Conference Boar d o f the Mathematica l Science s

CBMS Regional Conference Series in Mathematics

Number 10 4

Collisions, Rings , and Othe r Newtonia n

iV-Body Problem s

Donald G . Saar i

Published fo r th e Conference Boar d o f th e Mathematica l Science s

by th e American Mathematica l Societ y > p*£tl *p

Providence, Rhod e Islan d with suppor t fro m th e *~^x*^ >

National Scienc e Foundatio n °°ND£ S°

http://dx.doi.org/10.1090/cbms/104

NSF-CBMS Regiona l Researc h Conferenc e o n

The Dynamica l Behavio r o f th e Newtonia n iV-Bod y Proble m

held a t Easter n Illinoi s University , Jun e 9-11 , 200 2

Partially supporte d b y th e Nationa l Scienc e Foundatio n

2000 Mathematics Subject Classification. Primar y 70F10 ; Secondar y 70F15 .

Cover photograp h courtes y NASA/JPL-Caltech .

For addi t iona l informatio n an d upda te s o n thi s book , visi t w w w . a m s . o r g / b o o k p a g e s / c b m s - 1 0 4

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

Saari, D . (Donald ) Collisions, rings , an d other Newtonia n N-bod y problem s / Donal d G . Saari .

p. cm . — (CBM S regiona l conferenc e serie s i n mathematics, ISS N 0160-764 2 ; no. 104 ) Includes bibliographica l reference s an d index . ISBN 0-8218-3250- 6 (alk . paper) 1. Many-bod y problem—Congresses . 2 . Collision s (Astrophysics)—Congresses . I . Title .

II. Regiona l conferenc e serie s i n mathematics ; no. 104 .

QA1.R33 no . 10 4 [QC174.17.P7] 510 s—dc2 2 [530.14'4] 200504120 5

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Visit th e AMS home pag e a t http://www.ams.org /

10 9 8 7 6 5 4 3 2 1 1 0 09 08 07 06 0 5

For tw o grea t sons-in-la w

Adrian DufR n an d Eri k Sieberg ,

and al l of my "iV-body " student s represente d b y the firs t an d th e obvious ,

Neal Hulkowe r an d Zhihon g (Jeff ) Xi a

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Preface

This boo k i s the writte n versio n o f m y Conferenc e Boar d o f Mathematica l Sciences (CBMS ) lecture s presente d durin g th e wee k o f Jun e 10 , 2002 , a t Eastern Illinoi s Universit y i n Charleston , Illinois . Th e te n lecture s centere d on my firs t an d persisten t academi c love—th e Newtonia n iV-bod y problem .

While some experts activel y participate d i n the sessions , this conferenc e fully live d up to the inten t o f the CBM S series in that mos t o f the attendee s were graduate students , new-comer s to the field, o r curious mathematician s wishing t o lear n somethin g abou t thi s fascinatin g topic . Accordingly , th e goals o f th e lecture s quickl y change d fro m a technica l presentatio n appro -priate primaril y fo r "experts, " t o presentation s no w intende d t o introduc e everyone t o th e basi c structur e o f AT-bod y systems , t o identif y certai n per -sistent researc h themes , and , hopefully , t o recrui t activ e participants t o thi s fascinating researc h area . A s such , durin g eac h lectur e severa l unsolve d research problem s wer e described : som e o f them ar e include d here .

The ne w goal s fo r th e lecture s change d th e nature , content , expositor y tone, an d eve n th e subjec t matte r t o mak e th e presentation s mor e respon -sive to th e specifi c interest s o f the participant s whil e addressin g thei r man y questions, Fo r instance , I include d mor e introductor y materia l tha n origi -nally planned : i n retrospect , thi s wa s a n excellen t addition .

The conten t an d approac h o f thi s boo k mimi c th e change d goal s o f th e lectures; e.g. , i n addition t o new material , yo u will find discussions intende d to develo p intuition , introductor y material , occasiona l anecdotes , an d de -scriptions o f open problems . T o provide cohesio n fo r eac h chapter , som e of the materia l revolve s abou t unsolve d researc h problems—wher e th e moti -vating rol e o f th e proble m ma y b e o f mor e valu e tha n th e actua l problem . In Chap . 1 , for instance , much of the discussion i s intended t o lead to a n un -resolved issu e abou t th e weir d dynamic s exhibite d i n th e F-rin g o f Saturn . In Chap . 2 , th e discussio n i s tie d togethe r vi a a conjectur e involvin g th e diameter o f the AT-bod y system. I n Chap . 3 , the unifyin g problem s involv e the importan t issu e of finding certain AT-bod y configurations, whic h leads t o

v

VI PREFACE

a discussio n o f the ring s o f Saturn . I n Chap . 4 , the issu e involves collisions . The concludin g Chap . 5 discusses the likelihoo d o f "ba d thing s happening. " Everyone, fro m novice s t o experts , wil l find somethin g new .

