The Kepler Problem Group Theoretical Aspects, Regularization and Quantization, with Application to the Study of Perturbations

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.

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.