America n Mathematica l Societ y

Colloquiu m Publication s Volum e 9

Dynamica l System s

Georg e D. Birkhof f

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America n Mathematica l Societ y Providence , Rhod e Islan d

http://dx.doi.org/10.1090/coll/009

2000 Mathematics Subject Classification. P r i m a r y 34 -XX .

Library o f Congres s Cataloging- in-Publ icat io n D a t a

Birkhoff, Georg e David , 1884-1944 . Dynamical systems . New York , America n mathematica l society , 1927 . 1. Dynamics . I . Title .

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INTRODUCTION T O T H E 196 6 EDITIO N

Many mathematician s wil l welcom e th e ne w editio n o f G. D . BirkhofF s boo k o n Dynamica l Systems . I t represent s essentially a continuation of Poincare's profound an d extensiv e work on Celestial Mechanics . Altogether Birkhof f wa s strongly influenced b y Poincare and devote d a major par t o f his mathe-matical work to subjects arisin g fro m Poincare' s tradition . Th e present boo k contain s BirkhofF s view s and idea s o f hi s earlie r period o f life—it appeare d whe n Birkhof f wa s 43.

To the modern reade r th e styl e o f this book ma y appea r les s formal an d rigorou s tha n i t i s no w customary . Bu t jus t th e informal an d livel y manne r o f writin g ha s bee n inspirin g t o many mathematicians. The effect o f this inspiration i s visible in a number of later papers. For example, Morse's theory on geodes-ies on a closed manifol d originate d directl y i n BirkhofF s idea s in dynamical systems . The recen t wor k b y Anoso v o n [/-Sys -tems answers the question of ergodicity an d densit y o f periodi c solutions for a wid e clas s o f differentia l equations— a proble m which in BirkhofFs boo k was studied fo r a single model system . These and othe r example s (give n below ) justif y th e hop e tha t the reprinting of this book again will stimulate further progress .

Of course, after nearl y 4 0 years a number o f statement s ar e outdated. Fo r thi s reaso n I hav e selecte d a lis t o f reference s to pertinen t literatur e afte r 1927 . Naturally , th e lis t canno t be complete an d furthe r reference s ca n b e found i n th e book s and surve y article s quoted . I n th e Addendu m sectio n a t th e back o f thi s boo k I adde d som e genera l remark s t o variou s chapters. Mor e specific comment s ar e supplied a s footnote s a t the end of the book . References t o these footnote s ar e given a s small number s i n th e margi n o f th e text . * T x .

JURGEN MOSER *I am indebte d t o Dr . R . Sacke r fo r hi s assistance .

in

A PREFAC E T O TH E 196 6 EDITIO N

I met Georg e Birkhof f i n 1914 , th e yea r afte r h e publishe d his "Proof of Poincare's Geometric Theorem " [ i j. I n a pape r [2] in 1912 Poincare had enunciated a theorem of great impor-tance for the restricted proble m o f thre e bodies , bu t ha d suc -ceeded i n treatin g onl y a variet y o f specia l case s afte r lon g efforts. Poincar e ha d als o referre d t o thi s theore m i n lecture s in Gottingen . Birkhofff f formulate d thi s theore m i n [l ] a s fol -lows.

"Let us suppose that a continuous one-to-one transformatio n T take s th e rin g R forme d b y concentri c circle s C a and C b of radii a and 6 respectively (a > b > 0 ) int o itsel f i n suc h a way as t o advanc e th e point s o f C a i n a positiv e sense , an d th e points of C b in a negative sense and at th e same tim e preserv e areas. Then ther e are at least two invariant points.* 1

BirkhofFs proof of this theorem in 191 3 was one of the most exciting mathematica l event s o f th e er a an d wa s widel y ac -claimed.

In 1912 i n [3 ] Birkhoff outline d th e conjectur e o f Poincar e that the "general" motio n i n dynamic s i n phas e spac e wa s of the so-called "discontinuou s type" . I t wa s i n thi s pape r tha t Birkhoff introduce d hi s nove l an d beautifu l conception s o f "minimal" or equivalently "recurrent " sets of motions. A min-imal non-periodic se t o f motion s i s o f "discontinuous " typ e i f in phase spac e i t possesse s n o subset s whic h ar e continu a ex -cept arcs of motions. Birkhoff aske d me in 191 5 to examine the conjecture of Poincare. M y answe r was by way o f a "symbolic dynamics". A non-periodic recurrent symbol introduced a t tha t time was discovere d independentl y b y a Russian i n 193 4 an d used by Novikov in his disproof of the Frobenius-Burnside con-jecture in group theory.

The problem of the generality o f "non-degenerate " periodi c orbits, that is orbits whose equations of variation admit no non-

iv

PREFACE T O THE 196 6 EDITIO N V

trivial periodi c solution s wa s ofte n emphasize d b y Birkhoff . I t is no w o f majo r interes t t o differentia l topologist s o f who m Birkhoff an d Poincar e wer e amon g th e first , althoug h no t s o called. Th e wor k o f Moser , Arnol' d an d other s o n stabilit y belongs to a related field, on e clos e t o th e cente r o f BirkhofF s interest.

