Stability and Chaos in Celestial Mechanics (eBook)

Dedication 5
Table of Contents 6
Preface 10
Acknowledgments 13
1 Order and chaos 15
1.1 Continuous and discrete systems 15
1.2 Linear stability 18
1.3 Conservative and dissipative systems 20
1.4 The attractors and basins of attraction 21
1.5 The logistic map 23
1.6 The standard map 26
1.7 The dissipative standard map 29
1.8 Hénon’s mapping 31
2 Numerical dynamical methods 34
2.1 Poincaré map 34
2.2 Lyapunov exponents 36
2.3 The attractor’s dimension 38
2.4 Time series analysis 39
2.5 Fourier analysis 44
2.6 Frequency analysis 45
2.7 Hénon’s method 46
2.8 Fast Lyapunov Indicators 48
3 Kepler’s problem 51
3.1 The motion of the barycenter 52
3.2 The solution of Kepler’s problem 53
3.3 ˜ f and ˜g series 56
3.4 Elliptic motion 56
3.4.1 Mean and eccentric anomaly 58
3.4.2 Solution of Kepler’s equation 60
3.5 Parabolic motion 61
3.6 Hyperbolic motion 62
3.7 Classification of the orbits 63
3.8 Spacecraft transfers 65
3.9 Delaunay variables 65
3.10 The two–body problem with variable mass 69
3.10.1 The rocket equation 69
3.10.2 Gylden’s problem 70
4 The three–body problem and the Lagrangian solutions 74
4.1 The restricted three–body problem 74
4.1.1 The planar, circular, restricted three–body problem 74
4.1.2 Expansion of the perturbing function 76
4.1.3 The planar, elliptic, restricted three–body problem 78
4.1.4 The inclined, circular, restricted three–body problem 78
4.2 The circular, restricted Lagrangian solutions 79
4.3 The elliptic, restricted Lagrangian solutions 84
4.4 The elliptic, unrestricted triangular solutions 87
5 Rotational dynamics 93
5.1 Euler angles 93
5.2 Andoyer–Deprit variables 95
5.3 Free rigid body motion 97
5.4 Perturbed rigid body motion 99
5.5 The spin–orbit problem 101
5.5.1 The conservative spin–orbit problem 101
5.5.2 The averaged equation 104
5.5.3 The dissipative spin–orbit problem 105
5.5.4 The discrete spin–orbit problem 106
5.6 Motion around an oblate primary 107
5.7 Interaction between two bodies of finite dimensions 108
5.8 The tether satellite 109
5.9 The dumbbell satellite 113
6 Perturbation theory 117
6.1 Nearly–integrable Hamiltonian systems 117
6.2 Classical perturbation theory 118
6.2.1 An example 120
6.2.2 Computation of the precession of the perihelion 122
6.3 Resonant perturbation theory 122
6.3.1 Three–body resonance 124
6.4 Degenerate perturbation theory 125
6.4.1 The precession of the equinoxes 126
6.5 Birkhoff’s normal form 128
6.5.1 Normal form around an equilibrium position 128
6.5.2 Normal form around closed trajectories 131
6.6 The averaging theorem 131
6.6.1 An example 134
7 Invariant tori 136
7.1 The existence of KAM tori 136
7.2 KAM theory 140
7.2.1 The KAM theorem 140
7.2.2 The initial approximation and the estimate of the error term 149
7.2.3 Diophantine rotation numbers 152
7.2.4 Trapping diophantine numbers 154
7.2.5 Computer–assisted proofs 157
7.3 A survey of KAM results in Celestial Mechanics 158
7.3.1 Rotational tori in the spin–orbit problem 158
7.3.2 Librational invariant surfaces in the spin–orbit problem 159
7.3.3 The spatial planetary three–body problem 161
7.3.4 The circular, planar, restricted three–body problem 162
7.4 Greene’s method for the breakdown threshold 165
7.5 Low–dimensional tori 169
7.6 A dissipative KAM theorem 171
7.7 Converse KAM. 174
7.7.1 Conjugate points criterion 177
7.7.2 Cone-crossing criterion 178
7.7.3 Tangent orbit indicator 179
7.8 Cantori 182
8 Long–time stability 186
8.1 Arnold’s diffusion 186
8.2 Nekhoroshev’s theorem 187
8.3 Nekhoroshev’s estimates around elliptic equilibria 191
8.4 Effective estimates in the three–body problem 192
8.4.1 Exponential stability of a three–body problem 192
8.5 Effective stability of the Lagrangian points 196
9 Determination of periodic orbits . 200
9.1 Existence of periodic orbits 200
9.1.1 Existence of periodic orbits (conservative setting) 200
9.1.2 Computation of the libration in longitude 202
9.1.3 Existence of periodic orbits (dissipative setting) 203
9.1.4 Normal form around a periodic orbit 205
9.2 The Lindstedt–Poincar´e technique 207
9.3 The KBM method 208
9.4 Lyapunov’s theorem 209
9.4.1 Families of periodic orbits 209
9.4.2 An example: the J2–problem 211
9.4.3 Linearization of the Hamiltonian around the equilibrium point 212
9.4.4 Application of Lyapunov’s theorem 213
10 Regularization theory 215
10.1 The Levi–Civita transformation 215
10.1.1 The two–body problem 215
10.1.2 The planar, circular, restricted three–body problem 219
10.2 The Kustaanheimo–Stiefel regularization 222
10.2.1 The restricted, spatial three–body problem 222
10.2.2 The KS–transformation 223
10.2.3 Canonicity of the KS–transformation 226
10.3 The Birkhoff regularization 230
10.3.1 The B3 regularization 233
A Basics of Hamiltonian dynamics 235
A.1 The Hamiltonian setting 235
A.2 Canonical transformations 237
A.3 Integrable systems 240
A.4 Action–angle variables 241
B The sphere of influence 244
C Expansion of the perturbing function 246
D Floquet theory and Lyapunov exponents 247
E The planetary problem 248
F Yoshida’s symplectic integrator 250
G Astronomical data 251
References 254
Index 261