Celletti A. Stability and Chaos in Celestial Mechanics
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Stability and Chaos in Celestial Mechanics
Alessandra Celletti
Stability and Chaos in
Celestial Mechanics
Published in association with
PPraxisraxis PPublishingublishing
Chichester, UK
Professor Alessandra Celletti
Department of Mathematics
University of Rome ``Tor Vergata''
Rome
Italy
SPRINGER PRAXIS BOOKS IN ASTRONOMY AND PLANETARY SCIENCES
SUBJECT ADVISORY EDITORS: Philippe Blondel, C.Geol., F.G.S., Ph.D., M.Sc., Senior Scientist, Department
of Physics, University of Bath, UK; John Mason, M.B.E., B.Sc., M.Sc., Ph.D.
ISBN 978-3-540-85145-5 Springer-Verlag Berlin Heidelberg New York
Springer is part of Springer-Science + Business Media (
springer.com)
Library of Congress Control Number: 2009936867
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or review, as permitted under the Copyright, Designs and Patents Act 1988, this
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# Praxis Publishing Ltd, Chichester, UK, 2010
The use of general descriptive names, registered names, trademarks, etc. in this
publication does not imply, even in the absence of a speci®c statement, that such
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Cover design: Jim Wilkie
Project copy editor: Mike Shardlow
Printed in Germany on acid-free paper
Author-generated LaTex, processed by EDV-Beratung Herweg, Germany
To the memory of my mother Mirella,
my best teacher and friend,
and in fond remembrance
of my grandmother Emilia.
Without them I would never have had
the privilege of studying the sky.
Table of Contents
Preface ........................................................ XI
Acknowledgments............................................... XIV
1 Order and chaos .............................................. 1
1.1 Continuousanddiscrete systems ............................. 1
1.2 Linear stability ............................................. 4
1.3 Conservativeanddissipativesystems.......................... 6
1.4 Theattractors and basins of attraction........................ 7
1.5 Thelogistic map ........................................... 9
1.6 Thestandard map.......................................... 12
1.7 Thedissipative standard map ................................ 15
1.8 H´enon’smapping........................................... 17
2 Numerical dynamical methods ............................... 21
2.1 Poincar´emap .............................................. 21
2.2 Lyapunov exponents . . ...................................... 23
2.3 Theattractor’s dimension ................................... 25
2.4 Timeseries analysis......................................... 26
2.5 Fourieranalysis ............................................ 31
2.6 Frequencyanalysis.......................................... 32
2.7 H´enon’smethod............................................ 33
2.8 Fast Lyapunov Indicators . . .................................. 35
3 Kepler’s problem ............................................. 39
3.1 Themotion of the barycenter ................................ 40
3.2 Thesolution of Kepler’s problem ............................. 41
3.3
˜
f and ˜g series.............................................. 44
3.4 Elliptic motion . . . .......................................... 44
3.4.1 Mean and eccentric anomaly . . . ........................ 46
3.4.2 SolutionofKepler’s equation .......................... 48
3.5 Parabolicmotion........................................... 49
3.6 Hyperbolicmotion.......................................... 50
3.7 Classification of the orbits ................................... 51
3.8 Spacecraft transfers . . . ...................................... 53
3.9 Delaunayvariables.......................................... 53
3.10 Thetwo–bodyproblemwithvariablemass..................... 57
3.10.1 The rocket equation .................................. 57
3.10.2 Gylden’s problem .................................... 58
VIII Table of Contents
4 The three–body problem and the Lagrangian solutions ....... 63
4.1 Therestricted three–body problem ........................... 63
4.1.1 Theplanar,circular, restricted three–body problem ...... 63
4.1.2 Expansion of the perturbing function . . . ................ 65
4.1.3 The planar, elliptic, restricted three–body problem ....... 67
4.1.4 Theinclined, circular, restricted three–body problem ..... 67
4.2 The circular, restricted Lagrangian solutions . .................. 68
4.3 The elliptic, restricted Lagrangian solutions . . .................. 73
4.4 The elliptic, unrestricted triangular solutions ................... 76
5 Rotational dynamics .......................................... 83
5.1 Eulerangles ............................................... 83
5.2 Andoyer–Depritvariables.................................... 85
5.3 Freerigidbodymotion...................................... 87
5.4 Perturbedrigidbodymotion................................. 89
