American Mathematical Society Colloquium Publications Volume 9
Dynamical Systems George D. Birkhoff
American Mathematical Society Colloquium Publications Volume 9
Dynamical Systems George D. Birkhoff
American Mathematical Society Providence, Rhode Island
1991 Mathematics Subject Classification. Primary 58Fxx; Secondary 34Cxx.
International Standard Serial Number 0065-9258 International Standard Book Number 0-8218-1009-X Library of Congress Card Number 28-28411
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© Copyright 1927 by the American Mathematical Society.
All rights reserved. Revised edition, 1966. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America.
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INTRODUCTION TO THE 1966 EDITION Many mathematicians will welcome the new edition of G. D. Birkhoff's book on Dynamical Systems. It represents essentially a continuation of Poincare's profound and extensive work on Celestial Mechanics. Altogether Birkhoff was strongly influenced by Poincare and devoted a major part of his mathematical work to subjects arising from Poincare's tradition. The present book contains Birkhoff's views and ideas of his earlier period of life-it appeared when Birkhoff was 43. To the modern reader the style of this book may appear less formal and rigorous than it is now customary. But just the informal and lively manner of writing has been inspiring to many mathematicians. The effect of this inspiration is visible in a number oflater papers. For example, Morse's theory on geodesics on a closed manifold originated directly in Birkhotf's ideas in dynamical systems. The recent work by Anosov on U-Systems answers the question of ergodicity and density of periodic solutions for a wide class of differential equations-a problem which in Birkhoff's book was studied for a single model system. These and other examples (given below) justify the hope that the reprinting of this book again will stimulate further progress. Of course, after nearly 40 years a number of statements are outdated. For this reason I have selected a list of references to pertinent literature after 1927. Naturally, the list cannot be complete and further references can be found in the books and survey articles quoted. In the Addendum section at the back of this· book I added some general remarks to various chapters. More specific comments are supplied as footnotes at the end of the book. References to these footnotes are given as small numbers in the margin of the text. * JURGEN MOSER
•I am indebted to Dr. R. Sacker for his assistance.
1lI
A PREFACE TO THE 1966 EDITION I met George BirkhotI Birkhoff in 1914, the year after he published his "Proof of Poincare's Geometric Theorem" [1]. In a paper [2] in 1912 Poincare had enunciated a theorem of great importance for the restricted problem of three bodies, but had succeeded in treating only a variety of special cases after long etIorts. efforts. Poincare had also referred to this theorem in lectures in Gottingen. Birkhoffff formulated. this theorem in [1] as follows. "Let us suppose that a continuous one-to-one transfonnation T takes the ring R formed by concentric circles Co and Cb of radii a and b respectively (a > b > 0) into itself in such a way as to advance the points of Co in a positive sense, and the points of Cb in a negative sense and at the same time preserve areas. Then there are at least two invariant points." BirkhotI's Birkhoff's proof of this theorem in 1913 was one of the most exciting mathematical events of the era and was widely acclaimed. In 1912 in [3] Birkhoff outlined the conjecture of Poincare that the "general" motion in dynamics in phase space was of the so-called "discontinuous type". It was in this paper that BirkhotI Birkhoff introduced his novel and beautiful conceptions of "minimal" or equivalently "recurrent" sets of motions. A minimal non-periodic set of motions is of "discontinuous" type if in phase space it possesses no subsets which are continua except arcs of motions. Birkhoff asked me in 1915 to examine the conjecture of Poincare. My answer was by way of a "symbolic dynamics". A non-periodic recurrent symbol introduced at that time was discovered independently by a Russian in 1934 and used by Novikov in his disproof of the Frobenius-Burnside conjecture in group theory. The problem of the generality of "non-degenerate" periodic orbits, that is orbits whose equations of variation admit no noniv
PREFACE TO THE 1966 EDITION
V
trivial periodic solutions was often emphasized by Birkhoff. It is now of major interest to differential topologists of whom Birkhoff and Poincare were among the first, although not so ArnoJ'd and others on stability called. The work of Moser, Arnol'd belongs to a related field, one close to the center of Birkhoff's interest. The above merely samples the many aspects of dynamics in which Birkhoff introduced new ideas, new theorems and new questions. One of Birkhoff's theorems which has aroused the greatest interest was his so-called "Ergodic Theorem" with its subsequent variations, interpretations and consequences in measure theory and probability. Birkhotf once remarked that "it is fortunate that the world of mathematics is as large as it is". Only by bringing to bear the genius and imagination of many lands and intellectual origins can one be sure oJ adequate appreciation of new mathematical ideas and significant responses. History has responded to these pages on Dynamical Systems in an unmistakable way. For this we are more than content.
REFERENCES I. Birkhoff, George, Proof of Poincare's geometric thron!ffl. tMon!ffl. Trans. Amer. Math.
Soc. 14(1913), 14·22. 2. Poincare. H., Sur un theoreme de geometrie, Rend. del Cir. Mat. di Palenno 33(1912), 375-407. 3. Birkhoff, George, Quelques theoremes sur Ie mouvement des sysremes dynamiques, Bull. Soc. Math. France 40(1912), 305-323. MARSTON MORSE
PREFACE TO THE 1927 EDITION The Colloquium Lectures which I had the privilege of delivering at the University of Chicago before the American Mathematical Society, September 5-8, 1920, contained a large part of the material presented in the following pages. The delay in publication has been due to several causes, one of which has been my desire to wait until some of my own ideas had developed further. I have taken advantage of a wellestablished tradition of our Colloquia by giving particular emphasis to my own researches on dynamical systems. It is my earnest hope that the lectures may serve to stimulate others to investigate the outstanding problems in this most fascinating field. It is only necessary to recall the work of Galileo, Newton, . Laplace, Clausius, Rayleigh in the physical applications of dynamics, of Lagrange, W. R. Hamilton, Jacobi in its formal development, and of Hill and Poincare in the qualitative treatment of dynamical questions, in order to realize the remarkable significance of dynamics in the past for scientific thought. At a time when no physical theory can properly be termed fundamental-the known theories appear to be merely more or less fundamental in certain directions-it may be asserted with confidence that ordinary differential equations in the real domain, and particularly equations of dynamical origin, will continue to hold a position of the highest importance. In looking back over my own dynamical work, of which a certain period is finished with the publication of this book, I cannot but express my feeling of deep admiration and gratitude to Hadamard, Levi-Civita, Sundman and Whittaker, to whom many important recent advances in theoretical dynamics are due, and in whose work I have found especial inspiration. It is with much regret that I have been unable to give adequate space to their achievements. VI
PREFACE TO THE 1927 EDITION vii Professor Philip Franklin cooperated with me in a first rewriting of part of my notes on these lectures. lowe him cordial thanks for his help. November 18,1927. GEORGE D. BIRKHOFF.
TABLE OF CONTENTS INTRODUCTION TO THE
1966 EDITION
PREFACE TO THE
1966 EDITION
PREFACE TO THE
1927 EDITION
iii iv
;
vi
CHAPTER I PHYSICAL ASPECTS OF DYNAMICAL SYSTEMS PAGE
1. 2. 3. 4. 5. 6.
Introductory remarks An existence theorem.... ... .. . . . A uniqueness theorem.... .. .. Two continuity theorems..................................... Some extensions The principle of the conservation of energy. Conservation systems 7. Change of variables in conservative systems 8. Geometrical constraints 9. Internal characterization of Lagrangian systems........ 10. External characterization of Lagrangian systems 11. Dissipative systems
1 1 5 6 10 14 19 22 23 25 31
CHAPTER II VARIATIONAL PRINCIPLES AND APPLICATIONS
1. 2. 3. 4. 5. 6. 7. 8.
An algebraic variational principle........................... Hamilton's principle........................................... The priIiciple of least action. ..... .. .. ........... .......... ... Normal form (two degrees offreedom) Ignorable coodinates........................................... The method of multipliers.............. .. . .. The general integral linear in the velocities............... Conditional integrals linear in the velocities.............. ix
33 34 36 39 40 41 44 45
x
9. 10. 11. 12. 13.
CONTENTS
Integrals Quadratic in the velocities The Hamiltonian equations.................................. Transfonnation Transformation of the Hamiltonian equations........... The Pfa1tian equations........................................ On the significance of variational principles..............
48 50 53 55 55
CHAPTER III FORMAL ASPECTS OF DYNAMICS PAGE
1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.
Introductory remarks The formal group............................................... Formal solutions The equilibrium problem..................................... The generalized equilibrium problem...................... On the Hamiltonian multipliers Normalization of H 2 ••••••••••••••••••••••••••••••••••••••••••• Nonnalization The Hamiltonian equilibrium problem.................... Generalization of the Hamiltonian problem On the Pfaffian multipliers Preliminary nonnalization normalization in Pfa1tian problem.......... The Pfaffian equilibrium problem Generalization of the Pfa1tian problem....................
59 60 63 67 71 74 78 82 85 89 91 93 94
CHAPTER IV STABILITY OF PERIODIC MOTIONS
1. 2. 3. 4. 5. 6. 7. 8.
On the reduction to generalized equilibrium Stability of Pfaffian systems Instability of Pfaffian systems Complete stability Normal form Nonnal fonn for completely stable systems Proof of the lemma of section 5 Reversibility and complete stability Other types of stability
97 100 105 105 109 114 115 121
CHAPTER V EXISTENCE OF PERIODIC MOTIONS
1. 2.
Role of the periodic motions An example
123 124
CONTENTS
3. The minimum method 4. Application to symmetric case 5. Whittaker's criterion and analogous results 6. The minimax method 7. Application to exceptional case 8. The extensions by Morse 9. The method of analytic continuation 10. The transformation method of Poincare 11. An example
Xl
128 130 132 133 135 139 139 143 146
CHAPTER VI APPLICATION OF POINCARE'S GEOMETRIC THEOREM
1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.
Periodic motions near generalized equilibrium (m = 1) 150 Proof of the lemma of section 1 154 Periodic motions near a periodic motion (m = 2) 159 Some remarks. . .. . . . . . . . .. ... .. . .. . . . . .. ... . . . . .. . . . . . . . . . . 162 The geometric theorem of Poincare 165 The billiard ball problem 169 The corresponding transformation T 171 Area-preserving property of T 173 Applications to billiard ball problem 176 The geodesic problem. Construction of a transformation TTf< 180 Application of Poincare's theorem to geodesic problem 185 CHAPTER VII GENERAL THEORY OF DYNAMICAL SYSTEMS
1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.
Introductory remarks Wandering and non-wandering motions The sequence M, Mit M 2 , Some properties of the central motions Concerning the role of the central motions Groups of motions Recurrent motions '" Arbitrary motions and the recurrent motions Density of the special central motions Recurrent motions and semi-asymptotic central motions Transitivity and intransitivity
189 190 193 195 197 197 198 200 202 204 205
CONTENTS
xii
CHAPTER VIII THE CASE OF TWO DEGREES OF FREEDOM PAGE
1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.
Formal classification of invariant points Distribution of periodic motions of stable type Distribution of quasi-periodic motions " Stability and instability The stable case. Zones of instability A criterion for stability The problem of stability The unstable case. Asymptotic families Distribution of motions asymptotic to periodic motions On other types of motion A transitive dynamical problem An integrable case The concept ofintegrability
209 215 218 220 221 226 227 227 231 237 238 248 255
CHAPTER IX 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.
THE PROBLEM OF THREE BODIES Introductory remarks The equations of motion and the classical integrals Reduction to the 12th order Lagrange's equality Sundman's inequality The possibility of collision Indefinite continuation of the motions Further properties of the motions On a result ofSundman The reduced manifold M 7 of states of motion Types of motion in M 7 Extension to n > 3 bodies and more general laws offorce
260 261 263 264 265 2J37 270 275 283 283 288 291
ADDENDUM
293
FOOTNOTES
296
BIBLIOGRAPHY
300
INDEX
303
CHAPTER I PHYSICAL ASPECTS OF DYNAMICAL SYSTEMS I, Introductory remarks. In dynamics we deal with physical systems whose state at a time t is completely specified by the values of n real variables Xl'
X~,
•• "
Xn.
Accordingly the system is such that the rates of change of these variables, namely
merely depend upon the values of the variables themselves, so that the laws of motion can be expressed by means of n differential equations of the first order (1)
dxddt =
Xi (XI ,
. " , Xn)
(i =
1, "', n).
Thus, for a particle which falls in a vacuum at the surface of the earth, Xtaud ~ may denote distance fallen and velocity respectively. In this case the equations of motion take the typiclll form where g is the gravitational acceleration. 2. An existence theoret:n. We proceed first to formulate an existence theorem for a set of. differential equations of the general type (1).* The set of n functions Xi will be assumed to be real a:nd uniformly continuous in some open * In connection with the first five paragraphs the following general references may be given: E. Picard, Traite d'Analyse, d'A.nalyse, vol. 2, chap. 11, and vol. 3, chap. 8;E.Goursat, Cours d'Analyse mathematique, vol. 2, chap. 19; G. A. Bliss, Princeton Colloquium Lecturt:s, Lectun:s, .chap. 3. 1
DYNAMICAL SYSTEMS
2
finite n dimensional continuum R in the 'space' with rectangular coOrdinates Xl' " ' , Xn. A 'solution' x(t) of the equations (1) in the open interval t' < t < t" is defined to be a set of n functions Xi (t), all continuous together with their first derivatives and represented for any such t by a point X in R, such that the differential equations are satisfied by this set of functions. EXISTENCE THEOREM. If the point Xo is in R at a distance at least D from the boundary of R, and if M is an upper bound for the functions I Xi I in R, there exists a solution x(t) of the equations (1), defined in the interval
I t-to I < DI(Vn M) X(to) = xo.
and for which To establish this theoreD;l, we observe first that, for any solution of the type sought, the n equations (2)
Si
=
xi-x?-lt Xi (Xl' o
"',
x,) dt
=
0
hold. Conversely, any set of continuous functions x(t) in R, which make the expressions Si vanish -in an interval containing t = to as an interior point, will obviously reduce to Xo for t = to, and will satisfy the differential equations in question, as follows by direct differentiation. Now define the set of infinitely multiple-valued functions X;' (Xl> .. " xu) as that given by any set Xi (YI, .. " Yu) taken at a point y whose various coOrdinates differ from those of the point X by not more than 11m in numerical value. It is evident that with this definition the n components of Xm may be chosen as constant in any rectangular domain (i =
1, "', n),
namely as the component parts of X (aI' "', an). If the functions Xi be replaced by xi" and the functions Xi by x;', the expressions for Si become
1. PHYSICAL ASPECTS
3
We propose to show that these expressions can be made to vanish. Choose X'" as X (x~, "', x~) in the rectangular domain (i =
1, "', n).
The integrals in the above expressions for si" will then be linear functions of t, and hence xi' may be defined as
as long as the point x m continues to be in this domain. In geometrical terms, the expressions for xi (t) yield the coOrdinates of a straight line with t as parameter, which passes through the center of the domain for t = to. If the n functions xi" happen to vanish, the line reduces to the point xu. In case the line emerges from the domain for t = t1 > to at a point yO, we can take this point as the center of a second like rectangular domain. of the same dimensions, and take xi as in this second domain. The expressions Si" will then continue to vanish for t > t1 until the point x'" leaves this second domain at a point zoo Thus, by a succession of steps, the expressions si" can he made to vanish for t > to and likewise for t < to. The process can only terminate in case the broken line representing x m (t) passes a boundary point of R. Now, if t be taken as the time and xi as the n coOrdinates of a particle. its velocity [(x;n)t
+ ... + (X:,)t]1!2
is clearly not more than V n M. Hence the particle must remain inside of R at least in the interval
I t- to 1< DI(vn M). 1·
DYNAMICAL SYSTEMS
All of the functions x'!' are defined in this fixed t interval whatever be the value of m. As m takes on the values 1, 2, 3, "', there arises an infinite sequence of sets n (t) of functions defined in this interval. All of these sets lie in R, and so are uniformly bounded. Furthermore, since the vanish for all i and m, the inequality
x:
Sr
Ix,!, (t
+ h) -
x'!' (t)
I = IJ:t~h Xr (x~n,
"', x;:,) d t
I~ M h
obtains. Hence, by a special case of a well known theorem due to Ascoli, * there exists an infinite sequence of values of" m for which every element of the set x'!' approaches a function Xi of the set £ uniformly, these functions being' themsel yes continuous. It is easy to prove that the functions Xi so obtained satisfy the integral form (2) of the differential equations. In fact, since the vanish for every i and m, we have
st
"& - S~n
Si =
(Xi -
x'!') -
It
o
[Xi (Xl' .. " :en) -
Xr (X'{', . ",
X~)] d t. .
For m sufficiently large, the first term on the right becomes uniformly small inasmuch as each Xi is approached uniformly by the corresponding x'!' over the sequence under consideration. Also Xi(:.s, ",,~) will differ from Xi (x'{', ...• .x~') for any i by a uniformly small quantity, since Xi is uniformly continuous in R by hypothesis; and Xi (x'{', ... ,x;:) in turn will differ from Xr (x~, .. " .x:) by a uniformly small quantity, in virtue of the definition of the funct.ions xt. Hence the quantity under the integral sign on the right also becomes uniformly small as In increases and the expressions"&, which are independent of m, must vanish as stated, so t.hat x(t) yields the required solution of (1). By repeated use of the existence theorem, the given solution x(t) may be extended beyond its interval of definition unless
* For a brief statement and proof see W. F. Osgood, Annals of ·1rlathematics, vol. 14, series 2, pp. 152-153.
[)
1. PHYSICAL ASPECTS
as t approaches either end of the interval, the corresponding point x(t) approaches the boundary of R. Hence we infer the truth of the following statement: COROLLARY. The interval of definition for any solution x(t) of the equations (1) may be extended so as to take one of the followin,q four forms: -
00
< t < + 00;
-
00
< t < til;
t
f
< t < + 00;
t
f
< t < til ,
where, as t approaches t' or til, the point x approach& the boundary of R. 3. A uniqueness theorem. It may now be proved that there is only one solution of the type described in the existence theorem, in case the functions Xi possess continuous first partial derivatives. This last requirement may be lightened to a well known form given by Lipschitz. UNIQUENESS THEQREM. If for every i and for every pair of points x, y in R the functions Xi satisfy a Lipschitz condition, n
IXi (Xl' "', Xn) -
Xi (Yl' "', Yn)/
< j=l 2.: LjlXj- wi,
the quantities L l , •. " L n being fixed positive quantdies, then there is only one solution x(t) of(l) such that .x(to) = xo. For if two distinct solutions x(t) and yet) have the same values Xo for t = to, the corresponding integral forms of the differential equations give at once Xi -
Yi -
i:
[Xi (Xl' .. " x n) - Xi (Yl' .. " Yn)] dt =
0
for all values of i, and thence by the Lipschitz condition imposed,
Let L be the maximl!m of the n positive constants Li, and let Q be the maximum of any of the n quantities !xi-Yd in any closed interval within the interval It-tol
1 S Q/2. This proves that Q must be O. Hence the two solutions x{t), y (t) which coincide for t = to will continue to do so in any such interval. The theorem follows by repeated application of this result. The physical meaning of the existence and uniqueness theorems is evidently that the motion of a dynamical system is completely determined by the differential equationS and the initial values of the variables determining the state of the system-a fact which is intuitively obvious. Thus the treatment of a dynamical problem requires a formulation of the appropriate differential equations by means of physical principles, and a subsequent mathematical treatment of the properties of the motions on the basis of these equations. 4. Two continuity theorems. There are certain further continuity theorems which are closely allied to the two theorems established above. FIRST CONTINUITY TIiEOREM. If the functions Xi in (1) satisfy a Lipschitz condition in R, the unique solution x(t) for which x (to) = X O is a set of continuous functions of the n parameters x7 and of t- to· We observe first that, in changing the independent variable t to t f = t-to, the modified differential equations obtained differ from (1) only in that t is replaced by t', while in the initial conditions to is replaced by O. Hence the dependent variables Xi involve t and to in the combination t- to only, so that it will suffice to prove the functions Xi to be continuous in x~ and t in the case to = O. A slight extension of the method used in proving the uniqueness theorem may be employed. It is apparent that if Xi and Yi are two solutions of (1) which reduce to ~ and
I. PHYSICAL ASPECTS
7
!It
respectively for t = 0, then by subtraction of the corresponding integral equations there is obtained
provided that the value of t lies within the common interval of definition of Xi and Yi. Suppose that XO lies at a distance at least D from the boundary of R, and then yO at a distance not more than D/2 from xO. This requirement will be met if we take the maximum difference ly?-x~1 not more than D/(2Vn). Restrict t further to lie in the interval It I < 1/ (2n L). Under these circumstances if QO denotes the maximum difference Ix~ - y? I for any i, and Q the maximum difference IXi - Yi i in the t interval under consideration, we find for some value t* of t by means of the above integral equations,
Hence in the stated interval we have constantly Q < 2 QO, i. e., the difference Xi - Yi cannot exceed twice the maximum This means initial difference xJ - yJ in numerical value. that if yO approaches xO, then Y approaches X uniformly throughout the stated interval. Since: dXiI d t I < M ~ery where,the functions Xi must be continuous in x? and t in the restricted t interval. It remains only to remove the restriction upon the interval t. In any closed interval of definition 0 < t < T, the point x(t) is throughout at a distance exceeding a positive D from the boundary of R. Consequently in a t interval of fixed length about any point ( of the selected interval, each function Xi (t) and will vary continuously with Xi (t') and t - t'. It will then be possible to select points
such that t1 is in the interval about
to,
t! in the interval
DYNAMICAL SYSTEMS
8
about ti , and so on. obtain successively IXi(ti)-Yi(ti)
Thus if we take
I ;:;;
IYl-x~ I : :;: ;.
2q, "', IXi (tk)-Yi(tk) I
q we
2k q.
0, while the partial derivatives of the f"th order satisfy Lipschitz conditions. The method of proof
12
DYNAMICAL SYSTEMS
of the second continuity theorem given above shows that the given system (1) of differential equations can be replaced by a system of the same type of order 2 n with Xl, . . . , X n and YI, .. " Yn where Yi = 0 xii 0 Xi for instance, as dependent variables; this system of order 2 n consists of course of the n given equations and the n equations of variation. ~ow if we apply the second continuity theorem to this augmented system, we conclude at once that the second partial derivatives 0 2 Xi / 0 xj 0 Xi and likewise 02 Xi / 0 xj 0 xZ will exist and be continuous. In the augmented system, however, the righthand members will in general possess continuous first partial derivatives of order f.1' -1 which will satisfy a Lipschitz condition. Repeating the above process we obtain the existence of partial derivatives of Xl, .. " X n up to those of order f.1' which respect to the variables x~, ... , x~. In case the functions Xi of Xl, " ' , X'I possess continuous first partial derivatives of order f.1' while the partial derivatives of order f.1' satisfy Lipschitz conditions, the components Xl, ... , X n considered as functions of J.:~., • " x~, t - to will possess continuous partial derivatives in these uariables nf order f.1'. An important case, and the only one entering subsequently into consideration, is that in which the functions Xi admit continuous partial derivatives of all orders in the variables concerned. The components Xl, .. " X n will then necessarily possess continuous partial derivatives of all orders in Xi"",~, t - to' If, furtltennore, the junctions Xi are analytic in Xl, .• " XII' the components Xl, .. " X n considered as functions of Xi, "', x~, t - to' will be analytic in these variables. Let us indicate briefly a proof of this important fact. We observe first that it suffices to show that the unique solution of (1) for which X reduces to XO for t = 0 has components analytic in .-ti, ... ,~, t; here the device used in the proof of the second continuity theorem, namely the intrOduction of t'... = t - to in the differential equations, is applicable.
1.
