient choice of x and y real and arbitrary small in magnitude, the unperturbed motion is unstable. lf, however, V itself is a sign-definite function the sign of which is opposite to that of V’, the motion is stable. One can add, that in the latter case every perturbed motion corresponding to perturbations which are sufficiently small in magnitude, tends asymptotically to the unperturbed motion. In order to apply this statement to the case in which we are interested, let us return to the equation aF dF (x- + X ) - + [ Y - f ( x ) ] - = y@,

+

+

ax-

aY

in which @ is an arbitrary given holomorphic function of x and y , whereas F is an unknown function of the same variables. This equation yields for aF/ax an expression of the form (aF/ax) = P(aF/ay) Q, in which P and Q are holomorphic functions of x and y . It follows, according to a known theorem, that this equation always has a solution F, holomorphic too and, furthermore, for x = 0 it can be made equal to an arbitrary given holomorphic function of y . This last condition determines the function F entirely. Our problem is now to choose suitable assumptions for @ and this holomorphic function of y.

+

Investigation of One of the Singular Cases o f Theory of Stability of Motion

139

Put (I, = ay, where a is the coefficient of the lowest order term axa off(x), and require F to be equal to y for x = 0. Denoting by V the holomorphic function Fdetermined under these assumptions, we have in virtue of Eq. (10) I/' = ay*

+f(x)(av-/a.y).

(11)

From the definition of V it follows that aVjay is equal to 1 for CI is even the obtained expression of V' is a sign-definite function of x and y . Remarking that the function V itself can be made to take values of arbitrary sign, we conclude from here that when c i is even, the unperturbed motion is always unstable. Let us take now the same Q, as previously, but impose that F vanishes for x = 0. If V is the function F corresponding to these assumptions, we have as previously, equality (1 1). However, in the present case V does not contain terms lower than of second order with respect to x and y , the unique second order term being axy. Thus aV/ay, which vanishes for x = 0, is of the form aVj8y = ax( 1 u), where u vanishes for x = y = 0. It follows that for CI and a positive, the function V' considered here is sign-definite. The function V , however, is able to take values of any sign, as previously. We must conclude, therefore, that for c( odd, if a is positive, the unperturbed motion is unstable too. Suppose finally that CI is odd and a is negative. Assume, furthermore, that < CI. Let y Y(x, y ) be a holomorphic function of x and y , satisfying Eq. (10) in the assumption that (I, = 0, and becoming equal to y for x = 0. The function Y(x, y ) will not contain terms lower than second-order with respect to x and y . Moreover, it will be such that the lowest order term of Y(x, 0) is equal to - [b/(fl 1)]xp++'. Such will also be the lowest order term of the function $(x) if

x = 0. As a result, if

+

+

+

cy

+ Y(X, y)I[l +

= $(x)

+ y(1 + H ) ,

where H is a holomorphic function of x and y , vanishing for

140

Stability of Motion

x = y = O . Put now in Eq. (10) cD=by"+p-2(1 + H ) f ( x ) , and impdse on the holomorphic function F the condition that it vanishes for x = 0. The function F determined under these assumptions, and denoted by U, will obviously be of the form U = - [2a/(a l ) ] x a + ' u, where u does not contain terms lower than of the ( u 2)th order with respect to x and y. As a consequence, if we put V = [ y Y(x,y)]' U , this function V will be positive definite under the adopted assumptions. Take the derivative of V with respect to t by virtue of Eqs. (1). As V necessarily satisfies Eq. (10) for the chosen cD, we find the following expression for V':

+

+

+

+

+

By assumption p < a and the function f(x)(au/ay) does not contain terms lower than of the (212 1)th order with respect to x and y . It follows that the collection of lowest order terms in the expression for V' is reduced to +'- [2ab/(B l ) ] X a + p + l . For even p the function V' is therefore sign-definite. It has the same sign as 6. Taking into account the above-shown property of V we deduce that if, in our assumptions, p is even, the unperturbed motion is unstable when b is positive. When b is negative, it is stable and every perturbed motion corresponding to a perturbation which is sufficiently small in magnitude, tends asymptotically to the unperturbed motion. Of all the possible cases what remains to consider are only those when u being odd and a negative, either p is less than u and odd, or p >, u, the latter including also the case q(x) = 0. In order to investigate these cases we must have recourse to considerations different from the previous ones.

+

+

Investigation of One of the Singular Cases of Theory of Stability of Motion

141

7. Let us consider the equation aY (y+X)-=Y,

ax

deduced from system (1) by eliminating dt. Let us see whether it is possible, under the assumption that f ( x ) is not zero, to satisfy this equation supposing that y is a holomorphic function of x vanishing for x = O . Let hxk be the lowest order term in the expansion of this function in increasing powers of x , k being some positive integer and h, a constant different from zero. Since, by assumption, X does not contain terms independent of y , the lowest order term of the left-hand side of Eq. (12) will necessarily be kh2xZk-'. This also must be the case for the lowest order term of the righthand side of this equation. This being of the (2k - 1)th order, will coincide necessarily with the lowest order term of the function .f(x>+ rP(X>Y.

+

(13)

+

We have now to analyze three cases: (1) a > p k ; (2) a < p k [or q(x) = 01; and (3) a = k . In the first case, the lowest order term of the function (13) is bhxS+k,and therefore k = p 1, h = b/(p 1). This case is characterized by the relation

+

+

+

p < (a - 1y2.

(14)

I n the second case the lowest order term of the function (13) is axa, so that k

+ 1)/2,

= (a

h

= +[~u/(R

+ l)]'".

Therefore this case is possible only when is not zero, under the condition that

p > (.

- 1y2.

a

is odd and, if q(x) (15 )

If, however, we looked for real solutions of Eq. (12), i.e., t takes real values for real x, then we should also impose the condition a >O.

Stability of Motion

142

Finally, in the third case, we have two subcases corresponding to the assumptions: (1) 2k - 1 > a, and (2) 2k - 1 = a. We shall not discuss here the first subcase in which we would have k = c1- B, h = - a/b and, as in the first case, p < ( c t - 1)/2. This case is only possible when some known relations between the coefficients of the functionsf(x) and q(x) hold. As to the second subcase, we have k = (N 1)/2 = p 1, whereas h is determined from the equation ( p + l)h2 - bh - u = 0. (16) Thus, the second subcase is only possible if

+

+

p=- a - 1 2

and can lead to real solutions only if bZ + 4(8

+ 1)a 2 0.

(18)

Conditions (14), (17), and (15) or q(x) = 0 include all the possible cases. As we saw, if one of the two later conditions holds, the case of even LY must be excluded. If we do this, we can easily prove that in all the remaining cases our problem has always a positive answer. Suppose, for instance, that we deal with one of the cases when condition (14) or (17) holds. We transform Eq. (12) using the substitution y = x B + l ( h+ z ) , where h is equal to b/@ 1) when condition (14) holds, or to one of the roots of (16) in the case of condition (17). Remarking that in both cases f ( x ) hx8+lq(x) = (p 1) x h2xZBt1..-,where the omitted terms contain higher powers of x, we find that in the transformed equation

+

+

+

x

dz

- = -(p dx

+

+ l)(h + z ) + (Y +Yw x p

the right-hand side is a holomorphic function of x and z , vanishing for x = z = 0. This equation is therefore of the form dz

x-

dx

=l z +px

+ z,

where A and p are some constants and 2 a holomorphic function,

Investigation of One of the Singular Cases of Theory of Stability of Motion

143

not containing terms lower than second order with respect to x and z. Furthermore, one gets for A the following expression: A = [b- 2(8 l)hl/h. Thus, in the cases when condition (14) holds, A = - (p + l), whereas in the cases when condition (17) holds, A is obtained as a root of the equation aA2 [b2 4(p l)a]A (p l)[bZ 4 x (p l)a] = 0. When this equation has real roots, i.e., when condition (18) holds, it yields either 3, = 0, or two values for A, one of which being surely negative, Thus we can always assume that A is not a positive integer and as it is known, this is a sufficient condition for the existence of a holomorphic solution z of Eq. (19), vanishing for x = 0. The possibility of solving our problem in the cases when conditions (14) ar (17) hold can be considered therefore as proved. In a similar way its solubility is also proved in the case when the condition (15) is valid [or q(x) = 01, a being odd. We notice that in the cases when condition (14) holds, the considered holomorphic solution of equation (12) is always real, whereas when condition (17) is valid, it is only so when condition (18) holds. If a < 0, these cases are the unique ones in which our problem can admit real solutions.

+

+ + +

+

+ +

+

8. The analysis presented in the previous lines leadsimmediately to the solution of the stability problem in a number of cases, unconsidered so far. Assume that either condition (14), or conditions (17) and (18) hold. As we saw, in both cases, Eq. (12) admits a real holomorphic solution having the lowest order term of the form hxa ' . Consider the corresponding particular solution of system (1). In this solution, x as function of t is determined from the equation dxldt = hxP+' + ... , (20) +

the right-hand side of which is a real holomorphic function of x having the lowest order term h x a + ' .We can affirm therefore that, when condition (14) or conditions (17) and (18) hold and p is odd, the unperturbed motion is always unstable. The case of even p, under the same conditions, is contained

144

Stability of Motion

among those considered in page 140, and as it was shown, if c1 is odd and a is negative, all depends upon the sign of the constant b. We remark that by studying Eq. (20) we can deduce a number of cases of conditional stability but we do not treat this question here.

9. The cases not considered so far are all related to the assumption that c1 is odd and a is negative. For this reason we suppose in the sequel this hypothesis to be valid. We write c1 = 2n - 1, so that n is an integer not less than 2. All the cases considered pertain to two categories: (1) those, in which p > n - 1 or q(x) = 0, and (2) those in which j? = n - 1, whereas b satisfies the inequality b2

+ 4na < 0,

(21)

opposite to condition (18). Of the cases in which p < c1 (such are, for example, all the cases pertaining to the second category), it would be sufficient to consider only those in which p is odd. In our analysis, however, there is no need of such a restriction. The cases pertaining to the two mentioned categories differ essentially from those considered so far: while in the latter the solution of the stability problem was obtained at once, in the present cases its investigation needs, in general, a whole series of calculations. As to the character of these calculations, the cases of the two categories are perfectly analogous. However, in some respect, the cases pertaining to the first category are simpler, and therefore, we start our investigation with them. First of all, we point out the simplest case pertaining to the first category, in which X = 0, Y = ax2"-'. In this case the answer to the stability problem is always in the affirmative, since system (1) admits the first integral ny2 - axz", which, according to our assumption that a ( 0 , is a sign-definite function of the variables x and y. We insist on the integration of system (I) in this case, since it leads to functions playing in the sequel a very important role. This integration yields results that can be considered in some sense as

145

investigation of One of the Singular Cases of Theory of Stability of Motion

first approximations in the integration of any equation pertaining to the cases of the first category. 10. We make a = -1 since any other case can be reduced to this. In consequence, our equations are of the form (22) and their integration is immediately reduced to quadratures, if we use as a first integral the equation x2" ny2 = c2",following from them. Here c is an arbitrary constant. In order to define the functions in terms of which the problem will finally be solved, independently of any arbitrary constants, we replace t by the independent variable 9 = c"-l y, where y is a new arbitrary constant, and put x = cCs9, y = -cnSn9. Now, the problem is to investigate the properties of the functions Cs9 and Sn9 of the variable 9, related by the identity dxldt = y,

dyldt = - x ~ ~ - ~

+

+

CsZn9

+ n Sn2 9 = I

(23)

and satisfying the following differential equations :

In order to determine these functions uniquely, we can make CsO=1,

SnO=0,

which are compatible with (23). In this case Eqs. (24) determine our functions completely for all real values of 9. They are uniquely determined and are also continuous for values of 9 of the form ~9 = p CJ 4 - 1, where p and CJ are real numbers, the former being completely arbitrary, whereas the latter is submitted to the condition that its absolute value does not exceed some bound, the finding of which will not concern us here, Although, in some cases we consider complex values of 9, we almost always suppose that the argument of our functions remains real. The function Cs9 is even, whereas Sn9 is odd. Furthermore, both are periodic, as one can deduce from the equations

+

-Jq

cs 9

9=

1

dx 2n

(1-x)

Sn 9

(1 - nx2)('-2n)i2nd x,

= 0

146

Stability of Motion

which follow from (23) and (24), and can serve as definitions of our functions if one imposes some restrictions on the integration paths, and adopts corresponding definitions for the square roots appearing under the integral sign. If we put

assuming the variable x to be real in both integrals and dx positive, the number 2w is the period of both functions. This number can be expressed in terms of the r-function since from the formulas written we find, provided that the square roots are considered positive :

In order to deduce formulas determining our functions uniquely, at least for all real values of 8, we introduce the auxiliary variable cp determined as a function of 8 from the equation

Then, if restricting ourselves to real values of 8, we admit that cp is real, we find:

in the upper limit of the integral, as well as in the integrand, Cs 8 =cos cp,

Sn 8=

sincp

- (1

J.