The abov e merel y sample s th e man y aspect s o f dynamic s i n which Birkhof f introduce d ne w ideas , ne w theorem s an d ne w questions. On e o f BirkhofF s theorem s whic h ha s arouse d th e greatest interes t wa s hi s so-calle d "Ergodi c Theorem " wit h it s subsequent variations , interpretation s an d consequence s i n measure theor y an d probability .

Birkhoff onc e remarke d tha t "i t i s fortunat e tha t th e worl d of mathematic s i s a s larg e a s i t is" . Onl y b y bringin g t o bea r the genius an d imaginatio n o f man y land s an d intellectua l ori -gins ca n on e b e sur e o f adequat e appreciatio n o f ne w mathe -matical idea s an d significan t responses .

History ha s responded t o these page s o n Dynamica l System s in a n unmistakabl e way . Fo r thi s w e ar e mor e tha n content .

REFERENCES

1. Birkhoff, George , Proof of Poincart's geometric theorem, Trans. Amer . Math . Soc. 14(1913) , 14-22 .

2. Poincare , H. , Sur un thtoreme de gtometrie, Rend , de l Cir . Mat . d i Palermo 33(1912) , 375-407 .

3. Birkhoff , George , Quelques th&oremes sur le mouvement des systimes dyna-miques. Bull . Soc . Math . Franc e 40(1912) , 305-323 .

MARSTON MORS E

PREFACE T O TH E 192 7 EDITIO N

The Colloquiu m Lecture s whic h I ha d th e privileg e o f delivering a t th e Universit y o f Chicag o befor e th e America n Mathematical Society, September 5-8, 1920 , containe d a larg e part o f th e materia l presente d i n th e followin g pages . Th e delay i n publicatio n ha s bee n du e t o severa l causes , on e o f which has been my desir e to wait unti l som e o f m y ow n idea s had develope d further . I hav e take n advantag e o f a well -established traditio n o f ou r Colloqui a b y givin g particula r emphasis to my own researches on dynamical systems . I t i s my earnest hope that the lectures may serve to stimulate other s to investigate th e outstandin g problem s i n thi s mos t fascinatin g field.

It i s only necessar y t o recal l th e wor k o f Galileo , Newton , Laplace, Clausius , Rayleig h i n th e physica l application s o f dynamics, of Lagrange , W . R . Hamilton , Jacob i i n it s forma l development, and of Hill and Poincar6 in the qualitative treat -ment o f dynamica l questions , i n orde r t o realiz e th e remark -able significance o f dynamics in the past for scientific thought . At a tim e whe n n o physica l theor y ca n properl y b e terme d fundamental—the known theories appear to be merely more or less fundamental i n certain directions—it ma y b e asserted wit h confidence tha t ordinar y differentia l equation s i n th e rea l domain, an d particularl y equation s o f dynamica l origin , wil l continue to hold a position o f th e highes t importance .

In lookin g bac k ove r m y ow n dynamica l work , o f whic h a certain perio d i s finished wit h th e publicatio n o f thi s book , I canno t bu t expres s m y feelin g o f dee p admiratio n an d gratitude to Hadamard, Levi-Civita, Sundman and Whittaker, to whom man y importan t recen t advance s i n theoretica l dy -namics ar e due , an d i n whos e wor k I hav e foun d especia l inspiration. I t i s with much regret that I hav e bee n unabl e t o give adequat e spac e t o thei r achievements .

vi

PREFACE T O THE 192 7 EDITIO N vi i

Professor Phili p Frankli n cooperate d wit h m e i n a first re -writing of part of my notes on these lectures . I owe him cordial thanks for his help. November 18,1927 . GEORG E D . BlRKHOFF .

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TABLE O F CONTENT S

INTRODUCTION TO THE 1966 EDITIO N m

PREFACE TO THE 1966 EDITIO N i v

PREFACE TO THE 1927 EDITIO N v i

CHAPTER I PHYSICAL ASPECTS OF DYNAMICAL SYSTEMS

PAGE

1. Introductor y remarks 1 2. A n existence theorem 1 3. A uniqueness theorem 5 4. Tw o continuity theorems 6 5. Som e extensions . , 1 0 6. Th e principle of the conservation of energy. Con-

servation systems 1 4 7. Chang e of variables in conservative systems 1 9 8. Geometrica l constraints 2 2 9. Interna l characterization of Lagrangian systems 2 3

10. Externa l characterization of Lagrangian systems 2 5 11. Dissipativ e systems 3 1

CHAPTER I I

VARIATIONAL PRINCIPLES AND APPLICATIONS 1. A n algebraic variational principle 3 3 2. Hamilton' s principle 3 4 3. Th e principle of least action 3 6 4. Norma l form (two degrees of freedom) 3 9 5. Ignorabl e coodinates 4 0 6. Th e method of multipliers 4 1 7. Th e general integral linear in the velocities 4 4 8. Conditiona l integrals linear in the velocities 4 5

ix

X CONTENTS

9. Integral s quadratic in the velocities 4 8 10. Th e Hamiltonian equations 5 0 11. Transformatio n of the Hamiltonian equations 5 3 12. Th e Pfaffian equations 5 5 13. O n the significance of variational principles 5 5