PHYSICAL ASPECTS
13
Furthermore by writing
in these equations, it becomes clear that we need only prove the components Xl"'" X n to be analytic in x't, ... , x? in the neighborhood of the origin. .Now since Xi are then analytic in the neighborhood of the origin, we may write
Xi«
-,-M _ _-,-_
Xl
1
+... +Xn
(i
1, "', n),
r
where llJ£ is a sufficiently large positive quantity and r is a sufficiently' small positive quantity. The relations written mean that every coefficient in the series expansion of Xi in powers of Xl,"', .xn does not exceed the corresponding coefficient of the series on the right in numerical value. * Now consider the comparison differential system
11'£ XI+",+X,?l 1----
(i
=
1, "', n),
r
of which the unique solution which satisfies the conditions
for t = 0 is evidently given by Xi
=
x~+u
(i
1, "', n),
whereu is defined by the implicit equation
(1-
:X~+"'+X~). n~(! u - - - = Mt. r 2r
Xn
In this case Xl' •. " are clearly analytic in x~, .. " furthermore the explicit formulas obtained for Xl' •• "
x?, t; X'l
on
• Fora proof of this type of relation see, for instance, E. Picard, 7'raite !l'Analyse,vol. 2, chap. 9.
DYNAMICAL SYSTEMS
14
successive differentiation of the comparison system and setting ... = x? = t = 0 shows that the coefficients in the convergent power series for Xl, .. " X" in x~, .. " x~, tare all positive. But the inequality relations written above obviously imply similar relations between any partial derivatives of Xi and of the same partial derivative of the right-hand member of the comparison equations. Thus we see in succession that the formal- series made out with coefficients obtained by successive differentiation of the equations (1) as to x~, "', x~., t and setting = x~ = t = 0, must con verge since the coefficients are less than the corresponding coefficients of known convergent series. Thus these formal series define analytic functions Xl' .. " X", while the mode of determination of these functions renders it certain that every difference function
:tt =
:tt ...
(i =
1, .. " n),
considered as functions of x~" . " x~, t, vanishes together with all of its partial derivatives at the origin when these analytic functions are substituted in. Hence these difference functions must vanish identically. Thus Xl"'" x n obtained in this formal manner will constitute the unique solution satisfying the prescribed conditions, and the stated analyticity is proved. 6. The principle of the conservation of energy. * Conservative systems. In the case of many dynamical systems the geometric configuration is determined by m 'co6rdinates' qll .. " qm having a spar.ial nature, while the state of the system is fixed by the coordinates and the. velocities q;, . ", q~" where qj = dqi/dt. Such a system is said to have m 'degrees of freedom'. With these coOrdinates may be correlated 'generalized external forces' Qi so that by definition * For historical and critical remarks concerning this principle see the article by A. Voss in the Encyklopadie dtT mathematischen Wissenschaften, vol. 4 or in the French vergion by E. and F. Cosserat. I preRented the results here obtained at the Chicago Colloquium in 1920. The following treatment of the principle differs essentially from any other which I have seen.
1. PHYSICAL ASPECTS
16
the 'work' W done on the system is given by m
dW =
1: Qj rl q)
)=1
in which the differential symbols have their ordinary significance. We shall assume that the functions Qi are real, single-valued, analytic functions of the cOOrdinates, velocities, and accelerations; thus there is one and only one set of extRrnal forces Qi which yields a prescribed se~ of accelerations for a given set of coordinate values and velocities. In this case the variables determining the state of the system are clearly the 2 m coordinates and velocities. As a concrete model of such a dynamical system, we may think of a concealed mechani8m which is controlled by a set of m rods which project from a wall. If the rods project by distances qi, .. " qm, then QlI"" Qm are the ordinary forces applied to these rods in an outward direction. The fundamental hypothesis which embodies the principle of the conservation ot energy is that if, by any application of such external forces. the dynamical system is carried through a closed cycle, so that the set of 2 m final values of qi and q~ coincides with the set of initial values, the total amount of work done on the system during the cycle vanishes. Any system of this type will b.e called 'conservative'. Conservative dynamical systems can only be regarded as idealizations of the systems actually found in nature, but nevertheless they are of great importance. Let us now consider the properties of such a conservative system. If it is carried through a cycle ABCA and a modified cycle AB'CA, (which may be represented graphically by closed curves in the· 2 m dimensional space of the qi ~nd qi), the work done in parts ABC and AB'C is the same, namely the negative of that done along the common part CA. Thus the work done along the part AC is independent of the path taken, and so depends only upon the values of ql, .. " qm, qL . ", q:" at C:
DYNAMICAL SYSTEMS
16
By differentiation with respect to t we obtain the following fundamental identity in the 3 m variables qi, qi, qi' : (4)
This relation must subsist if the principle of the conservation of energy is to hold, and conversely it is easily seen to ensure that the principle is valid. It is possible to give the identity an interesting explicit form. Let us endeavor to determine a function L of the 2m variables qi, qi so that the following identity holds: ql!) ~[.i-(~) _~]ql.J-~ = ~ (oW ql.+ oW IJ-
~
j=l
dt
I
OV
oqj
Comparing the coefficients of conditions
j=l
qi'
IJ
oqj
OV
on both sides, we find
m
which hold if (5)
oL) -L l:m ( qj--, oV
W,
j=l
as follows by difl'erentiatio,n with respect to qi. Comparing the remaining terms which are independent of qi', we get the further condition
which is obviously satisfied if L satisfies (5). A value of L for which (5) holds can always be found. Observe first that if Ql, - . " Qm can be.expanded in ascending powers of the velocities q~, .. " q:n, then no first degree terms appear in W. That is, we have
I. PHYSICAL ASPECTS
17
where the subscript indicates the degree of the term in the velocities. In fact the presence of a term WI would lead to a term not involving the velocities on the right-hand side of the fundamental identity whereas there is no such term on the left. If we substitute the above expansion of Wand the cOlTesponding expansion of L, L =
Lo
+ £1 + L t + ...
in the partial differential equation (5) while noting that by Euler's theorem concerning homogeneous functions, ~ , a v" £.J qj -a-'-, = qJ
j=l
TT
(n = 0, 1, ...),
n • II
and if we equate terms of equal degree in the velocities, we find L _ n -
W"
n-l '
while L 1 is unrestricted. Any such function L may be called a 'principal function' associated with the arbitrary conservative system with which we started. When the linear terms in the velocities are lacking in L, a special function is obtained which has important properties. On defining the functions R; by means of the equations (6)
(i =
1, ... , n),
we observe that by the definition of L we have m
(7)
l;RjqJ
= o.
j=l
Conversely, if Ql' ... , Qm are of such a form that (7) obtains, the principle of the conservation of energy holds.
DYNAMICAL SYSTEMS
18
If W is the work function of a conservatzve dynamical system and if L is the associated principal function, then the generalized external forces Qi may be 'Ih-ritten in the form (6), (7). The italicized conclusion above may be expressed in a somewhat different fashion. As· is customary, let us call a dynamical svstem for which R i ~- 0 (i -'- 1, "', n) a 'Lagrangian' system. Also let us call a system W .. . 0 a 'non-energic' system. The appropriateness of the latter ne,,· term lies in the fact that whatever external forces may be applied no work can be done. In this case we may take L - 0 also. The alternative statement is the following: Any conservative dynamical system has external forces which a1'e the snm of the forces of a Lagrangian system and of a non-energic system. Before leaving this topic we may note that for unconstrained motion we have Ql = ... = Qm = 0 by definition. Here the equations of motion take the form (i
=
1. "', 111)
where the quantities R;, are subject to (7). Hence wehi'uy state the following conclusion: An unconstrained, conservative, dynamical system unde'r,qoes the same motion as a Lagmrtgwn system to which a set of non-energzc external f01'ces is applied. An unconstrained conservative dynamical system clearly admits an energy integral W = const., which by means of (5) can be written in the alternative form
~ k.I j=l
(,qj --~aL) aqJ
- L
= canst.
Lagrangian and non-energic systems have. been defined by means of the types of the external forces. These definitions are not mutually exclusive. In fact let us inquire when U' dynamiCal system, will be both non-energic and Lagrangian. Since it is non-energic; we have
I. PHYSICAL ASPECTS
19
and hence we find the most general function m
L
= L1 =
1:
a)
qj.
)=1
Since the system is Lagrangian, we may take Ri = 0 for every i, and thus find directly
Q,i
_ ~ (0
-
~
)=1
ai
0
a)) q),
~----
oq)
oqi
1, .. " n).
(i =
Hence for a system to be of both types the generalized external forces must have this specific form. It is worthy of note that in the case m = 1, equation (7) implies that R 1 is zero, so that any conservative dynamical system with a single degree of freedom is Lagrangian. Similarly in the case m = 2 the external forces may be represented in the form
where J. is an arbitrary function of the c06rdinates, velocities, and accelerations. 7. Change of variables in conservative systems. In the first instance the c06rdinates qi of a conservative dynamical system are a.ctual distances, while the Qi are forces which act in the direction of these collrdinates. But for most physical purposes it is not desirable to adhere to a single set of cOllrdinates. Let us now define the modified external forces Qi corresponding to the new c06rdinates qi by means of the equations (8)
1, ... ,n).
(i =
When this definition is adopted and a further change of variables from qi is made, while the new external forces are defined in terms of Qi by analogous formulas, it is found 2"
20
DYNAMICAL SYSTEMS
that the final functions Qi obtained have the same expressions as if only a single direct change of variables is made. This group property is an immediate consequence of the above definition. Hence the Qi are uniquely defined for any coordinate system whatsoever. It may be observed that for change from one rectangular system to another the above formula for the determination of the Qi in terms of Qi agrees with that obtained by the ordinary laws for the composition of forces. In more general cases the equations above define the generalized force components in the appropriate sense. Now on account of the identity rn
dTV=
L
m
QjdCJj
j=l
= L
Qjdqj,
j=l
it follows that the system will remain a conservative dynamical system according to our definition, in these new coordinates. Furthermore the modifif'd work function will be the same as before (up to an additive constant), and since the formulas of transformation of the velocities m
I
qi
=
aqi - I -=- qj j=l aCJj '"
~
(i
=
1, . ", m)
are linear and homogeneous in the velocities, it follows that the various components TVa, W" . .. in W will be unaltered; If we agree for the sake of definiteness always to choose the unique determination of L which lacks linear terms in the velocities, it follows that the prinCipal function L is the same in both problems. Suppose now that we define Hi by means of the equations
where L is the first given principal function, but expressed in terms of the new variables qt', fit.
21
I. PHYSICAL ASPECTS
It is easy to prove the formal identities
f[~(~) -~] oqj
j=. dt Oqj
OCJ! = _~ ( a.!,) oqi dt oqi (i
_ oqi a.! =
1, .... n),
where on the left side fJi is any function of qi, qi. To see this, we note that from the linear relationship written between the variables q~ and qf we have (i,
i-
1, .. " n).
Hence we deduce for any i
f
j=l
[~(~)] oqj = f[~(~ Oqj)_~ _~_(Oqj)] dt oq)
oqi
j=l
_
d (
dt . oq) oq[
afJi )
- di oq[ -
f
j=l
oqj dt oqi
afJi d (a qj) oqj di o~ .
Moreover we have also for any i
Subtracting the two identities thus obtained we obtain the specified identity. Thi~ identity with fJi = L shows of course that the functions Rt. as obtained from Rt. by the defining formulas have the same siructure as those which give ~ in terms of Qi, Hence we are led to the following general result:. If the variables 'ql, .. " qm of a conservative dynamical system are transformed to gi, ... , qm, the system remains conservative in the new variables with L, W unaltered, while Qi, R i a1'e both modified to corresponding new expressions obtained as in (8), In particular then if the' system is Lagran.qian or non-energic in the first set of variables, it remains so in the modified variables.
DYNAMICAL SYStEMS
22
8. Geometrical constraints. Weare now in a position to deal with the question of .geometrical constraints.' Let us suppose that various geometrical points of the given conservative system are fixed or are constrained to lie in smooth curves or surfaces or to move subject to connections by various rigid bars without mass. The effect of such constraints is to reduce the number of degrees of freedom. In fact, by properly taking coOrdinates ql, .. " qm, the k conditions of constraint may be made to take the form qp.+l = const., ... , qm = const. (ft = m-J.:). Now denote by L that which L becomes when the 7c constrained coOrdinates qp.H, ... , qm have these assigned constant values, while the corresponding qi, qi' vanish of course. Then it is clear that L
=
L,
oL
oL
oqi '
0 qi
(i
=
1, " '. ,II).
Hence we have the relations Qi =
d
dt
(oL) 0 qi -
of 0 qi
+ Ri
(i = 1, ... , ft),
where Qi and R i are defined as usual. But the original external forces Qi may be decomposed into a sum (i=l, .. ·,m), in which the 'forces of constraint' Pi can do no work for any possible displacement subject to the constraints. It follows that the functions Pi"", Pp. must vanish when the collrdinates are selected as above. Hence we may replace Qi by Qi in the formula above for i = 1, .. " /I. Thus we are led to the following conclusion: If a conservative dynamical system with m degrees offreedom is subject to k geometrical constraints, it may be treated as such a conservative system with m-k degrees of f1·eedom.
23
I. PHYSICAL ASPECTS
g. Internal characterization of Lagrangian systems. In. most of the dynamical applications, Lagrangian systems can be regarded as dealing' with a system of particles subject to certain forces and geometrical constraints. This type of internal charact~rization formed the basis of Lagrange's derivation of these equations, and is considered briefly in the present section. In the next section an external characterization is developed. We shall begin by considering three particular types of particles in ordinary space: (a) The inertial particle. Here if x, y, z are the rectangular co6rdinates of the particle, the external forces X, r, Z in the directions of the corresponding axes are proportional to the accelerations in these directions:
Y
X= mx",
== my".
Z =
mz",
where the constant of proportionality, tn, is terined the 'mass' of the particle. This is the case of an ordinary mass particle. The particle is seen to be of Lagrangian type with Lagrangi;m function 1 ('2 L = 2m x y·,2 z'2),
+ +
and L is its 'kinetic' energy. (b) The non-kinetic particle. Such a particle is subject to forces independent of the velocity and having the particular form:
x
=
-
aVlax,
y
= - aVlay,
Z
=
-av/az
where V depends on .the co6rdinates of the particle in space. Here the dynamical system is Lagrangian with L = V. The function -v is the 'potential' energy of the particle due to the field of force in which the particle moves. An electrified particle of slight mass moving in a static electric field is nearly of this type.
24
DYNAMICAL SYSTEMS
(c) The gyroscopic particle. By definition the gyroscopic particle is ·subject to forces with components of the type
. X = (a a I ay - aPI ax) y' + (a a I aZ" - ar I ax) z', so that the force vector is perpendicular to the velocity vector and therefore can do no work. Nevertheless the system is Lagrangian with L = ax'+Py'+rz'. It will be noted that this is the type of special Lagrangian system whicn is also non-energic. An electrically charged particle of negligible mass moving in a static magnetic field falls under this case. (d) The system of generalized particles. If a particle moves subject to a sum of forces of the inerual, non-kinetic, and gyroscopic types it may be termed a gen~ralized particle. Such a situation is realized for example when an ordinary mass particle moves in a gravitational field. It is evident that the resultant system will then be Lagrangian with a principal function merely the sum of the principal functions associated with the component forces. Consider further a set of such particles which do not at first interact in anyway. If we add together the Lagrangian functions for the several particles, there is obtained a single function L which can serve as a single function from which the equations of motion of the system of particles may be derived. It is necessary of course to use suitable variables (Xi, Yi, Zi) where i = 1, 2, .. " m to differentiate between the collrdinates of the various particles. Clearly this yields a principal function L which will be quadratic in the velocities. It is another step in the way of generalization to take L to be any quadratic polynomial in the velocities, in which the homogeneous quadratic part is the kinetic energy T, in which the term U independent of the velocities is the potential energy, and in which the
1. PHYSICAL ASPECTS
25
homogeneous term of first degree may be called the' gyroscopic energy'. Furthermore, as has been seen above, we may suppose the particles to be subject to certain types of geometrical constraints, thus diminishing the number of degrees of freedom, without affecting the Lagrangian character of the problem. It is such a system of generalized particles whjch suffices for most of the applications. (e) The .generalized particle in m-dimensional space. By an obvious extension into which we shall not enter here it appears that a single mass particle lying on an m-dimensional manifold defined by a quadratic differential form, subject to a field of force derived from a potential function in the surface, and furthermore to gyroscopic forces derived from some linear function of the velocities in the surfa~e, will be of Lagrangian type. The function L is quadratic in the velocities. Conversely any Lagrangian system with m degrees of freedom for which L is quadratic in the velocities is representable by the motion of a mass particle in such an m-dimensional manifold. Thus we may interpret the motion of any dynamical system with m degrees of freedom as isomorphic with the motion of a single generalized particle on a suitable m-dimensional surface. 10. External characterization of Lagrangian systems. * In this section we propose to characterize an important type ·of Lagrangian systems by means of certain simple properties ot the external forces. In fact, we shall char~cterize those 'regular' dynamical systems for which the Lagrangian function L is a quadratic function of the velocities, without first degree terms. These form an important class of dynamical systems in which L has the form T - U where T is homogeneous and quadratic * The material of this section was presented before the Chicago Colloquium in 1920. For an analytic characterization of the Lagrangian system in the case when the external forces are linear in the velocities see E. T. Whittaker, Analytical Dynamics, p. 45.
26
DYNAMICAL SYSTEMS
in the velocities, while U is· a function of the cordinates only. It will be observed that regular systems remain of this type under an arbitrary transformation of the coordinates q1, ... , qm'
The first of the characteristic properties which we shall state is the following: 1. The external forces vary linearly with the coordinate accelr:!rations.
Evidently this means that we may write. m
~ avq'/ + bi
Qi =
J=1
where av, bi do not involve the accelerations. II. PRINCIPLE OF RECIPROCITY.· The change in the acceleration q'/ due to a change in the i-th force Qi is the same as the change in the acceleration qt' d1w to an equal change in the J-tlt force Qj, (i, J = 1, .'., n). ~ In order to see what this means we suppose that Qk receives a certain acceleration increment Q, in which case the
above equations give m
QJik =
L: ltij.1
j=1
1
qj'
(i
=
1, .... m)
where .1 denotes the increment as usual and where Jik for i = k and Jik = 0 for i =t= k. Now suppose that· Ql receives the same increment. find similarly
=
1 'Ve
m
L: av.1 q'/.
Q Ju =
2
j=1
It -we assume that the determinant Iav I is not zero, we may solve these equations and obtain for all i, k, l m
.11 qi
=
L: av Q J
jk
j=1
=
aik Q,
.12qi
=
ail Q,
where aij is the cofactor of the element in the J-th row and i-th column of I av I, divided by this determinant. Putting • Compare Ra)·leigh,
Th~ory
of Sound, vol. 1, chap. 4:
1. PHYS ICAL ASPECTS
27
i = land i = k respectively, we obtain from IT, alk = iiAt. Thus the cofactors a-ij are symmetric in i and j, whence it follows that the elements aij are symmetric also, i. e. we must have (l,fJ = aji for all values of i and j. III. For U family of similar motions, the forces urequadratic functions of the speed. In other words, let qi = qi(t), (i = 1, .. " m) be a motion of the system. Suppose that the motion is speeded up in the ratio l to 1. The external forces become m
Qi =
.J; atj{ql, J=1
"', qm, l q~, "', l q'm) l B q'/
inasmuch as the coOrdinates qi are unaltered while the velqcities qi and the accelerations qi' are multiplied by). and l B respectively. If these expressions Qi are to be quadratic in l (qi, qi, qi' being entirely independent variables of course), the functions Uij cannot depend on the velocities, while bi will be quadratic in them. Thus in virtue of ITl we obtain more precisely m
.
Qi =
~ %
j=1
m
m
E bijk qJ qk +j=1 ~ bij qj + bi hk=l
q'/ +
where the functions uij, bijk, bij, bi depend only upon the coOrdinates ql, .. " qm. This form is rendered still more specific by the following hypothesis. IV. REVERSIBILITY. Any motion, under prescl"ibed p-xternal jorces ?11Uy equally be described in the J'et:ersed order of time. The meaning here is that the above relations continue to hold when t is replaced by - f. But this changes the velocities to their negatives while .leaving the coOrdinates qi' and the accelerations qi' unaltered. We infer that the functions bij must be lacking. Hence we may write
DYNAMICAL SYSTEMS
Qi -
... ... 1: av qJ' + 1: bijk qj q" + hi j=1 j,k=1
where av, bijk - ~ikj, bi involve only the co~rdinates. All of the properties I to IV so far employed are invariant under a change of coOrdinates qi and have to do with the nature of the external forces in the neighborhood of a set of values~, "', q~. By a suitable choice of coOrdinates at a pointlj{, .. " q~, we can reduce the expressions for Ql' ... , Q... to the simple form (i =
1, ... , m)
at that point. To· establish this fact we assume that Ij{, ... , the origin, and make a first lineal' transformation
...
qi =
1: IJij qj j=1
U=
~
is at
I, .. o,m),
where the IJij are constants with IJij I :j:: O. For the functions Qi we have then !
Qi =
f
/=1
Qj
~ .=
Q
oqi
f
j=1
Qj IJji.
Hence we find by substitution for every i . Qi
=
... 1: ajk IJji IJklqz" + terms independent j,k,I=1
of q~', ... , q',,;••
It follows that if we chQose oilr transformation so as· to transform the quadratic form
...
1: Ojk qjqk j,k=1 into the sum of squares
qi+ .. ·+q~, be o for i t-j and
the quantities % will 1 for i = j. Consequently it is at least legitimate to assume that av has
29
1. PHYSICAL ASPECTS
been reduced to oij at the origin by this preliminary transformation, so that we have m
Qi= qi'
+ 1:
j,k=l
bijk qj qk
+ b~
(i
1, "', m)
at the ongm. Now suppose that we write further
where the constants b~jk have the values specified in the equations above. We find by differentiation that the equations (i =
1, "', m)
hold at the ongm. 1t is then found at once that in these variables q" the formula for Qi has the stated form at the origin. V. CONSERVATION OF ENERGY. The dynamical system is conkervative. If W is the work function we have the fundamental relation m
dW -
~Qjqjdt, j=l
characteristic of conservative systems. But the sum on the right is linear in the accelerations, and comparing the coefficients of qJ obtained by employing the form for Q, above we find
(j
=
whence
w=
T+U
where
and" where U is a function of qh "', qm only.
1,.", m),
DYNAMICAL SYSTEMS
30
Employing this more specific form for W, we have of course L = T- U according to the earlier developments. and thence
d (oT) oU at 0 qi + 0 qi + R;
Qi =
(i =
1, . ", m)
where Hi satisfy (7). But the first two terms on the right yield expressions just like those for those derived above for Qi, the terms in q'J being ide~tical. It follows that we must have the differences Hi of the form
Hi
m
=
~
Ci.jk·
qj qk + Ci
(i
=
1, "', m).
j,k=t
Applying now the above condition (7) we conclude further that for all i, j, k the relations CVk
+ + Cjki
Ckij
=
0,
Ci
= 0
must obtain where. Ci,ik = tikj also of course. Hence. principles I-V lead to the type of external forces, .
Qi =
d (oT)+oU+ -0-'
dt
~
qa
qa
~
II
A:.J Cijk qj q,c,
j,k=l
where the Cijk - Cikj are functions of the coOrdinates such that for all i,j, k CVk
+ + Cjki
cki,j
=
O.
It remains to specify a final condition, as simple as possible, wbich will allow us to conclude Ci,jk = 0 for all i;j, k. VI. If by a particlllm' choice of coordinates, tlte kinetic energy T is made stationm'y in qt, .. " qm at a certain poinf ~, then the fm'ces Qi yildd accelemtions which m'e independent of the velocities. Suppose for a· moment that such a stationary T exists, so that we have at q~, .. " f m
it, .,.,
(i,j,k= 1,"',m).