+ cos' cp + ... + ~ o s ~ " - ~' c1 ' p. )

If we suppose that the square roots appearing here are taken positive, then, in virtue of (25), cp is an increasing function of 9 varying by 71 whenever 8 increases by w . In consequence, we have Cs (9 o)= -Cs 8, Sn (9 o)= -Sn8, whence we get among other things Cs (o- 9)= -Cs 9, Sn(w - 9) = Sn8. From these formulas we conclude that our functions are known for all real values of 8, as soon as their values for 8 situated between 0 and 4 2 are known. One can notice that for 9 increasing from

+

+

Investigation of One of the Singular Cases of Theory of Stability of Motion

147

zero to 4 2 , the function Cs 9 decreases, whereas Sn 9 increases. For 9 = 4 2 they reach the following values : Cs ( 4 2 ) = 0, Sn ( 4 2 ) = lid;. These properties of Cs 9 and Sn 9 remind us of the functions cos 9 and sin 9, to which they are reduced for n = 1 . In our investigation, however, the smallest possible value of n is 2. In this case they become elliptic functions, namely: Cs 9 = cn 9, Sn 9 = sn 9 dn 9 of modulus 1/d2 They are expressed in terms of elliptic functions also in the case n = 3. The expressions themselves are rather complicated and also include square roots. These expressions are not given here as they do not present any special interest, Let us return to system (22), the general integral of which is represented by x = cCs (c"-'t y), y = -cc,Sn (c"-'t y). If we suppose the constants c and y to be real and limit ourselves to real values o f t , these relations define real functions o f t , which are continuous and periodic. Their period is 2o/c"-' and depends therefore upon the initial values of the functions x and y. This dependence is such that, by making the initial values sufficiently small in magnitude, the period can be made arbitrarily large. We notice that if the constant c is supposed to be an infinitesimal quantity and its order is taken as unity, then, ingeneral, the function x is an infinitesimal quantity of the first order, whereas y is an infinitesimal quantity of the nth order.

+

+

11. Let us return now to our problem. Dealing with a case pertaining to the first category, we consider system (1) under our usual assumption that the function X vanishes for y = 0. Whatever the negative number a is, we can always reduce ourselves to the case a = - 1. In order to do this we have only to transform system (1) using the substitution = (.-u)-[1/2("-1)l

= (-u)-w2("-1)l

Yl.

Suppose therefore, as in the previous paragraph, that a = - 1. In the cases in which we are interested, if x is considered a firstorder quantity and y a quantity of order n, the lowest order term

148

Stability of Motion

of Y is x Z n - land X does not contain terms lower than of the (n 1)th order. As a consequence, supposing that x, y, dxldt, and dy/dt are infinitesimal quantities of order 1, n, n, and 2n - 1, respectively, and retaining in each of Eqs. (1) only the lowest order terms, these equations are reduced to those treated in the previous paragraph. It is therefore natural to integrate our equations using the method of variation of parameters. Taking for x and y the expressions deduced in the previous paragraph, we consider c and y as new unknown functions. It will be simpler to replace y by the y. To this end we perform the substitution combination c"-'t

+

+

x=rCs9, y = -r"Sn9, taking as new unknown functions r and 9. By virtue of (23), Eqs. (26) yield x2" + ny2 = r Z n ,

(26)

(27)

whence xz"-'(dx/dt)+y(dy/dt)= r2"-'(dr/dt).From the same equations we find, by virtue of (24) and (23) :ny(dx/dt)-- x(dy/dt)= r"+'(d9/dt). Replacing now the derivatives dxjdt and dy/dt by their expressions, following from Eqs. (I), we get the equations r2"-'(dr/dt) = xZn-'X + y(x2"-' + Y), r""(d9ldt) = r2" + n y X - x(xZn-l + Y), the right-hand sides of which, being expressed in terms of r and 9, are represented by series expansions in positive integer powers of r with coefficients periodic with respect to 9. By virtue of the properties of X and Y shown above, the righthand side of the first of these equations will not contain terms lower than of the 3nth order in r, whereas the lowest order term in the right-hand side of the second equation will be r2".These equations are therefore of the form drldt

=

d9jdt

= r"-'

r"'lR,

+ r"O,

(28)

where R and 0 are series expansions in positive integer powers of r with coefficients periodic with respect to 9.

Investigation of One of the Singular Cases of Theory of Stability of Motion 149

These coefficients are rational entire functions of Cs9 and Sn9, and hence they are functions of 9 defined, and continuous for all real values of this argument. The series R and@converge uniformly for such values of 9 for all values of r which are sufficiently small in magnitude.

12. From Eqs. (26) and (27) it follows at once that the stability problem in which we are interested is reduced to the problem of stability with respect to the variable r . The right-hand side of the first of Eqs. (28) vanishes for r = 0, and therefore, at least as long as it remains small in magnitude, Y conserves the sign of its initial value. We conclude that the problem can be divided into two problems of conditional stability, corresponding, one to the condition r > 0 and the other to r < 0. However, these two problems do not differ essentially, since replacing r by -r and 9 by o - ( - 1>”9, Eqs. (26) remain unchanged. We can, therefore, limit ourselves to solve only one of them, for instance, that corresponding to the condition r > 0. In order to solve this last problem we notice that the variable t can be replaced by 9, which will be considered the time variable. Indeed, in the problems of stability the time-variable can be replaced by any function of it which is continuous and increasing. The only condition is that if the time-variable increases beyond all bounds, this new function also increases beyond all bounds. Moreover, if it is desirable to reserve this role to a function not given a priori, but defined only by conditions depending upon the variables with respect to which the stability is investigated, then it is not necessary that this function always has these properties. It is sufficient to have these properties only as long as the variables in question remain smaller in magnitude than some arbitrary small bounds. As to our problem, it is obvious that there always exists a positive number 1, sufficiently small, so that whenever the following condition O
holds, the variable 9 increases permanently together with t.

(29)

150

Stability of Motion

The validity of our remark will be established if we will show that the number 1 can be taken sufficiently small so that in all the cases in which condition (29) holds for all the values of the timeparameter following the initial moment, if such cases are possible at all, the variable 9 increases beyond all bounds when t tends to infinity. As we shall see in a moment, this condition is satisfied if 1 is so small that : (1) series R and 0 are uniformly convergent for all real values of 9 provided that (29) holds, and (2) under the same assumption, for any real 9 the inequality 1 + rO > A, (30) is satisfied, where A is an arbitrary positive rational number. Indeed, if there exists a motion so that for such an I, condition (29) holds for all positive values of t, the initial moment being t = 0, this motion has the property that for positive t the following inequality holds: d9/dt > Arn-'. This inequality follows from the second of Eqs. (28) by virtue of (30). Denoting the value of r corresponding to t = 0 by y o , the first Equation (28) can be written in the form 1 -_--

1 ,.n-1

,.PI-~

0

- (n -

1) JtrR d t . 0

Denote by M a positive number such that the inequality rR > --M holds for every r satisfying condition (29) and for each real 9. By the manner the number I was chosen, such a number M always exists. From the last equation we get for the motion considered another inequality : 1

1

,.n-l

or

,.n-l

0

rn- 1

>

> ( n - 1)Mt

rz1 + ( n - 1)Mr;- t '

valid also for any positive t. We have therefore Arn- 1

d9 0 -> dt 1 + ( n - l)MrG-'t '

Investigation of One of the Singular Cases of Theory of Stability of Motion

151

and, hence, for every positive t, 8-90>

A ( n - l)M

log[l

+ (n - l)Mr;-lt],

where 9, is the value of 9 corresponding to t = 0. It is clear from this that the variable 9 increases beyond all bounds when t tends to infinity. Returning to our problem, we notice that system (28) leads to the equation dr - r 2 R d9 1 + r e '

+

the right-hand side of which is represented by the series R2r2 R,rZ ..., in positive integer powers of r. Being entire rational functions of Cs9 and Sn9, the coefficients R, are periodic with respect to 9.The series converges for sufficiently small r uniformly for all real values of 9. Considering in this equation 9 as the time-variable, we reduce our problem to one of the simplest cases of the stability problem for periodic motions. This case was considered in the abovementioned work. Following the rules presented there, one treats this case in the following manner. Let the function r, which satisfies Eq. (31), be written in the form of a series

+

r

=c

+ u2c2 + u3c3+ ... ,

(32)

in integer positive powers of the arbitrary constant c. Here u p ,u, , etc., are supposed to be functions of 9 not depending on c, and which are found successively from the equations (du,/d9) = R 2 , (du/d9)= R , 2 u 2 R 2 .Suppose that the first nonperiodic function in the sequence

+

u2

9

u3

,u 4 ,

**'

+

(33)

is u, . It will necessarily be of the form u, = g9 v , where g is a constant different from zero and v, some periodic function of 9. As soon as the constant g is determined, we can consider our problem as solved, since taking into account the condition r >0, we can affirm at once that for g > 0 the unperturbed motion is

152

Stability of Motion

unstable, whereas for g < 0 it is stable. In the last case, the perturbed motions, corresponding to sufficiently small perturbations, tend asymptotically to the unperturbed motion. It can happen, of course, that the sequence (33) contains only periodic functions. If one proves that we d'eal with such a case and the calculations are performed in such a way that for some real 9 all the functions us vanish, then the series in the right-hand side of Eq. (32) converges uniformly for all real values of 9, c being sufficiently small. By virtue of this equation, which represents the general integral of Eq. (31), one can deduce therefore that stability holds. Obviously, this stability is also conservative (page 136). One can notice that if the calculation of the functions us is not restricted by any special condition, the determination of each one of these functions will be accompanied by the introduction of an arbitrary constant. However, these constants will appear in the expressions of us in such a manner that if by one choice of them there will be nonperiodic functions among u s , the same will be true for any other choice of constants, the numbers rn and g being always the same. 13. The method presented, which involves the successive calculation of the functions u2 = 1 R 2 d9,

u 3 = u2'

+ S R 3 d9, ... ,

is reduced to performing a set of quadratures upon rational entire functions of Cs9, Sn9, and the functions us previously found, as long as nonperiodic functions do not occur. Let us consider the simplest of these quadratures, with which one starts the calculation. These are the integrals, the integrands of which are rational entire functions of Cs9 and Sn9. Their investigation is reduced to that of integrals of the form JSnP 9 Cs43 d9,

where p and q are nonnegative integers.

(34)

Investigation of One of the Singular Cases of Theory of Stability of Motion

153

These integrals can easily be transformed into integrals of differential binomials, but we will treat them by using the properties of the functions Sn9 and Cs9, expressed by Eqs. (23) and (24). Let us remark first of all, that Eqs. (24) yield immediately the following two integrals :

Knowing these two integrals, the integral (34) can be calculated in two cases: (1) when p is odd, and (2) when the remainder of the division of q by 2n is 2n - 1. In both cases the integral can be calculated by a simple transformation based on Eq. (23). It is expressed by a rational entire function of Cs9 in the first case and of Sn9 in the second one. In order to discover other cases of integrability we use the following reduction formulas /Snp 9 Cs49 d9

P-1 + + + ( p - l)n + q + 1

SnP-' 9 Csq++' 9 ( p - l)n q 1

= -

x JSnPZ 9 Csq9 d9,

2n+1 + ( p q--1)n + q + 1 JSnP 9

9 d9,

which are obtained by integration by parts together with a small transformation using Eq. (23). They can easily be verified by differentiation. Using successively these formulas, which allow to reduce the exponent p by two units and the exponent q by 2n units, we can calculate the integral (34) in the two cases shown above, and also

154

Stability of Motion

when q is a multiple of 2n. In two cases of the latter type p = 2, q = 0, and p = 0, q = 2n, these formulas yield directly : /Sn2 9 d9 = sCs2"9 d9 =

Sn9Cs9,

n + l

nSn9Cs9

n + l

+-n 9+ l + const.,

+-n +9 1 + const.

