Celletti Stability and Chaos in Celestial Mechanics

1. Auflage 2010
ISBN: 978-3-540-85146-2
Verlag: Springer Berlin Heidelberg
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 264 Seiten

Reihe: Springer Praxis Books

This overview of classical celestial mechanics focuses the interplay with dynamical systems. Paradigmatic models introduce key concepts - order, chaos, invariant curves and cantori - followed by the investigation of dynamical systems with numerical methods.

Autoren/Hrsg.

Celletti, Alessandra

Inhaltsverzeichnis

1;Dedication;5
2;Table of Contents;6
3;Preface;10
4;Acknowledgments;13
5;1 Order and chaos;15
5.1;1.1 Continuous and discrete systems;15
5.2;1.2 Linear stability;18
5.3;1.3 Conservative and dissipative systems;20
5.4;1.4 The attractors and basins of attraction;21
5.5;1.5 The logistic map;23
5.6;1.6 The standard map;26
5.7;1.7 The dissipative standard map;29
5.8;1.8 Hénon’s mapping;31
6;2 Numerical dynamical methods;34
6.1;2.1 Poincaré map;34
6.2;2.2 Lyapunov exponents;36
6.3;2.3 The attractor’s dimension;38
6.4;2.4 Time series analysis;39
6.5;2.5 Fourier analysis;44
6.6;2.6 Frequency analysis;45
6.7;2.7 Hénon’s method;46
6.8;2.8 Fast Lyapunov Indicators;48
7;3 Kepler’s problem;51
7.1;3.1 The motion of the barycenter;52
7.2;3.2 The solution of Kepler’s problem;53
7.3;3.3 ˜ f and ˜g series;56
7.4;3.4 Elliptic motion;56
7.4.1;3.4.1 Mean and eccentric anomaly;58
7.4.2;3.4.2 Solution of Kepler’s equation;60
7.5;3.5 Parabolic motion;61
7.6;3.6 Hyperbolic motion;62
7.7;3.7 Classification of the orbits;63
7.8;3.8 Spacecraft transfers;65
7.9;3.9 Delaunay variables;65
7.10;3.10 The two–body problem with variable mass;69
7.10.1;3.10.1 The rocket equation;69
7.10.2;3.10.2 Gylden’s problem;70
8;4 The three–body problem and the Lagrangian solutions;74
8.1;4.1 The restricted three–body problem;74
8.1.1;4.1.1 The planar, circular, restricted three–body problem;74
8.1.2;4.1.2 Expansion of the perturbing function;76
8.1.3;4.1.3 The planar, elliptic, restricted three–body problem;78
8.1.4;4.1.4 The inclined, circular, restricted three–body problem;78
8.2;4.2 The circular, restricted Lagrangian solutions;79
8.3;4.3 The elliptic, restricted Lagrangian solutions;84
8.4;4.4 The elliptic, unrestricted triangular solutions;87
9;5 Rotational dynamics;93
9.1;5.1 Euler angles;93
9.2;5.2 Andoyer–Deprit variables;95
9.3;5.3 Free rigid body motion;97
9.4;5.4 Perturbed rigid body motion;99
9.5;5.5 The spin–orbit problem;101
9.5.1;5.5.1 The conservative spin–orbit problem;101
9.5.2;5.5.2 The averaged equation;104
9.5.3;5.5.3 The dissipative spin–orbit problem;105
9.5.4;5.5.4 The discrete spin–orbit problem;106
9.6;5.6 Motion around an oblate primary;107
9.7;5.7 Interaction between two bodies of finite dimensions;108
9.8;5.8 The tether satellite;109
9.9;5.9 The dumbbell satellite;113
10;6 Perturbation theory;117
10.1;6.1 Nearly–integrable Hamiltonian systems;117
10.2;6.2 Classical perturbation theory;118
10.2.1;6.2.1 An example;120
10.2.2;6.2.2 Computation of the precession of the perihelion;122
10.3;6.3 Resonant perturbation theory;122
10.3.1;6.3.1 Three–body resonance;124
10.4;6.4 Degenerate perturbation theory;125