31
1. .PHYSICAL ASPECTS
The form for Qi
Qi
at this point becomes
~ au qj" + -au + .~ ~ CVk qj"qk. = j= ~ 1 a!]i j,k=l
It is to be observed that the form of
Qi employed holds for any eoordinate system. Now if these forces Qi are to be independent of the velocities, we must have Ci,jk = 0 for all i, j, k in the special cdordinate system and thus in the most general system by the known. law of ,transformation of the, terms Ri. Consequently the desired Lagrangian form of external forces is obtained. The hypothesis that a stationary T exists is justified by the wen~known fact that for any coOrdinates of geodesic type at fl':., .. " tm , the surface element ds 'where m
~
ds! =
i,j=l
au dqi dqj
has coefficients au stationary at this point. Conversely, it is readily seen that a regular Lagrangian system has external forces Qi which satisfy I-VI. II. Dissipative systems. Conservative systems are often limiting. cases of what is found in nature, since actual work is usually done on the system during a closed cycle. A system for which work is done may be called dissipative. More explicitly we shall define dissipative systems to be such that (i =
1"", m)
where m
~Rjqj
j=l
>
.-
O.
Furthermore we shall assume that the equality sign can only hold for motions in a manifold of dimensionality less than m in the m-dimensional coordinate space. Suppose now that such a system is unconstrained, or at least is subject to external forces which do no work, so that m
~Qjqj
j=l
=
O.
32
DYNAMICAL SYSTEMS
Because of the obvious relation dW.
m
/
-dt +.1:Rjqj = )=1
0,
where W denotes the work function associated with L, namely
j~ m
( ,
'OL)
qj 'Oqj -L,
we infer that W constantly diminishes toward some limiting value· W o • It is assumed that the work function· cannot diminish to - 00. Now consider the limiting motions of the given motion~ Along these motions W has this limiting value Wo , and of course the sum
vanishes. A dissipative system of this type tends in its unconstrained motion either toward equilibrium 01", more gfmeraUy, toward the motion of a conservative system with fewer degrees offreedom.
CHAPTER II VARIATIONAL PRINCIPLES AND APPLICATIONS I. An algebraic variational principle. On the formal side of dynamics it has proved to be· a fact of fundamental importance that the differential equations can in general be obtamed by. demanding that the 'variation'· of some definit~ integral vanishes. To make clear the essential nature of the variational method, we may consider an analogous question concerning ordinary maxima and minima. Let there be given 12 equations in 12 unknown quantities.
fi(xl' "', Xn)
=
0
(i = -I, "',
12),
in which the left hand-members are expressible as the partial derivatives of a single unknown real analytic function F, fi -
of/ox
i, (i = 1, . ", n), also yield a second set of such multipliers. Continuing in this way we infer that lni(>i' mi(?~' ••• must form further sets of multipliers. Hence by linear combination still more general sets of multipliers
are obtained. But the n quantities in parentheses here can be made to take'on n arbitrarily assigned values, just because the roots ~ are distinct. In particular the coefficient of any (?k:to 0 can be replaced by 1 while all the others are made to. vanish. .But this would necessitate that it"k = 0, (i = 1, "', n), contrary to hypothesis. Conseqnently IXi; I cannot vanish. It . is also obvious that we may take' the quantities lv so that in the transformation
..
Yi
= Y
~
lij Zj
(i'
=
1, "', n),
from Yi to Zi, the variables Zi and Zj corresponding to conjugate imaginary roots mi and mj will have conjugate imaginary values when Yl, .. " y .. are real, and conversely.
69
III. FORMAL ASPECTS
Now .let us introduce this change of variables in the special type of equations (1) under consideration. The equations obtained by sUbstitution in (1) of the linear expressions in Zi for Yi will then assume the form
where the· terms of higher than the first degree in Zl, •• " Zn in the right-hand members are only indicated; in thus writing the· first terms on the right, the characteristic property of the quantities lij is of course employed. It follows from these relations that the equations in Zi have the form (i = 1, "', n)
where only the linear terms are explicitly written. Thus we may take the system to be of the prepared form
in which we may write (i
=
1, "', n),
with
Fik a homogeneous polynomial of degree k in Xt, .. " x,.. W~ shall show next that we can obt;:iin formal series
€Pi(XI, "', x n )
=
€Pi2+
€pi!!
+...
(i = 1, ... , '11),
such that the transformation (i = 1,.•.. , '11)
reduces the differential equations to the form
This will be achieved, provided that the equations
70
DYNAMICAL SYSTEMS
follow in consequence of the differential equations in Xi. Using these equations to eliminate dXi/dt, we obtain as the desired relations
1ft
+I
n
(a pda Xj)
J=1
(mj Xj
+.Fj) = n~ Pi (i=l,.··,n) ..
On expanding 1ft and Pi in series, these take the form fOf i= 1, .. ·,n,
Fi2
~ api2 +~ -aj=1
Xj
mj Xj
=
mi Pi2,
Let us consider the first equation written for auy i, which obviously constitutes a partial differential equation for· Pi2. The coefficient Ci of· the term
in Pi2 is then eyideIitly determined in terms of the analogous. coefficient di ofFi2 be means' of the n equations
But the term in parenthesis -is riot zero by virtue of the hypothesis made concerning the quantities nli,SO that Ci can be determined as desired. Hence there is a unique set of homogeneous quadratic polynomials Pi2 satisfying the finlt set of the equations written above. In the same way the second set of equations determines Pi3 uniquely since the equation for determina.tion of the coefficients in Pia is of the same general type as above except that we have l1 In = 3 in this case. Ignoring then the questions of convergence of the series employed we arrive at the following conclusion:
+ ... +
71
III. FORMAL ASPECTS
By means of a formal tmnsformation n 1 n Xi = fa (Zl, ... , en) = 1: lv Zj + -2 1: lijk-zj Zk j=l
j,k=l
+ ...
( i - 1, ... , n)
with Ilvl t 0, the differential equations (1) with an ardina1y equilibrium point of general twe at the origin, can be reduced to the normal form (9) dzJdt = ffliZi (i = I, ... ,n), so that corresponding to conjugate roots ffli and mj there are conjugate variables Zi and Zj. Since the normal form just written is integrable with general solution (i = 1, ... , n)
we may state the following conclusion also: The corresponding formal sQlutian of (1) may be written in the·form I". (m,t (i = 1, .. ·,n) Xi = Ji Cl e , .•. , Cn em.. t) where the fa are the same formal power series as appear in the transformation to normal form. 5. The generalized equilibrium problem. It is not difficult to extend the above method to the generalized equilibrium problem in which we start with equations of the form (3). Here· the equations of variation form a system of n ordinary linear differential equations dyJdt -:
XI
n 8 1;_1
j=l 8xj
Yj
(i =
l, ... ,n)
t=o
with coefficients 8 XJ8 Xj/t=o which .are analytic periodic functions of t of period ~. Let Ylk, •.. , ynk, (k = 1, ... , n),. give for each k a solution such that the n solutions are linearly independent. Then the general solution is a linear combination of these particular solutions. When t is increased by 'to, the equations of variation are unaltered. Hence we have 0
o
0
DYNAMICAL SYSTEMS
72
n
Yilt (t
+~-) = l: Ytz (t)
Ckl
(i,k=I •...• n).
l=l
Now if we define ml, ...• mn as in (8) by means of the square array .cv thus obtained, we may choose another linearly independent set ofn solutions which will reduce the above relations to a normal form 1, ...• n).
(i, k -
We confine ourselves as before to the general case, in that we exclude linear commensurability relations between 11'10 ••• , 11'm and 2 'It V Ih. where 11'k = (log mk)h. In this event ~. ...• mn are ali distinct. Now let us write the elements appearing in the solution of the equations of variation in the form (i, k =
1•... , n).
when it· is apparent th8:t the functions Pik will be periodic of period '1".' 'Furthermore from a' familiar theorem we know that the determinant I Yij I = Ipij I e(,ul+·· +f.l..)t is nowhere variables
o.
Consequently the particular linear change of n
.Xi= ~ pf.jZj
(i -
1, ...• n)
)=1
fro~ Xh ••• ,Xn to Zh ••• , Zn is within the admitted groJlp. The equations of variation will have a solution
(i
==
1, ..., n)
for -k -.-:. 1,- ... , n, so that the new equations must be
dz./ d t =' 11'iZi
+ ...-
(i
=
1, ..., n).
Consequently, we .infer· that it· is. no restrictioJi to write the given equations in the prepared form
73
III. FORMAL ASPECTS
(i= 1, "', n)
where Fi is periodic in t of period ~ of course. It is obvious that if certain pairs of the quantities mi are conjugate imaginaries the transformation of variables employed may be taken' to be of the admitted conjugate type. Suppose that we continue by effecting a further transformation of the same type as in the ordinary equilibrium problem save that the coefficients in the selies Pi need not be constants but may be periodic analytic· functions of t of period 7:. If we endeavor to choose this set of functions Pi so as to normalize the transformed differential equations as in the special case of ·ordinary equilibrium we obtain analogous equations, namely Fi!
8 Pi! n 8 Pi! + -a t + ~ -8-' l~jXj = J=1
XJ
On considering a typical term in
~ifj!i!,
Pi2,
Ci.(t)X~ ... x~
we fbid as the required conditions ·di(t)+
~~
+ [l1~1+ · .. +(h-l)~i+ ... +~n]Ci = (i =
where di is the like coefficient in
Fi2.
0
1"", n)
Here the coefficient
l of Ci is not zerO', and it is immediately possible to solve for Ci,
Ci(t)
=
~e-A.t - e-J.tL d(t)tf t dt.
This solution Will be periodic in t of period ~ if and only if
lei (1 ~ tfT) =
f
a
4l(t) tft t.
DYNAMICAL SYSTEMS
74
It is possible to determine /£i in one and only one way 80 that this· equation holds, pr9vided thatlis not- an integral multiple of 21f v=T/ or. But'this relation· would require a commensutability relation between the' multipliers' ISl, •• " ~n and 2 1fY 1 h of the excluded type. Thus, as before, no difficulty arises in determining Pi2,P~, "', in succession in such fashion that the desired normal form is obtained.. By means of a formal tmnsJormatUm Xi
=ji(Zl, ~'"
Zn,
n 1 n t) =l;lij(t)Zj+-2-1:lv7c(t)ZjZk+'" )=1
hk=l
(i
=
1, ... ,n)
with lij analytit: in t and periodic ofperiod or, and i lij(t) ito, the differential equations (3) with a ge:neralized equilibrium point at the origin of general type can also be redur-ed to the not'mal form (10) dei/dt - lSiZi (i = 1" ", n). The wrresponding fot'mal solution of (3) is then eLidently (i
=
1, ... , n).
6. On the Hamiltonian multipliers. As a first step toward obtaining an analogous norma] form for a Hamiltonian system of equations at an equilibrium point, we demonstrate some fundamental well-knowu properties of the multipliers in this case. * Here the equations occur in the particular form (ll)
dqi dt
oR °Pi
(i
=
1, "', m)
where H is .a real analytic function of n = 2 m variables pl, "', gm. If these equations have an equilibrium point at • Cf. Poincare, La Mltlwdu fWUtJdUB de la Mkan~ dleBU, vol. 1, ehap. (. au' characteristic exponents' are our multipliers.
75
III. FORMAL ASPECTS
the ongm, then- evidently all of the first partial derivatives of H vanish at the origin and jf we ignore an additive eonstant in H we may write
H = H.+H,+ "', where Hk is a homogeneous polynomial of degree k in' these dependent variables and where in particqlar we have
Here we may take a'J = aji, Clj = Cji, but hij are in general distinct from bji. The equations of variation are obtained by replacing H by H 2 and pi, qi by Pi, Qi, and may be written in the, explicit form dPildt
=
-
m
m
m
m
I bji Pj- j=l 2; Cij~, j=l I t1ij Pj+ j=l I bij Qj j=l
dQ;/dt =
(i
=
1, "', m)
Which is a particular type of system of 2m linear differential equations of the first order with constant coefficients. Our first remark is merely to the effect that in gmu!ral the multipliers will be distinct. To verify this fact, .it is merely necessary to exhibit the 2m exponential solutions in a single special case. If we take
~ -2 1 p.'( p. + q. D.. -'j=l """ IT _
J
2
2)
J
J
I
the equations of variation reduce to
with 2 m particular solutions Pi =
dUe
e±P'v=t t ,
.t e-+JC.v=T.t Qi = -'- ClUe I
DYNAMICAL SYSTEMS
76
where oij has its usual significance. Hence the 2 m values of the multipliers are +!"k V -1, (k = 1, .. " m), and these will be distinct if fL'1' •• " fL'm are distinct positive numbers, for instance. Before leaving the special case just cited we note that if H! is of this special form, we can introduce conjugate variables ~i = Pi+ V -1 qi,
"Ii -
pi-
v=1 qi
(i
=
1, .'., m),
and then we find d~i
d"li
(it
at where fi
=
-
aH
= Hi
(i
=
1" . . ,.m)
2 v=I Hand fis takes the normal form m
fi! =
-
2: fL'j V -1 ~jfJj. j=l
Consequently under this type of change of variables the Hamiltonian form of the differential equations is maintained. In this event the equati.ons of variation are still simpler, namely of the form d'gddt -
fL'i
V -1 'gi,
dfjddt
= -
fL'i
V -1 "Ii
(i =
.1, "', m).
This type of conjugate variables plays an important role later on. Let us suppose then thatwe are confronted by the general . case in whIch the 2m multipliers are distinct. We propose to show that these quantities occur in m distinct pairs, each one the negative of the multiplier paired with it. This has already been seen above to be true in the special case cited. Since the multipliers are distinct by hypothesis, a complete set of solutions QIk, ' . ' , Qmk
(k
D-ik eA..t (.z = 1· " '.',
m~
P 1k , " ' , P mk , exists of the form
p,ik
= . ""ik eA.,.t ,.. Qi k = fl
= 1,.", 2m), k
1 2) =.,"', m,
17
Ill. FORMAL ASPECTS
where the corresponding determinant of constants of order 2 m formed from Cik , Dik is not O. Such a complete set of particular solutions has the property that the most general solution is expressible as a linear combination of these particular ones. But if Pi, Qi and Pt, Q7 are any two solutions of the equations of variation we have m
2: (Qj Pj* -
j=l
Pj Qj)
=
const.
This fact is readily verified by differentiation with respect to t, and use of the equations of variation, when it is seen that the derivative of the left-hand side reduces identically to zero. If then we substitute in this integral relation, pairs of the above particular solutions we find m
2: (Dik Cil -
Cik Dil) e().·+).llt =
const.
j=l
for all k and l. This clearly implies at once that either Ak + Al is 0 or that the constant on the right-hand side is O. A proper use" of this fact will lead us easily to the desired conclusion. If each Ai has a corresponding Aj such that ~ Aj = '0, then there is clearly only one such root and the property under consideration is proved. But in the contrary case some root as Ak has no value so paired with it. Hence, in the integral relations deduced above, the right-hand members must vanish for l = 1, .. " 2 m if k has this value, whence we find D1kCll+" .+D~Cml-Clk Dll-'" -CmkDml = 0 (l= 1, "', 2m).
+
These 2m equations are linear and homogeneous in D1k , "', Dmk, - Clk, "', - Cmk, so that the determinant of their coefficients would necessarily vanish. But this determinant is precisely the determinant of order 2m referred to above which cannot be zero. Hence there is no such root.
DYNAMICAL SYSTEMB
78
In general at an equilibrium point for the Hamiltonian equations the multipliers can be grouped in m pairs la, -.A.i' (i = 1, . ", m),and are all distinct. It is plain that in general the multipliers are real or are grouped. in conjugate imaginary pairs with distinct moduli. Consequently the conjugate of an imaginary multiplier· must coincide with its negative. By passing to the special cases by a limiting process we conclude further: The multipliers A.i are either real or pure imaginary qua'll tities. We define the general equilibrium point of this type as the one in which 11 , ••• , 1m are not subject to any linear commensurability relation of the type (7), and confine attention to this general case. .7. Normalization of H s• Assuming then that the equilibrium point of the Hamiltonian system under consideration is of this general type, we can effect a linear transformation of variables on
Pi
=
1: (dijPi+ev qj),
j=l
on
qi =
1: (jij pj +gij qj)
j=l
(i =
1,.", m),
which i"educes the corresponding equations of variation to the normal form dPildt = A.i Pi, dQddt = -la Qi (i = 1, .. " m). In fact this reduction. (see section 4) merely required that the roots of the characteristic equation (6) be distinct, as is here the case. Of course the associated pairs pi, q; are taken as corresponding to associated roots Ii, -la. If la is real, Pi, iii are real variables. If Ii is a pure imaginary, Pi, iii are conjugate variables. We propose to demonstrate that this linear transformation does not destroy the Hamiltonian form of the equations. To begin with we observe that the equations of variation can be written in the variational Hamiltonian form
79
III. FORMAL ASPECTS
in which H is replaced by its second degree terms. Under the above linear change of variables this evidently takes the form
l
t1 [
o.t.
4 (Ejk P.i Pic + LJk Q; Pic ".
},k=1
+ ~7< P.i Q" + Njk Q; QIc) -
H!] dt =
0,
where we may write ".
HI
= },~1 (R}k P.i P k + 8)"k Pj Qk + T)"k Qj Qk).
Here the dashes over the letters have been omitted, and we may obviously assume
Rv
=
Rji,
Tij =
(i,j =
l}i
1, ... , m).
Applying the ordinary Lagrangian rule this gives the equations of variation iu the new variables, dt
d
[i
)=1
(Xji
~ + Lji Qj)] -
1; (Xi) Pi + Mij QJ)
)=1
m
+4
j=l
d['rl
]
(2Rv P.i (i
+8 =
Q;) = 0, 1, ... , m),
i)
-dt ~ (¥ii Pj+ N}iQ;) - ~ (Lij Pj + Ni) Q) )=1
on
)=1
on .
+4
j=1
(8ji Pj
+ 2 TiJ 0)
= 0, 1, ... , m).
(i -
But the solutions of these equations are known. cular we have a solution
In parti-
=
1, ... ,m)
(i
-which when substituted in the first ot the above equations gives at once Interchanging i and k;and noting -that -~ further for any i and k
=
Rki we infer
DYNAMICAL SYSTEMS
80
so that Kki =00. $~ for i distinct from k as well as for i = k. It follows also' that Ene. vanishes for all i and k. Similarly we can infer that Nki = Nue and that Tik vanishes from the second set of equations. Thus the terms
...
~ K;'kPjPIc,
j,k=l
are exact derivatives, and may be omitted under the integral sign. The equations of variation are thus of the more special form d [
-.
at
-
d [
dt
In
]
m
·m
In
In
j=1
j=1
ILjiQj - '5"'MijQi+ ~SijQj=O J=1 ~ j=1
In.]
~ Mji Pj -
j=1
~·Lij Pi + ~ SjiPj = 0
(i = 1, ... , m), (i=I, ... ,m).
To determine these equations still more completely we substitute ~ -.I. C Pi = 0, (i = 1, ... , T!') Qi = (Jik e ~ in· the first set of these equations, and obtain immediately for all i and k Similarly from· the second set we obtain for all i and k,
Interchanging i and k· in this equation, and comparing the equation obtained with the preceding one we infer that for i :f k we have ' Mik - Lki, In consequence the sum m.
Y (Ljk QJPIc+ M.Jk P; Qfc) j~l
81
III. FORMAL ASPECTS
differs from the sum m
1: Ljj Qj PJ + MjjPjQ},
j=l
and so from m
1: (ME -
Ljj) PjQ5
j=l
by an exact differential. Thus it is legitimate to· write the principle of variation in the specific form J
itlL~ (Mjj---'- Ljj) Pj Q5 + j~ Sjj Pj Qi]
dt = 0
in such new variables with equations of variation* (Mii- Lii) Q5
+ Sii Qi =
0,
(Mii- Lii) Pi.,- Sii Pi = 0
(i = 1, "', m), so that we have necessarily (Mii- Lii) i..i =
Sii
(i = 1, "', m).
Consider now the simplest case when every root i..i is real. In this case if we replace the real variable Pi by (i = 1, "', m)
the variational principle is seen to have the form
This further change of variables is legitimate inasmuch as pi, qi were not determined up to real multipliers in this case. It appears then that the term m
I
J=l
P.i Qj
* The constants Mil - L u are not zero since the equations of variation do not degenerate. 6
82
DYNAMICAL SYSTEMS
remains of essentially the same form after the linear transformation. Another case is that in which every A.i is a pure imaginary quantity. Here by taking the. pairs pi, qi appr()priately we can clearly write the pure imaginary quantities Mii - Lit as f!i V· 1, fli> O. aere we may replace Pi, Qi by (i -,- 1,
0", m)
when a like variational form is obtained. It is apparent that this same linear change of variables must preserve the original Hamiltonian form since
is essentially unaltered by this transformation. By a suita.ble preliminary linear transformation with constant coefficients any Hamiltonian system with equilibrium point of general type at the origin may be taken in a normalized fmom in which
8. The Hamiltonian equilibrium problem. In order to further .normalize the Hamiltonian equations in the vicinity of a11 equilibrium point, we propose to apply a series of transformations (i=l,ooo,m)
with m
K= ~pjqj+Ks+K.+ ... j=l
where K s, K., ... are homogeneous functions of Pi, qi, (i = 1, ... ,m) of degree indicated by the subscripts. Such transformations have been seen to leave the Hamiltonian form undist~bed and to form a group. It will be observed that if K II = 0 for 8> 2, the transformation is the identity.
Ill. FORMAL ASPECTS
.83
We ·beginbytaking K s . 0, s > 3, and by attempting to select K.~ so. as to simplify H s as far as possible. Here we have Pi = pi+oKs/aqi, qi = qi+oKS/Opi (i= 1, .:., m). If we solve explicitly for pi, qi, (i = 1, .. " m) in terms of pi, gi, (i = 1... " 17/), we will clearly obtain
Pi
=
where in K s. up to of H,
jJi+ aKn*/oqi+ "',
qi =qi- aKS*/OPi+ '" (i = 1, ., " m),
Ka* denotes the function obtained by replacing qi by qi The terms explicitly written give the series expansion the terms of the third degree. The modified value obtained by direct substitution is
where the arguments of Hs, H" . .. are the same as those of Hz. To terms of the third degree inclusive we find then
Thus, as would be . expected, the form of Hz is unmodified while H s takes the form
in which K s is at our disposal. . be written
Now any term in K" may
The cOITesponding term in the modified H s has a coefficient
84
DYNAMICAL SYSTEMS
in which h is the coefficient analogous to c in the original H s• Moreov-er the coefficient of c cannot vanish unless
which is clearly not possible inasmuch as the sum of all the a/s and 11/s is 2. Thus by proper choice of each c we ean make the new H s vanish. If now we proceed to try to eliminate HI. as far as possible by a further transformation of the same type in which K s = 0 except for 8 = 4 we obtain a transformation Pi =
Pi+ 8 Ktl8qi+ "',
which does not affect H s or H s
qi =
qi - 8Kt18Pi+ .. ,
=
(i = 1, "', m) 0, but alters HI. to
Here we can eliminate the terms of HI. save those which eontain each pi, qi to the same degree, namely those of the forms (i,j = 1,· .. , m),
by the same method. a1
For we have
+ ... +,Bm
=
4
in this case, and all terms can be· made to disappear except those for which ai = ,Bi,(i = 1, "', m), i. e., those of the stated type. Thus it is readily seen that by an infinite series of steps we ean eliminate from Hall terms except the terms in the m products Pi qi. By suitable traniformations of the above types, a Hamiltonian system with equilibrium point of general type at the origin may be taken into a normal Hamiltonian form in which
85
III. FORMAL ASPECTS
only the m products PI ql, ... , pm qm appear in H while has the special fOrm
H~
It may be noted that the linear transformation employed is also a contact transformation* so that in reality this normal form may be obtained through a single formal contact transformation. In this normal form the genera] formal solution is at once obtainable. If we write 1fi = Pi qi, the normal Hamiltonian equations may be written dPi
oR .
dqi _
oH 0 1fi pi,
dt = -
dt -
0 1fi q.