Some result s ar e new , whil e other s hav e bee n presente d earlie r (e.g. , a t colloquia, Oberwolfac h meetings--particularl y severa l durin g th e 1970s — Midwest Dynamica l System s meetings, a 198 3 month lon g mini-course give n in Receife , Brazi l whil e visitin g Hildebert o Cabral , i n a serie s o f lecture s in Pari s ove r 1985-8 7 hoste d b y Michae l Herman , severa l informa l lecture s during 198 9 in Barcelona hosted b y Jaume Llibre , etc., ) an d eve n advertise d as "wil l appear " i n full y intende d bu t neve r complete d papers . I n othe r words, man y o f thes e result s hav e no t bee n previousl y published . A s mos t authors o f a boo k quickl y discover , th e har d par t i s no t t o decid e wha t to include , bu t wha t t o exclude—particularl y i f a boo k i s to b e eventuall y completed. (Som e o f the exclude d materia l probabl y wil l appea r i n [90]. )

Other result s describe d i n thi s boo k com e fro m m y earlie r papers . Th e particular journal s tha t publishe d thes e paper s ar e implicitl y acknowledge d and thanke d vi a th e references . Bu t m y expositor y pape r [88 ] "A visit to the Newtonian n-body via elementary complex variables" i s extensively use d to provid e structur e an d motivatio n fo r a coupl e o f th e chapters , particu -larly th e introductor y one , s o I want t o explicitl y than k th e MA A fo r thei r permission t o us e i t i n thi s manner .

My dee p thank s an d appreciatio n g o t o Patric k Coulton , th e chai r fo r this particula r CBM S conference , an d m y long-time frien d Gregor y Galpri n for invitin g m e t o b e th e CBM S lecture r an d fo r thei r effort s t o assembl e a successfu l CBM S application . I als o than k the m fo r thei r ful l an d activ e participation i n al l lectures an d extr a session s that the y helpe d t o organize , and for everything they did to make the stay so enjoyable fo r al l of us. I want to than k al l o f the participant s fo r keepin g th e worksho p session s s o lively ! My thank s t o th e Mathematic s Departmen t a t Easter n Illinoi s Universit y for thei r graciou s hospitality . M y thank s t o Ro n Rosie r an d th e CBM S for thei r progra m tha t make s thes e kind s o f lecture s possible . Thank s t o Neal Hulkower : twic e a t Northwester n h e too k m y yea r lon g cours e o n th e Newtonian iV-bod y problem (th e first i n 1969-70) , and h e still had both set s of lectur e notes ! Hi s note s prove d t o b e usefu l i n recoverin g som e o f m y earlier result s an d arguments . Als o thanks t o anothe r studen t (bu t I do no t recall wh o i t was ) wh o gav e m e a cop y o f hi s note s man y year s ago .

Irvine, Californi a January, 200 5

Contents

Preface v

1 Introductio n 1 1.1 Mar s 1

1.1.1 Motio n o f Mars 2 1.1.2 Th e a far out " planet s 4

1.2 Mercur y 6 1.3 Epicycle s 1 1 1.4 Chaoti c behavio r 1 3

1.4.1 Newton' s metho d 1 4 1.4.2 Perio d thre e an d circl e map s 1 9 1.4.3 Th e force d Va n de r Po l equation s 2 3

1.5 Th e ring s o f Satur n 2 6 1.5.1 Kink y behavio r 2 6 1.5.2 A model 2 7

2 Centra l configuration s 3 1 2.1 Equation s o f motion an d integral s 3 2 2.2 Centra l Configuration s 3 4

2.2.1 Wh y centra l configuration s ar e importan t 3 6 2.2.2 Valu e o f A 4 0 2.2.3 Equivalenc e classe s o f configurations 4 2