CHAPTER II I

FORMAL ASPECTS O F DYNAMIC S

PAGB

1. Introductor y remarks 5 9 2. Th e formal group 6 0 3. Forma l solutions 6 3 4. Th e equilibrium problem 6 7 5. Th e generalized equilibrium problem 7 1 6. O n the Hamiltonian multipliers 7 4 7. Normalizatio n of H2 7 8 8. Th e Hamiltonian equilibrium problem 8 2 9. Generalizatio n of the Hamiltonian problem 8 5

10. O n the Pfaffian multipliers 8 9 11. Preliminar y normalization in Pfaffian problem 9 1 12. Th e Pfaffian equilibrium problem 9 3 13. Generalizatio n of the Pfaffian problem 9 4

CHAPTER I V

STABILITY O F PERIODIC MOTION S

1. O n the reduction to generalized equilibrium 9 7 2. Stabilit y of Pfaffian systems 10 0 3. Instabilit y of Pfaffian systems 10 5 4. Complet e stability 10 5 5. Norma l form for completely stable systems 10 9 6. Proo f of the lemma of section 5 11 4 7. Reversibilit y and complete stability 11 5 8. Othe r types of stability 12 1

CHAPTER V

EXISTENCE O F PERIODIC MOTION S

1. Rol e of the periodic motions 12 3 2. A n example 12 4

CONTENTS XI

3. Th e minimum method 12 8 4. Applicatio n to symmetric case 13 0 5. Whittaker' s criterion and analogous results 13 2 6. Th e minima* method 13 3 7. Applicatio n to exceptional case 13 5 8. Th e extensions by Morse 13 9 9. Th e method of analytic continuation 13 9

10. Th e transformation method of Poincar6 14 3 11. A n example 14 6

CHAPTER V I

APPLICATION O F POINCARE'S GEOMETRI C THEOREM

1. Periodi c motions near generalized equilibrium (m = 1 ) 150 2. Proo f of the lemma of section 1 15 4 3. Periodi c motions near a periodic motion (m = 2 ) 15 9 4. Som e remarks 16 2 5. Th e geometric theorem of Poincare 16 5 6. Th e billiard ball problem 16 9 7. Th e corresponding transformation T 17 1 8. Area-preservin g property of T 17 3 9. Application s to billiard ball problem 17 6

10. Th e geodesic problem. Construction of a transform-ation TT* 18 0

11. Applicatio n of Poincare's theorem to geodesic problem 18 5

CHAPTER VI I

GENERAL THEOR Y O F DYNAMICAL SYSTEM S

1. Introductor y remarks 18 9 2. Wanderin g and non-wandering motions 19 0 3. Th e sequence M, Mu M 2, 19 3 4. Som e properties of the central motions 19 5 5. Concernin g the role of the central motions 19 7 6. Group s of motions 19 7 7. Recurren t motions 19 8 8. Arbitrar y motions and the recurrent motions 20 0 9. Densit y of the special central motions 20 2

10. Recurren t motions and semi-asymptotic central motions 20 4

11. Transitivit y and intransitivity 20 5

Xii CONTENT S

CHAPTER VII I

THE CAS E OF TWO DEGREES OF FREEDOM

PAGE

1. Forma l classification of invariant points 20 9 2. Distributio n of periodic motions of stable type 21 5 3. Distributio n of quasi-periodic motions 21 8 4. Stabilit y and instability 22 0 5. Th e stable case. Zones of instability 22 1 6. A criterion for stability 22 6 7. Th e problem of stability 22 7 8. Th e unstable case. Asymptotic families 22 7 9. Distributio n of motions asymptotic to periodic

motions 23 1 10. O n other types of motion 23 7 11. A transitive dynamical problem 23 8 12. A n integrable case 24 8 13. Th e concept of integrability 25 5

CHAPTER I X

THE PROBLEM OF THREE BODIES

1. Introductor y remarks 26 0 2. Th e equations of motion and the classical integrals 26 1 3. Reductio n to the 12th order 26 3 4. Lagrange' s equality 26 4 5. Sundman' s inequality 26 5 6. Th e possibility of collision 26 7 7. Indefinit e continuation of the motions 27 0 8. Furthe r properties of the motions 27 5 9. OnaresultofSundma n 28 3

10. Th e reduced manifold M7 of states of motion 28 3 11. Type s of motion in Af 7 28 8 12. Extensio n to n > 3 bodies and more general laws

offeree 29 1

ADDENDUM 29 3

FOOTNOTES 29 6

BIBLIOGRAPHY 30 0

INDEX 30 3

A D D E N D U M

General Remarks. Firs t w e wan t t o refe r t o th e boo k o f A . Wintner [3 ] whic h deal s wit h th e analyti c aspect s o f celestia l mechanics and contains a large number o f reference s t o ol d an d new literature . Secondl y Siegel' s boo k [2 ] contain s man y topics relate d t o BirkhofT s boo k an d i s a ver y valuabl e sourc e of information . Finall y th e secon d par t o f Nemytskii' s an d Stepanov's book [ l] deals with th e abstrac t aspect s o f dynami -cal systems and has much contac t wit h Chapte r 7 o f BirkhofT s book. I n [ l] th e reade r wil l find a larg e numbe r o f reference s about th e mor e recen t development s i n thi s area .