(i =
1, "', m),
(i =
1, "', m).
whence we find formally
Thus the series OH/01fi reduce to constants and we are led to the following conclusion in a purely formal manner: The general formal solution of the normal Hamiltonian equations near such an equilibrium point has the form
-where In terms of the m'iginal variables the corresponding solution may be obtained with the aid of the contact transformation "elating the git'en variables and the normal varitibles. g. Generalization of the Hamiltonian problem. . If Pi =
'Pdt),
is a periodic motion of period
* See Whittaker,
.A~ytical
qi T,
=
l/Ji (t)
(i =
and if we write
Dynamics, ehap.16.
1, "', m)
DYNAMICAL SYSTEMS
86
then the differential equations for the. modified variables are of Hamiltonian form with modified principal'function III
ii
=
H
+ j=l ~ (fJI) qj -
t/J) Pj)
where theaccerits dellOte differentiation. Furthermore, this transformation may be written as a contact transformation qt
with
= aX/api
(i
=
1,. ", m)
m
X =
~(Pjqj+fJljqj - t/Jjfjj).
j=l
In these new variablesH is a function of PI, .. " 7i m and t, periodic of period 7: in the last variable, with Pi = qt = 0, (i = 1, "', m), a solution corresponding to the given periodic motion. Thus, at least in a formal sense (see chapter IV~ section 1), the problem reduces to one of generalized equilibrium. It will be our aim here to show that a reduction to normal form for such a generalized equilibrium problem can be made which is altogether analogous to that made above in the case of ordinary equilibrium. We shall merely call attention to the modifications necessary in the argument in de8llng with this more general problem. The first difference to which attention needs to be called is to the obvious fact that in the equations of variation the constants aij, bij, Cij are replaced by periodic functions of t with period 7:. The second difference is that the constants Oij, Dij which appeared in the solutions of these equations are also such periodic functions. These modifications do not, however, interfere with the argument made. that the multipliers may be grouped in m pairs
2
where ll' "', Am are real or pure imaginary. The general equilibrium point may here be appropriately defined as that in which there are no linear commensur-
87
III. FORMAL ASPECTS
ability relations between the m +1 quantities A,1' ... , A,,,, aud 2rrV lh. In the corresponding linear transformation the coeffiCients dq, ev, f;,;, gv are periodic in t of period 't". Likewise in the variational principle the quantities K v' L~, Mq, N v, Rv, 8 v, Tv are similar functions. In determining the fonn of these functions one finds modified conditions such as that dKId A,i«KId -Kik)+de- +2R;k =
o.
If i and k be interchanged and the results subtracted, there is. obtained d dt (Kki - Kik) (A,le - A,i) (Kid - Kik) = O.
+
This differential equation in (Kki - Kik) has no periodic solution of period 'T: (other than 0), just because A,1' ... , A,m are of the general type assumed. Hence we infer as before that K ik, Kid are equal while 2~1£ is - dKik/dt. But in this case 1ft
I K;kPjPk i,k::1 is an exact differential if
t
~ dK,jk 2 ,j,t't:1 de- PjFk be added while the negative of this last expression may be incorporated in H. Thus it is clear that we may assume
. K v = Nv =
Rv =
Tv
=
0
as before. In fact similar. slight modifications show that the same nonnal form for He is obtained by this linear transfonnation in the generalized equilibrium problem as in the ordinary equilibrium problem. To make clear that· the analogy is. complete in dealing with H a, U" ..., let us consider the new Haobtained by a transformation
88
DYNAMICAL SYSTEMS
where X, is homogeneous of the third degree in Pa, qa, (i =1, "', m) with coefficients which are periodic in t of period
The new form of H. is
'Z".
ax· +I lj (oX· aK·) qj~-Pj~ +H,(P1' m
3
-;-t v
u~
;=1
u~
q,,.).
The terms c(t)p;'"
·1,:,
X:
in and HI respectively lead to a total con'esponding equation de
dt + c[ldP1 -
a1)
+... + lm (Pm -
am)]
+ It =
0,
in the. attempt to eliminate such a term. This ordinary non-homogeneous linear equation of the first order will have one and only one periodic solution inasmuch as the coefficient of c is incommensurable with 2n V -1 h, since we are in the general case. Thus H, can be made to disappear. Likewise all the terms of H, can be made to disappear save those in the products pa qa,(i = 1, "', m). The coefficients in these latter terms can be replaced by constants however; in fact this demand leads to an equation of the form de
d:t+h(t)
=
C
where C is an arbitrary constant at our disposal; thus we find
c
=
f(C~h(t»dt,
which is obviously periodic of period 'Z" if C is chosen. as the mean value of h(t) over a period. Hence we are led to the same conclusion as before. By means of such a series of transformations of the generalized Hamilto1lian equiUbrillm problem, "the IlamiltO'Jl,ian function.
III. FORMAL ASPECTS
89
may be given the same normal fO'rm as was obtained in the case of O'rdinary equilibrium. * 10. On the Pfaffian multipliers. Suppose now that we take an extended Pfaffiall variation problem
which leads at once to the system of ordinary equations of order 2m (13)
~(8Xt
j=l
OXj
_
8.¥i) 8Xi
dX.j _
dt
8Z =
0 (i
=
1, "', 2m).
8Xi
We propose to consider these equations in the case when there is an equilibrium point at the origin, under the assumption that the 2m analytic functions Xi are such that the skewsymmetric determinant
aXi - aXj 1 aXj
ax,
I
is not 0 at the origin. The constant terms in. the series for the functions Xi may obviously be omitted throughout. It is clear that the Hamiltonian equations appear as a special case of these Pfaffian equations (12). As will be shown in the following chapter, this generalization of the Hamiltonian equations possesses the same property of automatically fnlfilling all of the conditions for complete stability, once the obvious conditions for first order stability are satisfied. Hence from this point of view the Pfaffian equations seem as significant for dynamics as the Hamiltonian equations, although more general in type. Moreover they ,possess the additional advantage of maintaining their Pfaffian form under an arbitrary transformation of the formal group. * 'fhe results of this chapter were announced in my Chicago Colloquium lectures of 1920.· The material so far given is obviously in close-relation with previous work, and, in particular, the normal form in the Hamiltonian case is jn relation with .the form~l -trigonometric serieS' in dynamics treated"'fol"instance; in Whittaker; Amlytieal..Dynamicsj "Chap~lf).
DYNAMICAL SYSTEMS
90
In fact it is )only' necessary to substitute, the' new variables under' the integral sign in (12) to obtaiDth.e .transformed functions· Xi and Z. Asa first step'in the direction of obtaining a normal.form f9r the Pfaffian equations at an equilibrium poInt, we propose to prove that. for these equations just as for the Hamiltonian equations the multipliers are associated in pairs It, -li. To begiu with we obseI'Ve that in general these roots must be distinct since they are distinct in the Hamiltonian sub-ease. Now let us make the linear transformation with constailt coefficients which takes the equations of variation into normal form. This does not affect the Pfaffian form of course. The corresponding equations of variation obtained from (13) are
and these must ha v'Ef the particular solutions Yi =
~
Uik
e.l.k t
(i
=
1, "', 2m)
for k = 1, .. " 2 m. It is understood that the partiaJ, derivatives mvolved in the equations of variation are evaluated at the origin. Substituting in these particular solutions we obtain readily v X °•Z.= O~ (~-~)lkOXk oXi OXi OXk
If we interchange i and k here, and subtract the equation so obtained from the one last written, we find _ ( oXe OXk
OXk) O'k+lt) = oXi
0
(i, k =
1,; .. , m);
But if for each k we do not have lk+l, = 0 fOrsorne i~ it~ould followftom these equations that the skew-symmetric determinant sp.ecified above would necessarily vanish. This
91
III, FORMAL ASPECTS
is impossible since the equations·· of 'variation would then degenerate. Hence there is associated necessalily with the root A. a second root A.i = -A.k. This is precisely what we desired to prove. In the Pfaffian equilibrium problem the multipliers also occur in pairs; one of each pair being the negative of the other. These multipliers may be designated by AI' -ll' .. " A.m, -Am, and will be real or pure imaginary quantities. It. is clear that the general case is to be defined as that in which there are no linear relations of commensurability between AI' .. " Am just as was done in the special Hamiltonian case. We shall restrict ourselves to this general case. I I. Preliminary normalization in Pfaffian problem. It is very easy to establish the fact that the normalization used in the preceding section makes the first degree terms in Xl, .. " X 2m take essentially the Hamiltonian form. Indeed if we call the 2 m dependent variables PI' ... , p'm, qt, .. " qm, in such wise that pi, qi correspond to paired multipliers Ai, -Ai and if we let PI' .. " Qm denote the coefficients of 1J~, . ;', q'm respectively under the integral sign in (12), the previously obtained equations between the partial derivatives () Xi/a Xj at the origin take the form aPi _ aPj apj.- api' aPi aqj
(i,j = (i,j =
1, ... ,711.),
1"", mj i
:fJ).
The first sets of equations show that the linear terms in Pi involving PI' .. " pm correspond to an exact differential, as do the linear terms of Qi in ql' .. " qm. Similarly the second . last set of equations shows that the' term of Pi involving qj (i :f j) together with the corresponding termofQj involving Pi. combine in a like manner. All of these terms maybe omitted, and there remains for consideration ouly terms m
k
j=l
(CjPj dqj + dj (jj dpj),
3
92
DYNAMICAL SYSTEMS
which evidently may be replaced by IR
1: (Cj -
dj)pjdfJj.
j=1
In these terms no Ci - di can vanish, because of the hypothesis that the fund,amental skew-symmetric determinant uoes- not' vanish at the origin. If then pi, qi are real \'ariableswe may make the further linear transformation Pi =
pi,
~ =
(r-i-di)qi (i= 1, "', m)
to obt.ain the desired linear Hamiltonian term. On the other hand if pi, qi are conjugate variables then Ci and Iii are conjugate imaginaries and ci-Iii is a pure imaginary quantity f! V -1 . Here we may set
Pi =V{!pi, qi = 'V{!qi if f! > O. If f! < 0, we ma.y interchange the roles of pi and qi. Let us turn next to consider the function Z. Since we have an equilibrium point at the· origin, it is plaiA that aZlapi, aZlaqi, vanish there for i = 1, 2" ", m, i. e., that there are no linear terms in Z. The lowest terms in Z are then of the second degree. Thus it is apparent that the equations of variation, which depend only on these first degree terms in Xl' .. " X 2m , and upon the second degree terms in Z, are of the same type as in the Hamiltol1ian case. In consequence the same linear transformation employed to obtain a normal form for these lowest degree terms gives - ~ the form of HI in the Hamiltonian case. We can summalize our results as follows: By a preliminary linea,r tmn.1'ormation the Pfaffian equations with an equiUbrium point of genp.:1"al type at the origin t:an be w1'itten
93
I II. FORMAL ASPECTS
where Pi = Pi+ P i2+""
Qi= "'+Qi2+'"
(i=l .... ,m),
m.
R =
1; ljPHj+Rs+···.
j=1
12. The Pfaffian equilibrium problem. After this preparatory work it is a simple matter to establish the general result, which is that by means of point transformations (independent of t), it is possible to reduce the Pfaffian type of equations to Hamiltonian form. More precisely we propose to show that it is possible to reduce Qi, (i = 1~ "', m) to 0 by a suitable succession of such transformations without interfering with the normal form of Pi. When this has been accomplished, it is merely necessary to write
(i
=
1, "', m)
to obtain complete Hamiltonian form in the case when PI ~ "', qm
are real variables. A slight modification is necessary in case the variables pi, qi are not all real. In the real ease suppose that we write Pi = pi,
(i
=
1, "', m),
where, according to our usual notation, Gi2 is a homogeneons quadratic polynomial in its arguments. The variational principle takes a corresponding form in which the new coefficient Pi has also an initial first degree term Pi as desired, while for the new Qi we find readily expressions in series
. . . . aG· *+ Qi2 + 1: pj __ +.. . api j~i
J_2
(i =
1, ... ,7/1.1.
Here the linear· terms are lacking, as desired; and only the quadratic terms are written explicitly. Now we have d
(Ipj Gj2) J=1
94
DYNAMICAL SYSTEMS
This identity shows that, by subtracting an exact differential under the integral sign, we may modify the new Qi2 to the form Qi2 - Gi2 without introducing any first degree terms iii Pi. Hence if we take Gi2 = Qi2, (i = 1, "', m). the third degree terms in Qi2 will have been eliminated. Next by a' further transformati6n Pi
=
pi,
we can similarly eliminate the third degree term in Qi. Proceding thus indefinitely we alTive at 11 variational form
'Jt d
to
l [
m' ~ Pj qj
)=1
where Pi = Pi
+* -
R] d t =
+ P i2 +...
0
(i =
1, "', m),
which can evidently be given Hamiltonian form in the manner indicated. In case some of the pairs of variables pi, qi are conjugate imaginaries we may first perform the simple linear trans~ fl)rmations to corresponding real variables Pi
=
(Pi
+ q i V -1 ) / V 2,
qi
=
(Pi - qi V -1 )/ V 2
so that the term Pi qi is replaced by Pi qi except for an exact differential. Thus the normal form for Pi, Qi is maintained . for these real variables. Operating with them as indicated we can reach the same conclusion in this case also. By a suitable transformation. of the formal group, say
the .qeneral Pfaffian equilibrium problem may lie made to assume Hamiltionian form. 13. Generalization of the Pfaffian problem. Under the above circumstances it is natural to expect that Pfaffian
95
'III." FORMAL ASPECTS
equations containing the, time t,with .generalized" equilibrium point at the origin, admit of formal reduction to Hamiltonian form. It isnot difficult to establish thetrnth of this conjecture on the basis of cert~in slight modifications of the above discussion. In the case of such equilibrium the equations are defined by
with Xi, (i (15)
=
1, .. " 2m), and Z periodic in t, that is by
I (aaXjXi _
;=1
aX;) dXj + aXi _ aZ ax; dt at ox; (i
=
0
= 1,.", 2m).
In the first place it is obvious that the multipliers are in general distinct by consideration of the same special case. as was taken up in the ordinary equilibrium problem. Furthermore, the' equations of variation may again be normalized by a linear transformation in which the coefficients involved are periodic analytic functions of t of period ~, so as to have the solutions Yi =
~
(}ik
eAtk
(i= 1, "', 2m)
for k = 1, .. " 2m. It follows that the multipliers A.j occur in pairs, each one of a pair being the negative of the .other, by essentially the same argument as was used in the equilibrium problem. Moreover the same argument shows that the linear terms in Pi, Qi lead to certain exact differentials and terms which may be absorbed in R, so that the same normal form for the first order terms in Pj, Qi and for the second order terms . in R is obtained as before. Finally as in section 11 we write Pi = pi,' qi -:- qi
+ Gj2
(i= 1, .", m)
96
DYNAMICAL SYSTEMS
where now Gi2 has coefficients which are periodic in t of period 'l". Then by an obvious modification of the argument there made, we can make Qi2 = 0 for i = 1, .. " m, and then in succession QiS = 0, .. '. Bya suitable transformation of theIormal,qroup the generalized Pfaffian problem of:periodic motion may be made to assume Hamiltonian form. Hence the normal form in the Hamiltonian" case serve..s also in the Pfaffian case.
CHAPTER IV STABILITY OF PERIODIC MOTIONS I. On the reduction to generalized equilibrium. For motion near equilibrium of a Hamiltonian or, more generally, of a Pfaffian system, the stable calle is naturally defined as that in which the multipliers 11 , •• " 1m are pure imaginaries, at least provided that there are no linear commensurability relations between these multipliers. In this chapter, however, we shall limit attention to the analogous but somewhat more complicated question of stability lor motion near a periodic motion of such a system. * The method employ~d involves a reduction to the case of generalized equilibrium. In the more general Pfaffian case this can be accomplished by a change of variables
(i= 1, "', 2m),
in which the periodic functions Pi (t) of period 7: are the coOrdinates of the given periodic motion. By this means the functions Xl":'" X 2m , Z are modified (see (12), page 89), since they are no longer independent of t but periodic of period 7:; and the given motion now corresponds to generalized equilibrium at the origin in the new XJ, ., " X2m space. Hence we are led to consider the question of motion near such a point of generalized equilibrium. There is, however, a difficulty associated with this reduction to generalized equilibrium which was first signalized by Poincare for Hamiltoniim· systems, and which it is desirable to explain briefly. Following the analogy with the case of ordinary equilibrium; the stable case is defined as that in which the multipliers 1 1 , ., " 1m are pure imaginaries, at least provided that there • Cf. my article Stability and the Equations of Dynamics, Amer. Joum. Math., vol. 49 (1927) for a treatmeut of the equilibrium problem. 97
DYNAMICAL SYSTEMS
98
are no linear commensurability relations between these multipliers and 2 n V -1h:. If such relations exist the questions to be ~ousidered become mOre complicated in character. Unfortunately, .for a P{).int of generalized .equilibrium obtained by the ab~ve method of reduction, the multipliers will not satisfy this condition;. more specifically, there will always be a m~tiplier 0, which is double of course. This may be readily seen. ThePfaffian system admits of the integral Z.= const. in the original· variables, and therefore admits the integral Z(XI fj'1, •• " X2.,~ fj'2.n) = const.
+
+
in the modified variables.
By differentiation with respect to the 2 m arbitrary constants in the general solution XI' ; • " 3o'2m, it appears that the linear relation
az az- Y2m = --Yl+ ... + aXI aX2m
const.
subsists for 2 m linearly independent solutions yl, .. " Y2m of the .equations of variation, and so for the most general solution; it is understood that aZlaXi, (i = 1, "', 2m), have !PI, ' . ' , !p2m as arguments. Now if the 2m multipliers + ).,1, •• " + lm are distinct, a· complete set of 2m solutions Yi
..
=
A.t
pike'
(i'
=
1, .. . ,2m)
for k = 1,· .. , 2m exists (lm+i = -li), ih which pij are of period 'T in t. Since aZlaXi are also periodic, substitution of these solutions in the linear integral relations in the vi's leads immediately to the conclusion that the constants on the righthand side must vanish, at least for ).,k:f= O. But if these constants vanished for such a complete set of solutions, the constants would vanish for every solution yl, .. " Y2m. This cannot be the case since VI, "', Y2m can be taken arbitrarily for any particular value of t. * * It is not possible for az/ox, to vanish simultaneously for i = 1, '.', 2m along the original motion, since the Pfaffian equa.tions then yield (lx,/dt 0, ·(i 1, •", 2m) which is impossible, the case of ordina.ry eqnilibrinm being excluded.
=
=
IV. STABILITY OF PERIODIC MOTIONS
99
There is then a pair of solutions of the pqnations of variation which belong to the multiplier O.1'l'ow
f-or any" defines a ~olution of, the given equations after the. I'eduction, so that by' diff~rentiation with respect to. k, one solution of the equations of variation yl
=
,
fJ'1, •• " Y2...
=
,
fJ'2m
is obtained. This has periodic components and so belongs to the multiplier O. On the other hand the periodic motion with which we start is not isolated, but varies analytically ~ith the constant c in the known integral (i. e., with the energy constant in the Hamiltonian case). This yields t1le second periodic solution Yl =
afJ'~ ac' "', Y2m
=
afJ'~m
~,
belonging to the multiplier O. In general there· will be no others. The difficulty may be turned in the following manner. The variable Z may be taken as· one of the '. dependent variables Xl;"', X2m, say as X2m, in the original Xl.···, X2m space. Furthermor(:l the variable 8 = X2m-1 may be selected as the single ang1!J.ar.coOrd.i.nate, which increase!! by 271' when a circuit of the curve of periodic motion is made. .The remaining coOrdinates Xl, " ' , X2m-2 may. be made to vanish along this curve. Now let us restrict attention to those mo.tionsnear the given periodic motion for which
z=c has the same value as along this motion. With this under:.. standing, the Pfaffian system becomes of order 2 m - 1 in Xl, " ' , X2m-2, 8, lmd may be written in the variational form
4
100
DYNAMICAL SYSTEMS
tJ
i
tl fm-2
\,I Xjxj + X
.
to
) 211'-1
==
8' dt
)=1
0,
to which set of equations must be added the last equation of the first set. But the integrand is positively homogeneous of dimensions unity in xi, ."', X~m-2, 8', so that (J may be taken as parameter instead of t. Then the variational principle takes the fonn 0
i
t1
to
rm-2 I Xjxj+X2m- 1]d8
=
O.
)=1
Hence we obtain a Pfaffian system of even order 2 m - 2 only· instead of 2 m, in which the coefficients are periodic' in a variable (J of period 2", and the known periodic motion corresponds to the origin in Xlt " ' , X2111-2 space. By this second method of reduction to a generalized equilibrium problem, the formal difficulties refeiTed to above are avoided. For these reasons, in dealing with the applications we can restrict attention to the case of generalized equilibrium of stable type as above defined. 2. Stability of Pfaffian systems. Our starting point is furnished by the equations of motion, normalized to terms of an arbitrary degree s by means of an appropriate trans'" formation defined by convergent series, according to the method of the preceding chapter. The equations are thus given the form dPi (ft = -
(1)
(i=l,.·.m)
an
dqi
-a:t
aH
a" i Pi + Li,8+1,
=
a" i qi + Mi.8+l
where we may write m
H
=
1: ).jpjqj+H•... +Hs
)=1
(8 =
s or s
+ 1),
IV. STABILITY OF PERIODIC MOTIONS
101
in which Hk involves only the m products tri = Pi qi, of total degree k / 2 in 7ri, "', 7rm , while Li, 8+1, Mi,6+1 are convergent power series in PI' "', qm which commence with terms of degree not lower than s 1, the coefficients being of course analytic and periodic in t of period To Suppose that we write
+
m
u
l
l: pj (jj. j=l
=
Evidently u can be appropriately regarded as measuring t~e distance of a point from equilibrium at any instant t; for, in terms of the original real variables Xl' ; " , X!m, the function u! is given by a real power series in Xl' " ' , X2m which begins with a positive definite quadratic form in these variables, 2m
1£2
l: ajlc(t)XjXk+"" j,k=l
=
for all t, whence 2m
2m
"2
k~ XjO,
j=l
in a certain neighborhood of the origin. It is obvious then that we can choose N so large that
I Li,8+11, I Mi,.~+ll
s can also be exactly represented by. a sum of order N not exceeding 28.+1 • Hence the conditions of the definition are satisfied. The results of section 2 show that in the case of Hamiltonian or Pfaffian systems, there will be complete' stability if there is ordinary stability as defined earlier. This is obvious since the differences it. - l ; which enter in the· trigonometric sums of section 2 are nearly given by a certain limited number of integral linear combinations of the m 1 quantities;.JV-=-l, .. " ;."jV 1, 2 rth:, and no such combination vanishes. In case of complete stability, the solutions of the normalized eguationsof variation (chaptel; ill, section 5) are limits of trigonometric sums of the specified type, and are trigonometIic by the lemma on tIigonometric sums of sections 5,6. Hence the multipliers are pure imaginaries. It is important toestablisb' that this definition of co~ plete stability is independent of the particular cQOrdin,ates Xl, ••• , X2", selected. In fact, suppose that the given. system
+
107
IV. STABILITY OF PERIODIC MOTIONS
is completely stable. variables
Let us make the admissible change of 1, "', 2~)
(i =
in which Pi are analytic in Xl, .. " .X2m, t, vanish at the -origin, and are such that the. determinant 10 p;lo Xjl is not 0 there, while the coefficients in Pi are analytic periodic functions of t of period ~. Then the two variables
and E=
[x.21
+ ... + X':
]1/2/t=t
2m
o
evidently serve equally well to measure the distance from the origin att = to, since we have
O 1, the motions of minimum type remain so, but the motions of minimax type associated with them will not be the same as in the case k = 1, but will be distinct from these. The general situation here requires further study.