2.3 A conjecture an d a velocity decompositio n 4 7 2.3.1 Viria l Theore m an d th e conjectur e 4 7 2.3.2 Th e syste m velocit y decompositio n 5 1 2.3.3 Centra l configuration s an d th e velocit y decompositio n 5 4 2.3.4 Motio n preservin g a n Eule r similarit y clas s 5 6

vn

V l l l CONTENTS

2.3.5 Sundma n inequalit y 6 1 2.4 Mor e conjecture s 6 5

2.4.1 Anothe r conjectur e 6 5 2.4.2 Specia l case s 6 7

2.5 Jacob i coordinate s hel p "see " th e dynamic s 6 9 2.5.1 Velocit y decompositio n an d a basi s 7 1 2.5.2 Describin g p" wit h th e basi s 7 4 2.5.3 "Seeing " th e gradien t o f U 7 6 2.5.4 A n illustratin g exampl e 7 7 2.5.5 Findin g centra l an d othe r configuration s 7 9 2.5.6 Equation s o f motion fo r constan t / 8 0 2.5.7 Basi s fo r th e coplana r Af-bod y proble m 8 1

3 Findin g Centra l Configuration s 8 3 3.1 Fro m th e ancien t Greek s t o 8 4

3.1.1 Arithmeti c an d geometri c mean s 8 4 3.1.2 Connectio n wit h centra l configuration s 8 8

3.2 Constraint s 9 2 3.2.1 Singularit y structur e o f F 9 4 3.2.2 Som e dynamic s 9 7 3.2.3 Stratifie d structur e o f the imag e o f F 9 9

3.3 Geometri c approach—th e rul e o f signs 10 2 3.3.1 Th e "eonfigurationa l average d length " £CAL 10 3 3.3.2 Sign s o f gradients-coplana r configuration s 10 5 3.3.3 Sign s o f gradients-three-dimensiona l configuration s . . 10 6 3.3.4 Degenerat e configuration s 10 7

3.4 Consequence s fo r centra l configuration s 10 9 3.4.1 Surprisin g regularit y 10 9 3.4.2 Estimate s o n £CAL 11 2 3.4.3 Ar e ther e centra l configuration s o f these types ? . . . . 11 4

3.5 Wha t can , an d cannot , b e 11 5 3.5.1 Mor e centra l configuration s 11 5 3.5.2 Masse s an d collinea r centra l configuration s 11 9 3.5.3 Masse s an d coplana r configuration s 12 4

3.6 Ne w kinds o f constraint s 12 6 3.7 Ring s o f Satur n 13 0

3.7.1 Stabilit y 13 1 3.7.2 Mor e ring s 13 3 3.7.3 Saturn , an d som e problem s 13 6

CONTENTS i x

4 Collision s — bot h rea l an d imaginar y 13 7 4.1 On e bod y proble m 13 8

4.1.1 Levi-Civita' s approac h 14 0 4.1.2 Kustaanheim o an d Stiefel' s approac h 14 1 4.1.3 Topologica l obstruction s an d hair y ball s 14 3 4.1.4 Sundman' s solutio n o f the three-bod y proble m . . . . 14 4

4.2 Sundma n an d th e three-bod y proble m 14 7 4.2.1 Comple x singularities ? 14 7 4.2.2 Avoidin g comple x singularitie s 14 8 4.2.3 Singularitie s retaliat e 14 9

4.3 Generalize d Weierstrass-Sundma n theore m 15 0 4.3.1 A simple case-th e centra l forc e proble m 15 1 4.3.2 Large r p-value s an d "Blac k Holes " 15 1 4.3.3 Lagrange-Jacob i equatio n 15 3 4.3.4 Proo f o f the Weierstrass-Sundma n Theore m 15 4 4.3.5 Bounde d abov e mean s bounde d belo w 15 9 4.3.6 Problem s 16 1 4.3.7 A n interestin g historica l footnot e 16 1

4.4 Singularitie s - a n overvie w 16 2 4.4.1 Behavio r o f a singularit y 16 3 4.4.2 Non-collisio n singularitie s 16 5 4.4.3 Of f t o infinit y 16 6 4.4.4 Problem s 17 0

4.5 Rat e o f approac h o f collisions 17 2 4.5.1 Genera l collision s 17 2 4.5.2 Tauberia n Theorem s 17 3 4.5.3 Proo f o f the theore m 17 9

4.6 Sharpe r asymptoti c result s 18 4 4.7 Spin , o r n o spin? 18 5

4.7.1 Usin g the angula r momentu m 18 7 4.7.2 Th e collinea r cas e 18 9

4.8 Manifold s define d b y collision s 19 1 4.8.1 Structur e o f collision set s 19 2 4.8.2 Proo f o f Theorem 4.1 8 19 3

