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Full Description
Starting from ancient astronomy, this text follows the development of celestial mechanics culminating in applications of the most recent results concerning stability of planetary orbits: Kolmogorov's and Nekhoroshev's theorems. Key topics covered include: a historical introduction from ancient astronomy to Kepler and Newton; Lagrange's perturbation theory; the problem of three bodies, with a discussion of Levi-Civita regularization and of Sundman's theorem; methods of algebraic calculation of perturbation series, including a discussion of non-convergence due to the accumulation of small divisors; and a complete application of Kolmogorov's and Nekhoroshev's theorems. Written in an accessible, self-contained way with few prerequisites, this book can serve as an introductory text for senior undergraduate and graduate students, and for young researchers. Its approach allows students to learn about perturbation methods leading to advanced results.
Contents
Apology; Plan of the book; Part I. From Ancient Astronomy to Newton: 1. Ouverture; 2. Kinematics of the Keplerian model; 3. The gravitation of Newton; Part II. From Newton to Poincaré:4. Integrability and action-angle variables; 5. The perturbing function; 6. Classical perturbation methods; 7. The problem of three bodies; 8. Regularization of collisions; Part III. Modern Celestial Mechanics: 9. A toolbox of perturbation methods; 10. Computer-assisted methods for KAM theory; 11. Rotational dynamics of celestial bodies; 12. Stability of planetary systems; A. Analytical tools; B. Indexing functions; C. Validated numerics in CAP; D. Benchmarking CAP with a simple model; References; Index.