V. EXISTENCE OF PERIODIC MOTIONS
135
7. Application to excepti(mal case. The case of the m-dimensional Lagrangian problem when the characteristic surface can be set into one-to-one analytic correspondence with the hypersphere is of exceptional interest, but the mi~ nimax method outlined above is not directly applicable since there are no closed curves l, not deformable to a point, from which to start. Nevertheless the existence of a periodic motion of minimax type may be established. In order to make the reasoning as concrete as possible we shall direct attention to the reversible geodesic problem, although it is clear that the reasoning applies equally w.ell. for a Lagrangian problem ·of the kind treated in the preceding sections, with characteristic surface homeomorphic with the hypersphere. Our first step will be to define what is meant by a 'covering' of the surface. In the case of a two-dimensional surface let the surface of the sphere be set into one-to-one analytic correspondence with the given surface. The small circles on the sphere in planes perpendicular to some axis are evidently carried into a set of closed analytic curves covering the given surface, two of these being point-curves. Thus we may conceive of the spherical surface as being distorted analytically to form a covering of the given surface M by means of this set of closed curves. The points of the covering can then be specified by two angular COOrdinate functions 8, p on the surface where (J and p represent colatitude with respect to the given axis and longitude respectively. The given closed curves correspond to 8 = const., while p varies from o to 21T. The coOrdinate 8 ranges from 0 to 1T only, with the two extreme values corresponding to the point curves. Now conceive of this covering as continuously deformed. This means that each point of the covering is carried by . continuous variation into nearby points, while the curves of the covering go over into new curves. It is obvious that such a covering will always actually cover each point at least once and cannot reduce to a point. * ... The suggestion for a proof may be found in a footnote, p. 246 of my article, loco cit., and this proof extends readily to the m-dimensional case.
136
DYNAMICAL SYSTEMS
Similarly for the m-dimensional case, we introduce a system of small circles on the hypersphere
(Xl, ... , ;I'm+! rectangular coordinates) with equations
Here the null circles of the set are in one-to-one, continuous correspondence with an (m -I)-dimensional bypersphere. The image of this system of circles leads to an analytic covering of the given characteristic surface M. The points of the covering can then be specified by suitable coordinates, and we may conceive of the covering as continuously varied. It is obvious that such a covering will always cover each point of M at least once. Now there is a maximum length L * for any image of a circle, and there can be selected a distance d such that two points at geodesic distance not greater than d from each other in ¥ are connected by a unique minimizing geodesic of length 0 < d. Let n be the positive integer such that
L* 0 x'
=
p(x,
y) (x 2+ y2 -1)
within this circle, and if we write further =
Ypx',
y' =
Ypy',
z'
-Vpz',
=
the equation for S takes the form
X,2 + y,2 + Z'2 + x 2+ fr
1
(z' >
0)
which may be written
z'
=
(1- x2-
y2 _ X,2 _ y,2l/2.
Hence the interior and boundary of S are in oue-to-one, continuous correspondence with the interior and boundary of the four-dimensional hypersphere
~rhe transformation T defines a one-to-one, continuous, direct transformation of this hypersphere into itself. But, by a well-known theorem due to Brouwer, such a transformation leaves some point invariant. In the problem at
V. EXISTENCE OF PERIODIC MOTIONS
149
hand, we conclude that a periodic motion exists which intersects z = 0 twice, (case of an interior invariant point), or else a periodic motion z = 0 exists (case of an invariant boundary point). But this last case is that in which the· equations Xl = Xo, Yl = Yo, xi = xb, Yl = yb obtain. This clearly means that the equations of variation possess· a periodic solution along this plane periodic motion in which the z component is not O. Hence the periodic motion is multiple, and, in a certain sense there is still a periodic motion in the infinitesimal vicinity of z = 0, intersecting z = 0 twice. It seems highly probable that an actual periodic motion intersecting z = 0 twice must exist in all cases.
CHAPTER VI APPLICATION OF POINCARE'S GEOMETRIC THEOREM 1.
Periodic motions near generalized equilibrium
(m = 1). Poincare's last geometric theorem and modifications thereof* yield an additional instrument for establishing the existence of periodic motions. Up to the present time no proper generalization of this theorem to higher dimensions has been found, so that its application remains limited to dynamical systems with two degrees of freedom. It is our aim in this chapter to give some of the fundamental ideas involved in the theorem and its application. It will be remembered that motion near to a periodic motion of a Hamiltonian or Pfaffian system, with m degrees of freedom and not involving the time explicitly, can be reduced to that of a similar system with only m -1 degrees of freedom but with an independent variable involved of period 2 n. Here the periodic motion itself appears as generalized equilibrium. This reduction is accomplished by means of an analytic device (chapter IV, section 1). In the present section we shall take up the question of the existence of motions with period 2 k ,7 near the position of generalized equilibrium for a single degree of freedom. We shall prove the existence of infinitely many such nearby periodic motions in the general stable case by a process of reasoning which, while not employing Poincare's geometric theorem explicitly, is precisely that which establishes the theorem in certain simple cases. Later (section 3) these results are interpreted with reference to the original dynamical problem with two degrees of freedom.
* See my paper, An Extension of Poincare's Last Geometric Theorem, Acta Mathematica, vol. 47 (1926). 150
VI. POINCARE'S GEOMETRIC THEOREM
151
Let the single pair of variables be p, q, so that the Hamiltonian function H involves p, q, i, being periodic in t of period 2 rr, and vanishes at the origin p = q = 0, together with its first partial derivatives, for all values of t. If then (Po, qo)is any point near to the origin, there is a unique solution
which for t = 0 takes on the values Po, qo, and which is analytic in Po, qo, t for t arbitrarily large and Po, qo sufficiently small. Let PI' ql denote the values of p, q after a complete period 2 rr. Evidently we have
where fand 9 are analytic in Po, qo, and vanish ,vith these variables. In this way a transformation T is defined, of the same nature as the transformation of the surface of section obtained in the preceding chapter (section 10). For if we write r = t, the pair of Hamiltonian equations may be replaced by the equivalent set dp dr aH !!:g -= 1 ap , dt dt ' dt in which H is a function of P, q, l' of period 2rr in ". Here the manifold of states of motion is the three-dimensional iJ, q, r space in which the" axis represents a periodic motion, namely that corresponding to generalized equilibrium j it must not be forgotten that r is an angular variable. Now fjI = r = 0 will serve as a surface of section according to our earlier work, although here we are limited to a certain neighborhood. A point (Po, qo, 0) in this surface of section is taken along its stream line to (PI, ql, 2 rr), i. e., to (PI' ql, 0). Thus the transformation written above is indeed a transformation T of a surface of section S which is, however, only locally defined. Such' local surfaces of section' can of course be
152
DYNAMICAL SYSTEMS
constructed near a periodic motion in any dynamical problem by merely taking an element of surface which intersects but is not tangent to the corresponding stream line in the manifold of states of motion. Now the fluid motion defined by the above three equations is that of an incompressible fluid, since the divergence of the right-hand members is O. Consequently if we follow any tube of fluid made of sections of stream lines between the parallel planes r = 0 and r = 2 'if, which will move constantly with unit velocity in the l' direction, we infer that the loss of volume at one base in time L1 t is nearly 0'0 L1 t (0'0, area of first base), while the equal gain at the other is nearly 0'1 L1 t (0'1' area of second base). By allowing L1 t to approach 0 we infer that (10 = 0'1' Since 0'0 is an arbitrary area in r = 0, it is clear that T must be an area-preserving transformation of the variables Po, qo· This important property of T corresponds to a general property of the surface transformations associated with dynamical problems. It is necessary now to state the conditions to be imposed upon the generalized equilibrium, with the aid of which the conclusion stated may be established. We assume in the first place that the generalized equilibrium is of general stable type, and therefore completely stable. The significance of the normal form (chapter III, section 9) is that the solution may be written
in properly cnosen conjugate variables p, q. Here W, l.[1 are given as convergent power series in Po, qo with initial terms of arbitrarily high degree 2 fi' 1, and with all coefficients analytic functions of t; these series converge absolutely and uniformly for any fixed range of values for t, such as It I < 2 7l", when Po, qo are small. The function M can be taken as a polynomial of degree not more than fi' in the product Po qo, with pure imaginary constant coefficients, of the form
+
VI. POINCARE'S GEOMETRIC THEOREM
153
)..+lpoqo+""·+sp~q~
with ).. the multiplier. By the hypothesis of stability ).. / V -1 is not rational, and in particular is not 0. Our second assumption is that l is not 0. In case l vanishes but some other coefficient in M is not 0, essentially the same argument would apply. Thus the only case of failure is that in which the formal series M in the complete normal form reduces to a mere constant )... This is a highly degenerate case, and actual examples can be constructed to show that an analogous conclusion cannot then be drawn. The normal form gives a means of studying the transformation T. The property of the transformation T necessary for our present purposes is embodied in the following lemma whose proof is deferred to the next section: LEMMA. For l =t= 0, upon suitable choice of the variables p, q, the positive quantitye may be taken arbitrarily small, and then the integer n so large that any transformation TP (v < n) takes the circle r < e about the invariant point r ---: 0 into a region within the circle of radius 2 e, while the angular rotation effected by Tn increases from nUv=1 with r along any radial line for r < e, being at least 27T greater for r = e than for r = 0. Let r, 0 be polar collrdinates and let ("n, On) denote the . iterate of (r, 0) under Tn, where the rectangular collrdinates p, q, the radius e, and the integer n are selected as in the lemma. For any fixed' 00 the difference On - 00 will then increase from nUV-1 at '" = 0 to a quantity at least 2 IT greater at r = e. Hence there will be a unique solution of the equation On -
00 =
2kIT
along a fixed radius vector, where j? k IT is the least integral multiple of 2 IT exceeding nJ../V -1. Hence the analytic curve 0 given by this equation is met once and only once by each radius sector. Now let us consider the image On of this curve under the transformation Tn; the curve On intersects 0 at some point Q,
DYNAMICAL SYSTEMS
154
since if Cn is wholly within C or outside C, Tn would not be area-preserving in the original variables. The point Q is obtained from some pointP, also on C, by the transformation Tn. Moreover P and Q have the same (), by definition of C. Thus P 'and Q must coincide, and P is invariant under Tn. Since e is arbitrarily small, we obtain the result stated: In the case of generalized p-quilibrium of general stable type fm' a Hamiltonian problem with one dpgree offreedom (l =1= 0), there exist infinitely many periodic motions in the vicinity. 2. Proof of the lemma of section I. Let us define F(u) by the equation luF!(u)
=
M(u)- J•.
It is clear that F(u) is the square root of a real polynomial of degree fl' -1 and constant term 1. If then we write further
p=
F(pq)p,
q=
F(pq)q,
it is found that the above normal form for T is further simplified and may be written -
P
O.+IPoqo)t + rl. = poe 'V,
so that all of the terms of M except the first two disappear. When the associated real variables are introduced, and we let (1 and s denotfl the real constants 2 If U V -1 and '2 nll v=l respectively, we obtain formulas defining T,
where P, Q are real power series in Po, qo with initial terms of degree 2 fl' 1 at least. It is apparent then that with these variables T is an ordinary rotation through a variable angle (1 sr~, except for terms of order 2 fl' 1 in the distance from the origin. It is such a choice of variables that will be adopted. If l =1= 0, we may take s as positive.
+
+
+
VI. POINCARE'S GEOMETRIC THEOREM
155
Hence we have, in some fixed neighborhood of the origin and for a fixed K. (2)
From the above formulas we find at once 2 1'1
(3)
=
2
ro+R,
where the series R begins with terms of at least the degree 2/~ + 2. This gives, for the increment L1 r~ = 1,i - r~~ (4)
where L is a fixed constant. From (4) it is seen that successive iteration than if
r; increases
less rapidly upon
But this yields rn =
ro / ( 1 - L
2/J. ) 1/2/J. ~ro n .
Hence r n can only increase to twice the initial valuero after n > Y iterations; where
o
that is for n of at least the order of r 2f'. Likewise rn can only decrease to half the initial value for n of the same order. These· results may be combined in the form (5)
This is our first important conclusion. L~ewise since we have frem (1) qJPO-PlqO _
PI Po
+ qlli.:.
(p~+q~) sin (O'+8r~)+poQ-qoP (p~ q~) cos (0' 8r~) qo Q Po p'
+
+ + +
DYNAMICAL SYSTEMS
156
it follows that we have (6)
where e is a power series in 1'0 beginning with terms of degree 2!1' at least, with coefficients of simple trigonometric type in 00 , Moreover e is uniformly and absolutely convergent for 1'0 sufficiently small, and its partial derivatives are given by derived series of a similar sort. This iormula shows that for 1'0 - 0, we have . On = 00 n (J, while for To > 0, the difference On - 00 ~ n (J can be made arbitrarily large by taking n sufficiently large but in the range (5). By the aid of (3) and (6) we obtain
+
a < A 1 ~1 00
(7)
=
/l-j-l
(!o
I aa0 -1\ < ~Ap. Ol
=
0
where we have written (! chosen positive constant. But the identities
=
a(!n a (10
1'2,
and where A is a suitably
a On a (!n-l a(!n-l ~
+.
a on
a On-l
aOn-I~'
may be written Un
=
Vn
=
(1 (8
+ El)ttn-l + Vn-l, + n + (l + )Vn-l, E2
E3 )U -1
in which we put Un
(!o'
~ a(!n-l + ~ a On-l a (!n-l a (!o a On-l a (!o '
a On _ a(!o -
(8)
'
=
a(!n/ a(!o,
while, by (5) and (7), fOr n S; N(!op,
E4
VI. POINCARE'S GEOMETRIC THEOREM
157
These equations (8) enable us to determine Un, V,,· in succession for n = 1, 2, .. " with. the initial conditions Uo = 1, Co =0. Suppose now for a moment that the small terms be neglected in these equations (8). They then take the form v" =
SU,,-1
+ Vn-l,
whence, by elimination of u, we obtain
It is easily verified that the complete result of elimination
yields similarly (9)
L1!v" =
E5L1Vn+EaVn,
in which we have (10)
within a small region about the origin. initial conditions may be written
Furthermore, the
(11)
where E~ denotes the value of E8 when (1"-1' 0"-1 are replaced by eo, 00 respectively. It is obvious that VI - S + eg is positive and thence, by use of (9), V2, Va, .. , are also positive for n = 1,2, ... until n becomes large if eo is small enough, the approximate value of v" being ns. We desire to obtain a more definite idea of the range of values for eo and n for which Vn remains positive. During this· range the angular variable On increases with ro, for a fixed angle (Jo. Now the equation (9) is a homogeneous linear difference equation of the second order in v"' and we are considering the particular solution satisfying (11). Evidently v" will remain positive so long as L1 Vn continues positive. A first question is then. to determine the range of values of n throughout
DYNAMICAL SYSTEMS
158
which both Vn and L1 Vn necessarily remain positive. linear difference equation yields
so that clearly
Vn
But the
and L1 v" diminish less rapidly than if
while v" and L1 v" remain positive. Thus V n and L1 Vn will remain positive for n = 1,2, ... , at least as long as for the solution of the linear difference equation of the second order with constant coefficients
satisfying the initial conditions (11).
But this solution is
(12) where aI, a2 are defined by the equation
ai
=
log(l- ~. Be/~-l+ V-(Be
fl - 1 o
~B2e~"-2)) (i
=
1,2).
But Vn and L1 v" as thus determined will certainly remain positive until d vnl d n vanishes, i. e.
Since the leading term in a l
-
a2
is clearly
while the leading term in a2! a l is -1, this relationship shows that n must be of the reciprocal order eo(fl- 1)/2.
159
VI. POINCARE'S GEOMETRIC THEOREM
Hence we infer that so long as n < N* r (fl-1l (compare with (5», the angle On will increase with ro for fixed 00 in the prescri1?ed neighborhood. The nature of the inequalities derived above makes it clear that we can select a value of ro so small, and then of n so large, that the conditions laid down in the lemma are satisfied.
o
3. Periodic motions near a periodic motion (m
=
2).
We have already seen (chapter IV, section 1) how the general Pfaffian system in which the time t does not appear explicitly admits of arl'\duction to a similar system With one less degree of freedom, provided that weare considering motions near a given periodic motion. In the reduced equations, however, an angular variable of period 27f appears in the differential equations, and the given periodic motion takes the form of generalized equilibrium. In this section we propose to consider the periodic motions near a given periodic motion for· the special Hamiltonian case (m = 2) oH dqi dPi (13) (i = 1, 2), dt
dt
OPi
in which H is an analytic function of PI' ql, Pi' q!, not involving t. However such a periodic. motion admits of analytic continuation with variation of the energy constant H = h (chapter V, section 9), and so is not i.solated. Our aim then will be to consider only those nearly periodic motions which belong to the same value of h as the given periodic motion; this value may be taken to be h = O. The possibility of reduction to· a Pfaffian case m = 1, combined with the results of the preceding. section renders it highly probable at the outset that there will in general .be infinitely many nearby periodic ;Uotions,of. long period, provided that the given periodic motion is of stable type. In· considering this question, we shall make the further assumption that the given Hamiltonian problem is associated with an ordinary Lagrangian problem whose principal function L
DYNAMICAL SYSTEMS
160
is quadratic in the velocities. If ql1 q~ are the coordinates in this Lagrangian problem, the equations
Pi
aL = --, aq,
(i
1,2)
serve of course to define the variables PI' P2' Let ql = ql (t), q2 = q2 (t) be the equations yielding this periodic motion of period t, and consider the corresponding analytic curve in the ql' q2 plane. Evidently we can introduce a modified system of coordinates ql, q~ such that ~. vanishes along the motion, while ql increases by 2 rr: as a point makes a circuit of the motion. For instance, if the curve of motion is without double points, it may be deformed into a circle concentric with the origin, in which case ql and (b. may be taken as angle and radial displacement respectively. It is clear indeed that we may take fJI = 2 rr: tlr: along the periodic motion. Of course such a change of variables from ql' q2 to ql, q2 does not affect the Lagrangian character of the dynamical problem, although the new principal function L is periodic of period 2 rr:in the variable ql. The corresponding Hamiltonian problem will have the form (13) in which H is periodic of p~riod 2 rr: in ql, while for the periodic motion under consideration we have ql = 2 rr: tlr:, qt = O. From the Hamiltonian equations in these variables, we have also along the periodic motion 2rr:/r:
=
aHiapI,
It is obvious then that we may solve the equation H for PI in the form (14)
PI + K(ql' P2, q2, h)
=
= h
0
where K is a real, single-valued, analytic function of its .four arguments, periodic in ql of period 2 rr:. Furthermore we may regard h, ql, Pt, q2 as the dependent variables instead
161
VI. POINCARE'S GEOMETRIC THEOREM
of PI' ql, p!, q!; we observe that (14) may be solved explicitly for h, since from the 'relation H = h, we derive 1
aH apI = api a h
so that aPi/a h t 0 along the motion. When these variables are used instead of PI' ql, p!, q!, the variational principle (chapter II, section 10) takes the form (15)
<J
Jtr\o
K
qf + P2 q~ -
h) d t =
0,
which leads to the four equations
+
+
a K d ql 1 = 0 aK d h a K dp! ah d t ' ' a h d tap! d t dq! _
dt -
aK dql api at'
dp! dt
+
+ aK
aK dql aq! dt
a q!
=
d q! dt
=
0 '
O.
From these equations we infer directly h = const., which we know to be true of course. Now it is evident that near the given periodic motion qi can serve as independent variable as well as t. If we eliminate t in the above equations, we find (16) Here we are to set h = 0 in the function X, and K is periodic of period 2 n in qi' The given periodic motion corresponds to where fjJ is periodic of period 2n in qi' These .equations are clearly in Hamiltonian form (m = 1), With a generalized equilibrium point at the' origin, at least after the simple modification 11
DYNAMICAL SYSTEMS
162
in which we may take
Conversely, if we have a solution of (16), we can determine t from the equation dt dql
aK. -f)h'
and obtain a solution of the original system when t is taken as independent variable. Thus (13) and (16) are equivalent.* Periodic motions near the given periodic motion for (13) correspond to motions near the origin of period 2 k 7T for (16). For a Hamiltonian problem (13) which reduces to a ge11e1'alized equilibrium problem (16) of stable type (l to), there will £Xist infinitely many periodic motions in the vicinity of the given periodic motion, making in general many circuits of that motion before re-entering. This result is of course obtained as the direct application of section 1 for the reduced problem. 4. Some remarks. The general conclusion which appears in consequence of the preceding ~ections is that, for a given value of the energy constant, there exist in general periodic motions in the vicinity of a periodic motion of stable type, at least when the dynamical system has two degrees of freedom and is of ordinary type. The fact that there may exist isolated periodic motions of stable type, even for dynamical systems with two degrees of freedom, may be brought out by means of the following elementary example. Let us write
H
=
1 i( 2 2 1 i( 2 2 2 k Pl+q')+2l P2+qJ,
where the quantities k, l are incommensurable with one another, of which the general solution is * For the reduction employed, cf. Whittaker, Analytical Dynamics, chap. 12.
163
VI. POINCARE'S GEOMETRIC THEOREM
Pt = A cos kt+ B sin kt, Ps = C cos It+D sin It,
qt = qs =
-A sin kt+B cos kt, - C sin It D cos l t.
+
The energy constant h is defined by the relation
H
=
~
kS(AS+B S)
+ ~ lS(cs+D S) =
h.
The only periodic solutions are the two analytic families PI =
ql =
0 and ps
=
q2 =
0,
all of which are of stable type. For an assigned value of the energy constant, there are only two such periodic motions; thus all periodic motions of the second family with assigned A s+ B 2 represent the same closed curve in the three-dimensional manifold H = h in four-dimensional PI' ql, Ps, qs space. If the transformation T be set up for this case as in the preceding section 2, it is found to be essentially a rotation through an angle incommensurable with 2n, and so to correspond precisely to the highly degenerate case thete excluded from consideration. A first question as to a possible generalization of the above results in the case m =·2 is the following: Suppose the origin is a point of generalized equilibrium of general stable type for a given differential system which is, in addition, completely stable; if the constant l is not 0, does it follow that there will always exist infinitely many periodic motions in the vicinity of the origin? It seems to me very doubtful that the answer is in the affirmative. In the preceding argnment the area-preserving property played a· vital part. For this more general completely stable type, there is no reason to believe that this property continues to hold, even in the Pfaffian case. The example can be generalized so as to indicate a preliminary necessary requirement if the conclusion that there are infinitely many periodic motions near a given stable periodic motion is to hold for Hamiltonian systems with more than two degrees of freedom. 11*
DYNAMICAL SYSTEMS
164
In fact, consider the case of a dynamical system dp; dt
aH
dqi dt
(i=l,"',m)
in which H is a function of the m products PI ql, "', pm qm with pi, qi conjugate imaginary variables, namely
Here the coefficients Ci, di) are periodic of period ,,; in t. The origin is a point of generalized equilibrium with multipliers (i =
1"", m),
so that it will be of general stable type if these m quantities and 2 n V -1/ r have no linear commensurability relations. If for the sake of brevity we write
Xi
i
Jo,/[Ci +)=1
=
d .. p~ tJ
J
ri] d t, J
the general solution is at once found to be Pi =
p~ e-:r:"
qi =
q~ eX,
(i = 1, "', m).
Moreover if we write Ci
=
Jojt" Ci d t..