In general, when p is even, the formulas considered lead finally to the integrals J Csq9d9,with q < 2n. Of these, besides the integrals corresponding to q = 0 and q = 2n - 1, we can only express one more, in terms of our functions, namely, that corresponding to q = n - 1. One calculates this integral by means of the relation 1 JCs"-' 9 d9 = -arc cos Cs" 9 + const.,

Jn

if we define arc cos such that sin arc cos Cs"9 = Jn Sn 9. The remaining 2n - 3 integrals, corresponding to q = 1, 2, ... , n - 2, n, n 1, ... , 2n - 2, need a special investigation.* Putting together all that was said above, we conclude that the integral (34) can be calculated in all the cases when p is odd, or when q satisfies one of the following three relations: q = 0, q = n - 1, q E 2n - 1 (mod 2n). In all the other cases, all we can do is to bring it to the form:? JSnPCs49 d9 = E(Sn 9, Cs 9) A J Cs'9 d9, where F is some rational entire function of Sn9 and Cs9, A is a constant different from zero, and r the remainder in the division of q by 2n. If p is even, our formulas always permit bringing the integral considered to such a form. Since for r < 2n - 1 the integral appearing in the right-hand side of the last equality will surely not be expressed as a rational entire function of Sn9 and Cs9, one can affirm that the integral (34) is represented by such a function only in the two

+

+

* One can remark that if we take Cs8 as the independent variable, in the differential binomials to which the integrands of these integrals are reduced, not one of the known conditions of integrability holds. t This is a misprint in Liapunov's collected works. It must be F( ) and not E( ). In the original paper this is printed correctly (Translators' note).

Investigation of One of t h e Singular Cases of Theory of Stability of Motion

155

cases mentioned at the very beginning, when p is odd or when q satisfies the relation q 2 n 3 - 1 (mod 2n). For our problem it is important to treat the case when the considered integrals are periodic functions of 9. It is not difficult to see that this is the case whenever at least one of the numbers p and q is odd. Furthermore, if these numbers are both odd, for the period which in general is equal to 2w, the number w itself can serve. This follows at once from the fundamental properties of the functions Sn9 and Cs9, appearing when the argument is varied by o.If p and q are both even, our integral will not be a periodic function. It can be written in the form: SnPSCs49d9 = Q(9) G9, where Q(9) is a periodic function of 9 of period o and G a constant, the value of which is determined by G = 2/w J;'' SnP9 Cs49 d9. If one likes, this constant can be expressed in terms of the r-function, since as it is not difficult to convince ourselves, the following formula holds :

+

s

14. Supposing that among the functions us there are nonperiodic ones, we denoted by in the index of the first such function in the sequence u 2 , u 3 , u 4 , ... . Let us exhibit some properties of this number rn that appear by considering Eq. (31) more closely. We want to show first of all that when n is even, m is even too, and if I I is odd, the same is true form. To this end we notice that Eq. (31) is invariant when r is replaced by -r and 9 by o - ( - 1>"9,since by such a substitution, as it was already shown, Eqs. (26) are invariant. We can affirm therefore, that if in (32) we replace 9 by o - ( - 1)"9 and change the sign of all the terms, the new series satisfy formally Eq. (31), just as the old one. However, the role of the constant c will be played by -c and in order to obtain a series of the same form as (32), we have to replace c by -c. Suppose that the series obtained in this way is c + v 2 c 2 + u3c3 + ... , (35) where 0 2 , u 3 , etc., are functions of 9 independent of c.

156

Stability of Motion

We can affirm a priori that the first nonperiodic function in the sequence u 2 , uj , u 4 , ... will be u, and that this function is of the form g9 + a periodic function. From the manner in which we obtained series (35) follows that, in general, us = - (- l)"u,[o - (- 1)"9] where us($) designates us as function of 9. From here we find u, = (- l)"+'g$ a periodic function. We must conclude therefore, that (- ,),+' = 1, i.e., rn n is even. Let us show now that there exists some lower bound for the numbers m, depending upon the numbers n and p. To this end let us return to the expressions of the functions R and 0 in terms of the functions X and Y. These expressions, which follow from the formulas of page 148, can be written in the form:

+

+

X

--

,n+

+ Y 1

c s 9.

We remark now that from the assumption that X vanishes for y=O, it follows that in the expansion of X / r f l t l in increasing powers of r, all the terms lower than of the (n - 1)th power in r will have coefficients of the form KCs"9Sn9, where K is some constant and k is a positive integer. We notice further that if we mean by N the number n - 1 when p 3 2 n - 1 or q(x) =0, and p - n when p <2n - 1, then in the expansion (x2"-' Y)/r" all the terms of order less than the Nth, relative to r will have coefficients which are rational entire functions of cs9. Denoting by E ( x ) a rational entire function of x in general, we can affirm, therefore, that the coefficients of the terms of order less than the Nth, with respect to r in the expansion of R, are of the form E(CsS)SnS, and in that of 0 they are of the form E(Cs9). Therefore, of the coefficients R, of the expansion of the right-hand side of Eq. (31), all those for which s < N + 2 are of the form E(Cs9)SnS. From here it is not difficult to deduce that all u s , for which s < N + 2 , are rational entire functions of Cs9. Thus, the

+

investigation of One of the Singular Cases of Theory of Stability of Motion

157

+

number m can not be less than N 2. Taking into account that m n is always even, we arrive at the following conclusion. Whenever p 2n - 1 or q(x) = 0 identically, the number m, if it exists, satisfies the inequality m > n 1, and whenever p is less than 2n - 1 and is odd, the number m satisfies the inequality m > p - n 2. In addition, if p, being less than 2n - 1, is even, we always have m = p - n 2. Indeed, it is not difficult to see that in this case the function Rp-n + is of the form b Csp 9 SnZ9 E(Cs 9)Sn 9. To the same form is also brought the right-hand side of the equation d ~ g - n i 2- R P - n + 2+ F ( u 2 , u 3 , ..., Cs9)Sn9,

+

+

+

+

+

d9

which serves as a definition for the function , since F represents here a rational entire function of the quantities in brackets, which, as it was proved, will also be rational entire functions with respect to Cs9. We see therefore that for /3 even, the function is surely not periodic. upConsidering the expression for the constant g, which is given in this case by the formula b u g = - j Csa9Sn29d9, 0

0

we arrive at the conclusion that when b is positive, the unperturbed motion is unstable, and when b is negative, it is stable. This result appeared already among the conclusions of page 140, which were obtained by other methods. 15. We have shown above how the problem of stability is solved in the case when all the functions us are periodic, as well as in the case when among them there are also nonperiodic ones. However, we have not shown any criterion, which would permit us to say a priovi which of the two cases we deal with. The absence of such criteria causes, of course, very serious difficulties. These difficulties are closely related to the essence of the problem itself. Because of the general form in which the problem is stated these difficulties cannot be removed.

158

Stability of Motion

All we can do is indicate a number of conditions that, when fulfilled, can state with certitude that the functions us are all periodic. Among these conditions are obviously included those which ensure the existence of a first integral for system (I), independent of t , and holomorphic with respect to x and y . Whenever such a first integral exists, all the functions us are rational entire functions of Cs9 and Sn9. For instance, such is the case when system (1) is canonical, i.e., when the functions X and Y satisfy the relation axlax a y p y = 0. Of the other cases of this type, we mention the one when system (1) is invariant when y is replaced by --y and t by -t. In this case the functions X and Y are necessarily of the form X = yf (x,y’), Y = q(x, y’), where f and q are holomorphic functions of x and y 2 , vanishing for x = y = 0. Therefore, the right-hand side of the equation dy2/dx= 2y Y / ( y X ) is holomorphic with respect to x and y’, whence it is not difficult to deduce the existence of first integrals for system (I), independent of t, and holomorphic with respect to x and y’. The existence of first integrals of this type is not a necessary condition for the periodicity of the functions u s ,since one can show a case when such first integrals do not exist, although all these functions are periodic. To these cases belongs, in general, the case when system (1) is invariant if x is replaced by -x and t by -t. In this case, as we can easily convince ourselves by examples, the functions usare not necessarily rational entire functions of Cs9 and Sn9 and, therefore, system (1) does not necessarily possess holomorphic first integrals not depending on t, although all the functions us are periodic. In order to prove this, we remark that in the present case, Eqs. (31), and therefore also the differential equations which define the functions u s , are invariant when 9 is replaced by o - 9. Thus, if these equations are satisfied by putting u 2 = Q2(9), u3 = ~ ~ ( 9 ... ) ,, then they are also satisfied by u2 = Q2(o- 9),

+

+

u3 = Q3(o - 9).

...

The functions Q,(9) and Qs(w - 9), however, taking on the same

Investigation of One of the Singular Cases of Theory of Stability of Motion

159

value for 9= 4 2 , must be identical, as it follows from the equations they satisfy. We can therefore write 02(0- 9) = 02(9), 03(o- 9)= Q3(9), ... for every real 9.From here we find, among other things :

);

s2(3

= Q2(

-;),

Q3(3

);

= Q3(

)-;

... .

We remark further that our equations can be satisfied also by taking u2 = 02(9 20), u3 = 03(9 20), ... , and that, by virtue of (36), the functions O,(9) and O,(9 2 0 ) take on the same values for 9= -0/2. We can affirm therefore, that OD,($ 2 0 ) = Q2(9), ~ ~ 2( 0 )9= 03(9), ... for every real 9. To the three cases just shown we can add those which can be reduced to them by changing the independent variable using an equation of the form dt = (1 H)dt,. Here t , is the new independent variable and H a holomorphic function of x and y , vanishing for x = y = 0 . Let us remark that whenever all us are periodic, the variables x and y have the following property. When their initial values are sufficiently small in magnitude, they are periodic functions of t with a period T depending upon the initial values in such a way that by taking these values sufficiently small, T can be made as large as one desires. This period is found from the formula

+ +

+

+

+

+

[see Eq. (28)], where Y is the solution of Eq. (31), corresponding to the initial values of the functions x and y. The period T is a perfectly determined function of the value c of the function Y corresponding to 9 = 0 . If c is supposed to be sufficiently small in magnitude, T is represented by a series expansion in increasing integer powers of c with the lowest order term 20/c"-'. Let r = (2wt/T) y , where y is a constant determined by the condition that z = 0 for 9 = 0. Then, for IcI sufficiently small, the functions x and y are represented by series expansions in positive integer powers of c with coefficients depending only upon r and periodic with respect to it, with a period 20.

+

160

Stability of Motion

Writing only the lowest order terms of these series, we have = -c" Snz ....

+

x = c Csz + ..., y

16. Consider now the cases pertaining to the second category, characterized by the simultaneous validity of equality /i'= n - 1 and inequality (21). Assuming as previously that a = - 1, we transform system (1) using again the substitution (26). Under the adopted conditions we obtain, in this way, equations of the form: dr dt

- = br" Sn2 9 CS"-'

9 + r""R,

where R and 0 are functions of the same character as in the system (28). As previously, our question is reduced to the stability problem with respect to the variable Y and for solving it, we can settle, as before, the condition Y > 0. Furthermore, taking 9 as the independent variable, provided that the last condition holds, this variable can be considered the time-parameter, as before. This affirmation can be proved as in 12, since the proof given there was based on two properties of system (28) which also hold for system (37) and which consist in the following: (1) in the equation containing drldt, the lowest order term with respect to Y is higher than the lowest order term in the equation containing d9/dt, and (2) the coefficient of the lowest power of Y in the equation containing d9/dt never vanishes, since it is always positive. We convince ourselves that this last property is valid for system (37) if we remark that the smallest value of the function 1 + bSn9Csf19,

(38)

equals 1 - (lbl/2Ji), and is positive, in view of condition (21) in which we make a = - 1 . The solution of our problem depends upon the investigation of the equation dr d9

- --

br Sn2 9 Cs"-' 9 + r2R 1+bSn9Csf19+rO'

(39)

Investigation of One of t h e Singular Cases of Theory of Stability of Motion

161

The right-hand side of the equation is represented by the series expansion R,r R2r2 R3r3 ..-, in positive integer powers of r with coefficients R,,periodic with respect to 9. For sufficiently small r this series is uniformly convergent for all real values of 9.As to the form of the coefficients R, , they are rational functions of Sn9 and Cs9 with denominators equal to different powers of the function (38). The coefficient R, is determined from the following formula:

+

+

+

R '-1

b Sn29 Csn-' 9 + bSn9Csn9'

Note now that the characteristic equation of (39), corresponding to the period 201, has the root exp[Ji"R, d9] and that, therefore, whenever the integral in the exponent is different from zero, the stability problem depends upon the sign of this integral. When the integral is negative the answer is in the affirmative and when it is positive, the answer is negative. Such is the situation when n is odd. We can affirm then that for b < 0 the unperturbed motion is stable, and for b > 0 it is unstable. This result, similar to that shown at the end of 14, page 157, is only a particular case of that described in page 140. Suppose now that n is even. In this case the considered integral vanishes and, therefore, the integral J R1q9 is a periodic function of 9.Putting r = p exp[J:R, d9], we find from (39) the equation d p l d 9 = P 2 p 2 + P 3 p 3 + .'. , (40) Here the coefficients P, = R, exp[s - 1, J t R , d9] are also periodic functions of 9. The equation obtained in this way is of the same character as (31). As a consequence, our problem, which obviously can be considered as the problem of stability with respect to the variable p , will be solved by the method presented in pages 151-152. To this end we build successively the functions of 9 u 2 , u 3 , u4,

...)