4.9 Proo f o f the slowl y varying assertio n 19 5 4.9.1 Center s o f mass 19 8 4.9.2 Bac k t o th e proo f 20 1 4.9.3 Th e las t step s 20 3

x CONTENTS

5 Ho w likel y i s it ? 20 7 5.1 Motivatio n 20 8

5.1.1 Ide a o f proof 20 9 5.1.2 Wh y do we need th e Baire categor y statement ? . . . . 21 0

5.2 Proof : C is of first Bair e categor y 21 0 5.2.1 Findin g a n appropriat e C subset 21 1 5.2.2 A comment abou t th e set of singularities 21 3

5.3 Proof : C is of Lebesgue measur e zer o 21 4 5.3.1 A common collisio n fo r k > 3 particles 21 4 5.3.2 Lowe r dimensions , binar y collisions , an d othe r forc e

laws 21 7 5.4 Likelihoo d o f non-collision singularitie s 22 0

Bibliography 22 3

Index 23 2

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[42] Mather , J . an d R . McGehee , Solution s o f the collinea r four-bod y prob -lem which become unbounded i n finite time , Lecture Notices in Physics 38 (1975) , 573-597 .

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[78] Saari , D. G., Improbability o f collisions in Newtonian gravitationa l sys -tems II , Trans. Amer. Math. Soc. 181 (1973), 351-368 .

[79] Saari , D . G. , Singularitie s an d collision s o f Newtonia n gravitationa l systems, Arch. Rat. Mech. & Math. Anal. 4 9 (1973) , 311-320 .

[80] Saari , D . G., On oscillatory motio n i n gravitationa l systems , J. Diff. Equations 1 4 (1973), 275-292 .

[81] Saari , D. G., A Tauberian theore m fo r absolutely continuou s function s and fo r series, SI AM J. Math. Anal. 5 (1974) , 649-662 .

[82] Saari , D. G., Dynamics and clusters of galaxies, pp 273-284 in Stability of the Solar System and of Small Stellar Systems, ed . Y. Kozai, IAU, 1974.

[83] Saari , D. G., Collisions are of first category , Proc. Amer. Math. Soc. 47 (1975), 442-445 .

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[84

[85;

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[87;

[89;

[90

[91

[92;

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[96

Saari, D . G., A global existenc e theore m fo r the four bod y proble m of Newtonian mechanics , Jour. Diff. Eqs. 26 (1977) , 80-111 .

Saari, D . G., On the role an d properties o f N bod y centra l configura -tions, Celestial Mechanics 2 1 (1980), 9-20.

Saari, D . G. , Manifol d structur e fo r collision s an d fo r hyperbolic -parabolic orbit s i n the n-body problem , J. Diff. Eqs. 55 (1984) , 300-329.

Saari, D . G., Fro m rotation s an d inclinations t o zer o configurationa l velocity surface s I , a natural rotatin g coordinat e system , Celestial Me-chanics 3 3 (1985), 299-318 .

Saari, D. G., A visit t o the Newtonian n-bod y vi a elementary comple x variables, Math.Monthly, 97 , (Feb 1990), 105-119 .

Saari, D . G. , A chaoti c exploratio n o f aggregatio n paradoxes , SIAM Review 3 7 (1995), 37-52.

Saari, D . G. , Chaotic Expansion of the Newtonian N-body problem, book manuscript , 2004 .

Saari, D. , G., The Way it Was, America n Mathematica l Society , Prov -idence, R. L, 2004.

Saari, D. G., and N. Hulkower, On the manifolds o f total collapse orbit s and of completely parabolic orbit s for the n-body problem , J. Diff. Eqs. 41 (1981) , 27-43 .

Saari, D. G., and J. Urenko, Newton's method, circl e maps, and chaotic motion, Amer. Math. Monthly 9 1 (1984) , 3-17 .

Saari, D . G. , and Z. Xia, The existence o f oscillatory an d super hy-perbolic motio n i n Newtonian Systems , Journ. of Diff. Eq. 82 (1989) , 342-355.

Saari, D. G., and Z. Xia, Off to infinity i n finite time. Notices of AMS, 42 (May , 1995) , 538-546 . (Translate d versio n i n th e Czec h journa l " Pokroky matematiky , fyzik y a astronomie" ("Advance s i n Mathemat -ics, Physic s an d Astronomy").)

Saari, D. G., and Z. Xia, Hamiltonian Dynamics and Celestial Mechan-ics, Contemporar y Mathematics , 198 , AMS, Providence , RI , 1996 .