D··IJ =
Jojt" UriJ dt , d ..
the condition that the solution is periodic of period k,,; is m
Ci
+ .2: D i) P~ qJ = J= 1
2n k i V -11k
(i= 1, "', m),
where k 1 , . " , k,n are integers. But these form m linear, nonhomogeneous, algebraic equations in P~ q~, (i = 1, .. " m),
VI. POINCARE'S GEOMETRIC THEOREM
165
which can be solved if the determinant I Dij I is not O. Moreover by making the ratios kd k small, the periodic motion can be taken near the origin. On the other hand if IDij I = 0, . such a determination will be impossible in general. In the particular case when the C/S and dij's are constants, the system appears in complete normal form, and the c/s are the multipliers, while the dils are invariants analogous to l in the case m = 1. Hence at least the condition Idij I =1= 0 must be imposed in the case m > 1, as analogous to the condition l =1= 0 in the case m = 1, if an infinitude of nearly periodic motions is to be anticipated in all cases. Any generalization must of course take proper account of the uniform analytic integrals (such as the energy integral) which exist. In fact, if there are k of these integrals which are independent, the given stable periodic motion will admit of k-fold analytic continuation. Evidently it is not such periodic motions of the same analytic family as the given motion which interest us, but rather nearby periodic motions for the same values of the constants of integration as the given periodic motion, and making many circuits of it in a period. 5. The geometric theorem of Poincare.* Poincare showed that -the existence of an infinite number of periodic orbits in the restricted problem of three bodies and other , dynamical problems would follow at once from a certain geometric theorem to which the lemma of section 1 is intimately related. For convenience we shall first state: POINCARE'S THEOREM. Given a ring 0 < a < l' :S b in the 1', () plane (r, () being polar CoOrdinates), and a one-to-one, continuous, area-preserving transformation T of the ling into itself, which advances points on r = a and regresses points on r = b. Then there will exist at least two points of the ring invariant under T.
* This section is essentially the same as section 34 of my paper, Dynamical Systems with Two Degrees of Freedom" Trans. Amer. Math. Soc., vol. 18 (1917).
6
166
DYNAMICAL SYSTEMS
We will indicate briefly the proof of this theorem. Let us take x = fl, Y = r 2 as the rectangular coordinates . of a point in the x, y plane. The ring then appears as a strip a 2 < y < bZ• The transformation T of this strip advances points of the boundary y = a 2 to the right, and moves points on y = b2 to the left. Moreover T is areapreserving in the x, y plane (for we have 2rdrdO = dxdy), and displaees any two points which have the same ordinate and whose abscissas differ by a multiple of 27r in the same way. Let us combine T with a further transformation T.. which effects a translation of the x, y plane in the direction of the y axis through a distance £ > O. The transformation T followed by T.. yields an area-preserving transformation T T.. which shifts the given strip into the strip a 2 £ ::;; Y .:; b2 £. Suppose if possible that there exists no invariant point of 7. There exists then a positive quantity d such that all points are displaced at least a distance d by the transformation T. Choose £ less than d. Consider now the narrow strip a2 < y < a Z £. By the transformation TT.. the lower edge of this strip is carried into the upper edge and the strip is carried into a second strip lying wholly above the first one save along the common edge. By a repetition of the transformation T T.. the second strip goes into a third, and so on. By a continuation of this process, a series of strips is obtained forming consecutive strata. Each of these strata is unaltered by a shift of 2 n to the right. This follows from the fact that T and T e are single-valued over the ring. The images of these strata on the ring are a set of closed strata about the ring, all having equal area of course Rince T T.. is an area-preserving transformation in the r, 0 as well as in the x, y plane. Consequently some one of the strata on the infinite strip, say the k-th. must overlap the upper edge y = bZ• In the x, y plane let Q be a point of the upper edge of the k-th stratum for which y is a maximum. Let P be the point of y = a2 from which Q is derived by k-fold repe-
+
+
+
VI. POINCARE'S GEOMETRIC THEOREM
167
tition of T T e , and let pI, P", ... , P(k) = Q denote the successive images of P under the 'iteration of TT~. Draw the straight line P p' which will obviously lie on the first stratum. The successive images of this line, P pI, P' P", ' .. , P J). But, interpreted for
DYNAMICAL SYSTEMS
174
the billiard ball problem, this means that the ball can be projected very nearly with arbitrary position and direction to return subsequently to nearly the same position and direction. As elaborated by Poincare, * this chain of reasoning leads to the conclusion that the 'probability' is unity that an arbitrary motion returns infinitely often to the neighborhood of its initial state. He called this· property of the dynamical system 'stability in the sense of Poisson'. The proof that the double integral is invariant depends on an explicit evaluation of the determinant J= ~ ~ _ ~ Brpl. BO Brp Brp BO
II II
In fact, if
M t8, rp) dOd rp is invariant we have
ll-I(Ol,rpl) dOl drl
=
II
M(O,rp) dOdrp
where the variables 0 1 , rpl range over the region ai, just as 0, rp do over a. But according to the fundamental theorem for change of variables, the change of variables T gives the integral on the left the form
iI
M(Ol, PI) J dO dr·
Comparing this expression and the integral on the right, which are both integrals over the same arbitrary region a, we deduce the functional relation
as the necessary and also sufficient condition for invariance.
II
Hence to establish that sin 0 dO drp is invariant, we need only prove J = sin Olsin 0 1 , Let y =
G(r!')
.. See his Methodes nouvelles de la Mecanique celeste, vol. 3, chap. 26.
175
VI. POINCARE'S GEOMETRIC THEOREM
be the equations of C in rectangular coordinates, so that, if t" denotes the angle between the positive x axis and the positive tangential direction at a point of C, we have t"
= tan-1
G' (p)
F' (p) .
Similarly let t"1 denote the like angle at the transformed point, which will be given by the same expression save that P is replaced by Pr. Finally let a designate the angle between the positively directed axis and the direction of initial projection (figure). It is evident that the following two relations will hold
o= Or =
a -r:, t"1-
a.
Substituting in the above value for t" and the analogous value for '(;1, and also· substituting in for a the value
evident by inspection, we obtain the explicit formulas 0 = T: { 0 = 1
tan-1 G(Pr) - G(p)
F(Pl)-F(p)
t,an-1 G' (PI) F'(Pl)
tan-1--:G='(.o;:-p)7-
F'(p) -
tan-1 G (PI) - G (p)
F(Pl)-F(p)
=
L(m n
M(
,n ) 71
,
) P,Pl'
These two equations define the transformation T from (0, '1') to (0 11 PI)' Taking differentials, we find
176
DYNAMICAL SYSTEMS
This gives J=_M
0 is analytic in P, q, r, h. Hence P dl) dq ell' is invariant in .M. It follows further that T will leave a double integral
If1/J
du dv invariant, where
1l,
v are coordinates in S, and
VIII. TWO DEGREES OF FREEDOM
213
1/1> 0 is analytic in
11 and v. The argument is essentially that of chapter VI, section 1. This fact alone suffices to lead to the normal forms (2), (3), and to the conclusions cited above. * Hence the Hamiltonian problem need not be restricted in this manner. In fact it is found that for the most general transformation T
with such an invariant double integral
ff l/J
du du, there
is always a formally invariant function g (1;, u) given by it formal power series in u, v. We may define the case in which the equation g = 0 yields real formal invariant curves as of unstable type. In this case there are always asymptotic invariant analytic families of motions (or analytic families of periodic motions containing the given periodic moticn). All other nearby motions approach and then recede from the given motion. Thus there are no nearby periodic motions, except those that bdong to the same analytic family as the given periodic motion, if there are such. If SJ = 0 yields no real formal invariant curve of this kind, the periodic motion may be called of stable type. In the general stable ~ase treated above, g is ;-2, to terms of higher order. When (f is incommensurable with 2 n, while s, together with some but not all of the set of analogous constants perhaps, vanishes, no essential modification is required except that the term sr~ in (2) is replaced by a term S(k) r~k. If, however, all of these constants vanish, the normal form (2) holds with s = 0, and for these irregular periodic motions it is no longer possible to apply the reasoning by which the existence of infinitely many nearby periodic motions was established. On the' other hand, no essential difficulty arises in the case of stable type when (J is 0 or + n or, more generally, is commensurable with 2 n; this is the case when the given periodic motion is multiple, at least when taken as
* See my paper (Ioc. cit.) for justificatiol) of the fact stated as well as of what follows.
9
214
10
DYNAMICAL SYSTEMS
described a certain number, Ie, of times. It is only necessary to consider Tk in place of T, for which the number () is also O. Here the invariant function Q starts off with higher degree terms than the second, and casual inspection indicates that T is analogous to a rotation through an angle which vanishes at the origin but increases (or decreases) with distance from the origin. It would therefore seem highly probable that in this case too there must be infinitely many neighboring periodic motions, although the analytic details need to be carried through. Consequently it appears that in very general stable cases. and probably in all cases except the highly exceptional case when T is equivalent formally to a pure rotation through an angle incommensurable with 2 n, this property will continue to hold. This exceptional case is that in which the function M in the formal solution reduces to its first term ).. Hence £n the most ,r;eneral .c((se of nnstable type (m = 2) the phenomenon 4' asymptotic analytic families of motions (01' at least of analytic families of periodic motions containill1J the given motion) is characteristic, 'Other nearby motions approach and then }'(xedp from the gii'en periodic motion, In the most ,r;eneral stable casco e,Tr:ept the highly degenerate case whe'rc 6 is incommensurable with 2 n and the formal series im'olve no variable periods, therell'ill be neiqltllOring periodic motions. It is to be emphasized that the second of these conclusions has been formulated without completion of a detailed proof, such as I have not yet had the opportunity to effect. The degenerate case of stable type includes a real exception, as the example of chapter VI, section 4, shows, and should be further studied. Moreover the formal series break down in the stable case when J.. is commensur'able with V'="""1. 'fhese must be replaced by much more complicated types of series, a suggestion for the structure of which may perhaps be found in my paper referred to above; the earlier definition of complete formal stability will need to be extended so as to permit of indefinitely larg'e periods.
VIII. TWO DEGREES OF FREEDOM
215
Between the non-specialized dynamical problem and the highly exceptional integrable case, there exists an enormous variety of intermediate cases. In order to possess the analytical weapons with which to treat all cases whatsoever, it will undoubtedly be necessary to treat the question of the stability and instability of analytic families of periodic motions in much the same way as that which is outlined for the periodic motions above. While the individual periodic motions in such a family are to be regarded as unstable, this fact yields no information as to how nearby motions behave with respect to the family of motions as a whole. In order to avoid complication then. rather than because of any essential mathematical difficulty, we propose to deal mainly with the class of dynamical problems for which every periodic motion and its multiples are simple "vith l :f O. Such systems will be termed I non - integrable systems of general type'. The integrable case will be treated separately (section 13), while indications as to the nature of the result in the intermediate cases will be given. 2.
Distribution of periodic motions of stable type.
Our first aim will be to establish the following result: FO'T non,-integ1'able Hamiltonian systems of general type (m = 2), the set of periodic motions of general stable type is dense on itself in M. It will be observed that this result constitutes a slight improvement o,-er the result of chapter VI, sections 1-3, according to which other periodic motions, stable or unstable, lie near such a periodic motion of stable type. To begin with, we recall the facts developed in the lemma of chapter VI, section 1. It was found there that an arbitrarily small vicinity of the origin, l' < e (", 0, polar coOrdinates) can be selected at pleasure, and then an integer n such that (1) all the points of .,. < e remain in the region 1';; 2 e, under T, T 2, •• " Tn and (2) a Onla ro is positive for l' < e, On being at least 27T greater for l' = ' e than for -/" = O. It is easy to extend the argument to show that a1'"la 1'0' a on/a 00 are positive under the same circumstances.
II
216
DYNAMICAL SYSTEMS
In this way the curve On -
80
-
2 k n: =
0,
where k is so chosen that the left-hand member is negative but not less than ~ 2 n: for r = 0, will have one and only one point (r, 0) on each radius vector, with r < Q. Thus the equation· written defines an analytic curve C encircling the origin and meeting each radius vector only once. But, by the defining property of C, each point P of C goes into a point P" on the same radius vector; thus the curve C" is also met only once by any radius vector. Also, because of the area-preserving property of Tn, Cn and C will intersect ill at least two points, and these are obviously invariant points of Tn. In the case under consideration Cn and C cannot coincide, for C would then correRpond to an analytic family of multiple periodic motions. We propose to conr sider more closely the indices of these invaL M riant points. Let us i-----~ regard T, 0 as rectanR gular coordinates and consider the adjoining figure in which I is an Q invariant point at which K N the curve passes from within C to outside -+---------7- 8 of e, as a moving point describes in the sense of increasing O. If a point P makes a positive circuit of I; for instance around a rectangle KLMN, the vector P P n will have a component to the right above C, n component to the left below C, as follows from the facts noted above. At the points Q and R the vector P P n is directed upwards and downwards respectively. It is therefore apparent that during the circuit, the vector P P n rotates through an angle + 2n:,
c
en e
V I I I. TWO DEGREES OF FREEDOM
217
so that the index of I is + 1; while the invariant point J. at which Cn crosses C in the opposite sense has an index - l . Now by hypothesis the periodic motions corresponding to I and J are not multiple. If of stable type, the number if is not commensurable with 2 n. The corresponding normal forms are either of the type (3), in which f~ is positive or negative but not -+- 1, or of the type (2). By the aid of these forms it is easy to determine the respective indices. In the first and second cases, the slope of the vector P P Il is
v,,- v
1
v+ .. ·
lin - / I
!1'
u+·.·'
where only the first order terms are indicated explicitly in the numerator and denominator. It thus appears that if 1/ i'" negative the rotation is the same as that of the vectol' drawn from the invariant point I at the origin to the point (/(, v), i. e., 2n. Hence the index is + 1 if f1 is negative; likewise the index is obviously - 1 if !1' is positive. Moreover, in the third case of stable type when the number r; i::; by hypothesis incommensurable with 2rr in (2), Til is approximately a rotation through an angle incommensurable with 2 n near the origin, so that the vector P P n rotates through 2n during such a circuit. Hence the index is + 1 if the motion is of stable type. We infer then that J corresponds to a periodic motion of nnstable type, while it is not clear as yet whether 1 is of stable or unstable type. As a matter of fact, however, under the conditions stated I must be stable. The numbers !1' are the roots of tilt' characteristic equation, which takes the form
or" 08 0
=0
12
DYNAMICAL SYSTEMS
218
when the variables r, 0 are used. Since the roots are the reciprocals of one another, this equation reduces to
'" 3
!_(Brn Bro
+
BOn) BO o '"
+1
=
0
,
in which we are assured that the coefficient of '" is negative. Consequently '" is positive and I corresponds to a periodic motion of stable type. This disposes of the general case when there are no multiple periodic motions and l :f O. If the original motion is of stable type but not of that highly exceptional type when there are no variable periods in the formal series, it seems to me that analogous results are to be expected, i. e. that there will exist nearby periodic motions of stable type. This exceptional case merits particular attention; it is conceivable that it can only arise for integrable dynamical problems. 3. Distribution of quasi-periodic motions. Let us suppose that there exists at least one periodic motion of stable type for the Hamiltonian system under consideration, taken of non-integrable general type. This motion is represented by a closed curve C in the manifold :M of states of motion. Now select any such closed curve C1 Of motion of staole type. Very near to it can be found a closed curve C! of motion of stable type which makes k1 circuits of C before closing. Next choose a closed curve of motion Cs of stable type very near to C2 and making k 2 . circuits of C!, and so k 1 k2 circuits of C1 , before closing. Thus we obtain a sequence of closed curves Cn, (n = 1, 2, ...), which can evidently be chosen so as to tend toward a definite geometric limiting set C as n becomes infinite, merely by restricting sufficiently the successive !leighborhoods of C1 , C2 , •••• Furthermore, we can prevent C from being itself one of the numerable set of closed curves of motion by the same process. For instance
VIII. TWO DEGREES OF FREEDO!>1
219
at the nth stage we might confine attention to a neighborhood of which is so small as to contain no closed -curve of motion (other than en) of length less than n; there are only a finite number of such motions of course. It is interesting to inquire into the analytic form of the .set C. Let ql be the angular coordinate in M which increases by 2 rr when a single circuit of C is made. Then PI' p~, qt may be thought of as appropriate coOrdinates of this motioll (section 1), and we may write
en
lh = f.(ql) ,
JJ~
=
gl (ql),
q2 = hi (ql),
t = fk l (ql)dql
as the equations of the periodic motion C1 , in which fl' (jl , hi, k l > 0 are analytic periodic functions of ql of periodic 2 iT. For C2 we have likewise }J2
=ft (ql),
P2
=
92 (ql),
1J2
=
h2 (qd,
t = Jk2(ql)dIJI
where f2' 92, h 2, k 2, are analytic periodic functions of ql of period 2 kl 'Jr. Thus we form in succession a sequence of functions t" .fl", h", kit, periodic of period 2k l . , . k"-1 rr in f][, corresponding to the periodic motioll~ Cit, (n = 1, 2, ...). If we take points ql = 0 so as to approach a limit, it b obvious that fn, .fJn, hn , Oil approach limits f,.'/, h, 0, where the limits are approached uniformly for all values of ql' 1f there e:J:ists a single periodic motion of stable type for a non-integrablr: Hamiltonian problem of general type, there /(:ill exist infinitely many nearby motions, quasi-periodic but not periodic, with coordinates of the form PI =
lim j;l ('11),
u~oo
]12
=
lim .rI" (If I),
J~~OO
where 1", .r!n, h n , kIt are analytic periodic functions of ql with pet'iods 2 rr k l ••• k"-I, ki, .. " k n being positive integers which may be taken g'reater than 1. . The conver.r;ence is uniform f01' all values of ql'
220
DYNAMICAL SYSTEMS
Evidently there is a non-denumerable number of such quasi-periodic motions, and the coordinates are functions of the type treated by Bohr. It is clear that they constitute a class of recurrent motions of a new type. 4. Stability and instability. For the consideration of the periodic motions of stable type in dynamical problems, a fundamental division of cases must be made. It may happen that all motions sufficiently near the given periodic motion remain in a small neighborhood for all time. This is the simpler of the two cases, in which case the periodic motion in question may be termed 'stable'. The other possibility is that for some small fixed neighborhood of the given periodi(~ motion, there may be found motions which are arbitrarily near the given periodic motion at the outset but ultimately pass out of the fixed neighborhood. In this case the periodic motion in question may be termed 'unstable'. Evidently the classification here effected may be made not only for periodic motions but also for recurrent motions of any type. Stability in this fundamental qualitative sense is not to be confused with the 'complete formal stability' introduced earlier, and a periodic motion 'of stable type' mayor may not be stable. The transformation T of the surface H yields an immediate simple condition for stability. Consider a small region s of S about the invariant point. and its images SI, 82, . . . under successive applications of the transformation T. All of these contain the invariant point as interior point. The infinite set of regions 8, 81, ... will lie in the vicinity of the invariant point, according to the hypothesis of stability. These regions taken together occlude a certain neighborhood '8 of the origin, which is taken into all or part of itself .by the transformation T since the set S, SI , 82, .. , is taken into 8t, 82, .. '. But cannot be taken into a part of itself because of the existence of an invariant area integral. Hence s yields an invariant area in H, corresponding to which there is an invariant torus-shaped region of }.[ enclosing the curve of the given periodic motion in its interior.
s
VII1. TWO DEGREES OF FREEDOM
221
A -necessary and sufficient condition for stability is the existence of infinitely many inl'ariant torus-shaped re,qions dosing down upon the CUTve of the gi1:en periodic motion in the manifold M of states of motion. 5. The stable case. Zones of instability. What then is the nature of the boundary of such an invariant torus-shaped region in lvI surrounding the given closed curve of 5table periodic motion? In answering this question we llaturally turn to consider the nature of the corresponding invariant closed curvc in S forming the outer boundary of invaria nt region s. Let u>; assume that the periodic motion, although of general stab/P type is not such that the formal series involve no variable periods (section I), In this case a normal form (2) with s t- 0 (or a similar form) can be used. We may assume that s is positive, for if s is negative for T the corresponding' quantity, - s, is positive for T-l. This normal form (2) shows that the counter-clockwisc rotation about the invariant point incrcases with radial distance if l' is sufficiently small. No' region s lying nry near to the invariant point can be met more than once by some radial line. In fact let s* denote the part of the plane formed by the radial lines extended to the most distant points on the boundary of s. The regions of s* not forming part of s are of one of two possible types: either they are bounded by the boundary of s and a piece of a radial line on the left, or by the boundary of s and a piece of a radial line on the right. But the transformation T evidently takes a region bounded on the left by such a radial line l into a region bounded by s and the image II of this radial line l. Since the angular coordinate increases with the radius, it is geometrically evident (see figure) that
222
DYNAMICAL SYSTEMS
the image of such a region will necessarily fall within a region of the same type. But this situa tion is not possible since the transformation admits an invariant area integral, and consequently no set of regions can be taken into part of itself by T. Thus there exist no regions bounded on the left by such radial lines. Similarly the use of the inverse transformation T-l shows that there are no such regions bounded on the right by SlICh radial lines. Hence the boundary of the invariant region s is met only once by any radius sector. The impossibility of a radial segment forming part of the boundary of s is obvious, so that there is actually only onepoint of intersection with each radial line. A more elaborate consideration based on the normal form shows that the boundary curve r = f(O) is one for which the difference quotient
4
is bounded and indeed small for invariant regions near enough to the invariant point. * The truth of the fact seems almost obvious if one observes that T rotates positively directionsdiffering by any considerable amount from the directions perpendicular to the radial direction into other directions differing still more from that direction. 0nr conclusion may thus be summarized in the form: For a stable periodic motion of general stable typc and with vm'iable periods in the formal series, the invariant torns-shaped regions are such that their intersections I("ith the analytic s~lIface of section S may be represented in the forni r = f(O), where 1', 0 ((rc polar coiirdinates with the invariant point at the on:ql:n, and wherc f is ((. continnous pc-riodic function of (j of pc-riod 2 n for which the dijfere-nce quotie-nt is bounded. The curves of motion on the boundary of such a torusshaped region form a closed invariant family. In any such * For details see my paper (loc, cit.), sections 42-48.
223
VIII. TWO DEGREES OF FREEDOM
closed invariant family of motions near the given stable periodic motion, there is of course at least one recurrent motion. If the rotation number x along the corresponding invariant curve on 8 is incommensurable with 2 rc the surface of the torus may represent a single minimal set of recurrent motions. Jll this case the coordinates and the time can be expressed in terms of continuous doubly periodic functions. In order to make this clear, let us first select angular coordinates 0, rp on the torus as follows. The coordinate rp will be taken to vanish in 8, and to increase proportionately with the time along each curve of motion, the factor of proportionality being so taken as to increase Cf' by 21C between successive intersections with 8. The coordinate 0 will be defined along the invariant curve on S so that the transformation T takes the form OI = 0 where x is the rotation number specified. Elsewhere on the torus the variable 0 may be defined as the (f of the corresponding point on 8 diminished by x rp / 2 rc in order to make 0 single-valued on the torus. In these coordinates the equation of a curve of motion is 0 - 00 = x rp/2 JT. Moreover the coordinates PI' ql, P~, q~ are doubly-periodic continuous functions of 0, rp, and d t/ d rp is also. Hence we may write
+"
PI =j(xrp/2-rr, rp),
q~ =
ql
=
k(xrp/2n, 9'),
.r;(xrp/2rc, rp),
Pt
=
t = JZ(Xrp/21C,
h(xrp/2rc, rp),
9') drp,
where 1, g, h, k, Z are continuous doubly periodic functions. of period 2 rc in their two arguments. There is a second possibility to be considered also. The minimal set of curves of motion may correspond to a perfect nowhere dense set of points on the invariant curve; all other curves of motion on the surface of the torus-shaped region will then approach this minimal set of recurrent motions. asymptotically as the time t either increases or decreases. * * For proof of these facts and reference to the prior work of Poincare, see my paper Quelques theoremes generales sur le mouvement des systeme$ dynamiques, Bull. Soc. Math. France, vol. 40 (1912).