(41)

determined from the condition that the expression p = c u2c* u3c3 * f .

+

+

+

Stability of Motion

162

satisfies Eq. (40),whatever the constant c is. If among these functions there are nonperiodic ones, the first of this type in the sequence (41), denoted by u,, is of the form u,, = g9 a periodic function, where g is some constant. The whole question is now to determine the sign of g since if g > 0 we have instability and if g < 0, stability. It can happen that sequence (41), however far we continue it, contains only periodic functions. If we deal with such a case, we can affirm that stability holds. This stability is conservative.

+

17. As we saw in the above considerations, when n is even, the cases pertaining to the second category are completely similar to those pertaining to the first one. As in the latter cases, the problem is reduced in general to a number of quadratures. In the cases pertaining to the category under consideration, these quadratures are rather complicated. The simplest of these integrals which can be met here and with the investigation of which we have to start the calculations, are of the following form: SnP9 Csq 9 d9 d9] (1 + bSn9Cs"9)"'

jexp[k~osR1

Here k, p , q, s are negative integers, the first of which is always different from zero. The integrand depends upon the integral Rld9, which can be expressed, for convenience, in terms of Cs9 and Sn9. This integral is equal to

ly

b SnZ9 CS"-' b (4n2 - bZ)l/'Sn 9 --9 d9 arctan n(4n - bZ)l/' 1 b Sn 9 Cs" 9 2Cs"9 b S n 9

+

+

1 2n

- - log (1 + b Sn 9 Cs" 9) where arctan is defined so that it vanishes at the same time as 9. Without getting into a detailed investigation of the integrals of the form (42), we want to put down a number of formulas which allow to reduce all the integrals of this type to a predetermined number of some of them.

Investigation of One of the Singular Cases of Theory of Stability of Motion

163

Denoting the integral (42) by Ip,q,s,we can write at once the following two equalities: 1p.q.s tllp,q,s

Further, putting exp[k

= Ip,q,s+

= Ip-2,q.s

lo'

Rid8

1

(1

+ bIp+

1

l,q+n,s+

(43)

13

- Ip-2,q+~n,s.

(44)

9 Csq9 +SnP b Sn 9 Cs" 9)" = Lp*q*s

and taking into account that d

d3 Lp+1 , q +

1,s- 1 = ( P

- s + 2)Lp,q,s1

+ Ck - 4 - 1 + (2s - P - %lLp+2,q,s-l + (s - l)Lp,q,s - C(k + 2)s - l)nlLp+z,q,s 9

we have Ck

+ 2(s - l)nllp+2,q,s- (s - l)Ip,q,s

=

Ck - 4 - 1 + ( 2 s - P - 3)nlIp+2,,,,-1

+ (P - s + 2 ) I p . q . s - 1

(45)

- Lp+l,q+l,s-l.

As we shall show in a moment, Eqs. (43), (44),and (45) permit expression of all our integrals as functions of those for which s has any given value. Since Eq. (43) yields an expression of Zp,q,sin terms of integrals for which s is larger by one, it will obviously be sufficient to have formulas expressing Ip,q,sin terms of integrals for which s is smaIler by one. Such formulas can be obtained in the following manner. From Eqs. (43) and (44) one deduces easily the equation: nb21p+4,q,s- b21p+2,q,s+ I p . q , s = I p , q , s -

1

-

bIp+l,q+n,s-1 .

Associating to it Eq. (45) and the one which is obtained from it by replacing p by p 2, get a system of three linear equations with respect to our integrals. Namely, the following integrals are contained in this system:

+

I,+

l,q,s,

Ip,q.s-l,

I P + 2,q.s

,

Ip+2,q,s-19

IP.4,S

(46)

3

Ip+4,q,s-l,

Ip+l,q+n,s-l.

(47)

164

Stability of Motion

These equations can always be solved with respect to the quantities (46), since the determinant

1

k

- b2

nb2

+ 2(;-

1)n

-(s

k

- 1)

+ 2 ( -~ l

1 ) ~-(s O- 1)

I

formed with the coefficients of these quantities, which can be brought into the form [k 2(s - 1)nI2 - (s - l)b2[k 2(s - l)n] n(s - 1)2b2,can not vanish when b2 < 4n, and k is different from zero. Thus, these equations always permit us to express the integral Zp,q,s in terms of the integrals (47) and some functions of 9 which we consider as known. Having shown the possibility of reducing all our integrals to some of them, for which s is given arbitrarily, we want to show now relations between integrals corresponding to the same s. A series of such relations is obtained from Eq. (44). They permit us to express all our integrals in terms of those for whichp = 0 o r p = 1. Between these integrals there exists, however, another series of relations, permitting us, as we shall show, to reduce the case p = 1 to the case p = 0. These relations follow from the equations obtained from (45) replacing the integrals Zp,q,s-l and Zp+2,q,s- by their expressions in terms of I p , q , s ,I q + l , q + n , Is p, + 2 , q , sI, p + 3 , q + n , sfollowing , from (43). In this way one obtains the following equation:

+

+

E4

+

+ 1 + ( P + l)nllp+2,q,s- ( P + l)lp,q,s = Ck - 4 - 1 + (2s - p - 3 ) ~ l b ~ p + 3 , q + f l , s +(P- s + 'p+l,q+l,r-2* 2)blp+l,q+n,s-

Putting here successively p = - 1 and p = 0, and afterwards eliminating by (44)the integrals, the first index of which is greater than one, we find n(q + l)zl,q,s= Ck - 4 - 1 + (s - ~ ) ~ l ~ ~ o , q + f l , s - Ck - 4 - 1 + 2(s - l)nlblO,q+zn,s- n'o,q+l,s-

1,

(48)

( 4 + 1)10,q,s - ( 4 + 12 + 1 ) ' 0 , q + 3 n , s

+ Ck - 4 - 1 + (s - l)nlbll,q+",s - Ck - 4 - 1 + (2s - 3)nlbJl,,+

= -nLl,q+l,s-l

3n,s *

(49)

Investigation of One of the Singular Cases of Theory of Stability of Motion

165

Equation (48) permits to express all the considered integrals of the form Zl,q,s in terms of integrals of the form Zo,,,s. The problem is therefore reduced to the investigation of the latter. Put k - 1 (2s - 3)n = x and introduce, instead of Zo,,,s, the simpler notation Zq . We get from (48) and (49)

+

+ c q z q + 2 n + Dqzq + L q ,

Aqzq+6n=Bqzq+4n

where A, B,

(50)

+ n + l)(x - q)(x - 2n - q)b2, = ( q + n + l)(x - q ) [ x - (s + l ) n - q ] b 2 = (q

+ ( q + 3n + l)(x - q)[x - (s - 2)n - q ] b 2 ,

+ +

+ +

-(q 3n l ) { n ( q n 1)' + [x - (s - I)n - q ] [ x - (s - 2)n - q ] b z > , D, = n(q l)(q n l ) ( q 3n l), C,

=

+

Lq = n(q + 3n

+ +

+ 1"

+ +

- (s - 2)n - ~ l b L O , q + n + l , s - l

+ n(q + + l ) { N q + 3n + l ) J L l + l , s - l - (x - q ) b ~ 0 , q + 3 n + I , s - l ) .

As we will show in a moment, Eq. (50) permits expression of all integrals of the form I, (we consider only those with q 3 0), in terms of 6n - 2 of them. Suppose first that x < 0. Remarking that for q 3 0 the coefficient A , can not be zero in this case, and using expression (50), all our integrals can be expressed in terms of the integrals Z, for which q < 6n. However, these last integrals are related by two relations, one of which is obtained from (50) by putting q = -1, the other follows from the same equation by putting q = - n - 1 or q = - 3n - 1 [and also from (48) by putting there q = - I]. These relations are

+ (2s - 5)n]b216n-1 = [ k + (2s - 3 ) n ] [ 4 k + (4s - 7)n]b214,-

[ k f (2s - 3 ) n ] [ k

1

+ [k + (s - 1)n][li + (s - 2 ) n l b Z } 1 2 , - , + 3n3Ll,o,,- + 3n[k + (s - l)n]bLo,,,,s- 3{n3

- nCk + (2s - 3 ) n w 0 , 3 " , , - 1 , [ k + 2(s - l ) n ] b 1 3 n - l = [ k + (s - l ) n ] b l n - l - nLo,o,s-l.

(51)

(52)

166

Stability of Motion

Therefore, the integrals in terms of which one can express all the others are just 6n - 2 in number. Every integral I, with q not satisfying 4 + 1 = 0 (mod n), (53) is expressed in this case in terms of the following three integrals: I,+ ~n

(54) where r is the remainder of the division of q by 2n. An integral for which this relation is satisfied is expressed in terms of two of the integrals (54). Suppose now that x 3 0. Every integral I,, for which the number q - x is not divisible by 2n, is expressed in this case like in the previous one. However, we cannot express the integrals for which q - 2 x is divisible by 2n in the same way since A , = 0. Instead, these integrals can always be expressed in terms of the following three integrals : Irt4n>

9

I r

9

1x+6n, Ix+4n, Ix+Zn. (55) In addition, the integrals for which q <x, can only be expressed in terms of the integral IxfZn only. Indeed, by replacing q in Eq. (50) by the sequence of negative numbers x,x - 2n, x - 4n, ... ,we get the equations

+

Cxlx+Zn D,I, Bx-Zn1,+2n

+ Cx-Znlx

+ L, = 0,

...... + Dx-ZnJx-2n + L x - Z n

= 0,

(56)

+

in which the symbol I will not have indices larger than x 2n. The coefficients D,, D,- 2 n , etc., are different from zero, since D, cannot be zero for 4 3 0 . These equations allow therefore to etc. successively in terms of express the integrals I,, When neither of the two relations x + 1 = 0, x + 1 = n (mod 2n), (57) is satisfied, Eq. (50) does not yield any relations between the integrals (55). However, when at least one of them is satisfied, then as we shall show next, the integral I, + 2 n can be expressed in terms of known functions. Let us start by supposing that the first relation is satisfied. We

Investigation of One of the Singular Cases of Theory of Stability of Motion

167

form the system (56) and add to it Eq. (51), which contains the same integrals, as a consequence of the adopted assumption. In this way weget a system containing as many equations as integrals. This system can always be solved with respect to integrals since between the free terms of the equations L,, L,-z,, ... , L 2 n - 1 , ( 1 / n ) L l , there is no linear relation, as is sufficiently clear from the expressions of L, . This system allows us therefore to find all the integrals it contains, namely, I, + Z n , I,, ... , 14,- 1 , Izn Suppose now that the second relation holds. Writing system (56) and adding to it Eq. (52), we get, as previously, a system containing as many equations as integrals. Looking at the free terms, we see again that this system can be solved with respect to the integrals it contains : I, + 2 n , I,, ... , 13n- , I, - . These integrals are therefore expressed in terms of known functions of 9. From these considerations one sees t'hat for x 3 0 as well as for x < 0, as long as relation (53) holds, the integral I, can be expressed in terms of two integrals of the type (54)and (55). When this relation is not satisfied, the integral can be expressed in terms of three such integrals. The integrals in terms of which all I, are expressed are as previously 6n - 2 in number. Also all the other integrals of the form (42), corresponding to the same k , are expressed as functions of the same 6n - 2 integrals. We have shown above a class of integrals of the type I,, which can be expressed in terms of the functions Cs9 and Sn9. These are the integrals for which q is one of the positive numbers of the sequence x 2n, x, x - 2n, ... , and for which one of the relations (57) holds. Remarking that in virtue of the equality x 1 = k (2s - 3)n, the last condition is equivalent to the divisibility of k by n, we can affirm that any integral of the type

+

+

+

can be expressed in terms of our functions. Here I is any integer larger than 1 - 2s and q a positive number of the form: q = (20 1 - 1)n - 1, provided that the integer 0 does not exceed s.

+

168

Stability of Motion

So, for instance, we have jexp[2n J:Rl

d$]Cs"-' 9 d9

1 b

d9](1

= - exp[

2n J:R1

Jexp[ 3n I:Rl -

+ b Sn 9 Cs" 9) + const.,

d9]Cs2"-l 9 d9

R , d9 " ( n + 2b2)

[

exp 3n

(11

Sn 9

+ 2b Cs"9)(1 + b Sn 9 Cs" 9) + const.

