The Kepler Problem

Because of the correspondences existing among all levels of reality, truths pertaining to a lower level can be considered as symbols of truths at a higher level and can therefore be the "foundation" or support leading by analogy to a knowledge of the latter. This confers to every science a superior or "elevating" meaning, far deeper than its own original one. - R. GUENON, The Crisis of Modern World Having been interested in the Kepler Problem for a long time, I have al ways found it astonishing that no book has been written yet that would address all aspects of the problem. Besides hundreds of articles, at least three books (to my knowledge) have indeed been published al ready on the subject, namely Englefield (1972), Stiefel & Scheifele (1971) and Guillemin & Sternberg (1990). Each of these three books deals only with one or another aspect of the problem, though. For example, En glefield (1972) treats only the quantum aspects, and that in a local way. Similarly, Stiefel & Scheifele (1971) only considers the linearization of the equations of motion with application to the perturbations of celes tial mechanics. Finally, Guillemin & Sternberg (1990) is devoted to the group theoretical and geometrical structure.

"This is an interesting book, which well organizes the group-geometric aspects of the Kepler problem on which a great number of articles have been published along with the advance of symmetry theory. . . . a nice reference not only for graduate students but also for scientists who are interested in dynamical systems with symmetry." --MathSciNet

Texte du rabat

This book contains a comprehensive treatment of the Kepler problem, i.e., the two body problem. It is divided into four parts. In the first part, written at an undergraduate student level, the arguments are presented in an elementary fashion, and the properties of the problem are demonstrated in a purely computational manner. In the second part a unifying point of view, original to the author, is presented which centers the exposition on the intrinsic group-geometrical aspects. This part requires more mathematical background, which the reader will find in the fourth part, in particular, the basic tools of differential geometry and analytical mechanics used in the book. The third part exploits some results of the second part to give a geometrical description of the perturbation theory of the Kepler problem.

Each of the four parts, which are to some extent independent, could itself form the basis for a one-semester course. The accompanying CD contains mainly the Microsoft Windows program KEPLER developed by the author. This program calculates the effects of any perturbation of the Kepler problem and plots the resulting trajectories.

Contenu
Preface.- List of Figures.- 1 Introductory Survey.- 1.1 Part I - Elementary Theory.- 1.2 Part II - Group-Geometric Theory.- 1.3 Part III - Perturbation Theory.- 1.4 Part IV - Appendices.- I Elementary Theory 17.- 2 Basic Facts.- 3 Separation of Variables and Action-Angle Coordinates.- 4 Quantization of the Kepler Problem.- 5 Regularization and Symmetry.- II Group-Geometric Theory 109.- 6 Conformal Regularization.- 7 Spinorial Regularization.- 8 Return to Separation of Variables.- 9 Geometric Quantization.- 10 Kepler Problem with Magnetic Monopole.- III Perturbation Theory 235.- 11 General Perturbation Theory.- 12 Perturbations of the Kepler Problem.- 13 Perturbations with Axial Symmetry.- IV Appendices 321.- A Differential Geometry.- B Lie Groups and Lie Algebras.- C Lagrangian Dynamics.- D Hamiltonian Dynamics.