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[97] Santoprete , M. , A counterexample t o a Generalize d Saari' s Conjectur e with a continuu m o f centra l configurations , 2004 , preprint .

[98] Saslaw , W . C , Gravitational physics of stellar and galactic systems, Cambridge U . Press , Cambridge , 1985 .

[99] Sarkovskii , A . N. , Coexistenc e o f cycle s o f a continuou s ma p o f a lin e into itself , Ukrain. Math. Z. 1 6 (1964) , 61-71 .

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[111] Widder , D . V. , The Laplace Transform, Princeto n Universit y Press , 1946.

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Index

Fs, 21 3 A, 16 4 Ur<9£, 7 1

Usca/, 7 1 *»con fig i ^

W r o t , 5 3 W s c a / , 5 3 V s , 5 2

I , 9 2

[/, 9 2 R^, 8 6

R, 5 1 V, 5 1

Alligood, K. , 14 , 22 3 Almagest, 1 1 almost periodic , 1 3 angular momentum , 8 , 3 3 Anosov, D. , 169 , 22 3 aphelion, 7 Arecibo, 7 Aristotle, 11 , 3 5

Baire category , 21 0 Barrow-Green, J. , 137 , 22 3 Birkhoff, G . D. , 149 , 15 9 black holes , 15 2 Boas, R. , 176 , 22 3 Bohr, H. , 13 , 22 3 Bohr, N. , 1 3

Cabral, H. , vi , 51 , 22 3 CAL, 104 , 11 0

Caloris Basin , 6

Cantor set , 1 8

Carlson, D. , 2 6

Cauchy, A. , 85 , 16 2

CBMS, v center o f mass , 33 , 5 3 central configuratio n average d length ,

104

central configuration s

collinear, 90 , 95 , 11 9

collisions, 3 7 definition, 35 , 4 0

dengerate, 4 6

Euler similarit y classes , 4 3

expansion o f galaxies , 3 8

N-l dimensional , 8 9

nonhomogeneous potentials , 4 5

relative equilibria , 3 8

relative equilibrium , 3 1

Chazy, J. , 22 3

Chenciner, A. , 223 , 23 1

Collins, S. , 2 6

configurational measure , 40 , 42, 43, 55, 62 , 65 , 76 , 156 , 15 7

conjecture, 49 , 6 5 conservation o f energy , 3 4 constraint

actual values , 12 6

using area , 12 8 using volume , 12 7

constraints

232

INDEX 233

degenerate pentahedron , 103 , 106

degenerate tetrahedron, 101 , 106 degenerate triangle , 10 0 number, 10 3

Coulton, P. , v i cracks, 4 5 Cushman, R. , 223 , 231

Delgado, J. , 22 4 Devaney, R. , 14 , 224 Diacu, F. , 43 , 50, 68 , 137 , 224 Dziobek, O. , 101 , 224

Earth Lagrange points , 8 9

Eastern Illinoi s University , v i Easton, R. , 36 , 224 Einstein, A. , 11 , 43, 139 epicycles, 1 1 Euler's Theorem , 4 0 Euler, L. , 9 0 Eureka, 8 9

F ring , 2 6 Fang-Yen, C , 9 Feller, W. , 168 , 224

galaxy expansion , 3 8 Galileo, 2 6 Galprin, G. , v i Gerver, J. , 171 , 224

half-astronomical units , 2 Hamilton, W. , 14 2 Hampton, M. , 22 4 Hardy, G. , 84 , 173 , 224 Hasselblatt, B. , 189 , 225 Helsinki Observatory , 13 8 Herman, M. , v i Hernandez-Garduno, A. , 50 , 224

Hilbert space , non-separable , 1 3 Hill curves , 8 9 Hille, E , 19 5 Holmes, P. , 137 , 224 homographic motion , 3 9 homothetic solution , 4 0 Hopf maps , 14 4 Hulkower, N. , iii , vi, 192 , 225, 229

Inequalities, 8 4 invariable plane , 33 , 158 iterates; convergence , 1 6 itinerary, 1 6

Katok, A. , 189 , 225 Kepler

third law , 5 second law , 8

Kepler equations , 14 8 Kovalevsky, S. , 14 7 Kustaanheimo, P. , 141 , 142, 225 Kyner, T. , 9 , 22 5