224
DYNAMICAL SYSTEMS
When, however, the rotation number is not incommensurable with 2Tr, but is 2p Trlq (p, q, relatively prime integers), there will of necessity exist points of the invariant curve which are invariant under Tq. It call be proved that the entire curve is then made up of analytic arcs terminated by points invariant under Tq, while the interior points of such arcs tend asymptotically towards these invariant points upon iteration of T or its inverse. * Vi'e are thus led to the following conclusion. Any such dosed invariant family of motions near the !licen stable periodic motion of general stable type and Icith variable periods in the formal scn'es is cha'l'acterized by a 'rota,tion number. If this numbe1' is 'incommensurable with 2 Tr, either the family consists of a single minimal set of FellI/Tent motions of continuous type, or it contains a ]JC'lject 1l0/clwre dense minimal set of recurrent motions of discontillolls type Ichich all other motions of the family approach asymptotically as t increases or decreases. If this number is commens'll1'able ll:ith ~ Tr, there exists one or morc closed periodic motions in the family, while the other motions form rrnalytic branches asymptotic to these periodic motions. It may be observed that this is a result! concerning invariant sub~manifolds of the manifold ill and, in particular, concerning the central motions in this sub-manifold. Incidentally it appears that, although in dynamical systems of classical type, all the motions are central with reference to the whole manifold, the same is not necessarily true of invariant submanifolds, so that the concept of central motions continues to play a part even in the problems of classical dynamics. Any two of these closed families must be entirely distinct from one another, except when both have the same rotation JlUmber, commensurable with 2Tr. Clearly the rotation numher. which measures the mean angular rotation, must be the same for two intersecting families. '}'o establish that this number must be commensurable with 2 Tr, we note that since the two families have at least one motion in common although they do not coincide. the two corresponding curves. l' = J,. (0), * See my paper in the Acta Mathematica. loco cit., sections 42-48.
225
VIII. TWO DEGREES OF FREEDOM
= j~ (0) in S will enclose one or more areas oetween them, each bounded by a single arc of either curve. Under iteration of T this area must ultimately overlap itself and so coinci 1) 2 2 * Le.s surfaces Ii courbures opposces et leur lignes gcodesiques, Journ. de Math., ser.5, vol. 4 (1898). t Recurrent Geodesics on a Surface of Negative Curvature, Trans. Amer. Math. Soc., vol. 2~ (1921); A Fundamental Class of Geodesics on Any Closed Surface of Genus Greater Than One, Trans. Amer. J.lath. Soc., 'Vol. 26 (1924).
VIII. TWO DEGREES OF FREEDOM
239
in which x, y, z are rectangular coordinates, and where we shall make the convention that aU points (x ± 2 k n, Y + 2l n)
(k, Z= 0, 1,2, ...)
correspond to the same point of the surface. This convention is legitimate inasmuch as the linear group of translations
x=
x+2kn,
y=
takes the surface into itself, is then the square given by
z = z,
y+2ln,
A fundamental domain for x, y
o 0, and then from a region z > 0 to a region z < 0. To every allowable deformation of the path AB (.4. and B being fixed of course) wi'U correspond a modification of the symbol. The class of symbols obtained from one another in this way may be caned 'equivalent'. It thus becomes important to determine the legitimate type of modification of a symbol, and a normal form of each class of equivalent symbols.
°
16
242
DYNAMICAL SYSTEMS
The legitimate operations are of two types. 'l'he first allows us to insert or remove any pail' of elements a a-lor a-I a in the symbol, since this corresponds to a deformation over a boundary. The second operation allows us to replace such a symbol as W s y by Y w2 ; or U'a l y by y 1/;:;\ or w 2 y-l uy y-Iw a, or 1l,:;ly-1 by y-Iw a- I ; these are the changes possible in the symbol when a point P of A B is deformed through the point common to the quadrants W2, Ws of a geodesic circle. There will be similar operations at the point common to Ws, W4, at the point common to W4, WI' and at t he point common to WI, W2' In order to obtain a normal form we reduce the number of elements in the symbol as far as possible by the following three processes. First we strike out any pair aa- l or a-I ((. Secondly we replace any triple such as y~1;2 y--l by its equivalent Ws; here we have YW2y-1 =
/Ua
(yy-I)
=
Wa
of course. For each of the sixteen operations of the second type specified above there will be two corresponding triples of the form p Wi p-I or plOi! p-I where ]J is X, X-I, Y or y-t, which can be replaced by a single letter LLj or Wjl. Thirdly we replace O. If the invariant area integral is
in these variables (see section 1), we will have I (Ci'l , 'l/Jl) of/arp = I(rp, 'l/J).
f
so that the integral of I drp over any arc 'l/J over the transformed arc has the same value. Now write x/2 n
=
L'"
IL
=
const., and
2
I(rp, 'l/J) drp
I (rp, l/J) drp,
7l
thereby introducing a new analytic angular variable x of period 2 n which can replace rp. The transformation T will take the form
in these special variables.
Here a ('l/J) is an analytic function of l/J. Hence the transformation of each invariant curve of the analytic family which abuts on 0 = 0 is essentially a rotation of that curve into itself through an angle a which varies analytically with the curve and increases from 0 along 0 = 0 toward a limiting value n. But this variablel/J is not to
VIII. TWO DEGREES OF FREEDOM
253
be regarded as defined along the limiting non-analytic r,uryc of course. Consequently, if a (l./J) is commensurable with 2 IT, say a = 2 n p / q, every point of the in variant curve corresponds to a polygonal periodic motion of the billiar2 .If,
R'2 , =
(m1' 2 +,P,(!2) (m1',2 + p,e12) - mp, ( 1'e' - e1',")2"
which may be written R' 2
= m l'12+ p, e12 -
mp, R! ( r e' -
el' ')2 .
Furthermore we have the obvious identities x ,2+ y '2+ Z,2 ' ')2 + ( ')2J, =1',2 + Sr1 [("yz-zy')2 + ( x'y - y x 'zx-xz ~,2
+ "1'2 + ,'2 =
(!''J +
-; [("'~' _ (}
~ 1/')2 + (~~' - ~ ")2 + (~1/' -fJ ~'n.
Multiplying these last two equations through by m and p, respectively, and subtracting them, member for member, from the preceding equation, there results the equation
266
SYSTEMS
DY~AMICAL
R,2+ p
(16)
2(U-K)
=
where P (to be minimized) is a sum of seven squares, p (I7)
=
n: [(yz'-zy'/+(zX'-Xz')2 + (xy'-yx'/J
r
+ e/l'2 [(1J" -
'1]')2
R (' + m/l'
er) .
Z
+ ('~' -
H')2
+ (h' -1J ~')2]
,2
1'(1 -
Here the energy integral (12) has been made use of. From this relation due to Sundman we may derive the inequality which plays a fundamental part in his work and in the present chapter. If we write U = yz'-zy'. V = 1J"-'1J', it will be observed that there are two terms in P of the form
while the first integral of angular momentum yields
mU+" V
=
a.
It is easily found that the minimum value of S when U and V vary subject to the restriction just written, while l' and e remain fixed, is a 2 / R 2 • Similarly there are two other analogous pairs of terms with minimum values 11 2 / R 2 , ell / R ll respectively. Hence· we conclude that we have (18) (19)
p fll =
>
fll/R ll ,
a +b +c ll
2
ll
•
Suppose now that we eliminate U between Sundman's equality (16) and Lagrange's equality (15). This gives us 2
2RR"+R: +2K = P,
267
IX. THE PROBLEM OF THREE BODIES
whence, by using (18), we obtain the inequality refen'ed to:
(20)
2RR"+R,2+ 2K
> ~:.
If we define the auxiliary function of Sundman, (21)
the inequality (20) enables us to infer the relation (22)
H'=FR'
(F
>
0).
Hence H increases (or at least does not decrease) as R increases, and decreases (or at least does not increase) as R decreases. This is the consequence which is of fundamental inportance in what follows. 6. The. possibility of collision. Thus far we have been taking for granted the existence of solutions in the ordinary sense. In fact, inspection of the differential equations shows the existence of" a unique analytic solution for which the coordinates and velocities have assigned values at t = to, provided that the bodies Po, Pi' P" are geometrically distinct. In the case of the coincidence of two or three of these bodies, the right-hand members of the differential equations are no longer analytic, or even defined, so that the existence theorems of chapter I fail to apply. But, according to the results there obtained, either these solutions can be continued f{)r all values of ~he time, or (fo~ example), as t increases, continuation is only possible up to t. Let us consider this possibility in the light of the elementary existence theorems. In the 18~dirn.ensional manifold of states of motion associated with the 18 dependent variables Xi, Yi, Zi,
I
Xi,
I
Yi,
I
Zj
(i
=
0, 1, 2),
we need to exclude the three 15-dimensional analytic manifolds (i
= 0, 1, 2).
DYNAMICAL SYSTEMS
268
The remammg region is open towards infinity and along these excluded boundary manifolds. According to the results obtained, indefinite analytic extension of a particular motion will be possible unless as t approaches a certain critical value t, the corresponding point P approaches the boundary of the open region specified. Now suppose if possible that the least of the three mutual here it is distances does not approach 0 as t approaches not implied that a specific mutual distance such as Po PI remains least neal' to We can thcn find positions of the three bodies for t arbitrarily near to for which the three mutual distances exceed a definite positive constant d. But by the energy integral relation (4), in which
t;
t.
t,
it is clear that the velocities xi, yj, zi are uniformly limited. It is physically obvious that for such an initial condition, (~ontinuation of the motion is possible for an interval of time independent of the particular mutual distances or velocities, because of the character of the forces which enter; "\ve shall not stop to obtain an explicit expression for such an interval on the basis of oUl' first existence theorem. Thus a contradiction rcsults. Analytic continuation of a partiClllar motion in the jJroblem of three bodies lUill be possible unless as t approaches (( certain raluet, the least of the three mutual distances approaches O. At this stage it is desirable to revert to Lagrange's equality (15). As t approaches U becomes positively infinite of course. Hence if we represent R 2 as a function of t in the plane by taking t and R 2 as rectangular coordinates, the corresponding curve will be concave upwards for t suffiTherefore R~ either becomes infinite, or tends ciently near toward a finite positive value, or approaches O. The first case is manifestly impossible, since one of the bodies would then recede indefinitely far from the two which approach coincidence as t approaches t; and such a state of affairs
t,
t.
IX. THE PROBLEM OF THREE BODIES
269
is impossible because of the fact that the forces on the distant body are bounded in magnitude. In the second case it is clear that a particular distance approaches 0, for instance 1'2, while the other two approach definite equal limiting values. This is the case of double collision. Since the forces on the non-colliding body are finite near collision, it approaches a definite limiting position; and thus the other two colliding bodies approach a corresponding limiting position, since the center of gravity may be taken fixed and at the origin in the space of the three bodies. In the third case we have triple collision of course, and this takes place at the origin. However if the constant f is not 0, triple collision cannot take place, as follows from (22) immediately. For it is seen that d R 2 /d t will be negative for t near t in the case of triple collision, since d2 R 2 /d t 2 is positive by Lagrange's equality (15), Hence H \vill decrease with R (or at least not increases) as t approaches T But inspection of H shows that H becomes positively infinite as R approaches 0. Thus a contradiction is reached. As t approaches i, there is either double collision bet/ceen a definite pair of the bodies at a definite point, while the third body approaches a definite distinct point, or there is triple collision at the common center of ,gravity. If, however, f is not 0, i. e., if the angular momentum of the tlu'ee bodies about . every a.TlS in space js not constantly 0, triple collision can never take place at t. Henceforth we shall make the assumption f> 0, thereby eliminating the possibility of triple collision in the sense above specified. This assumption may be looked upon as merely confining attention to the general case. In fact it is readily proved that in the case f = 0, th'e motion is essentially in a fixed plane. Thus immediate reduction of the problem is possible. M-oreover in the case f = 0 the angular momentum about a perpendicular to the plane of motion at the center of gravity vanishes. Thus we are only excluding a special case of motion in a plane. The case excluded is of great inter-
270
DYNAMICAL SYSTEMS
est and should be g'iven thorough consideration on its own account. 7. Indefinite continuation of the motions. III the general case under consideration it is thus plain that any motion can be continued up to a double collision. We propose now to take up briefly the case of double collision in order to render it physically plausible that the motion admits of continuation beyond such· a double collision in a certain definite manner. Analytic weapons sufficiently powerful to deal with the singularity of double collision were first developed by Sundman (loc. cit.). A different method of attack, not going outside of the domain of equations of usual dynamical type, has since been obtained by Levi-Civita. * A rigorous treatment of the question ,,'ill not be attempted here, but the analytic details can be supplied without difficulty on the basis of the researches of Sundman or Levi-Civita. Let us suppose that the bodies Po and PI collide for instance, while P2 is at a distance away. The motion of Po and PI near collision will clearly be essentially as in the two body problem. What propose to do is to ignore the disturbing forces due to, P2 during the near approach of Po and PI to collision, i. e, to replace U by its single component 17lij ml / r2, and then to take it for granted that the situation is of essentially the same nature in the actual case. But if the motion of Po and PI were just as in the two body problem, their center of gravity would move with uniform velocity in a straight line, while, relative to this point, Po and PI would move in a fixed straight line until they collide. More precisely, Po and PI will be at distances inversely proportional to their masses from the center of gravity, while their squared relative velocity is 2 (mil ml)lr2 increased by a certain constant whose value depends on the total energy relative to the center of gravity. The motion relative to the center of gravity will be thought of as merely
we
+
• Sur'la regulm'izatioll lilt probUme des trois corps, Acta Mathematicn, vol. 42 (1921).
IX. THE PROBLEM OF. THREE BODIES
271
reversed in direction after collision. In the original reference system the bodies Po and P l will describe two cusped cnrves, and will collide at the ,common cusp; the cuspidal tangents of the two curves are of course oppositely directed, and it would be easy to specify the precise motion near collision by giving the e'xplicit formulas. Evidently such a motion of collision in the two body problem is completely characterized by the followingquantities: (1) the three coOrdinates of the point of collision; (2) the three velocity components of the center of gravity at collision; (3) the two angular coOrdinates 0,
0, which is under consideration. In the first place we observe that not only R but also R' must be continuous at double collision. In fact the differential equations themselves show that dg'§ldt!, d!"Ildt!, d 2 'ldt! are continuous at collision so that (I' as well as (I must be continuous. On the other hand r' will not be; but, since we have r 2r '2
(x x'
+ y y' + Z/)2
+ yE +z!) (X'2 + y'2 + /2)
:::;:; (x E
< ;~! (U + IK/) on account of the energy integral (12), it is clear that r r' is continuous and vanishes at collision. Hence R ' is continuous at collision, having the value !' (I (I' / R, as follows from (13). Secondly, as t approaches t, the least ri must approach 0. Otherwise we should have r; > d > (i = 0, 1, 2) indefinitely near T. We have already seen that, because of the energy integral, this would require x', y', z', r, 1/, " to be limited, so that continuation of the motion during a definite interval of time, dependent only on d, would be possible without collision. This is absurd. Thirdly, R must approach a finite limit as t approaches t, as follows from Lagrange's equality (15), just as in the case of approach to double collision, inasmuch as R' and Rare both continuous at double collision. Reasoning on the basis of Sundman's inequality (22) in the same way as befor~ we infer also that R cannot approach as t approaches t. Hence we conclude that as t approaches t, the body P E approaches a definite limiting position distinct· from the corresponding definite limiting coincident position of Po and Pl' But it is physically obvious, and might readily be established analytically, that there can only be a finite number of collisions for t < t in such a case. Thus a contradiction arises.
°
°
275
IX. THE PROBLEM OF THREE BODIES
In the augmented manifold of states of motion MIs, indefinite continuation of every motion for which f> 0 is possible in either sense of time. In the case f= 0, continuation can only be terminated by triple collision. Hitherto we have dealt with only the I8-dimensional manifold MIs, It is easy to modify the above results so as to apply to the manifold M 12 , obtained when only those motions are considered for which the center of gravity of po. PI' P 2 lies at the origin. In this case the six coOrdinates fixing the position and velocity of the center of gravity of Po and PI' for instance, determine these coOrdinates for P 2 • Entirely similar results obtain in the 12-dimensional manifold M 12 obtained by fixing upon those motions for which the center of gravity of the three bodies lies at the origin. As remarked earlier, these results can be fully established by use of the explicit regularizations effected by Sundman or Levi-Civita. An inspection of the formulas leads to the following additional conclusion: In the augmented manifold MIs not only are the states of motion at collision to be 1'egarded as constituted by three 15-dimensional analytic manifolds, but the curves of motion are also to be regarded as analytic and as varying analytically with the initial point and interval, provided this interval be measured by such a parameter as u where
8. Further properties of the motions. The case K < 0 is immediately disposed of, so far as the general qualitative character of the motions are concerned. Lagrange's equality (15) insures that d! R!/d t 2 will then exceed 41 K I. Hence R 2 , when plotted as a funct~on of t in the t, R! plane of rectangular coOrdinates, yields a curve with a single minimum which is everywhere concave upwards and rises indefinitely. Evidently the same conclusion holds for K = 0, at least unless U approaches O. But this can only happen if all three mutual distances increase indefinitely. IS'
276
DYNAMICAL SYSTEMS
In the case K S 0, f> 0, at least two, if not all three, of the mutual distances increase indefinitely as time increases and decreases. In the case K < 0, f = 0, tlte same is true unless the motion terminates in triple collision in one direction of the time. A fuller qualitative consideration of the motions K < is obviously desirable. But on account of the results just stated it seems proper to consider this case as 'solved' in the qualitative sense. Henceforth we shall confine attention to the case f> 0, K> 0, i. e. to the case when the angular momentum of the three bodies about every line through the center of gravity is not constantly 0, and the potential energy is insufficient to allow all three mutual distances to increase indefinitely. thus remains. Here the motion The case f = 0, K> is essentially in one plane, and it may be possible to obtain results similar to those here obtained in the case f> 0, K> 0 by suitable refinement of Sundman's inequality. We proceed to develop some of the simple and important properties of the motion in the case f> 0, K> O. In the case f> 0, K> the least of the three mutual distanc:'s cannot exceed M 2/(3 K). The proof is immediate. By the energy integral (12), U is at least as great as K. But ro, r 1 , r2 are at least as great as r, the least distance. Hence we obtain
°
°
°
The numerator on the left is not more than 11ft-/3, whence the stated inequality follows at once. In the case f> 0, K> 0, the largest distance ri will necessarily exceed k times the smallest distance rj, provided that
where m* denotes the least of the three masses mo, m 1, m2' To establish this fact, let k 1 denote the actual ratio of the largest to the smallest distance. Then we have at once
IX, THE PROBLEM OF THREE BODIES
where r denotes the smallest distance, a similar calculation
U
0, K> 0, any paJ·tof the curve R = R(t), (t,R, rectan.fJu1ar coordinates)jor which R<jj(21 /% Kl/2) tongisls
DYNAMICAL SYSTEMS
278
of a finite m'e, concave upwards and with a single minimum. If R = R o ,qives this minimum, the curve rises on either side until R> PI(2KRo), with correspondin,q slope R: at least as great as demanded by the inequality
at every intermediate stage. To prove this statement, we observe first that when R is restricted as in the first part, R cannot be a constant. In fact if it were, Lagrange's equality (15) would yield U = 2K. But the combination of Sundman's equality (16) and of (18) with the equation U = 2 K would give
in contradiction with the limitation imposed upon R. The same kind of argument shows that if H' vanishes when R is so restricted, then R" must be positive, For otherwise, by using Lagrange's equality, we find U < 2 K, and thence by using Sundman's equality (16) and (18) we are led to the contradictory conclusion written above. If there is a point R' = 0 along the arc under consideration, it corresponds to a proper minimum. On either side of it H (section 5) will increase (or at least not decrease) with R, until a second point R: = 0 reached for R = R I • Hence we obtain 2 K RI
+ pR > 2 K Ro + P R o I
whence, since R I
> Ro,
P
2K> RoR I
•
In this case R does increase until the specified value is passed. Furthermore until this happens, H is as great as H o •
IX. THE PROBLEM OF THREE BODIES
279
This fact demonstrates that R '2 is as great at every stage as stated, so that R must finally so increase. The case when R':f 0 anywhere along the arc can be eliminated. Here H must decrease (or at least not increase) with decreasing R. Consequently R cannot approach 0, since H then becomes infinite. As R approaches its lower limit R o , 11' will approach O. Consequently we infer that the inequality of the statement for 11'2 continues to hold if Ro be defined in this manner. But this kind of asymptotic approach to R = R o as t increases (or decreases) indefinitely is impossible. This impossibility may be made evident as follows. In the inequality H :2: H o we may replace the inequality sign by the equality sign. Thereby we define a new curve R = R(t) whose slope for any R is not greater in numerical value than that along the actual curve under consideration. Hence the new curve so defined approaches the t axis less rapidly, and must also approach R = R o asymptotically as follows from the equation H = H o . But, by differentiation of this equation as to t, there results
Hence as t approaches infinity, and R, R ' approach R o , 0, it is clear that R" would approach a definite positive quantity, which is absurd. The results thus far obtained may be regarded as concerned with motions in which the three bodies are all near together at some instant t = to, the amount of separation being measured by R. The bodies will separate in such a way that R increases, and very rapidly as long as R is not too large or small, until R has become very large. \Ve turn next to derive somewhat analogous results when at least one of the three mutual distances is large. Here it is convenient to use the quantity {! instead of R, but it is to be borne in mind that r denotes the smallest of the three distances in what follows.
DYNAMICAL SYSTEMS
280
In the mse f> 0, K>
°
as long as
(! ~
2Jf!/(3K), one
and the same distance ri is the least distance.
Under this condition it follows that (! is at least twice the least of the distances r = T!. Hence To and 1'1 exceed r, since (! is the distance from p! to the center of gravity of Po and PI' But when To and Tt are greater than T!, one and the same distance r! remains least. In the casef> 0, K>O,for (! ~ 2M!/(3K), the inequality
obtains.
(!
If for any such valuA of
e,
we have
will constantly increase without bound. We begin with the identity
The last three terms on the right give the square of the velocity of the point (~, 1], '), while (!/2 is the square of the radial velocity and is therefore not greater. By virtue of this fact and the differential equations (10) we obtain nn" ""
> ~ (:: ~ + I au... + t• au. f1' - a~ 01] a; ). 11
=
But the terms in parenthesis on the right are
preci~ely
eaUI an where p! is taken to vary by a distance n along the straight line which joins p! to the center of gravity of Po and PI' Clearly the rate of change of 1'0 and 1'1 with respect to n cannot. exceed 1 in absolute value, and we infer
,e => _
o "
~ f1'
( nl\ m! 'J
~
+ mo n~ ) ---. _ 2
~
----
M
(_1_ + _1_)
.). ( ! . ,
~
"
~
(see (7). Now in the case under consideration 1'0 and 1'1 exceed e- r and therefore eI 2 . This leads to the first inequality to be proved.
IX. THE PROBLEM OF THREE BODIES
281
Instead of continuing analytically we need simply observe that this inequality may be looked upon as requiring that a particle moves along a (! axis acted upon by a force towards the origin which does not exceed the gravitational for~e due to a mass 8M. But in this case it is obvious that the particle will recede indefinitely provided that the initial velocity outward is as great as the veloeity of fall from infinity under the attraction of such a mass. This is precisely the fact stated. It should be noted that since the initial value of e is as great as 2M i /(3K), e continues greater than this quantity, and accordingly one and the same distance r is the least of the three distances always. We propose next to combine these results in order to show that, for the minimum R o sufficiently small, Rand e increase indefinitely. The qualitative basis of the reasoning is obvious. According to what has been proved, for R* and R*' arbitrarily large a positive R o can be chosen so small that all motions for which the minimum R is not more than R o correspond to an R which increases from the minimum to R* and has, for R = R*, a derivative R' which is at least as great as R*'. This means of course that e* is arbitrarily large since lim RI e = (mo mi + ml m2)1/2 R=ot:J
uniformly.
Furthermore since the relation
RR"= Inrr'+fI'el obtains, it is clear that lee' must be large, and in particular must be large, provided that 1r r' I is uniformly bounded. But we have 1
ill
by the energy integral (12).