To these integrals we can associate the following one:

-

1b exp[ n J:Rl

d9](l

+ b Sn 9 Cs" 9) + const.,

not pertaining to the mentioned type. Of the integrals corresponding to an arbitrary k, we mention the following:

18. In the case when n is even (the only case which will be considered here), the integral (42) can be represented in the form: @(9) G9, where @(9) is a periodic function of 9 of period 2w, whereas G is a constant given by

+

'=-J01

2w

2w

SnP9 Cs4 9 d9 exp~k~~R'd3](l+bSn9Cr"9)'~

Consider now the function R,.Denoting by R,'the expression for R , in which b is replaced by -b, we get from the equality Jt-' R,d9 = J: R,d9, which is easy to prove, the relations

Investigation of One of the Singular Cases of Theory of Stability of Motion

169

J;+$ R, d9 = SOs R,d9 = Jg R,'d9. Using these equalities and putting

Ja"i:xp[k ?b:R1

"1

SnP 9 Csq 9 d9 (1 + b Sn 9Cs" 9)"= F(b);

(58)

+

+

we find easily: 2wG = [l (- l)q][F(b)(- 1)P+4F(-b)]. We see from here that whenever q is odd, the constant C is zero and, therefore, the integral (42) represents a periodic function of 9. When q is even, we have wC = F(b) (- l)PF(-b), which for p even is surely different from zero, whereas for p odd it can vanish, but only for some particular values of b. Thus, when q is even and we do not deal with some exceptional values of b that exist only when p is odd, the integral (42) is not a periodic function. We suppose number b to be within the bounds -2dn and + 2 h , without attaining them. Under this assumption the function F(b), as it is not difficult to see from its expression (58), can be represented in the form of a series expansion in positive integer powers of b. The first term of this series, equal to F(O), is given by the integral f;'2 SnP9Cs9qd9,whichcan beexpressed in terms of the r-function, as was shown in page 155. The following terms, however, will depend upon integrals of a more general type:

+

Joa'2cp1 SnP'9Csq'9d9,

(59)

where cp = dn Sics"- 9 d9 = arc cos Cs"9 (see 13). Here I, p', q' are some negative integers. For instance, the coefficient of the first power of b is given by the expression:

Taking cp as an independent variable, we transform the integral (59) into n - ( P ' + 1 ) ' 2J;i2cp1 sinp'cp cosq"cp dcp, where q" = [(q' l)/n] -1. Integrating by parts, such integrals can be reduced to integrals of the same type with p' = 0 and q" = r / n , Y being a nonnegative integer less than 2n. This expansion can serve as a definition of

+

170

Stability of M o t i o n

the constant G, for values of b which are sufficiently small in magnitude. In order to get some idea about the value of this constant that corresponds to b near the bounds - 2 4 ; and 2&, we give in the following, the corresponding values of the function F(b). As to the value corresponding to b = 2dn, it is found without any difficulties, since for any positive b the integrand of (58) is defined for all the values of 9 situated between the limits of integration. Therefore, we find this value immediately by putting b = 2dn in (58). Remarking that for this value of b, Sn 3

9

J0 R ,

d9 =

1

- - log(1 +2J&

Ji(csn9 + Ji sn 9)

2n

9 CS" 9),

and taking as independent variable the function

Ji Sn 9 CS"9 4-Jn Sn 9 = x,

we get after some manipulations

where q ' = - -q + l

n

1,

s' =

q+l-k

2n

p+l 2

+--s.

In order to obtain the value corresponding to b = - 2 ~ 5 , we remark that by putting b Cs"-' 9[(k + 2ns)Sn2 9 - s] 1 + bSn 9 CS"9

we can write F(b) = Jm'2enp[ 0

JI

= u,

9 d9]Snp 9Cs4 9 d9.

Let 9, and 9' be numbers situated between 0 and 4 2 , such that It is not Sn29, = s/(k 2ns) and dn Sn 9' = Cs"9' = 1/d2.

+

Investigation of One of the Singular Cases o f Theory of Stability of Motion

171

difficult to see that 9, <9'. Giving b some value between 0 and - 2 d n exclusively, we find that the integral J I u d9

(61)

increases when 9 increases from 0 to 9,,and decreases when 9 keeps increasing from 9,to 4 2 . For such values of b we can have therefore: Jtvd9 < Ji'vdS for any 9 situated between 9' and 4 2 . However, the integral

(where arctan is defined as being between 0 and n/2) is negative and arbitrarily large in magnitude, when b is sufficiently near to - 2 4 ; . As a consequence, when b tends to - 2 d n , lim exp [J9,ud9]= 0 for any 9 situated between 9' and 4 2 . Thus, denoting this value by F(-22/;), we have F( - 2 J n )

= Iim

]:'exp[

/099 d9]Snp 9 Cs4 9 d3.

From here, remarking that the integral Ji'vd9, which represents the maximum value of the integral (61) between 0 and 9',tends to some limit when b tends to -2&, and denoting by uo the value of the function u corresponding to b = - 2 h , we find F( - 2J'n)

=

/oexp[ o! /

d9]Sn' 9 Csq9 d9.

If we remark now that for 0 < 9 < 9'

and we take as independent variable the function

Jl Sn 9

=x

CS"9 - J n S"9

we get after some manipulations

172

Stability of Motion

As previously, the numbers q' and s' are determined by formulas (60). We have for F(b) an expression in the form of a definite integral, the integrand of which contains the functions Sn9 and Cs9. This integral, however, can be transformed, if we like, into another, the integrand of which contains (instead of these functions) trigonometric functions. In this connection we take as independent variable the angle cp, defined by cos cp

=

+

2 Cs" 9 b Sn 9 2(1 + b Sn 9 Cs" 9)lj2'

sin cp

=

(4n - b' Sn 9)'" 2(1 b Sn 9 Cs" 9)'" '

+

together with the condition that for 0 < 9 < 4 2 , it is situated between 0 and n. We have in this case 9

Iovd9 =

n(4n

k + 2ns kbcp -_ _ _log(1 b')l/' 2n

+

+ 6 Sn 9 Cs" 9),

and if we put b

ZJn

(4n - b')'/' = cos €'

2JT

= sin E ,

where E is an angle between 0 and n,we get, after some manipulations, ek'p/ (n tan 6 )

sinP cp sinY'(€- cp) dcp [sin' cp + sin2(€- cp)]"

Here q' and s' are the previous numbers (60), and 2s-

1.

CJ =

(kin)

+

19. Let us return now to our problem. It was shown in page 156 for the cases pertaining to the first category, that m f n is always an even number. The proof given there can, obviously, be applied also to the cases considered here. Thus, n being supposed here to be even, we can affirm that m is even too. Let us consider the simplest case, m = 2 , and let us write the corresponding general expression for the constant 9. In

Investigation of One of the Singular Cases of Theory of Stability of Motion

173

the present case even the first function in the sequence (41) is not periodic, and the constant g is given by

,J=, 1

2o

exp[

J: R, d9J R, d9.

Considering the expansion in powers of r of the right-hand side of Eq. (39), we find R,

=

(1 + b Sn 9 Cs" 9)R, - b Sn2 9 CS"-' 90, (1 + b Sn 9 Cs" 9 ) '

5

where R, and 0, are the functions to which R and 0 are reduced for r = 0. They are found, therefore, as the terms independent of r in the expansions of

br Sn2 9 csn-19 and 1 r2n+l {

n y -~ x(xZn-'

+ Y ) } - -bY Sn 9 cs" 9.

Let the expansion of Y in increasing powers of y be Y =f(x)

+ cp(x)y + $(x)y2 + ... .

According to our assumptions, we have here f(x) = -X2"-1 V(X) = bx"-'

+ a$" + ... ,

+ blx" +

* * *

If we now put $(O) = c, and denote the value of the partial derivative, a'Xjaxay, corresponding to x = y = 0 by D, then, according to the above remark, we have R,

=

-(D

+ aJSn

9 CsZn9 + b , Sn23 Cs" 9 - c Sn39,

O0 = (no- c)Sn2 9 Cs" 9 + b, Sn 9 Cs"+l9 - a , CsZn+l 9.

Using these formulas, we find R2 =

(b, - Db)Sn2 9 Cs" 9 - ( D + a,)Sn 9 CS'" 9 - c Sn3 9 (1 + b Sn 9 C S " ~ ) ~

Stability of Motion

174

From here we get where GP,Y,smeans the constant G from the previous paragraph, considered as a function of the numbers p , q, s. Between the quantities G P , 4 ,corresponding s to different p , q, s, there are valid relations which can be obtained from those found in page 163, replacing I by G, and making the quantities Lp,4,s equal to zero. From these relations, in which in our case we must take k = 1, it follows easily that the quantities G 2 , n , 2 ,G, , 2 n , 2 , and G3,0,2, upon which the constant g depends, can be expressed in terms of one of the quantities Gp,q,s.Thus, for instance, they can be expressed in terms of G 0 , n , 2 . Indeed, we find from (45) by putting p = O , q = n , and = 2: (2n f l I G 2 , f l , 2 = G 0 , n , 2 whenceG0,3n,2 = G 0 , n , 2 - nG2,n,2 = (n 1)/(2n W O , n , 2 Further, from (48) by putting successively q = 217, s = 2, and q = 0, s = 2, we deduce

+

+

From here, remarking that G3,0,2= (l/n)G1,0,2- (l/n)G1,2n,2 we find

6

GO,",2, (2n + Thus, we get the following expression for the constant g: G3,0,2 =

g = {[c

Since

1

=

-

+ ( n + l)a, - nDl6 + (2n + l)b,} (2n +

G0,n,2

20 1^,

20

exp[

loR , 9

d9

1

Cs"9 d9 (1 + b Sn 9 Cs" 9)2

investigation of One of t h e Singular Cases of Theory of Stability of Motion

175

is a positive quantity, the sign of g coincides with that of the expression [c (n l ) a , - nD]b (2n l)b, . Whenever this expression is negative, the unperturbed motion is stable and when it is positive, the unperturbed motion is unstable.

+ +

+ +

20. In page 158, considering cases pertaining to the first category, we indicated some criteria which certainly ensure the periodicity of the functions us. One of these criteria for system (1) can also be used in the cases considered here, namely, that implying the invariance of the system when x is replaced by -x and t by -t. Whenever system (1) has this property, the functions us are all periodic also in the cases pertaining to the second category. This fact can be proved as in the cases pertaining to the first category, taking into account that the integral

is invariant when 9 is replaced by w - 9. As to the criteria implying the existence of holomorphic first integrals of (l), independent of t, they are superfluous in the cases considered, since now system (1) cannot have such first integrals at all. Indeed, the function r, which satisfies Eq. (39), and takes the value c for 9 = 0, can be represented by a series expansion in powers of c for all c sufficiently small, at least when 9 varies between some known limits. It is not difficult to see that if system (1) had a holomorphic first integral independent o f t , then all the coefficients of this series would be algebraic functions of Cs9 or Sn9. But this is impossible since the coefficient of the first power of c, equal to exp[J:R,d$], is necessarily a transcendental function of Cs9, as can be seen from the expression of integral (62), given in page 162. The periodicity of the functions us will cause in each case the periodicity of x and y as functions of the variable t, satisfying Eqs. (l), if only the initial values of these functions are sufficiently small in magnitude.

176

Stability of Motion

The period of these functions in the cases considered here is given by the expression T=

d9

2w

Jo

(1

+ b Sn 9 Cs" 9)r"-' + Or"

[see Eq. (37)], where r is the solution of Eq. (39) corresponding to the initial values adopted for x and y . This period is a function of c, the value of Y corresponding to 9 = 0 . For sufficiently small c it is represented by a series expansion in increasing integer powers of c with the lowest order term cR('"-'),where

If we put z = R(t - t,)/T, where to means the value of t corresponding to 9 = 0, x and y are completely determined functions of z and c. For sufficiently small c, these functions can be expanded in series in positive integer powers of c. The coefficients of these series are periodic functions of z, of period R. Writing only the lowest order terms of these series, we have

Here ( is a function of

T,

defined by

21. So far we have considered Eqs. (1) under the assumption that the function X vanishes for y = 0. In the cases considered in the last paragraphs, we have used substitution (26) under this assumption, and in this way reduced the problem to the investigation of equations of type (28) or (37). However, it is easy to see that this assumption is not necessary. In order to obtain equations of the same type when system (1) is transformed by substitution (26), it is sufficient that, after making

Investigation of One of the Singular Cases of Theory of Stability of Motion

177

+

y = 0, no terms lower than of the (n 1)th order with respect to x remain in the expression of X.Indeed, if y = F(x) is a holomorphic solution of the equation y+x=o,

(63)

vanishing for x = 0, then, under the above condition, the function F(x) does not contain terms lower than of the (n 1)th order with respect to x. Therefore, if by expanding Y in increasing powers of y we have Y =f,(x) cpl(x)y ..., the function

+

+

+

does not contain terms lower than of the nth order, and the function

f ( x ) - f , ( X ) = ( y ) y = F ( x-) f l ( X ) does not contain terms lower than the 2nth power with respect to x . For this reason, the function cpl(x), in the cases pertaining to the first category, does not contain terms lower than of the nth power with respect to x. In the cases pertaining to the second category its lowest order term is bx"-I. The function f l ( x ) , however, has, in the cases pertaining to both categories, ax2"-' as its lowest order term. From here it is not difficult to deduce that if a = - 1, substitution (26) leads as previously to equations of the form (28) in the cases pertaining to the first category, and to equations of the form (37) in the cases pertaining to the second one. If system (1) does not satisfy the above-mentioned condition, then, in order to bring it to a form for which this condition is fulfilled, it is obviously sufficient to replace y by the variable y , putting y = y , A 2 x 2 A , x 3 ... A,x". Here A , , A , , etc., are the coefficients of the expansion A 2 x 2 + A , x 3 ... of the function y = F(x) which is the solution of Eq. (63) vanishing for x = 0. Furthermore, if it is desirable to have a = - 1, one can use the transformation

+

+

= (-u)-1/[2(n-l)lx

y

+ +

1 3

= (-u)-1"2("-l)1y1

+

+ A , X 2 + A , X 3 + ..* + A,x".