Lacomba, E. , 22 4 Lagrange multipliers , 10 3 Lagrange point s

Earth, 8 9 Mars, 8 9 Venus, 89

Lagrange, P. , 89 , 16 2 Lagrange-Jacobi equation , 48 , 49,

153, 16 5 generalized, 15 4

Law o f cosines , 13 0 Lawson, J. , 50 , 224 Le Verrier, U. , 6 Lehmann-Filhes, R. , 89 , 225 Lennard-Jones force law , 4 4 Levi, M. , 23 , 225 Levi-Civita, T. , 139 , 225 Li, T.Y. , 19 , 225

234 INDEX

linear stability , 13 2 Littlewood, J . E., 84, 173, 207, 224,

225 Llibre, J. , vi , 50 , 225

Marchal, C , 149 , 225 Mariner 10 , 6 Mars

perceived motion , 2 Lagrange points , 8 9

Marsden, J. , 50 , 224 mass; negative , 11 7 masses; negative , 11 2 Mather, J. , 169 , 226 maximum principle , 16 2 Maxwell, J . C , 3 1 McCord, C , 50 , 226 McGehee, R. , 169 , 171 , 226 mean

arithmetic, 8 5 geometric, 8 5 weighted, 8 6

mean motion , 8 measure preserving , 20 9 Mercury

advance o f perihelion , 6 , 43 , 153

orbit, 6-1 0 Mingarelli, A. , 22 4 minimum spacing ; r m^n, 16 3 Mioc, V. , 22 4 Mittag-Leffler, G. , 147 , 226 Moeckel, R. , 46 , 50 , 224 , 226 Moulton, F . R. , 90 , 226 Murdock, J. , 9 , 226 , 228

National Burea u o f Standards , 4 4 Neptune, 6 Newcomb, S. , 34 , 226 Newton's method , 1 4

Newton, L , 1 1

Oberwolfach conference , v i 1970s, 84 1964, 14 2 1978, 14 4

osculating plane , 6 5

Polya, G. , 84 , 224 Painleve, P. , 164 , 165 , 227 Palmore, J. , 13 3 Pandora, 2 7 parabolic motion , 18 9 Perez, E. , 43 , 50, 68, 227 perihelion, 6 Peterson, L , 137 , 227 Petit, F. , 8 9 Pina, E. , 50 , 225 Pioneer 11 , 26 Pizzetti, P. , 56 , 227 polar momen t o f inertia; I , 40, 179 Pollard, H. , 7 , 44 , 167 , 172 , 173 ,

175, 177 , 227 Prometheus, 2 6 Ptolemy, 1 1

quasi-periodic, 1 3 quaternions, 14 2

relative equilibria , 3 8 relative equilibrium , 3 9 Ricc-Curbatro, 13 9 Roberts, G. , 50 , 227 Robinson, C. , 9 , 14 , 186 , 189 , 223,

227, 23 1 Rosier, R. , v i Rouche's theorem , 16 2 Routh, E. , 3 1 rule o f signs , 102 , 109 , 11 2

Santoprete, M. , 50 , 68

INDEX 235

Sarkovskii sequence , 1 9 Sarkovskii, A. , 19 , 230 Saslaw, W. , 23 0 Sauer, T. , 14 , 223 Scheifel, G. , 23 0 self-potential, 3 4 sensitivity o f initia l conditions , 1 8 Siegel, C , 172 , 192 , 230 Simon, C , 20 8 singularity

algebraic branc h point , 19 3 characterization, 16 3 collision, 163 , 164, 208 complex valued , 14 8 definition, 16 3 non-collision, 164 , 208 other forc e laws , 21 9

slowly varying , 16 8 Smale, S. , 35 , 45, 230 SMD, 9 2 Somigliana-Pizzetti model , 5 6 space o f mutua l distances , 9 2 spectral stability , 13 2 Sperling, H. , 23 0 spinors, 14 2 Stiefel, E. , 141 , 142, 225, 230 Stoica, C , 22 4 stratified structure , 10 0 Sundman inequality , 61 , 156, 15 8 Sundman, K. , 140 , 172 , 177 , 178 ,

207, 23 0 Susin, G. , 22 7 system

configurational velocity , 5 3 decomposition o f W c o n ^ , 7 1 gradient, 5 2 inner product , 5 1 postion, 5 1 rotational configurationa l ve-

locity, 6 4

rotational velocity , 52 , 53 scalar configurational velocity ,

64 scalar velocity , 52 , 53 velocity, 5 1

Tauberian Theorem, 173 , 174, 177, 185

Trojan asteroids , 8 9

universal set , 1 6 Uranus, 6 Urenko, J. , 13 , 16, 208, 229 , 230

Van de r Po l equations , 2 3 Van de r Pol , B. , 2 3 Venus, 9

Lagrange points , 8 9 virial theorem , 47 , 17 5 von Zeipel , H. , 166 , 168 , 230 Voyager, 2 6 Vulcan, 6