Chapter III : I n thi s chapte r th e forma l aspect s o f trigono -metrical expansions of solutions is discussed. The "Hamiltonia n multipliers" are of basic importanc e an d i t i s shown (se e p . 78 ) that a t a n equilibriu m o f a Hamiltonia n syste m thes e multi -pliers occu r i n pair s o f A „ - A , (whic h wa s als o prove n b y Liapounov). A similar statemen t hold s nea r a periodic solutio n for th e so-calle d Floque t exponent s (se e Chapte r III , Sectio n 9). I t i s remarkabl e tha t anothe r restrictio n o n th e Floque t exponents was overlooked and onl y discovere d b y M . G . Krei n in 195 0 (se e [22] , [23]) , namel y tha t fo r th e pair s X it — A , on the circle |A | = 1 there i s a n orderin g whic h i s invariant unde r canonical transformations . I n othe r words , i f

A, = exp ( — l) iau>i

one ca n associat e a sig n wit h th e frequencie s <*>, . This fac t i s of importanc e fo r th e stabilit y theor y o f periodi c solution s (see Gelfand an d Lidski i [ 12} for th e linea r theor y an d Mose r [2S] for th e nonlinea r theory) . Thi s phenomeno n i s relate d t o the differen t behavio r betwee n "differenc e an d su m reso -nances".

Chapter V : Contain s a discussio n o f BirkhofT s minima x method and an "extension o f Morse " (se e Sectio n 8) . Thi s are a expanded t o a vas t theory , th e no w well-know n Mors e theor y

293

294 ADDENDUM

for whic h w e refe r t o Morse' s boo k [25] . Thi s theory , whic h started wit h Poincare' s stud y o f close d orbit s fo r th e thre e body problem ha s now take n man y ne w direction s an d prove d s o successful i n topolog y (see , fo r example , Milnor' s boo k [ 24]). We mentio n som e furthe r development s o f th e geodesie s prob -lem. After Morse' s an d Lusterni k an d Schnirelman' s stud y o f this proble m ther e appeare d recentl y a lon g pape r b y Albe r [4] estimating th e minimal numbe r o f close d geodesie s o n a n n-dimensional spher e whic h contain s furthe r reference s (se e als o Klingenberg [ 19]). However , i t shoul d b e mentione d als o tha t the general Mors e theor y ha s no t ye t bee n successfull y applie d to the problems of dynamics. Eve n fo r the restricted thre e bod y problem such an applicatio n woul d b e o f grea t interest .

Chapter VI : Thi s chapte r contain s a discussio n o f th e cele -brated "Poincare' s geometric theorem" , th e proo f o f whic h wa s BirkhofFs first work in this subject (1915) . Thi s beautifu l theo -rem withstood al l attempt s o f generalization s an d stil l i t i s no t clear whethe r i t ha s a n analogu e i n highe r dimensions , fo r say , canonical transformations .

We mentio n a ne w applicatio n o f thi s theore m t o th e re -stricted thre e bod y problem . I n [ l l ] Conle y establishe d th e existence o f infinitely man y periodi c solution s aroun d th e smal l mass poin t (luna r orbits) . Thi s i s a nontrivia l extensio n o f BirkhofFs stud y o f 191 5 mentione d i n th e footnot e o n p . 177 .

Chapter VII : Th e subjec t o f thi s chapte r ha s becom e a basi s of a ver y abstrac t formulatio n i n th e boo k b y Gottschal k an d Hedlund [14] . Anothe r sourc e o f reference s relate d t o Chap -ter 7 i s th e secon d par t o f Nemytskii' s an d Stepanov' s boo k on Differentia l Equation s [ l ]. W e mak e specia l mentio n o f a paper by S. Schwartzman [33 ] in whic h th e concep t o f rotatio n numbers i s generalized t o flows o n a compac t manifold .

Chapter VIII : I n th e problem s discusse d i n Chapte r VII I many advance s hav e bee n made . A numbe r o f question s hav e been settle d an d other s hav e expande d int o theorie s o f thei r own. Th e exampl e o f Sectio n 1 1 illustrate s a transitiv e flow. The study o f the geodesic flow on a manifold o f negativ e curva -ture ha s bee n studie d thoroughl y i n ergodi c theor y an d w e refer to Hopf s boo k [16] , hi s fundamenta l paper s [ 17] an d t o Hedlund's pape r [15] . Recentl y Anoso v [5 ] generalize d thes e ideas considerably an d studie d a clas s o f differentia l equation s (so-called {/-systems ) fo r whic h h e prove s transitivity . Fo r recent survey s i n thi s directio n se e Sina i [37] , [38] .