178

Stability of Motion

After performing this transformation the role of the number b will be played by bjv':. In page 174 we found the general expression of the constant g for the simplest of the cases pertaining to the second category, when m = 2. This expression was found under the assumption that the function X vanishes for y = 0 and that a = - 1. It is not difficult to see that it is valid for any equations of the form (l), for which p-1 + a1x2n + ... , (y)y=F(x) = -

with b2 < 4r1, and such that c is the coefficient of y 2 in the expansion of Y in powers of x and y , and D is the coefficient of x y in the expansion of X. For the cases when a is a negative number different from - 1, the expression of the constant g is obtained from the previous one replacing by

-a)-(Zn-l)/(2n-2)

b

by

6,

by by

C(-a)-1/(2n-2)

by

D(-a)-1/("-2)

D

b(-a)-(n-1)/(2n-2)

,

b1(-a)-"/(2n-2), 3

It follows that in this case the condition for stability is

i

c -(n

1

a + 1)2 - ?ID b + (2n + l ) b l < 0. a

The condition for instability is expressed by the opposite inequality,

22. All the possible cases of system (1) being examined, we consider it opportune to put together here the most important of the results obtained. Supposing that the differential equations of the perturbed motion are of the form: dxjdt = y X , dyjdt = Y, where X and Y are arbitrary holomorphic functions of the variables x and y , not containing terms lower than second order with respect to x and y , we consider the equation y X = 0. From this equation we

+

+

Investigation of One of t h e Singular Cases of Theory of Stability of Motion

179

deduce an expression for y in the form of a series expansion y = A 2 x 2 A , x 3 + A,x4 --.,in increasing powers of x. Let f(x) be the result of replacing y by this series in the expression of Yand ~(x),the result of the same substitution in the expression of axjax a Y/ay. In the case whenf(x) is not identically zero, let ax* be the lowest order term of its expansion in increasing powers of x. Suppose also that bxP is the lowest order term in the expansion of cp(x), i n case it does not vanish identically. These being settled, we can distinguish ten cases that we enumerate below, indicating at the same time for each one how the problem of stability is solved.

+

+

+

I. a is even. 11. a is odd, a > 0. In both cases the unperturbed motion is unstable. 111. CI is odd, a < 0 ; p is even and less than CI, b < 0. The unperturbed motion is stable. IV. p is even, b > 0 ; the function f(x) does not contain terms lower than of the l)th order. V. /3 is odd; the ,function f (x)does not contain terms lower than of the 2(/3 1)th order. VI . /3 iseven; a = 2 8 + 1 ; b 2 + 4 ( p + l ) a > 0 . In all these three cases the unperturbed motion is unstable. VII. ci is odd, a t o ; the function p(x) does not contain terms lower than of the [(a 1)/2]th order. Putting CI = 2n - 1, we first transform our differential equations by the substitution

(a +

+

+

= (-a)-1/(2n-2)

= (--a)-’/(2n-2)

X1,

y,

+

A,X2

+ A3X3 + ... + A,x”:

and afterwards by x1= rCs 9, y , = - rnSn9, where Cs9 and Sn9 are the functions of the variable 9, determined from the equations -d Cs 9

d9

- -Sn

dSn9 cs2n-l 9,___ 9 d9

provided that CsO= 1, SnO=O.

Stabilitv of Motion

180

Eliminating dt from the transformed equations, we obtain a differential equation of the form dr/d9= R2r2 R,r3 .-., where R 2 , R , , etc., are some periodic functions of 9. We integrate this equation using the expansion r = c u2c2 u,c3 in powers of the arbitrary constant c. The coefficients us of this series are found successively in order of increasing s, by quadratures. If among the us there are nonperiodic ones with respect to 9, the first such coefficient will be of the form g9 a periodic function where g is some constant, Then, for g > O the unperturbed motion is unstable and for g (0, it is stable. In the case when all the coefficients us are periodic functions of 9, the unperturbed motion is always stable. VIII. p is even, CI = 2p 1 ; b2 4(p 1)a < 0. Putting a = 2n - 1 and ' b Sn2 9 Csn-' d9 J = 0,

+

+

+

+

..a,

+

+

jb

a

+ +

+ b Sii 9 Cs "9

we transform our differential equations by the double substitution = (-

*)- 1 / ( 2 n - 2 )

= (-*)-1/(2n-2)

x1 = pee Cs 9,

X1,

y,

+ A2X2 +

A,X3

+ ... + A,x";

y , = -pnen@Sn 9;

and afterwards eliminate dt from the transformed equations. In this way we get an equation of the form dpld9 = P2p2+P3p3 in which P,, P,, etc., are periodic functions of 9. In order to solve the problem of stability, this equation is treated similarly to the equation of the previous case. IX. f ( x ) = 0 ; /3 is even, b < 0. The unperturbed motion is stable. X. f ( x ) = 0; cp(x)= 0. The unperturbed motion is unstable.

+

1 s . )

Investigation of One of the Singular Cases of Theory of Stability of Motion

23.

181

Let us consider three examples.

EXAMPLE 1. Suppose that the differential equations of the perturbed motion are the following: dx dt

+ Ax2 + Bxy + c y 2 ,

dY

+ M x y + Ny2,

-=y

d t = Lx2

where A , B, C, L, M , N are constants. We easily find

J ( x ) = Lx2 - A M x 3 + ... , ~ ( x= ) (2A

+ M ) x + ...

We see from this that if L is different from zero, we have case I. If L = 0, whereas A and M are of opposite sign, we have case 11. If for L = 0, A and M are of the same sign, then, remarking that in this case

+

8=1,

a = -AM,

+

b=2A+M,

we find b2 4@ 1)a = (2A and, therefore, we have case VI. If for L = 0 one of the numbers A and A4 is zero and the other is different from zero, we have case V. Finally, if L = A 4 = A = 0, we have case X. In conclusion, whatever the coefficients of our equations are, the unperturbed motion is unstable.

EXAMPLE 2. Given the equation d2x dt2

the right-hand side of which is an entire homogeneous function of any degree k of the quantities appearing in brackets. Suppose that it is requested to investigate the stability of the motion x = 0 with respect to x and dxldt. Replacing this equation by the system

_ dx -

dt - y ,

dY

dt = F(x, y),

182

Stability of Motion

+

+

and putting F(x, y ) =Loxk+L,xk-ly + L , - , x ~ ~Lkyk, -~ we find :f ( x )= Loxk,q(x) = Lkxk - '. Suppose first that k is even. In this case, if Lo is different from zero, we have case I and if Lo = 0, we have either case V or case X. We must conclude therefore that the investigated motion is unstable. Suppose now that k is odd. In this case, for Lo > 0 we have case 11, for L1 > 0 case IV, and for Lo = L , = 0 case X, and therefore our motion is unstable. If L , < 0, Lo < 0, we have cases I11 and IX, and, therefore, themotion is stable. Finally, for Lo < 0, L1= 0 we have case VII. If in this case not only L,, but also all L, corresponding to odd s are equal to zero, system (64) has obviously a holomorphic first integral independent of t (holomorphic with respect to x and y'), and our motion is stable. Suppose now that among the coefficients mentioned there are some which are different from zero. Putting k = 2n - 1, and denoting by p one of the numbers 2, 3, ... , n, suppose that L1 - L3 - ... = L z p - 3

. a .

=o,

(65)

but L2,-,* is different from zero. Supposing, in order to simplify the writing, Lo = -1, and transforming our equation by means of the substitution x =rCs 9, y = - rnSn9, we find dr dt

- = -Sn 9 ~

Here

r

~

~

u = 1 (-1)"LS CS2n-s-19 snsgr'n-

-

~

,

2n- 1 s=

1)(.7-2)

2

Eliminating dt from these equations, we obtain the equation

* In Liapunov's collected works L,, note).

-I

is written instead (Translators'

investigation of One o f t h e Singular Cases o f Theory o f Stability of Motion

183

the right-hand side of which can be represented in the form of a series expansion R Z n - 1 r 2 n - ' R 3 n - 2 r 3 n - 2 ..., in powers of Y, increasing by n - 1. The lowest order term is of the (2n - 1)th order. Thus we can seek Y as a function of 9 in the form of a series expansion

+

r

=c

+

U2n- 1 2 - l

+

+ U 3 n - 2 C 3 n - 2 + ... ,

in powers of the arbitrary constant c. Here the powers of c also increase in steps of n - 1. The coefficients us of this series are found successively from equations of the form

where Us is zero for s = 2n - 1. For s > 2n - 1, Usis a known polynomial formed with those u, and R, for which c < s. We remark now that by assumption (65), our formulas yield, for all R, corresponding to s < (n - l)(2p - 1) I , the same values as in the case when all Li,corresponding to odd i, are equal to zero. in consequence, all us corresponding to s < (n - 1)(2p - 1) -11 are necessarily periodic functions of 9. Let us now consider those u , corresponding ~ to s = (n - 1)(2p - 1) 1. Denoting by RS0' the value of R, after making L,,-, = 0, we can affirm that for the values of s shown right now, the integral J(Rio) Us)& is a periodic function of 9. However, for the same s our formulas yield

+

+

+

R, = L z p - l CS""") 9S I I9~+~Rlo'

Thus, for L,,-, different from zero, the function u, = J ( R , is not periodic and L2p- 1

w

g = 7 J 0

+ UJd9

SnZP9 C S ~ ( ~9 d9. -~)

From this we deduce that for L 2 , - , > 0 the investigated motion is unstable and for L,,-, < 0 it is stable. We supposed Lo = - 1 . But it is not difficult to see that the conclusion formulated is valid for any negative L o .

184

Stability of Motion

We can affirm, therefore, that in our example, stability is possible only when k is odd and it actually takes place in three cases: (1) when Lo = 0, L, < 0; (2) when Lo < 0, and the first coefficient different from zero in the sequence L,, L, , L , , is negative, and (3) when Lo ( 0 and all the other coefficients are zero. In all the other cases the investigated motion is unstable.

EXAMPLE 3. Finally we consider the case of a canonical system dx aH _ -y+--,

dt

ay

dy aH - -- -dt

ax ’

where H i s a holomorphic function of x and y , not containing terms lower than of the third order. Since for this system we find q(x) = 0, we can meet only one of the following four cases: I, 11, VII, or X. Of these cases, stability is possible only in case VII. Whenever this case is present we surely have stability, since our system has the holomorphic first integral independent of t, y2

+ 2H.

(66)

In case VII, this first integral is necessarily a sign-definitefunction of the variables x and y , since, by means of the substitutions shown in the previous paragraph, it is being transformed into a function of the variables r and 9, and it becomes a series expansion in powers of r, the lowest of which has necessarily a constant coefficient. One can affirm apriori that whenever the integral (66) is a signdefinite function (i.e., for x and y which are sufficiently small in magnitude it vanishes only for x = y = 0), the unperturbed motion is stable. To this we can now add that in all the other cases the unperturbed motion is unstable. Thus, for a canonical system the problem is reduced to the investigation of the function (66) and requires, for its solution, the same analysis as does the problem of maximum and minimum for the functions of two independent variables.

An Investigation of a TranscendentaJ Case of the Theory of StabiJity of Motion In the present work we treat that critical case of stability theory in which there are two zero roots corresponding to a multiple elementary divisor, and the number of equations is more than two. In this case, the stability problem is solved completely.

In reference [ 11 Liapunov studied the stability problem for the null solution of systems of differential equations of the form dx

-=Y

dt

+ X ( x , y , z),

2 dt = Y ( x , y ,z),

dz dt

- = Az

+ Z(x, y , z), (0.1)

where x, y , and t are scalar variables, z is an n-dimensional vector of components zl, z 2 , ... , z, ,and A is a constant quadratic n-order matrix of elements aij. (For the sake of convenience we made some changes in Liapunov’s notations.) The eigenvalues of A have negative real parts. is a vector function of components 2,. The functions X , Y, and 2, are power series in x,y , zl, ... , z, , with no term lower than the second order being present. Kamenkov [2], considering the same system, obtained independently of Liapunov* a series of the same results. Liapunov’s studies were not exhaustive. One transcendental case was not fully examined (following Liapunov, we call transcendental the case where the solution of the stability problem depends on all the terms in the expansion of the right-hand sides into power series).