Wagon, S. , 9 Weierstrass, K. , 14 7 Weierstrass-Sundman theorem , 147 ,

150 proof, 15 4

Widder, D. , 174 , 23 1 Wintner, A. , 45, 89, 162 , 172 , 176,

231 Woerner, K. , 23 0 Wolf, M. , 8 9 word, 1 6

Xia, Z. , hi , 49 , 169 , 171 , 201, 207, 223, 229 , 23 1

Yan, J. , 22 7 Yorke, J. , 14 , 19 , 223, 225

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Titles i n Thi s Serie s

104 Donal d G . Saari , Collisions , rings , an d othe r Newtonia n TV-bod y problems , 200 5

103 Iai n Raeburn , Grap h algebras , 200 5

102 Ke n Ono , Th e we b o f modularity : Arithmeti c o f th e coefficient s o f modula r form s an d q

series, 200 4

101 Henr i Darmon , Rationa l point s o n modula r ellipti c curves , 200 4

100 Alexande r Volberg , C alder on-Zygmund capacitie s an d operator s o n nonhomogeneou s

spaces, 200 3

99 Alai n Lascoux , Symmetri c function s an d combinatoria l operator s o n polynomials , 200 3

98 Alexande r Varchenko , Specia l functions , K Z typ e equations , an d representatio n theory ,

2003

97 Bern d Sturmfels , Solvin g system s o f polynomia l equations , 200 2

96 N ik y Kamran , Selecte d topic s i n th e geometrica l stud y o f differentia l equations , 200 2

95 Benjami n Weiss , Singl e orbi t dynamics , 200 0

94 Davi d J . Saltman , Lecture s o n divisio n algebras , 199 9

93 Gor o Shimura , Eule r product s an d Eisenstei n series , 199 7

92 Fa n R . K . Chung , Spectra l grap h theory , 199 7

91 J . P . Ma y e t al . , Equivarian t homotop y an d cohomolog y theory , dedicate d t o th e

memory o f Rober t J . Piacenza , 199 6

90 Joh n R o e , Inde x theory , coars e geometry , an d topolog y o f manifolds , 199 6

89 Cliffor d Henr y Taubes , Metrics , connection s an d gluin g theorems , 199 6

88 Crai g Huneke , Tigh t closur e an d it s applications , 199 6

87 Joh n Eri k Fornaess , Dynamic s i n severa l comple x variables , 199 6

86 Sori n Popa , Classificatio n o f subfactor s an d thei r endomorphisms , 199 5

85 Michi o J imb o an d Tetsuj i Miwa , Algebrai c analysi s o f solvabl e lattic e models , 199 4

84 Hug h L . Montgomery , Te n lecture s o n th e interfac e betwee n analyti c numbe r theor y an d

harmonic analysis , 199 4

83 Carlo s E . Kenig , Harmoni c analysi s technique s fo r secon d orde r ellipti c boundar y valu e

problems, 199 4

82 Susa n Montgomery , Hop f algebra s an d thei r action s o n rings , 199 3

81 Steve n G . Krantz , Geometri c analysi s an d functio n spaces , 199 3

80 Vaugha n F . R . Jones , Subfactor s an d knots , 199 1

79 Michae l Frazier , Bjor n Jawerth , an d Guid o Weiss , Littlewood-Pale y theor y an d th e

study o f functio n spaces , 199 1

78 Edwar d Formanek , Th e polynomia l identitie s an d variant s o f n x n matrices , 199 1

77 Michae l Christ , Lecture s o n singula r integra l operators , 199 0

76 Klau s Schmidt , Algebrai c idea s i n ergodi c theory , 199 0

75 F . Thoma s Farrel l an d L . Edwi n Jones , Classica l aspherica l manifolds , 199 0

74 Lawrenc e C . Evans , Wea k convergenc e method s fo r nonlinea r partia l differentia l

equations, 199 0

73 Walte r A . Strauss , Nonlinea r wav e equations , 198 9

72 Pete r Orlik , Introductio n t o arrangements , 198 9

71 Harr y D y m , J contractiv e matri x functions , reproducin g kerne l Hilber t space s an d

interpolation, 198 9

70 Richar d F . Gundy , Som e topic s i n probabilit y an d analysis , 198 9

69 Fran k D . Grosshans , Gian-Carl o Rota , an d Joe l A . Stein , Invarian t theor y an d

super algebras, 198 7

68 J . Wil l ia m Helton , Josep h A . Ball , Charle s R . Johnson , an d Joh n N . Palmer , Operator theory , analyti c functions , matrices , an d electrica l engineering , 198 7