ADDENDUM 295

The questio n o f stabilit y raise d i n Sectio n 7 ha s bee n an -swered and i t i s known tha t ever y fixed point o f genera l stabl e type (i n th e terminolog y o f this book ) i s stable i n th e sens e o f Liapounov. This assertion i s contained i n th e wor k o f Arnol' d [6], [7] , [8 ] and [9] , Kolmogorov [20] , [21] and Mose r [31] . The problem i s intimately connecte d wit h th e difficult y o f th e small divisors. The first definitive result s concerning suc h smal l divisor problem s were foun d b y C . L. Siegel [34] , [35 ] bu t hi s approach di d no t cove r th e cas e i n questio n here . I n 195 4 Kolmogorov suggested an approach which ultimately le d t o th e stability proo f o f periodi c solution s o f genera l stabl e typ e fo r systems o f tw o degree s o f freedom . However , th e result s o f Arnol'd reac h muc h furthe r coverin g Hamiltonia n system s o f several degrees of freedom, althoug h in this case stability i n th e sense o f Liapouno v canno t b e inferred . I n fact , Arnol' d [9a ] proves instability for a system of 3 degrees of freedom and one can say tha t th e concep t o f stabilit y fo r Hamiltonia n system s ha s been clarified t o a large extent . I n [7] , [9] Arnol'd give s a mos t remarkable applicatio n o f these result s t o th e n-body problem .

Chapter IX: Concernin g Sundman' s result s w e mentio n th e clear and complet e expositio n i n Siegel's book [2] . Als o Wint -ner's book on celestial mechanics [ 3] contains a wealth o f infor -mation o n the n-body problem .

FOOTNOTES

1. Pag e 78 ; Lin e 9 afte r "quantities. " This statemen t i s certainl y incorrec t a s i t stands . I t ca n

occur tha t A , X, — X , — X ar e fou r distinc t number s a s i n th e example H = viPiQi + P2Q2) + v(PiQ2 — P2Q1)' Incidentally , a n equilibrium o f thi s type occur s fo r th e equilatera l solutio n (o f Lagrange) o f th e restricte d thre e bod y problem , a t leas t fo r appropriate mas s ratio s (se e Wintne r [3 , §476]) .

2. Page 86 ; Third lin e fro m below , afte r "pur e imaginary/ ' The remark of Footnote 1 , p. 78, applies her e too .

3. Pag e 91 ; Lin e 8 , afte r "quantities. " See Footnote 1 .

4. Pag e 99 ; Lin e 13 , after "solution. " This solution will be periodic i f the perio d T(c) o f the famil y

of reference solution s i s independen t o f c . Otherwis e th e solu -tion in question involves a term linea r i n t t bu t stil l contribute s a secon d multiplie r zero .

5. Pag e 116 ; Lin e 13 , after "definition. " The investigation o f complete stabilit y ha s bee n carrie d fur -

ther b y J . Glim m f 13]. H e considere d a n equilibriu m (o r a periodic solution ) als o i n th e cas e wher e th e X , ar e rationall y dependent. H e replace d th e powe r serie s expansio n b y expan -sions i n term s o f rationa l functions .

6. Page 165 ; At the en d o f Section 4 , afte r "period. " This questio n wa s pursue d furthe r b y G. D . Birkhof f him -

self i n "Un e generalisatio n a ^-dimension s du e dernie r theo -reme de geometri e d e Poincare, " Compt . Ren d de s Science s d e FAcad. d. S . 192 , p . 196 , 1931.

7. Page 211 ; Line 8 fro m below , afte r "o r ? A - % / - 1. " This cas e distinctio n refer s t o Floque t theory : i f th e eigen -

296

FOOTNOTES 297

values o f linearize d mappin g ove r on e perio d ar e rea l an d denoted b y e ±2*\ the n fo r e 2'x > 0 on e ca n choos e \ rea l an d for e2'x < 0 one ca n choos e 2 X - ( - l ) l / 2 real .

8. Page 211; A t th e end o f formul a (3) , lin e 4 from below , afte r "(/* * ± D. "

In this case one ca n actuall y tak e $ = * = 0 a s was prove n in Mose r [28] . Th e transformatio n int o th e norma l for m i s indeed convergent- This point was left ope n in BirkhofFs paper of 1920 (which is cited o n p . 211) .

9. Page 213 ; Line 14 , afte r "motion). " The contents of this parenthesis apparently refer s t o degen -

erate cases, illustrated b y a transformation u x = u Q + v 0, vt = v0

where v0 = 0 represents a family o f fixed points.

10. Pag e 214 ; Lin e 7 afte r "fro m th e origin. " This remark ha s t o b e qualified . I f ajr i s rationa l on e ca n

easily produc e unstabl e example s an d fo r integra l 3<T/2 T in -stability i s the generic case, see for example [26].