* The work of Liapunov [l] was found in the archives of the Academy of Sciences of USSR by academician V. I. Smirnov. It was first published very recently. 185

186

Stability of Motion

This case occurs when the right-hand sides of the system (0.1) satisfy the following conditions : (I) The function X vanishes at x=O, y = O and at y=O, z = 0; i.e., X ( 0 , 0, 2 ) = 0,

X ( x , 0,O) = 0 ;

(0.2)

(IT) The function Y is of the form Y ( x , y, z ) = - x m + g l x m + l + g 2 x m + 2 + .*.

+ ~ ( u x +" u ~ x " ' + + Yl(x, y , z)),

(0.3)

where Yl(x, 0, 0) =O, rn = 2 q - 1 is an odd number larger than 1 and ci > (m- 1)/2 or ci = co,i.e., the function Y does not contain terms of the form yxk in its expansion. (111) For z=O, the components Z,* of the vector function 2 contain powers of x not less than (rn 1) in the terms independent of y . In the terms containing the first power of y , there are no powers of x less than (ci 1) for ci < m , and not smaller than rn for ct >, m. For other assumptions regarding X, Y, 2, Liapunov either gave the exhaustive solution to the stability problem, or indicated how the problem is reduced to the case shown above.

+

+

1. Let us put down briefly some of Liapunov's results related to the above case. As in [3], Liapunov introduces the functions C(9) and S(9) as solutions of the system of equations dCjd9 = - S ,

dSld9 = C2q-

(1.1)

with the initial conditions C(0) = 1,

S(0) = 0.

(1 4

Different properties of the functions C(9) and S(9) are treated in detail in [ l ] and [3]. It is shown, in particular, that C(9) and S(9) have the same period o > 0. The change of variables x = rC(9),

*

y = -rqS(9)

In the original work Z , was used, which was a misprint.

(1.3)

Investigation of a Transcendental Case of the Theory of Stability of Motion

187

brings the system (0.1) to the form dr dt

- = r q + ' R l ( r ,9)

d9 dt

- -- f

dz dt

- 1

- = AZ

+ rR2(r, z , 9)

+ rYO,(r,9) + @,(r, z, 9),

(1.4)

+ Z(r, z , 9>,

where the functions R, and 0, are power series in r with coefficients of period o with respect to 9, and which converge absolutely and uniformly for sufficiently small r and for all 9. The functions R, and 0, are the same kind of power series in r, zl, ... , z, , and vanish for z = 0. The components 2, of the vector function 2 are also power series in r , z,, ... , z,, , with coefficients of period o with respect to 9, but these series do not contain terms of an order smaller than two in r and zk . In terms independent of z k ,the power of r is at least 2q. Thus the problem is reduced to the study of the stability with respect to r, zk. According to the first equation of (1.4), r is either identically equal to zero, or it conserves the sign of its initial value as long as IrI and I IzI I are sufficiently small. Without loss of generality we can consider that r 2 0.

(1.5)

The solution of the stability problem for the null solution of the system (0.1) depends to a large degree on the existence of periodic solutions in a sufficiently small neighborhood of the origin. To find such solutions and to study their properties, Liapunov made one more change of variables: r = c(1

+p),

z

=2

5,

where c is a sufficiently small positive constant and number satisfying the inequality q

< f l d 2 q - 1.

Denote by C, the components of the vector C.

(1.6)

a natural (1.7)

Stability of Motion

188

By putting (1.6) into (1.4) and eliminating t we get dP d9

- = cP(c, p,

C,9),

cq-1

dl=

d9

+ cF(c, p, C,9),

(1.8)

where P is a scalar function and F a vectorial one. For all 9 and for sufficiently small c, IpI and 11C11*, both P and Fcan be expanded absolutely and uniformly into power series of c, p , 5,. Together with (1.8) one considers the system

where q(c), defined for c > 0, is chosen so that the system (1.9) has an w-periodic solution p = po(c, 9), 5 = C0(c, 9), with the initial condition po(c, 0) = 0, for all sufficiently small positive c. Liapunov proved that such a function q(c) does exist, gave an algorithm for its construction and studied its properties in detail. In particular, he proved that q(c) is a continuous differentiable function for sufficiently small nonnegative values of c, has an asymptotic expansion in power series for c 4 0 , and q(0) = 0. Furthermore, Liapunov studied in great detail the properties of the w-periodic solutions p = po(c, 9), i= Co(c, 9) of the system (1.9). From his results follow thc estimates lpo(c, 911 < Dc, Ili0(c, 9)ll < Dc24-8, ( I .lo) where D is some positive number. From the way the function q(c) was introduced into (1.9) it follows that if q(c) = 0 for some c, the corresponding periodic solution p = po(c, 9), 5 = io(c, 9) is a solution of the system (1.8). Therefore, for the values of c for which q(c) vanishes, the system (1.8) has a periodic solution, satisfying the inequalities (1. 10). The solution of the stability problem depends on the behavior of the function q(c) in the neighborhood of c = 0. Liapunov proved the following four statements. (1) If q(c) = 0 for all nonnegative c which are sufficiently small, then the null solution of the system (1.4) is stable.

* This is an inaccuracy of Liapunov’s. In order to get (1.8) it is necessary to make the supplementary substitution c = cIz,p = c,pl.

Investigation of a Transcendental Case of the Theory of Stability of Motion 189

(2) If there exists c' > 0, such that q(c) > 0 for 0 < c < c', then the null solution of the system (1.4) is asymptotically stable. (3) If there exists c' > 0, such that q(c) t O for 0 < c < c', then the null solution of (1.4) is unstable. (4) If the equation q(c) = 0 has an infinite set of arbitrary small roots, and q(c) is nonnegative for sufficiently small c, then the null solution of (1.4) is stable. The one case which remained unconsidered was that in which equation q(c) = 0 has an infinite set of arbitrary small roots, and the function q(c) takes negative values for arbitrary small c. We shall prove that the null solution of (1.4) is stable under the condition that q ( c ) = O has roots in each neighborhood of the point c = 0. Upon doing this, the stability investigation will be completed. Let us introduce the quadratic form V(zl, the following partial differential equation : 2.

... , zn), satisfying

where ~~z~~ is the euclidean norm of the vector z. It is well known (see reference [4]) that the form V(z) exists, is unique, and is positive-definite. We notice that if for some numbers B and c the inequality V(z) < B 2 ~ 4isq satisfied, then ~~z~~ < Blc2q, where B, is a positive number depending only on B. Lemma 1. There exists a number B such that,for suficiently small r and llzll and if the inequality V(Z) 2 ~

~

r

~

4

(2.2)

is satisfied, the following inequality holds:

3=

av C - (aslzl + n

s = l aZ,

+ asnz,+ Zs(rr z , 9)) < 0.

PROOF. From (2.1) follows the equality

c av

3= - 11zl12 + s = l az, Zs(r,z,9), n

(2.3)

190

Stability of Motion

or

As we assumed that the series representing ZS(r,0, 9) begins with terms of order 2q at least, it is clear that there exist a B* > 0, so that for sufficiently small r, llzll, the inequality

is satisfied. As the difference &(r, z, 9)- &(Y, 0, 9)vanishes for z = 0, then for sufficiently small r and I/z//we have

+

B*Ilzllr2q. The relations (2.5), (2.6), and (2.7) give V < -&llzll From this follows that if llzlj 3 3B*rZq,and Y and llzll are sufficiently small, then V
3. We shall suppose in the following that there exists a set C of positive numbers, having zero as a point of accumulation and so that q(c) = 0 for each c E C. For each c E C the system (1 .S) has a periodic solution P = PO(G

a),

r=

CO(C,

$1,

Po = (c, 0 ) = 0,

and this solution satisfies the inequalities (1.10). Let us study the neighborhood of such solutions, corresponding to sufficiently small c E C. I n order to do this we shall make the following change of variables in (1.8): P = PO(G 9) + a,

i= CO(C, 9) + *.

(3.1)

Investigation of a Transcendental Case of the Theory of Stability of Motion 191

The equations for a and $ are of the form

where

Q = P(c, ~ o ( c9) , + a,5 o ( ~9) + II/, 9) - P(C, PO(C, 91, CO(C, 91, $1,

(3.3)

G = F(c, PO(G 9) + a,CO(C, 9) + *, 9)

(3.4)

- q c , P O ( C , 9,5o(c, 9>,9>. From these equalities it follows that Q(c, 0, 0,9) =0, G(c,0, 0,s) = 0 and that G and Q have a period w with$respect to 9. Moreover, from these equalities and the inequality (1.10) it follows that the functions Q and G satisfy a Lipschitz condition with respect to the variables a and $: L

9) - Q(c,

a2

3

$2

5

$199) - G(c,

a2

7

*2

I

IQ(c,

~ 1 $ ,1 3

IIG(c,

a19

$11 < %( I ~ I L 9)Il < 5 (1%

+ II$1 -

- a21 + Il*l

$2II)9

(3.5)

- $211).

(3.6)

These conditions are satisfied for sufficiently small c, (a1and 11$11 and for all 9. The constant L can be taken as the same for all sufficiently small c E C (and accordingly for all periodic solutions P = PO(G 9>, 5 = io(c, 9)). Lemma 1. There exist constants E > 0, A > 0, co > 0, a, > 0 and a scalar function f ( c , a, O), where c and O are scalars and a an n-dimensional vector, such that (1) The function f is defined for each c E C, c < co for all 0 E ( - 0 0 , +a)and IlaII
+

192

Stability of Motion

( 5 ) f o r c E C, c < c o , the solution of the system (3.2) with the initial conditions

9 = 0, $ = a, satisJes the inequalities

for

a =f(c, a, 01,

llall

d a,

(3.7)

1.(9)1 < l / a l l ~ exp ~-~

(3.8)

ll$(9>ll < Elall exp

(3.9)

all 9 3 8.

PROOF. Consider the system of integral equations +m

a = --c

J9

Q
x G(c, a, $,

2)

dt,

(3.10)

dz,

where a is an n-dimensional vector. Using the fact that all the eigenvalues of the matrix A have negative real parts and the functions Q and C satisfy the Lipschitz condition (3.5) and (3.6) with the constant L independent of c, we shall prove that the system (3.10) has a solution for each vector a with a sufficiently small norm and that this solution satisfies the conditions enumerated in the lemma. W e shall construct the solution of the system (3.10) by the method of successive approximations : a.

A

= 0, $, = exp[ ,-1(9 +m

arn(9,0) = --c

j”

9

- 011 a ,

01,

Q(C, a m - l ( ~ , $ r n - l ( z ,

(3.11)

01,

dz,

Investigation of a Transcendental Case of t h e Theory of Stability of Motion

Let p

193

> 0, M > 0 be such constants that for 9 >, 8

where IlAIl is the euclidian norm of the matrix A and 1is an arbitrary number satisfying the inequalities 0 < R < p. Let us show that if c E C, c < c, , /la/l< a , , the constants a , and c, being sufficiently small, then all the successive approximations, i.e., the functions a,($, 0) and $n,(9,8) for 9 >, 0, are defined and satisfy the inequalities :

A

- o)].

(3.13)

< 2MIl4lexp -F (9- U)].

(3.14)

l a m ( 9 , ~ > 1G 11a11cq-1 exp[ - F ( 9

ll$m(9, 0)Il

i 1

For m = 0 these inequalities are satisfied. Suppose that the successive approximations al, a 2 , $ 2 , ... , $,,- are defined and satisfy the inequalities (3.13), (3.14). We shall show that in this are also defined and satisfy the same inequalities. case a, and From (3.1 l), and taking into account the identity Q(c, 0, 0, 9) = 0 and the inequality (3.5), we get

Considering that cq-' < 2 M , we get from the last inequality and from (3.13), (3.14):

for 9 3 0. We consider in the following the constant c, to be so small that [2LM/R]c, < 1 ; in this case we get (3.13) from the last inequality for c c, .

<

194

Stability of Motion

We deduce from (3.12) in the same manner as above, the estimate $rn(g, 0)ll Q M l l a l l e x ~

Considering as above that cq-l < 2M, we get from here, because of (3.13), (3.14): II$rnII

Q Mllallex~

or

We shall suppose that [2ML/(p - 2)]co < 1 ; in this case (3.14) follows from the last inequality. In all the following we shall take, of course, the constants a , and co to be so small that for llall
+ W , Q + 0)= a,($, el,

urn($

$,,,($

+ 0,o + w ) = $,(s, 0).

