Selected Title s i n This Serie s Volume 11 Jun e Barrow-Gree n Poincare an d th e thre e bod y proble m 1997 10 Joh n Stillwel l Sources o f hyperbolic geometr y 1996 9 Bruc e C . Berndt an d Robert A . Ranki n Ramanujan: Letter s an d commentar y 1995 8 Kare n Hunger Parshal l and David E. Rowe The emergenc e o f the American mathematica l researc h community , 1876-1900 : J . J. Sylvester , Feli x Klein , an d E . H . Moor e 1994 7 Hen k J. M . Bo s Lectures i n the histor y o f mathematic s 1993 6 Smilk a Zdravkovsk a and Peter L . Duren, Editor s Golden years o f Mosco w mathematic s 1993 5 Georg e W. Macke y The scop e and histor y o f commutative an d noncommutativ e harmoni c analysi s 1992 4 Charle s W. McArthu r Operations analysi s i n the U.S . Army Eight h Ai r Forc e i n World Wa r I I 1990 3 Pete r L . Duren, editor, e t al . A century o f mathematics i n America , par t II I 1989 2 Pete r L . Duren, editor, et al . A century o f mathematics i n America, par t I I 1989 1 Pete r L . Duren, editor, et al . A century o f mathematics i n America , par t I 1988 This page intentionally left blank Poincare and the Three Body Problem This page intentionally left blank https://doi.org/10.1090/hmath/011 History o f Mathematic s Volume 1 1 Poincare and the Three Body Problem June Barrow-Green American Mathematical Societ y London Mathematica l Societ y Editorial Boar d American Mathematica l Societ y Londo n Mathematica l Societ y George E . Andrew s Davi d Fowle r Bruce Chandle r Jerem y J . Gray . Chairma n Paul R . Halmos , Chairma n S . J . Patterso n George B . Seligma n 1991 Mathematics Subject Classification. Primar y 01 ; Secondary 70 . Photographs o n th e cove r ar e Henr i Poincar e (inset ) an d Osca r II , Kin g o f Sweden an d Norway (background) . A lis t o f photograp h an d figure credit s i s included a t th e beginnin g o f thi s volume . Library o f Congres s Cataloging-in-Publicatio n Dat a Barrow-Green, June , 1953 - Poincare an d th e thre e bod y proble m / Jun e Barrow-Green . p. cm . — (Histor y o f mathematics, ISS N 0899-2428 ; v. 11 ) Includes bibliographica l reference s (p . - ) and index . ISBN 0-8218-0367- 0 (acid-fre e paper ) 1. Three-body problem . 2 . Hamiltonian systems . 3 . Poincare , Henri , 1854-1912—Contribu - tions i n dynamics. I . Title. II . Series . QA852.B37 199 6 515'.352-<lc20 96-1111 2 CIP Copying an d reprinting . Individua l reader s o f this publication , an d nonprofi t librarie s actin g for them , ar e permitte d t o mak e fai r us e o f th e material , suc h a s t o cop y a chapte r fo r us e in teachin g o r research . Permissio n i s grante d t o quot e brie f passage s fro m thi s publicatio n i n reviews, provide d th e customar y acknowledgmen t o f the sourc e i s given . Republication, systemati c copying, or multiple reproduction o f any material i n this publicatio n (including abstracts ) i s permitte d onl y unde r licens e fro m th e America n Mathematica l Society . Requests fo r suc h permissio n shoul d b e addresse d t o th e Assistan t t o th e Publisher , America n Mathematical Society , P . O. Bo x 6248 , Providence , Rhod e Islan d 02940-6248 . Request s ca n als o be mad e b y e-mai l t o reprint-permissionQams.org . © 199 7 by th e America n Mathematica l Society . Al l right s reserved . Printed i n the Unite d State s o f America . Reprinted wit h correction s 1997 . The America n Mathematica l Societ y retain s al l right s except thos e granted t o th e Unite d State s Government . @ Th e pape r use d i n this boo k i s acid-free an d fall s withi n th e guideline s established t o ensur e permanenc e an d durability . The Londo n Mathematica l Societ y i s incorporated unde r Roya l Charte r and i s registered wit h th e Charit y Commissioners . 