11. Page 215 ; Line 1 5 after "multiple s ar e simple wit h / ^ 0. " This means that the number a appearing i n equatio n (2 ) o f

p. 211 is assumed to be incommensurable wit h * . The number / was defined o n p . 21 1 a s / = ( — l) l/2s/2w. Fo r th e definitio n of "simple" and "multiple" see p. 14 2 bottom .

12. Page 217 ; The sentence startin g o n line 5 is misleadin g an d should be replaced by:

In thi s cas e th e correspondin g norma l for m i s o f typ e (2 ) (see p . 211) . I n th e unstabl e cas e th e norma l for m i s o f th e form (3) wher e v i s positive o r negative bu t no t rf c 1.

13. Page 218; Line 4, replace period after "negative" by a comma. Replace lines 5 an d 6 by: as wa s mentione d o n p . 21 5 bottom . Consequentl y a rea l negative root /i i s not possible , and the case (3) ca n occur only with M > 0 , i.e . / i s a fixed point o f stable type.

14. Page 222 ; Lin e 3 from bottom , afte r "i s bounded." The statement s o f thi s sectio n ar e no t sufficientl y prove n

and i t seem s impossibl e t o suppl y th e necessar y arguments . It i s quite conceivable that suc h a n invarian t "curve " i s ver y pathological making the geometrical considerations inadequate .

298 FOOTNOTES

In fact , N . Levinso n f 18] constructed a secon d orde r differen -tial equatio n wher e suc h a pathologica l invarian t se t occurs . In Levinson' s exampl e th e rotatio n numbe r (whic h o n a n in -variant curv e shoul d b e constant ) take s o n variou s value s o n the invarian t se t whic h i s o f measur e zero . Bu t i t ha s t o b e mentioned tha t Levinson' s differentia l equatio n i s no t con -servative an d ca n onl y b e considere d a s a n illustration , no t a s a counterexample .

15. Pag e 227 ; Lin e 1 2 fro m belo w afte r "suc h actua l stability? " This questio n ha s bee n answere d i n th e affirmativ e a s wa s

mentioned i n m y Genera l Remark s t o Chapte r VIII , (se e [9] , [31]).

16. Pag e 227 ; Las t lin e afte r "variabl e periods. " According t o th e previou s footnot e a periodi c motio n o f

general stable typ e i s stable , whic h make s th e presen t assump -tion a s wel l a s Sectio n 8 vacuous !

17. Pag e 237 ; Afte r first paragraph , i.e . afte r "stabl e type . " Recent wor k b y Smal e [3 9 J extends thes e results , t o whic h

Birkhoff alludes , considerably . Smal e finds infinitel y man y periodic motions , and eve n a perfec t minima l Canto r se t nea r a "homoclinic" motion , eve n fo r severa l dimensiona l systems . Unfortunately, hi s result s ar e no t applicabl e t o Hamiltonia n systems o f mor e tha n tw o degree s o f freedom , du e t o som e assumption whic h fail s fo r Hamiltonia n systems .

18. Pag e 238 ; Lin e 3 afte r titl e o f Section : A ver y interestin g exampl e o f thi s typ e ha d bee n discusse d

already i n 192 4 b y E . Arti n [ 10] (followin g a suggestio n o f Herglotz). I t als o deal s wit h th e geodesi c flow o n a manifol d of two dimensions (th e modula r regio n i n th e uppe r hal f plane ) and a symbolism fo r thes e geodesie s i s pu t int o correspondenc e with th e continue d fractio n expansions .

19. Pag e 245 ; Lin e 5 afte r "o f motions. " Extensions o f suc h result s ar e containe d i n th e recen t wor k

by Anoso v [5 j . H e consider s system s o f differentia l equation s (so-called {/-systems ) i n severa l dimension s whos e solution s have a simila r behavio r a s th e geodesi c flow o n a manifol d with negativ e curvature .

FOOTNOTES 299

20. Page 257 ; Line 1 2 afte r "fo r m = 1. " The following argumen t leave s a numbe r o f point s unclear .

Careful proof s an d sharpe r result s hav e bee n give n b y Siege l [36], Russmann [32] and Mose r [27] . The pape r [30] contains an explicit clas s o f nonintegrabl e polynomia l transformations .

21. Pag e 259; Line 1 3 after "analyti c families.' ' These families lie on the level surface of the Hamiltonian and

of the integral / whic h was assumed to exist . The family coul d be parametrized by the canonically conjugate variable of / . Fo r this purpos e on e would hav e t o introduc e ne w variables , sa y **i, ui, v u *>2 b y a canonica l transformatio n suc h tha t u x = / , say, which can be done. Then v x would b e a famil y parameter .

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ton Universit y Press , Princeton , N . J. , 1947 . 4. Alber , S . I. , On the Periodic Problem of Global Variational Calculus,

Uspehi Mat . Nauk 12(1957) , no. 4(76) 57-124 , M R 19 , 751 . 5. Anosov , D . V. , Structural Stability of Geodesic Flows on Compact Rieman-

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6. Arnot'd, V. I., "Small divisors" . I , On Mappings of a Circle onto Itself, Izv. Akad. Nau k Ser . Mat . 25(1961), 21-86.