(3.15)

The functions a. and $o have this property. Suppose now that the relations (3.15) are valid for a m - l , $ m - l . From (3.11) we get urn($

+ 0,0 +

1

+m

W)

= -C

9fw

Q(c, U , - ~ ( T , 8

+ w ) , I ) r n - l ( ~0, + w ) , z) Liz

Investigation of a Transcendental Case of the Theory of Stability of Motion

195

In the integral in the right-hand side of this expression we shall perform a change of variable t l = z - w ; we then get am(3

+ 0,f3+ w ) = +m

+

Q(c, a , , - l ( ~ l w , 0

-cJ,

+ w),

$ m - * ( ~ l

+ w , 0 + w ) , z1 + w ) d ~ ~ .

Since by assumption the functions a,,,-l and $ , , - 1 satisfy (3.15), and the function Q is of period w , we get from the last equality ~

~+ 0, ( 03+ 0) = --c

+a0

J9

= LY,(3,

Q(c, a m - ~ ( r l01, , $ , , - 1 ( 7 1 ~ 01, 7 i ) d ~ l

0).

From (3.12) it follows that

x G ( c , M,,-

By changing here

Z~

=

z -w

$A3

0

1(~,

+ w), $,,-

,(Z,

0

+ o),

Z)

dZ.

as we did above, we get

+ w , 0 + 0) = $,,(3,0>.

Therefore, the validity of (3.15) is established for each m. Let us prove the convergence of the chosen process of successive approximations. Estimate first the quantities c(l - a. and t+h1- t+b0. We have

whence, by (3.5), we find

Stability of M o t i o n

196

or

(3.16) x G(c, 0, exp

I c4-1A 1

- 0) a, z) d.r;

(Z

L A

1

x LMaoexp - , s - l ( t - Q ) ds

or ll$l

- $011

<

M~LU,C ~

p-2

exp[

A

-F (9 - 011.

(3.17)

Let us prove for all m the validity of the inequalities:

ll$m

Mrnt 1

<

lam - am-11

LmcOm

(P -

- $m-111 <

M m + 1

a, exp[

LrnCOrn

(P -

a , exp[

A

-F (8 A

@], (3.18)

-?(9- 011.

(3.19)

As M 3 1, we see from (3.16), (3.17) that these inequalities are valid for m = 1. Suppose that the inequalities (3.18) and (3.19) are valid and let us estimate the quantities c(, + l - a#, and lc/,n+l - $, . We have +co am+1-am=

-cJ

9

C Q ( c ~ ~ m , ~ m , ~ > - Q < c , ~ m - 1 , $ m - 1 , z > l d z .

Investigation of a Transcendental Case of the Theory of Stability of Motion

197

From here, from (3.9, and from our assumption that (3.18) and (3.19) are valid, it follows that + Iarni1-amI

OD

M m+lLmCOm

GCJ

( P - 4"

9

aoL exp[

R 7- O ) ] (5

We shall consider, that co is so small, that c$-' <MA/(p this case we have Mm+2

lam+

1

- am1 <

L

m+l

- A);

in

A

m+l

co

( P -+,),

dz

1

(3.20) x CG(c, a m

3

$m

- G(c, am - 1 , $ m - 19z)l dz.

Thus follows from inequalities (3.6), (3.18) and (3.19):

and

From (3.20) and (3.21) it follows that the inequalities (3.18) and (3.19) are valid for each m. If we choose co so small that the inequality MLc, <1

u-A

(3.22)

Stability of Motion

198

is valid, then the sequences ~(~(9, 0) and I),($, convergent for c E C, c < co , llall
0) are uniformly

a , 9, 8) = a(c, a, 9, d),

(3.23)

m-100

lim $m(c, a , 9, 0) = $(c, a , 9,Q).

rn-m

(3.24)

The sequences a, and $, are uniformly convergent; i.e., the functions a(c, a, 9,8) and $(c, a, 9, 0) are continuous for fixed c E C, c < c, when llall < a,, 9 > 0. From (3.15) it follows that the functions a(c, a, 9, 0) and $(c, a, 9, 0) have the following property : a(c, a , 9 + 0,8 + 0)= a(c, a, 9,0), (3.25) $(c, a, 9

+ 0,0 + 0)= $(c, a, 9,0).

As the sequences a, and $, are uniformly convergent and conditiQns (3.13), (3.14) are satisfied, one can go to the limit under the integral sign in (3.1 1) and (3.12). From this it follows that the functions a(c, a, 9,8) and $ (c,a,9,0) represent the solution of the system of integral equations (3.10). Taking the derivative of (3.10) with respect to 9, we get that a(c, a, 9, 8), $(c, a, 9, Q) represent the solution of the system of differential equations (3.2) with the initial conditions : 9 = 0, $ = a, a = j ( c , a , el, (3.26) wheref(c, a, 0) = a(c, a, 8, 8). The function f ( c , a, 0) is defined for c E C, c < c,, llall < a o and for all 0 E (-00, 00). From the continuity of a(c, a, 9, 0) it follows that f ( c , a, 0) is also continuous for a fixed c. Finally, the functionf(c, a, 0) has the period 0 with respect to 0, as is shown by (3.25). By taking the limit for m -+ co in (3.13) and (3.14), II (3.27) MC, a , 9,e)i G i i w - 1 exp[ -c4-1 (9- 011

+

ll$(c, a , 9,Q)ll G 2MllallexP[

2

--+ (9 - 011.

(3.28)

From (3.27) it follows that statement (4) of the lemma is valid.

Investigation of a Transcendental Case of the Theory of Stability of Motion

199

Putting E = 2M, the inequalities (3.27) and (3.28) become (3.8) and (3.9). From this, and from the uniqueness of the solution of the system (3.2), follows the final affirmation of the lemma. The lemma has been proved. 4. Let us now prove our main statement.

Theorem. Suppose that there exists a set C of positive numbers, having zero as a point of accun~ulationand that q(c) = 0 f o r each c E C ; in this case the null solution of the system (1.4) is stable. PROOF. Take an arbitrary 6 > 0 and choose a value c E C, c < co , so that the inequalities c < 4 2 , B 1 2 2 4 ~< 2 q2c, are satisfied, where B1 is such a number that llzll < BlcZq for V(z)< B2c4q, whiIe B is given by Lemma 1 and the inequalities

< a, DcZ4-O + B 1 2 2 q ~ 2 4<- 8a,, Dc

+

(4.1) (44

and D is associated with the inequalities (1.10). Moreover, we shall consider the number c sufficiently small so that for c/2 < r < 2c, l/z/I< B 1 2 2 4 ~the 2 q inequality r4-l

+ r q @ , + O,

> (c/4)4-'

(4.3)

is valid. According to the chosen c, take 6 > 0 so that 6 < c/2 and

vyZ

z 2 , ..

. , zn) < ~2244~44

(4.4)

for IJzIl< 6. Let us now show that for each solution of (1.4) for which for some value t = to the inequalities r < 6, J/z/J < 6 hold, the relations

are valid for all t 3 t o . As c < t/2, we have thus also proved the stability. Let us suppose our claim to be wrong and that there exist solutions r(t), z(t) of the system (1.4) the initial values of which

200

Stability of Motion

satisfy the inequalities r(to) < 6 and [lz(to)(j< 8, and that there is a value of the time parameter T > to such that for t E [ t o ,T) the inequalities (4.5) and (4.6) are valid, and for t = T a t least one of them becomes an equality. We shall now show that for the assumed solution

v(z(t))< ~ 2 2 4 4 ~ 4 4

(4.7)

for t E [ t o , TI. According to (4.4),this inequality is satisfied for t = t o . Suppose that for the first time it is violated for t = t* € ( t o , TI. From inequality (4.5) it follows that V((z)t*))2 B2r44,

and then from Lemma 1 it follows that P(z(t*))< 0, which contradicts the definition of t*. The inequality (4.7) is therefore established. From this inequality it follows that for t E [ t o , TI, /1zil < B122g~2q < 2c. From this and the definition of T i t follows that for t = T, r = 2c. Let tl E ( t o , T ) be such a value of the time-parameter that for the assumed solution, r(tJ = c/2 and c/2 < ~ ( t<) 2c for t , < t < T. From inequality (4.3)it follows that for t E [ t l , T our solution satisfies the relation ;.1

;,

3

(t' i;. 9; 1 I_

(;)"I.

.,;'

""

72&$."

,&

(4.8)

Therefore in the assumed solution, 9 is a monotonic increasing function o f t f o r t E [tl,TI; it follows, that r and z can be considered, throughout our solution, as functions of 9 in the interval 9, < 9 < 9,,where g1 = 9(tl), 9,= 9(T). Let us construct the function

49) = cc1 + P O ( C , 9)+f(c, c-Pz(9) - io(c, 9),9>1- 491, (4.9) where r(9) and z(9) are the functions r and z in the assumed solution and p a , l o and f are as defined above. By the choice of c [inequality (4.2)]and the above proven inequality 11z(9)11 < B 1 2 2 q ~ 2we q , have for 9, \< 9 < 9,, ( I C - ~ Z ( ~ ) c0(c, 9)li < a o , and therefore the function u(8) is defined and is continuous for 9, < 9 < 9,.

Investigation of a Transcendental Case of the Theory of Stability o f Motion

201

From inequality (4.l), the first of the inequalities (I. 10) and Lemma 2 it follows that (4.10) cC1 + po(C, 91) +f(c, c - ’ z ( ~ I ) - CO(C,$ I > , 3111> t c . From here and from the relation ~(9~) = cj2 follows the relation .(9J > 0. (4.1 1) Analogously, using the same relations, we get

+ PO(C, 9 2 ) + f(C,

c-Pz(9,) - io(c, QZ), 9211 < 3 , and from this and from the relation ~ ( 9= ~) 2c follows: U ( Q 2 ) < 0. 4-1

(4.12) From the inequalities (4.11) and (4.12) follows the existence of such a value 0 E (91, S2), so that u(0) = 0, i.e., for the considered solution, the equality (4.13) r(e) = C c i + po(c, 0) +f(c, C - P Z ( O ) - i0(c, 01, 011 is valid. Pass now from the variables r(9), z(9) to a($), $(9) : N(9) = cC’r(9) - 1 - po(c, 9), (4.14) (4.15) $(9) = c-Pz(9) - iO(C,9). From (4.13) it follows that for our solution one gets the equalities $ = c-Pz(0) - io(c, 0) = a, a =jyC,

z,e)

(4.16) (4.17)

for 9 = 0. As it was shown above, for 9 E [sl,Q 2 ] we get the inequality J ~ C - ~ Z (~ )io(c, 9)ll < a o , and therefore in particular, the inequality jl5lj < a o and then, according to Lemma 2, the estimate 1a(3)1 < uocq(4.18) is valid for all 9 2 0. From this, by the first of the inequalities (1.10) and the inequality (3.1), we get the result that for our solution, we have the inequality r(9) = cC1 + po(c, 9) 49)l < $c, for all 9 3 0 in contradiction to the fact that r 2c for 9 = 9, > 0. The obtained contradiction proves the theorem.

+

V. A. PLISS

202

References

References 1 . Liapunov, A. M., Issledovanie odnogo iz osobennych slutaev zadati ob ustoitivosti dviieniya, Leningrad State Univ. Press, Leningrad, 1963. 2. Kamenkov, G. V., Ob ustoizivosti dviieniya, Sbornik trudov Kazanskogo Aviats. Instituta, No. 9, 1939. 3. Liapunov, A. M., lssledovanie odnogo iz osobennych slutaev zadati ob ustoitivosti dviieniya, “ Collected Works,” Vol. 11, 1956. 4. Liapunov, A. M., Obshtaya zadata ob ustoitivosti dviieniya, “Collected Works,” Vol. 11, 1956.

Index A

N

Algebraic cases of stability, 91 Asymptotic representation, 67 Asymptotically tending motion, 28 Autonomic system, 1

Null solution, I88 P Perturbation, 128 Perturbed motion, 123 PoincarC, H., 62, 67

C Canonical system, 184 Cetaev’s theorem, 4 Characteristic equation, 13 Conditional stability, 129 Conservative stability, 136 Continuous set, 75

R Rational entire function, 49

S

D

Secular terms, 48 Sign-definite function, 135 Smirnov, V. I., 2, 185 Singular cases of stability, 13 Stability problem, 124 Stable motion, 15

Discrete set. 75

E Elliptic functions, 45

F First integral, 33

T

I Infinitesimal upper limit, 24

Time-variable, 149 Transcendental cases of stability, 91

K Kamenkov, G. V., 11, 185

L Lipschitz condition, 191

U Unconditional stability, 129 Uniformly holomorphic, 51 Unperturbed motion, 13 Unstable motion, 15

M Material system, 123 Multiple elementary divisor, 185

V Variation of parameters, 148 203