10.4.1;6.4.1 The precession of the equinoxes;126
10.5;6.5 Birkhoff’s normal form;128
10.5.1;6.5.1 Normal form around an equilibrium position;128
10.5.2;6.5.2 Normal form around closed trajectories;131
10.6;6.6 The averaging theorem;131
10.6.1;6.6.1 An example;134
11;7 Invariant tori;136
11.1;7.1 The existence of KAM tori;136
11.2;7.2 KAM theory;140
11.2.1;7.2.1 The KAM theorem;140
11.2.2;7.2.2 The initial approximation and the estimate of the error term;149
11.2.3;7.2.3 Diophantine rotation numbers;152
11.2.4;7.2.4 Trapping diophantine numbers;154
11.2.5;7.2.5 Computer–assisted proofs;157
11.3;7.3 A survey of KAM results in Celestial Mechanics;158
11.3.1;7.3.1 Rotational tori in the spin–orbit problem;158
11.3.2;7.3.2 Librational invariant surfaces in the spin–orbit problem;159
11.3.3;7.3.3 The spatial planetary three–body problem;161
11.3.4;7.3.4 The circular, planar, restricted three–body problem;162
11.4;7.4 Greene’s method for the breakdown threshold;165
11.5;7.5 Low–dimensional tori;169
11.6;7.6 A dissipative KAM theorem;171
11.7;7.7 Converse KAM.;174
11.7.1;7.7.1 Conjugate points criterion;177
11.7.2;7.7.2 Cone-crossing criterion;178
11.7.3;7.7.3 Tangent orbit indicator;179
11.8;7.8 Cantori;182
12;8 Long–time stability;186
12.1;8.1 Arnold’s diffusion;186
12.2;8.2 Nekhoroshev’s theorem;187
12.3;8.3 Nekhoroshev’s estimates around elliptic equilibria;191
12.4;8.4 Effective estimates in the three–body problem;192
12.4.1;8.4.1 Exponential stability of a three–body problem;192
12.5;8.5 Effective stability of the Lagrangian points;196
13;9 Determination of periodic orbits .;200
13.1;9.1 Existence of periodic orbits;200
13.1.1;9.1.1 Existence of periodic orbits (conservative setting);200
13.1.2;9.1.2 Computation of the libration in longitude;202
13.1.3;9.1.3 Existence of periodic orbits (dissipative setting);203
13.1.4;9.1.4 Normal form around a periodic orbit;205
13.2;9.2 The Lindstedt–Poincar´e technique;207
13.3;9.3 The KBM method;208
13.4;9.4 Lyapunov’s theorem;209
13.4.1;9.4.1 Families of periodic orbits;209
13.4.2;9.4.2 An example: the J2–problem;211
13.4.3;9.4.3 Linearization of the Hamiltonian around the equilibrium point;212
13.4.4;9.4.4 Application of Lyapunov’s theorem;213
14;10 Regularization theory;215
14.1;10.1 The Levi–Civita transformation;215
14.1.1;10.1.1 The two–body problem;215
14.1.2;10.1.2 The planar, circular, restricted three–body problem;219
14.2;10.2 The Kustaanheimo–Stiefel regularization;222
14.2.1;10.2.1 The restricted, spatial three–body problem;222
14.2.2;10.2.2 The KS–transformation;223
14.2.3;10.2.3 Canonicity of the KS–transformation;226
14.3;10.3 The Birkhoff regularization;230
14.3.1;10.3.1 The B3 regularization;233
15;A Basics of Hamiltonian dynamics;235
15.1;A.1 The Hamiltonian setting;235
15.2;A.2 Canonical transformations;237
15.3;A.3 Integrable systems;240
15.4;A.4 Action–angle variables;241
16;B The sphere of influence;244
17;C Expansion of the perturbing function;246
18;D Floquet theory and Lyapunov exponents;247
19;E The planetary problem;248
20;F Yoshida’s symplectic integrator;250
21;G Astronomical data;251
22;References;254
23;Index;261