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13. Glimm, J. , Formal Stability of Hamiltonian Systems, Comm . Pur e Appl . Math. 16(1963) , 509-526 .

14. Gottschalk , W . H. , Hedlund , G . A. , Topological Dynamics, Amer . Math . Soc. Colloq . PubL , Vol . 36, Amer. Math . Soc., Providence, R. I. , 1955.

15. Hedlund , G . A. , The Dynamics of Geodesic Fbws, Bull . Amer . Math . Soc. 45(1939) , 241-260 .

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17. , a. Statistik der geoddtischen Linien in Mannigfaltigkeiten negativer Krummung, Verhandlunge n sachs . Akad . Wiss . Leipzi g 91(1939) , 261-304 ; b. Statistik der Lbsungen Geodatischer Probleme vom unstabilen Typus II , Math . Ann. 117(1940) , 590-608 .

18. Levinson , N. , A Second Order Differential Equation with Singular Solu-tions, Ann . o f Math . 50(1949) , 127-153 .

19. Klingenberg , W. , On the Number of Closed Geodesies on a Riemannian Manifold, Bull . Amer . Math . Soc . 70(1964) , 279-282 .

20. Kolmogorov, A . N. , General Theory of Dynamical Systems and Classical Mechanics, Proc . O f Intl . Congres s o f Math. , Amsterdam , 1954 : (Amsterdam : Ervin P . Nordhoff , 1957 ) 1 , pp . 315-333 .

21. , The Conservation of Conditionally Periodic Motions under Small Perturbations of the Hamiltonian, Dokl . Akad . Nau k SSS R 98(1954) , 527 -530.

22. Krein , M . G. , Generalization of Certain Investigations of A. M. Liapunov on Linear Differential Equations with Periodic Coefficients, Dokl . Akad . Nau k SSSR 73(1950) , 445-448 .

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24. Milnor , J. , Morse Theory, Annal s o f Mathematic s Studie s No . 51 , Princeton University , Princeton , N . J. , 1963 .

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32. Russmann, H. , Uber die Existenz einer Normal form inhalstreuer Trans for-mationen, Math . Ann . 137(1959) , 64-77 .

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33. Schwartzman, S. , Asymptotic Cycles, Ann . o f Math. , 66(1957) , 270-284 . 34.Siegel, C . L. Iteration of Analytic Functions, Ann . o f Math . 43(1942) , 607 -

612. 35. , Xjber die Normalform analytischer Differentialgleichungen in der

Ndhe einer Gleichgewichtsldsung, Nachr. Akad . Wiss . Gottingen , Math . Phys . Kl, Il a (1952) , 21-30 .

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37. Sinai, Y. G. t Geodesic Flows on Compact Surfaces of Negative Curvature, DokL Akad . Nau k SSS R 136(1961) , 549-552 .

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39. Smale , S. , Diffeomorphisms with Many Periodic Points, Differential and Combinatorial Topology, A Symposium in Honor o f Marsto n Morse , edite d b y Steward Cairns, Princeton University Press , Princeton, N . J . 1965 .

INDEX

Analyticity o f solutions , 12-1 4 Ascoli, 4 Asymptotic motio n t o recurren t

motions, 205-G ; t o periodi c motions, 211-2 , 227-3 7

Billiard bal l problem , 169-79 ; fo r ellipse, 248-5 5

Bliss, 1 Bohr, 22 0 Bolza, 35 , 3 7 Brouwer, 148 , 23 4 Cantor, G., . 19 4 Central motions, 190-97, 202-4; spe-

cial, 202-4 ; transitivit y an d in -transitivity of , 205- 8

Characteristic surface , 12 9 Chazy, 26 0 Clausius, ii i Closed geodesies , 130 , 135-9 ; o n

symmetric surfaces , 130-2 ; o n open surfaces , 132-4 ; o n convex surfaces, 135-9 ; o n a specia l closed surface , 244- 5

Conservation o f energy , 14-1 9 Conservative systems, 14-19; change

of variables , 19-21 ; subjec t t o constraints, 2 2

Contact transformations , 53-5 5 Continuity theorems , 6-10 , 10-1 2 Coordinates, 14-1 5 Cosserat, E . and F. , 1 4 Degrees o f freedom , 1 4 Dissipative systems , 31-3 2

I Energy , 23-2 5 Equations o f variation , 10 , 57-5 8 Equilibrium problem , 59 , 60, 67-71;

Hamiltonian cas e of , 74-85 ; I Pfaffia n cas e of , 89-9 4

Equivalence, 5 6 I Euler , 3 5

Existence theorem , 1-5 , 10-1 2 External forces , 14-1 5 Formal group , 60-6 3 Formal solutions, 63-67 ; containin g

a parameter , 14 3 Galileo, ii i Generalized equilibrium , 60 ; norma l

form for, 71-74; H ainiltonian case of, 85-89; Pfaffian cas e of, 94-96; reduction to , 97-10 0