TITLES I N THI S SERIE S

67 Haral d Upmeier , Jorda n algebra s i n analysis , operato r theory , an d quantu m mechanics , 1987

66 G . Andrews , (/-Series : Thei r developmen t an d applicatio n i n analysis , numbe r theory , combinatorics, physic s an d compute r algebra , 198 6

65 Pau l H . Rabinowitz , Minima x method s i n critica l poin t theor y wit h application s t o differential equations , 198 6

64 Donal d S . Passman , Grou p rings , crosse d product s an d Galoi s theory , 198 6

63 Walte r Rudin , Ne w construction s o f function s holomorphi c i n th e uni t bal l o f C n , 198 6

62 Bel a Bollobas , Extrema l grap h theor y wit h emphasi s o n probabilisti c methods , 198 6

61 M o g e n s F lensted-Jensen , Analysi s o n non-Riemannia n symmetri c spaces , 198 6

60 Gille s Pisier , Factorizatio n o f linea r operator s an d geometr y o f Banac h spaces , 198 6

59 Roge r How e an d Al le n Moy , Harish-Chandr a homomorphism s fo r p-adi c groups , 198 5

58 H . Blain e Lawson , Jr. , Th e theor y o f gaug e field s i n fou r dimensions , 198 5

57 Jerr y L . Kazdan , Prescribin g th e curvatur e o f a Riemannia n manifold , 198 5

56 Har i Bercovici , Cipria n Foia§ , an d Car l Pearcy , Dua l algebra s wit h application s t o

invariant subspace s an d dilatio n theory , 198 5

55 Wil l ia m Arveson , Te n lecture s o n operato r algebras , 198 4

54 Wil l ia m Fulton , Introductio n t o intersectio n theor y i n algebrai c geometry , 198 4

53 Wi lhe l m Klingenberg , Close d geodesie s o n Riemannia n manifolds , 198 3

52 Ts i t -Yue n Lam , Orderings , valuation s an d quadrati c forms , 198 3

51 Masamich i Takesaki , Structur e o f factor s an d automorphis m groups , 198 3

50 Jame s Eell s an d Lu c Lemaire , Selecte d topic s i n harmoni c maps , 198 3

49 Joh n M . Franks , Homolog y an d dynamica l systems , 198 2

48 W . Stephe n Wilson , Brown-Peterso n homology : a n introductio n an d sampler , 198 2

47 Jac k K . Hale , Topic s i n dynami c bifurcatio n theory , 198 1

46 Edwar d G . Effros , Dimension s an d C* -algebras, 198 1

45 Ronal d L . Graham , Rudiment s o f Ramse y theory , 198 1

44 Phil l i p A . Griffiths , A n introductio n t o th e theor y o f specia l divisor s o n algebrai c curves ,

1980

43 Wil l ia m Jaco , Lecture s o n three-manifol d topology , 198 0

42 Jea n Dieudonne , Specia l function s an d linea r representation s o f Li e groups , 198 0

41 D . J . N e w m a n , Approximatio n wit h rationa l functions , 197 9

40 Jea n Mawhin , Topologica l degre e method s i n nonlinea r boundar y valu e problems , 197 9

39 Georg e Lusztig , Representation s o f finite Chevalle y groups , 197 8

38 Charle s Conley , Isolate d invarian t set s an d th e Mors e index , 197 8

37 Masayosh i Nagata , Polynomia l ring s an d affin e spaces , 197 8

36 Car l M . Pearcy , Som e recen t development s i n operato r theory , 197 8

35 R . Bowen , O n Axio m A diffeomorphisms , 197 8

34 L . Auslander , Lectur e note s o n nil-thet a functions , 197 7

33 G . Glauberman , Factorization s i n loca l subgroup s o f finit e groups , 197 7

32 W . M . Schmidt , Smal l fractiona l part s o f polynomials , 197 7

31 R . R . Coifma n an d G . Weiss , Transferenc e method s i n analysis , 197 7

30 A . Pelczyriski , Banac h space s o f analyti c function s an d absolutel y summin g operators , 1977

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