10 9 8 7 6 5 4 3 0 2 01 00 9 9 For m y Mother an d Siste r and i n memory o f my Fathe r an d Brothe r with love and thanks This page intentionally left blank Contents Acknowledgements x i Photograph an d Figur e Credit s xii i Chapter 1 . Introductio n 1 Chapter 2 . Historica l Backgroun d 7 2.1. Introductio n 7 2.2. Mathematica l descriptio n o f the three bod y proble m 8 2.3. Histor y o f the three body proble m 1 4 Chapter 3 . Poincare' s Wor k befor e 188 9 2 9 3.1. Introductio n 2 9 3.2. Th e qualitativ e theor y o f differential equation s 2 9 3.3. Celestia l mechanic s an d the three body proble m 4 1 3.4. Othe r paper s 4 4 Chapter 4 . Osca r IPs 60t h Birthda y Competitio n 4 9 4.1. Introductio n 4 9 4.2. Organisatio n o f the competitio n 5 1 4.3. Kronecker' s criticis m 5 9 4.4. Th e entrie s i n the competitio n 6 1 4.5. Judgin g th e entrie s 6 3 4.6. Th e announcemen t o f the resul t 6 5 4.7. Discover y o f the erro r 6 7 4.8. Publicatio n o f the winnin g entrie s 6 9 Chapter 5 . Poincare' s Memoi r o n the Three Bod y Proble m 7 1 5.1. Introductio n 7 1 5.2. Table s o f contents 7 2 5.3. Poincare' s introduction s 7 3 5.4. Genera l propertie s o f differential equation s 7 5 5.5. Theor y o f invariant integral s 8 3 5.6. Theor y o f periodic solution s 9 1 5.7. Stud y o f the cas e with tw o degrees o f freedom 10 4 5.8. Stud y o f asymptotic surface s 10 8 5.9. Furthe r result s 12 2 5.10. Attempt s a t generalisatio n 13 0 Chapter 6 . Receptio n o f Poincare's Memoi r 13 3 6.1. Introductio n 13 3 x CONTENT S 6.2. Th e view s o f the priz e commissio n 6.3. Gylde n 6.4. Minkowsk i 6.5. Hil l 6.6. Whittake r 6.7. Othe r commentator s Chapter 7 . Poincare' s Relate d Wor k afte r 188 9 7.1. Introductio n 7.2. "Le s Methodes Nouvelle s d e la Mecaniqu e Celeste " 7.3. Th e three body proble m an d celestia l mechanic s 7.4. Genera l dynamic s an d "Th e Last Geometri c Theore m Chapter 8 . Associate d Mathematica l Activit y 8.1. Introductio n 8.2. Stabilit y 8.3. Singularitie s an d regularisatio n 8.4. Numerica l investigation s int o periodic solution s Chapter 9 . Hadamar d an d Birkhof f 9.1. Introductio n 9.2. Hadamar d an d geodesie s 9.3. Birkhof f an d dynamica l system s Chapter 10 . Epilogu e 10.1. Introductio n 10.2. Mors e 10.3. KA M theor y Appendix 1 . A letter fro m Gost a Mittag-Leffle r to Sony a Kovalevskay a Appendix 2 . Announcemen t o f the Osca r Competitio n Appendix 3 . Entrie s receive d i n the Osca r Competitio n Appendix 4 . Repor t o f the Priz e Commissio n Appendix 5 . Titl e Page s an d Table s o f Content s 5.1. Poincare' s Unpublishe d Memoi r 5.2. Poincare' s Publishe d Memoi r Appendix 6 . Theorem s i n [PI ] no t include d i n [P2 ] References Acknowledgements This boo k derive s fro m th e Ph D thesi s I prepare d a t th e Ope n Universit y between 1989-1993 , and I am very grateful t o Jeremy Gray, who suggested the topic and patientl y supervise d th e wor k involved . Hi s help , enthusiasm , an d kindnes s were unfailing an d hi s scholarship a n inspiration . Several peopl e bot h i n thi s countr y an d abroa d hav e gon e ou t o f thei r wa y on m y behal f an d I exten d thank s t o the m all . I particularl y wis h t o than k th e Institut Mittag-Leffle r (show n below ) fo r allowin g m e t o us e thei r archives , an d whose staf f provide d m e wit h ever y possibl e assistanc e durin g th e tim e I spen t there; Jespe r Lutzen , Roge r Cooke , an d Sergue i Demido v fo r helpin g t o mak e m y visit to the Institut Mittag-Leffle r s o rewarding and enjoyable ; an d Stee n Norgaar d for providin g insigh t int o Scandinavian culture , and whos e skill s o f translation an d sense o f humour wer e a constant sourc e o f delight .