! Geodesies , 38-39, 180-8 ; i n a tran-I sitiv e case , 238-48 ; i n a n inte -

grable case , 248-55 . Se e Closed geodesies

Goursat, 1 Hadamard, iv, 128,130, 170, 211, 238 Hamilton, ii i Hamiltonian principa l function , 5 2 Hamiltonian systems , 50-53 ; La -

grangian and , 50-53 ; transfor -mations of , 53-55 ; norma l forms of, 74-85 , 85-8 9

I Hamilton' s principle , 34-3 6 j Hilbert , 13 0 ! Hill , iii , 139 , 26 0 j Ignorabl e coordinates , 40-4 4

304 INDEX

Index o f invarian t point , 17 6 Instability, 105 , 220-31 ; zone s of ,

221-6 Integrability, 255- 9 Integral o f energy, 18 , 52; linea r i n

velocities, 44-45; conditiona l li -near, 45-47, quadratic , 48-50; in the problem of three bodies, 261-3

Jacobi, iii , 170 , 24 8 Koopman, 14 5 Lagrange, iii , 263 , 264 , 278 , 28 5 Lagrangian systems, 18-19 ; internal

characterization of , 23-25 ; re -gular, 25 ; externa l characteri -zation of, 25-31 ; transformatio n of, 36-39; normal form of, 39-40 ; reduction o f orde r of , 40-41 ; integrals of , 41-50 ; Hamiltonia n and, 50-5 3

Laplace, ii i Lebesgue, 24 8 Levi-Civita, iv , 270 , 273 , 27 5 Liapounoff, 12 2 Liouville, 4 8 Lipschitz, 5 Manifold o f state s o f motion , 143 ;

in proble m o f thre e bodies , 270-5; 283- 8

Mass, 2 3 Morse, 139 , 170 , 238 , 246 , 24 7 Multipliers, 74; Hamiltonian, 74-78 ;

Pfaffian, 89-9 1 Newton, ii i Non-energic systems , 1 8 Osgood, 4 , 13 8 Painleve\ 26 1 Particle, 23-5; i n force field , 124-8 ,

146-9 Periodic motions, 59; role of, 123-4 ;

minimum type of , 128-32 ; mini -mal typ e of , 133-9 ; simpl e an d multiple. 142 ; analyti c conti -nuation of , 139-43 ; obtaine d b y the transformation method, 143-9;

near generalize d equilibrium , 150-4; nea r a periodi c motion , 159-65; stabl e an d unstable , 209-15

Pfaffian systems , 55; variational prin-ciple for, 55; multipliers of,89-91; normal for m of , 91-96 ; genera -lized equilibriu m of, 97-100 ; sta-bility of, 100-4; instability of, 105

Picard, 1 , 13 , 105 , 12 2 Poincave, iii, 74 , 97 , 105 , 123 , 139 ,

143, 190 , 194 , 223, 237, 255, 257, 260, 28 8

PoincarG's geometric theorem, 165-9; application of , 150-8 8

Principal functio n o f conservativ e systems, 1 7

Principle o f leas t action , 36-3 9 Principle o f reciprocity , 2 6 Problem o f thre e bodies , 260-1 ;

equations of, 261-2 ; integral s of , 262-3; reduction of, 263-4,283-4; Lagrange's equalit y in , 264-5 ; Sundraan's inequalit y in , 265-7 ; collision in, 267-70 ; manifol d o f states of motion in, 270-5, 283-8*, properties o f motions of , 275-83 , 288-91; generalization of , 291-2 . See Restricted Proble m o f Three Bodies

Quasi-periodic motions , 218-2 0 Rayleigh, iii , 2 6 Recurrent motions , 198-201 , 204-5 ,

223-4 Restricted proble m o f thre e bodies ,

145, 171 , 26 0 Reversibility, 27 , 115 ; an d stability .

115-21 Rotation number , 18 4 Signorini, 13 0 Solution, 2 Stability, 97-122 ; o f Hamiltonia n

and Pfaffia n systems , 97 ; com -plete, 105-15 ; reversibilit y and ,

INDI

115-20; proble m of , 121 , 227 ; permanent, 121 ; unilateral , 122; in the sense o f Poisson , 174 , 190, 19-7; i n cas e o f tw o degree s o f freedom, 220-7; criterion of, 226-7

State o f motion , 1 Suudman, iv . 260 , 261 , 265 , 270 ,

278, 28 3 Surface of section, 143-5; local, 151-2 Systems wit h on e degre e o f free -

dom, 1 9 Systems with two degrees of freedom,

19; norma l for m for , 39-40 ; in -

EX 305

tegrals of , 45-50 ; motion s of , 150-85, 209-25 5

Transitivity, 205- 8 Uniqueness theorems , 5-6 . 10-1 2 Variational principles , algebraic .

33-34; of dynamics, 34-39. 55-5S Voss, 1 4 Wandering motions , 190- 5 Weierstrass, 26 1 Whittaker, iv , 25 , 55 , 89 , 130, 132.

162, 170 , 248 , 28 4 Work, 1 4