5.5 Thespin–orbit problem ..................................... 91
5.5.1 Theconservativespin–orbit problem.................... 91
5.5.2 Theaveragedequation................................ 94
5.5.3 Thedissipative spin–orbit problem ..................... 95
5.5.4 Thediscrete spin–orbit problem........................ 96
5.6 Motionaroundanoblate primary............................. 97
5.7 Interactionbetween two bodiesoffinitedimensions ............. 98
5.8 The tether satellite ......................................... 99
5.9 The dumbbell satellite . . . ................................... 103
6 Perturbation theory .......................................... 107
6.1 Nearly–integrableHamiltonian systems........................ 107
6.2 Classical perturbationtheory ................................ 108
6.2.1 Anexample ......................................... 110
6.2.2 Computation of the precession of the perihelion . . . ....... 112
6.3 Resonantperturbationtheory................................ 112
6.3.1 Three–bodyresonance................................ 114
6.4 Degenerateperturbation theory .............................. 115
6.4.1 The precession of the equinoxes . . ...................... 116
6.5 Birkhoff’s normal form...................................... 118
6.5.1 Normal form around an equilibrium position ............. 118
6.5.2 Normalformaround closed trajectories ................. 121
6.6 Theaveragingtheorem...................................... 121
6.6.1 Anexample ......................................... 124
7 Inv ariant tori ................................................. 127
7.1 Theexistence of KAM tori .................................. 127
7.2 KAMtheory............................................... 131
7.2.1 TheKAMtheorem................................... 131
7.2.2 The initial approximation and the estimate
oftheerror term..................................... 140
7.2.3 Diophantinerotationnumbers ......................... 143
Table of Contents IX
7.2.4 Trappingdiophantinenumbers......................... 145
7.2.5 Computer–assisted proofs ............................. 148
7.3 Asurveyof KAM results in CelestialMechanics................ 149
7.3.1 Rotationaltoriin the spin–orbit problem................ 149
7.3.2 Librational invariant surfaces in the spin–orbit problem . . . 150
7.3.3 Thespatial planetary three–body problem .............. 152
7.3.4 Thecircular, planar, restricted three–body problem ...... 153
7.4 Greene’smethodforthe breakdown threshold.................. 156
7.5 Low–dimensionaltori ....................................... 160
7.6 AdissipativeKAM theorem ................................. 162
7.7 ConverseKAM............................................. 165
7.7.1 Conjugatepointscriterion............................. 168
7.7.2 Cone-crossing criterion................................ 169
7.7.3 Tangentorbitindicator ............................... 170
7.8 Cantori ................................................... 173
8 Long–time stability ........................................... 177
8.1 Arnold’sdiffusion........................................... 177
8.2 Nekhoroshev’stheorem...................................... 178
8.3 Nekhoroshev’s estimates around elliptic equilibria . ............. 182
8.4 Effectiveestimates in the three–body problem.................. 183
8.4.1 Exponential stability of a three–body problem . . ......... 183
8.5 Effective stability of the Lagrangian points . . . ................. 187
9 Determination of periodic orbits .............................. 191
9.1 Existenceof periodic orbits .................................. 191
9.1.1 Existenceof periodicorbits(conservativesetting) ........ 191
9.1.2 Computationofthe libration in longitude ............... 193
9.1.3 Existenceof periodicorbits(dissipativesetting).......... 194
9.1.4 Normalformaround a periodicorbit ................... 196
9.2 The Lindstedt–Poincar´etechnique ............................ 198
9.3 TheKBM method.......................................... 199
9.4 Lyapunov’s theorem . ....................................... 200
9.4.1 Families of periodic orbits ............................. 200
9.4.2 An example: the J
2
–problem .......................... 202
9.4.3 Linearization of the Hamiltonian around the
equilibrium point . . . ................................. 203
9.4.4 Application of Lyapunov’s theorem . .................... 204
10 Regularization theory ......................................... 207
10.1 The Levi–Civita transformation . . . ........................... 207
10.1.1 The two–body problem ............................... 207
10.1.2 The planar, circular, restrictedthree–bodyproblem ...... 211
10.2 TheKustaanheimo–Stiefel regularization ...................... 214
10.2.1 The restricted, spatialthree–bodyproblem.............. 214
10.2.2 The KS–transformation............................... 215
10.2.3 Canonicity of the KS–transformation ................... 218