Stability and Chaos in Celestial Mechanics

Cover......Page 1
Title: Stability and Chaos in Celestial Mechanics......Page 4
Copyright - ISBN: 3540851453......Page 5
Dedication......Page 6
Table of Contents......Page 8
Preface......Page 12
Acknowledgments......Page 15
1.1 Continuous and discrete systems......Page 18
1.2 Linear stability......Page 21
1.3 Conservative and dissipative systems......Page 23
1.4 The attractors and basins of attraction......Page 24
1.5 The logistic map......Page 26
1.6 The standard map......Page 29
1.7 The dissipative standard map......Page 32
1.8 Hénon’s mapping......Page 34
2.1 Poincaré map......Page 38
2.2 Lyapunov exponents......Page 40
2.3 The attractor’s dimension......Page 42
2.4 Time series analysis......Page 43
2.5 Fourier analysis......Page 48
2.6 Frequency analysis......Page 49
2.7 Hénon’s method......Page 50
2.8 Fast Lyapunov Indicators......Page 52
3 Kepler’s problem......Page 56
3.1 The motion of the barycenter......Page 57
3.2 The solution of Kepler’s problem......Page 58
3.4 Elliptic motion......Page 61
3.4.1 Mean and eccentric anomaly......Page 63
3.4.2 Solution of Kepler’s equation......Page 65
3.5 Parabolic motion......Page 66
3.6 Hyperbolic motion......Page 67
3.7 Classification of the orbits......Page 68
3.9 Delaunay variables......Page 70
3.10.1 The rocket equation......Page 74
3.10.2 Gylden’s problem......Page 75
4.1.1 The planar, circular, restricted three–body problem......Page 80
4.1.2 Expansion of the perturbing function......Page 82
4.1.4 The inclined, circular, restricted three–body problem......Page 84
4.2 The circular, restricted Lagrangian solutions......Page 85
4.3 The elliptic, restricted Lagrangian solutions......Page 90
4.4 The elliptic, unrestricted triangular solutions......Page 93
5.1 Euler angles......Page 100
5.2 Andoyer–Deprit variables......Page 102
5.3 Free rigid body motion......Page 104
5.4 Perturbed rigid body motion......Page 106
5.5.1 The conservative spin–orbit problem......Page 108
5.5.2 The averaged equation......Page 111
5.5.3 The dissipative spin–orbit problem......Page 112
5.5.4 The discrete spin–orbit problem......Page 113
5.6 Motion around an oblate primary......Page 114
5.7 Interaction between two bodies of finite dimensions......Page 115
5.8 The tether satellite......Page 116
5.9 The dumbbell satellite......Page 120
6.1 Nearly–integrable Hamiltonian systems......Page 124
6.2 Classical perturbation theory......Page 125
6.2.1 An example......Page 127
6.3 Resonant perturbation theory......Page 129
6.3.1 Three–body resonance......Page 131
6.4 Degenerate perturbation theory......Page 132
6.4.1 The precession of the equinoxes......Page 133
6.5.1 Normal form around an equilibrium position......Page 135
6.6 The averaging theorem......Page 138
6.6.1 An example......Page 141
7.1 The existence of KAM tori......Page 144
7.2.1 The KAM theorem......Page 148
7.2.2 The initial approximation and the estimate of the error term......Page 157
7.2.3 Diophantine rotation numbers......Page 160
7.2.4 Trapping diophantine numbers......Page 162
7.2.5 Computer–assisted proofs......Page 165
7.3.1 Rotational tori in the spin–orbit problem......Page 166
7.3.2 Librational invariant surfaces in the spin–orbit problem......Page 167
7.3.3 The spatial planetary three–body problem......Page 169
7.3.4 The circular, planar, restricted three–body problem......Page 170
7.4 Greene’s method for the breakdown threshold......Page 173
7.5 Low–dimensional tori......Page 177
7.6 A dissipative KAM theorem......Page 179
7.7 Converse KAM.......Page 182
7.7.1 Conjugate points criterion......Page 185
7.7.2 Cone-crossing criterion......Page 186
7.7.3 Tangent orbit indicator......Page 187
7.8 Cantori......Page 190
8.1 Arnold’s diffusion......Page 194
8.2 Nekhoroshev’s theorem......Page 195
8.3 Nekhoroshev’s estimates around elliptic equilibria......Page 199
8.4.1 Exponential stability of a three–body problem......Page 200
8.5 Effective stability of the Lagrangian points......Page 204
9.1.1 Existence of periodic orbits (conservative setting)......Page 208
9.1.2 Computation of the libration in longitude......Page 210
9.1.3 Existence of periodic orbits (dissipative setting)......Page 211
9.1.4 Normal form around a periodic orbit......Page 213
9.2 The Lindstedt–Poincaré technique......Page 215
9.3 The KBM method......Page 216
9.4.1 Families of periodic orbits......Page 217
9.4.2 An example: the J_2–problem......Page 219
9.4.3 Linearization of the Hamiltonian around the equilibrium point......Page 220
9.4.4 Application of Lyapunov’s theorem......Page 221
10.1.1 The two–body problem......Page 224
10.1.2 The planar, circular, restricted three–body problem......Page 228
10.2.1 The restricted, spatial three–body problem......Page 231
10.2.2 The KS–transformation......Page 232
10.2.3 Canonicity of the KS–transformation......Page 235
10.3 The Birkhoff regularization......Page 239
10.3.1 The B3 regularization......Page 242
A.1 The Hamiltonian setting......Page 244
A.2 Canonical transformations......Page 246
A.3 Integrable systems......Page 249
A.4 Action–angle variables......Page 250
B The sphere of influence......Page 254
C Expansion of the perturbing function......Page 256
D Floquet theory and Lyapunov exponents......Page 258
E The planetary problem......Page 260
F Yoshida’s symplectic integrator......Page 262
G Astronomical data......Page 264
References......Page 267
Index......Page 274