Hence
282
DYNAMICAL SYSTEMS
since m exceeds one half of the least mass m*. find
Thus we
and thereby establish the fact that 11'1.' I is uniformly bounded. For f> 0, K> 0, if R o is taken sufficiently small, every motion for which the three bodies approach so dosely that R S Ro at some instant is SUdl that two of the distances rl). 1'1 become infinite with t lI;hile r~ remains less than llf!/(3K). 'Ve shall not pause to develop an analytic formula -which yields a suitable R o, although the specific results found above would supply the basis for such a computation. There is an interesting question to which we wish to refer briefly in conclusion. Which one of the three bodies will recede indefinitely from the other two nearby bodies, in the case of a near approach to triple collision? The answer is to be found in the folIo wing statement: A.ny motion of the a/Jove type is characterized by the property that one and the same body P 2 remains relatively remote fr'om the two nearest bodies Po, PI throughout the enti1'e motion. The truth of this fact is readily inferred~ At the beginning of this section it was shown that, for R greater or less than fixed values, the ratio of the largest to the smallest distance would be arbitrarily large. Hence we need only consider this intermediate range of values of R. But in such a range, if the ratio of the largest to the smallest side did not remain large for R o sufficiently small, there would be configurations of the' three bodies in which the distances 1'i and the ratios 1';1 rj lie between fixed bounds, no matter how small Ro is chosen. However, the value of U does not exceed an assignable quantity in such configurations, and thus, by the energy integral (12), the same would be true of the velocities x', y', z', ~', "J', ~'. Finally it is clear that RR' would not exceed an assignable quantity. But we have established that R' becomes arbitrarily large in such a definite range of values of R, so that this conclusion is absurd.
IX. THE PROBLEM OF THREE BODIES
283
Evidently there is further work to be done in the more precise determination of the motions on the quantitative side, but the facts developed above are sufficient to show that the only possibility of simultaneous near approach of the three bodies for given f> 0, K > 0, is that in which the three bodies act as a pair of bodies, one member of which corresponds to a close double pair Po, PI' while the second is P,. The motions of P 2 and the center of gravity of Po, PI are then along nearly hyperbolic paths, while Po, PI move in nearly elliptic paths relative to their center of gravity. g. On a result of Sundman. Sundman established (loc. cit.) that for given initial coordinates and velocities with f> 0, K> 0, the quantity H (t) for the corresponding motion will always exceed a specifiable positive constant. This fact is at once evident from the analysis of section 8. In the contrary case we should have indefinitely near approach to triple collision, and thus a motion for which H' is arbitrarily large for the given initial value of H, which is of course absurd. 10. The reduced manifold M 7 of states of motion. Let us turn next to the consideration of the problem of three bodies after use has been made of the 10 known integrals to reduce the system of differential equations from the 18th to the 8th order. In other words the 10 corresponding constants of integration are given fixed values, and attention is directed towards the 00 7 motions which correspond to the given set of constants. In what follows we shall suppose that not all the constants of angular momentum vanish, and that the constant of energy is positive, i. e. we take f> 0, K > 0. The angular momentum vector with components a, b, c will define a spatial direction which plays an important role in the sequel. Evidently two motions which cOlTespond to the same configuration of positions and velocities at some instant, aside from mere angular orientation relative to this axis of angular momentum, will continue to differ merely in this respect. In other words, if ep denotes any angular
284
DYNAMICAL SYSTEMS
coordinate which fixes the orientation about the axis of angular momentum, while UI, .. " 111 are any set of relative coordinates which do not involve gl, the differential equations defining the 00' motions take the form du;/ dt
Ui (UI,···,1l7)
d pi d t
(]) (UI , .. " 'Ut) .
(i
L .· .. 7),
The first set of equations constitutes a system of the 7th order in the coordinates lit, .. " 10, while the last equation enables one to determine p by a further integration. If it be desired, the time t can be eliminated, and the system becomes of the 6th order, du i ld1l1 =
U;/U t
(i
=
2,3" .. ,7).
Thus from the purely formal standpoint the system of the 18th order can be 'reduced' to one of the 6th order. From the point of view which we shall adopt, there is no essential gain in actually carrying through such a reduction which can be accomplished without affecting the Hamiltonian form.* Let us consider the augmented manifold .il'II8 of states of motion, in which the singularities corresponding to double collision have been removed by the method indicated in section 7. The boundary of jVII8 is to be reg'arded as made up of states of motion specified by one of the following possibilities: one of the coordinates Xi, Vi, . 0 and K> 0 there can be no singularities whatsoever in J.lfB• Let us choose coordinate axes so that x = y = 1J ~= 0 at the instant under consideration, i. e. the particle PI lies in the z direction from Po, while the line from P 2 to the center of gravity of Po and PI lies in the x, z plane. Let us attempt to solve the 4 equations for x', y', z', 1J' as functions of the other variables. The condition that this be .possible will be satisfied if tlie corresponding Jacobian determinant
IX. THE PROBLEM OF THREE BODIES
0
0
o -z z x
0 I
y
I
287
; 0 0-' 0 0 , , z "I
does not vanish; here we have removed an obvious factor m from the first three columns, and a factor f" from the last
column. Thus Ms is analytic at this point provided that the inequality - ; z! z':f 0 holds. But it has been pointed out that z is not O. Furthermore, we can take ~:f 0 unless p! is on the straight line Po P 1 constantly. And we can take z' :f 0 unless the distance Po P 1 (and similarly any other distance Pi Pj) is a constant. Hence we infer that either M s is analytic along the particular streamline under consideration, or the three bodies lie upon a straight line, or at a constant distance from each other, but not in the same straight line. In the latter case the bodies Po, P 1 , P t are known to lie at the vertices of an equilateral triangle in a plane perpendicular to the angular momentum vector; this triangle rotates at a constant angular velocity about its center of gravity. Furthermore it is known that there is one and only one size of triangle of tbis kind for an assigned angular velocity. Thus there will be in general no such motion for which f and K have the preassigned values. Similarly in the first case 'furtber examination shows that the distances are inalterable. It is known that there are three solutions for an assigned angular velocity, and thus in general no solution for general values of f and k. In any case the manifold jlf7 can only have a singularity at a point correspondin.fJ to an equilateral triangle solution or to a straight line solution at constant mutual distances. These llOssiMlities will only arise when certain analytic relations between f and K are satisfied. It is only as f and K vary through these critical values that the nature of M 7 from the standpoint of analysis situs can change.
288
DYNAMICAL SYSTEMS
The manifold M; has fundamental importance for the problem of three bodies, but. so far as I know, it has nowhere been studied even with respect to the elementary question of connecti\'ity. The work of Poincare refers to the existence of certain periodic motions, i. e. of certain closed stream lines in jl-J;, obtained by the method of analytic continuation from a limiting integrable case of thp. problem of three bodies; nearby motions, i. e., stream lines in the torus-shaped neighborhood of such a closed stream line, are also considered in relation to the formal series; but he does not consider lrJ7 ill the large. In conclusion it may be observed that the states of motion in which the three bodies move constantly in a plane through the center of gravity perpendicular to the angular momentum vector, correspond to an invariant sub-manifold M 5 within jl{7, which contains the exceptional singularities when these exist. So far as dimensionality is concerned, this manifold .1[5 would be suited to form the complete boundary of a surface of section (chapter V) of properly extended type. II. Types of motion in l;f;. The problem of three bodies is distinguished from the type of non-singular problem which \I'e have considered earlier, in that the manifold of states of motion is not closed. The singularity along the boundary cannot be removed by any exercise of analytic ingenuity. In fact consider a tube of stream lines in M 7 described by a 'molecule' of states of motion near triple collision at t = O. It is clear that the molecule tends toward the boundary of ill; as t increases. since we have then lim R = 00 according to the results decluced above (section 8). The half tube so generated is then carried into part of itself, and would have to correspond to an infinite value of the invariant 7-dimensionaf volume integral. This situation does not arise when the manifold of states of motion is closed and non-singular. More precisely, the stream lines corresponding to motions of near approach to triple collision not only lie wholly near the boundary of 111" and approach it as t increases or decreases indefinitely, but they fill out three entirely distinct regions
289
IL THE PROBLEM OF THREE BODIES
of ]1" since for every such motion there is a particular one of the three bodies which recedes indefinitely from the other two bodies. The stream lines correspondin,q to near approach to triple collision thus fill three distinct 7-dimensional continua of 111" corresponding to the fact that Po, PI' or P t may be the relatit"ely distant body during such a motion. These continua lie near to the boundary of lVI" and evei'y stream line in them approaches tl/(' boundary in either sense of time. Of course these continua are not precisely defined until the degree to which triple collision is approached is precisely specified. It is natural to believe that in this case of indefinite recession, the two nearby bodies have a definite limiting energy constant, orientation of plane of motion, eccentricity, and a limiting linear and angular momentum with reference to the center of gravity of the three bodies. In any case these motions may properly be regarded as to a large extent' known'. The very interesting question now arises: Do the motions for which lim R = 00 in one or both directions of the time fill 1"[, densely or only in part? It is important to understand the nature of the difficulty inherent in this question. By actual computation of the motions, it can doubtlesR be established whether or not a specific motion belongs to one of these continua or not. Certainly, for I K [ small, almost all of J.1£7 would be filled by these continua in consequence of the results obtained in the case K < O. Nevertheless when there exists a single periodic motion in M 7 of stable type, it will not be possible to determine whether or: not nearby motions belong to these continua without solving the fundamental problem of stability in this particular case. We have already alluded to the highly difficult character of the problem of stability (chapter VIII), which arises precisely because in a dynamical problem such as the problem of three bodies, formal stability of the first order insures the satisfaction of all the infinitely many 'further more delicate conditions for complete formal stability. 19
290
DYNAMICAL SYSTEMS
The question can, however, be put in a very suggestive form, which in my opinion renders it probable that the motions for which lim R = 00 for lim t = 00 fill up M densely, as do those for which lim R = 00 for lim t = - 00; because of the reversibility of the system of differential equations, both conjectures must be either true of false. The manifold M 7 has already been conceived of as a 7-dimensional fluid in steady motion. This fluid must be thought of as having infinite extent and as incompressible, in consequence of the existence of a 7-dimensional volume invariant integral. The three types of motion with near approach to triple collision correspond to three streams which enter M 7 from the infinite region and leave it there. What is likely to happen to an arbitrary point of the fluid? It seems to me probable that in general such a point will move about until it is caught up by one of these streams and carried away. It may, however, be anticipated that there will be found certain points which remain at rest or move in closed stream lines, and so are not carried off. In conformity with the results of chapter VII, there must then necessarily· exist other stream lines which remain near to the closed stream line as time increases or as time decreases. More generally, there will exist recurrent types of stream lines corresponding to recurrent motions, and various other stream lines which remain in their vicinity as time increases or decreases. The stream lines corresponding to such recurrent motions and nearby motions cannot of course approach the boundary of M 7 • For the determination of the distribution of such periodic motions, recurrent motions, and motions in their vicinity, it obvious that elaborate detailed analysis would be necessary. In conclusion we shall merely effect an obvious classification based on the function R(t): An arbitrary motion in the problem of three bodies for the case f> 0, K> 0 is of one of the following types as t increases: (1) Rincrcases toward 00, in which case one body 1"er;edes indefinitely from the othe1' two, while the near pair remain within finite distance of one another;
+
+
IX. THE PROBLEM OF THREE BODIES
291
(2) R tends toward a value R while U approaches 2 K, in which case the limiting motion is of special determinable type as in La,qrange's equilateral triangle solution; (3) R(t) is uniformly_bounded as in case (2) but oscillatory. Here the motion is wholly one offinite distances and velocities except possibly for occasional do-uble collisions 01' appj'oach to such collisions, and there nece-ssarily exist periodic or other recurrent motions among the limit motions; (4) R(t) is oscillatory with upper bound + 00 and a positive lower bound. This is an intermediate case in which the motion is one with finite velocities except near occasional do'uble collision or approach to double but not triple collision, while from time to time one of the three bodies recedes arbitrarily far from the near pair only to approach them again later. Similar results obviously hold as t decreases. The only part of this statement calling for any explanation is that if R approaches ii, U approaches 2 K. But this can be proved to follow from Lagrange's equality (15).
Extension to n> 3 bodies and more general laws of force. In indicating the possibility of generalizing 12.
the above results, both in respect to the number of bodies and the law of force, we shall entirely put to one side the question of collision. It would suffice for our purpose, however, if any kind of continuation after multiple collision were possible in which the constants of linear and angular momentum as well as of energy are the same after as before collision, and if also R', where
may be regarded as continuous at collision; here the masses of PI, .. " P n are ml, .. " mn respectively, while M is the sum of these masses, and rij denotes the distance Pi Pj. Let the function U of forces be any function of the mutual distances rij, of dimensions - 1 in these distances. For a function U of this type, the original form of differential equations, of the 10 integrals, and of Lagrange's equality
292
DYNAMICAL SYSTEMS
(15) and of the inequality (20) due to Sundman will subsist, provided that f denotes the total angular momentum of the system about the center of gravity. Our main reasoning above was essentially based upon this analytical framework. Hence we can state the following result: Let U be any analytic function dependin,r; on the mutual distances between n bodies Pi, (i = 1, .. " n), with coordinates (Xi, Yi, Zi) and masses mi respectively; let U be furthermore homo.qeneous of dimensions - 1 in these distance.s. If the n bodies are sufficiently near to,r;ethel', with assigned positive values of the total angular momentum f and the constant of ener,r;y K, at least two of the mutual distances will become very lar.qe in either sen.se of the time. FUTther consideration shows that the condition of homo,geneity upon U can be lightened to the form of an inequality
wheTe 0 < d < 2, without affecting the argument that at leost two of the mutual distances become very large. In this argument the function H has to be generalized to the form
I have not attempted to ascertain conditions under which at least two of the mutual distances become infinite.
ADDENDUM General Remarks. First we want to refer to the book of A. Wintner [31 which deals with the analytic aspects of celestial mechanics and contains a large number of references to old and new literature. Secondly Siegel's book [2] contains many topics related to Birkhoff's book and is a very valuable source of information. Finally the second part of Nemytskii's and Stepanov's book [ 1J deals with the abstract aspects of dynamical systems and has much contact with Chapter 7 of Birkhoff's book. In [ 1] the reader will find a large number of references about the more recent developments in this area. Chapter III: In this chapter the formal aspects of trigonometrical expansions of solutions is discussed. The "Hamiltonian multipliers" are of basic importance and it is shown (see p: 78) that at an equilibrium of a Hamiltonian system these multipliers occur in pairs of Ai, - Ai (which was also proven by Liapounov). A similar statement holds near a periodic solution for the so-called Floquet exponents (see Chapter III, Section 9). It is remarkable that another restriction on the Floquet exponents was overlooked and only discovered by M. G. Krein in 1950 (see [22], [23 J), namely that for the pairs Ai, - Aj on the circle I AI = 1 there is an ordering which is invariant under canonical transformations. In other words, if Ai = exp ( - 1) l,l Wj
one can associate a sign with the frequencies Wi. This fact is of importance for the stability theory of periodic solutions (see Gelfand and Lidskii [ 12] for the linear theory and Moser [29 J for the nonlinear theory). This phenomenon is related to the different behavior between "difference and sum resonances". Chapter V: Contains a discussion of Birkhoff's minimax method and an "extension of Morse" (see Section 8). This area expanded to a vast theory, the now well-known Morse theory 293
294
ADDENDUM
for which we refer to Morse's book [25]. This theory, which started with Poincare's study of closed orbits for the three body problem has now taken many new directions and proved so successful in topology (see, for example, Milnor's book [24]). We mention some further developments of the geodesics problem. After Morse's and Lusternik and SchnireIman's study of this problem there appeared recently a long paper by Alber [ 4] estimating the minimal number of closed geodesics on an ndimensional sphere which contains further references (see also Klingenberg [19]). However. it should be mentioned also that the general Morse theory has not yet been successfully applied to the problems of dynamics. Even for the restricted three body problem such an application would be of great interest. Chapter VI: Th"is chapter con tains a discussion of the celebrated "Poincare's geometric theorem", the proof of which was Birkhoff's first work in this subject (1915). This beautiful theorem withstood all attempts of generalizations and still it is not clear whether it has an analogue in higher dimensions, for say, canonical transformations. We mention a new application of this theorem to the restricted three body problem. In [Ill Conley established the existence of infinitely many periodic solutions around the small mass point (lunar orbits). This is a nontrivial extension of· Birkhoff's study of 1915 mentioned in the footnote on p. 177.. Chapter VII: The subject of this chapter has become a basis of a very abstract formulation in the book by Gottschalk and Hedlund [ 14]. Another source of references related to Chapter 7 is the second part of Nemytskii's and Stepanov's book on Differential Equations [1]. We make special mention of a paper by S. Schwartzman [ 33] in which the concept of rotation numbers is generalized to flows on a compact manifold. Chapter VIII: In the problems discussed in Chapter VIII many advances have been made. A number of questions have been settled and others have expanded into theories of their own. The example of Section 11 illustrates a transitive flow. The study of the geodesic flow on a manifold of negative curvature has been studied thoroughly in ergodic theory and we refer to Hopfs book [16], his fundamental papers [17] and to Hedlund's paper [ 15]. Recently Anosov [5] generalized these ideas considerably and studied a class of differential equations (so-called V-systems) for which he proves transitivity. For recent surveys in this direction see Sinai [37], [38].
ADDENDUM
295
The question of stability raised in Section 7 has been answered and it is known that every fixed point of general stable type (in the tenninology of this book) is stable in the sense of Liapounov. This assertion is contained in the work of Arnol'd [6], [7], [8] and [9], Kolmogorov [20], [21] and Moser [31]. The problem is intimately connected with the difficulty of the small divisors. The first definitive results concerning such small divisor problems were found by C. L. Siegel [34], [35] but his approach did not cover the case in question here. In 1954 Kolmogorov suggested an approach which ultimately led to the stability proof of periodic solutions of general stable type for systems of two degrees of freedom. However, the results of Arnol'd reach much further covering Hamiltonian systems of several degrees of freedom, although in this case stability in the sense of Liapounov cannot be inferred. In fact, Arnol'd [9a] proves instability for a system of3 degrees of freedom and one can say that the concept of stability for Hamiltonian systems has been clarified to a large extent. In [7], [9] Arnol'd gives a most remarkable application of these results to the nobody problem. Chapter IX: Concerning Sundman's results we mention the clear and complete exposition in Siegel's book (2]. Also Wint· ner's book on celestial mechanics [3] contains a wealth of information on the nobody problem.
FOOTNOTES 1. Page 78; Line 9 after "quantities."
This statement is certainly incorrect as it stands. It can occur that A, >;, - A, - X are four distinct numbers as in the example H = /l(Plql + pzqz) + I'(PlqZ - pzqt). Incidentally, an equilibrium of this type occurs for the equilateral solution (of Lagrange) of the restricted three body problem, at least for appropriate mass ratios (see Wintner [3, §476]). 2. Page 86; Third line from below, after "pure imaginary." The remark of Footnote 1, p. 78, applies here too. 3. Page 91; Line 8, after "quantities." See Footnote 1. 4. Page 99; Line 13, after "solution." This solution will be periodic if the period T(c) of the family of reference solutions is independent of c. Otherwise the solution in question involves a term linear in t, but still contributes a second multiplier zero. 5. Page 116; Line 13, after "definition." The investigation of complete stability has been carried further by J. G limm [13]. He considered an equilibrium (or a periodic solution) also in the case where the Al are rationally dependent. He replaced the power series expansion by expansions in terms of rational functions. 6. Page 165; At the end of Section 4, after "period." This question was pursued further by G. D. Birkhoff himself in "Une generalisation a n-dimensions due dernier theoreme de geometrie de Poincare," Compt. Rend des Sciences de l'Acad. d. S. 192, p. 196, 1931. 7. Page 211; Line 8 from below, after "or 21. - v'=T" This case distinction refers to Floquet theory: if the eigen296
FOOTNOTES
297
values of linearized mapping over one period are real and denoted by e±M, then for e 1rX > 0 one can choose >. real and for e 2d < 0 one can choose 2>. - ( - 1) 1/2 real. 8. Page 211; At the end of formula (3), line 4 from below, after ± 1)." In this case one can actually take 4> = 'It = 0 as was proven in Moser [28]. The transformation into the normal form is indeed convergent. This point was left open in Birkhoff's paper of 1920 (which is cited on p. 211). "(Jl ¢
9. Page 213; Line 14, after "motion)." The contents of this parenthesis apparently refers to degenerate cases, illustrated by a transformation u\ = Uo + VO, VI = Vo where Vo = 0 represents a family of fixed points. 10. Page 214; Line 7 after "from the origin." This remark has to be qualified. If (1/1f is rational one can easily produce unstable examples and for integral 3(1/211" Instability is the generic case, see for example [26]. 11. Page 215; Line 15 after "multiples are simple with I ;c 0." This means that the number (1 appearing in equation (2) of p. 211 is assumed to be incommensurable with 1r. The number l was defined on p. 211 as I = ( - 1)1/2 s /211". For the definition of "simple" and "multiple" see p. 142 bottom. 12. Page 217; The sentence starting on line 5 is misleading and should be replaced by: In this case the corresponding normal form is of type (2) (see p. 211). In the unstable case the normal form is of the form (3) where Jl is positive or negative but not ± 1. 13. Page 218; Line 4, replace period after "negative" by a comma. Replace lines 5 and 6 by: as was mentioned on p. 215 bottom. Consequently a real negative root Jl is not possible, and the case (3) can occur only with Jl > 0, i.e. I is a fixed point of stable type. 14. Page 222; Line 3 from bottom, after "is bounded." The statements of this section are not sufficiently proven and it seems impossible to· supply the necessary arguments. It is quite conceivable that such an invariant "curve" is very pathological making the geometrical considerations inadequate.
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r
In fact, N. Levinson 181 constructed a second order differential equation where such a pathological invariant set occurs. In Levinson's example the rotation number (which on an invariant curve should be constant) takes on various values on the invariant set which is of measure zero. But it has to be mentioned that Levinson's differential equation is not conservative and can only be considered as an illustration, not as a counterexample. 15. Page 227; Line 12 from below after "such actual stability?" This question has been answered in the affirmative as was mentioned in my General Remarks to Chapter VIII, (see [9], [31 j). 16. Page 227; Last line after "variable periods." According to the previous footnote a periodic motion of general stable type is stable, which makes the present assumption as well as Section 8 vacuous! 17. Page 237; After first paragraph, i.e. after "stable type. " Recent work by Smale [39 J extends these results, to which Birkhoff alludes, considerably. Smale finds infinitely many periodic motions, and even a perfect minimal Cantor set near a "homoclinic" motion, even for several dimensional systems. Unfortunately, his results are not applicable to Hamiltonian' systems of more. than two degrees of freedom, due to some assumption which fails for Hamiltonian systems. 18. Page 238; Line 3 after title of Section: A very interesting example of this type had been discussed already in 1924 by E. Artin [to I (following a suggestion of Herglotz). It also deals with the geodesic flow on a manifold of two dimensions (the modular region in the upper half plane) and a symbolism for these geodesics is put into correspondence with the continued fraction expansions. 19. Page 245; Line 5 after "of motions." Extensions of such results are contained in the recent work by Anosov [5 J. He considers systems of differential equations (so-called V-systems) in several dimensions whose solutions have a similar behavior as the geodesic flow on a manifold with negative curvature.
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20. Page 257; Line 12 after "for m = 1." The following argument leaves a number of points unclear. Careful proofs and sharper results have been given by Siegel [36], Riissmann [32] and Moser [27]. The paper [30] contains an explicit class of nonintegrable polynomial transformations. 21. Page 259; Line 13 after "analytic families." These families lie on the level surface of the Hamiltonian and of the integral I which was assumed to exist. The family could be parametrized by the canonically conjugate variable of I. For this purpose one would have to introduce new variables, say U b UZ, VIJ V2 by a canonical transformation such that UI = I, say, which can be done. Then VI would be a family parameter.
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