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Celestial Dynamics: Chaoticity and Dynamics of Celestial Systems

ISBN: 978-3-527-40977-8
320 pages
May 2013
Celestial Dynamics: Chaoticity and Dynamics of Celestial Systems (3527409777) cover image
Written by an internationally renowned expert author and researcher, this monograph fills the need for a book conveying the sophisticated tools needed to calculate exo-planet motion and interplanetary space flight. It is unique in considering the critical problems of dynamics and stability, making use of the software Mathematica, including supplements for practical use of the formulae.
A must-have for astronomers and applied mathematicians alike.
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Preface XI

1 Introduction: the Challenge of Science 7

2 Hamiltonian Mechanics 7

2.1 Hamilton’s Equations from Hamiltonian Principle

2.2 Poissen Brackets 11

2.3 Canonical Transformations 13

2.4 Hamilton-Jacobi Theory 19

2.5 Action-Angle Variables 23

3 Numerical and Analytical Tools 27

3.1 Mappings 27

3.2 Lie-Series Numerical Integration 41

3.3 Chaos Indicators 48

3.4 Perturbation Theory 52

4 The Stability Problem 69

4.1 Review on Different Concepts of Stability 69

4.2 Integrable Systems 72

4.3 Nearly Integrable Systems 78

4.4 Resonance Dynamics 80

4.5 KAM Theorem 86

4.6 Nekhoroshev Theorem 91

4.7 The Froeschle-Guzzo-Lega Hamiltonian 99

5 The Two Body Problem 105

5.1 From Newton to Kepler 106

5.2 Unperturbed Kepler Motion 108

5.3 Classifications of Orbits: Ellipses, Hyperbolae and Parabolae 110

5.4 Kepler Equation 112

5.5 Complex Description 115

5.6 Motion in Space and the Keplerian Elements 118

5.7 Astronomical Determination of the Gravitational Constant 120

5.8 Solution of the Kepler Equation 120


6 The Restricted Three-Body Problem 123

6.1 Set-Up and Formulation 124

6.2 Equilibria of the System 127

6.3 Motion Close to L4 and L5 131

6.4 Motion Close to L1, L2, L3 134

6.5 Potential and the Zero Velocity Curves 136

6.6 Spatial Restricted Three-Body Problem 141

6.7. Tisserand Criterion 144

6.8 Elliptic Restricted Three- Body Problem 145

6.9 Dissipative Restricted Three Body Problem 146

7 The Sitnikov Problem 149

7.1 Circular Case: the MacMillan Problem 150

7.2 motion of the Planet off the z-Axes 153

7.3 Elliptic Case 157

7.4 The Vrabec Mapping 176

7.5 General Sitnikov Problem 180

8 Planetary Theory 185

8.1 Planetary Perturbation Theory 185

8.2 Equations of Motion for n Bodies 194

8.3 Lagrange Equations of the Planetary n Body Problem 198

8.4 The Perturbing Function in Elliptic Orbital Elements 203

8.5 Explicit First-Order Planetary Theory for the Osculating Elements 207

8.6 Small Divisors 211

8.7 Long-Term Evolution of Our Planetary System 213

9 Resonances 215

9.1 Mean Motion Resonances in Our Planetary System 215

9.2 Method of Laplace-Lagrange 223

9.3 Secular Resonances 231

9.4 Three-Body Resonances 239

10 Lunar Theory 249

10.1 Hill’s Lunar Theory 250

10.2 Classical Lunar Theory 261

10.3 Principal Inequalities 264

11 Concluding Remarks 271

Appendix A Important Persons in the Field 277

Appendix B Formulae 281

B.1 Hansen Coefficients 281

B.2 Laplace Coefficients 283

B.3 Bessel Functions 284

B.4 Expansions in the Two-Body Problem 289

Acknowledgement 293

References 295

Index 305

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Rudolf Dvorak, now serving at the Institute of Astronomy in Vienna, has gained research experience in planet research at international institutions including the Bureau des Longitudes in Paris. More than 130 publications provide proof.
Dr. Christoph Lhotka is member of Rudolf Dvorak's workgroup, and an expert with the software Mathematica.
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“In summary, despite the omissions mentioned above, this book is an interesting addition to the subject because the presentation is systematic and self-contained, and it contains a valuable material that would be useful to postgraduate students interested in the field and also to some people working in celestial mechanics.”  (Zentralblatt MATH, 1 March 2014)

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