The Kepler Problem: Group Theoretical Aspects, Regularization and Quantization, with Application to the Study of Perturbations (Progress in Mathematical Physics)

This is a book dealing with the subject in an original way, presenting the many aspects of the modern Kepler Problem, and is subdivided in four parts: Elementary Theory, Group-Geometric Theory, Perturbation Theory and Appendices.The non-specialist should read first the Appendices, which effectively outline topics as Differential Geometry, Lie Groups and Algebras and Lagrangian and Hamiltonian Dynamics, in order to fully understand the more advanced points in the book.The first part, Elementary Theory, deals with the classic issues relevant to the Kepler Problem: from the basic facts on the conics, the Kepler Equation, to the elements of the orbit for the most interesting case of negative energy. Other modern, and more advanced, related topics are also treated here concerning, e.g., the Dealaunay and the Pauli variables, the Schrödinger Equation for the Kepler Problem and three regularization methods, which help to exhibit the various aspects of the symmetries of this Problem.The second part, Group-Geometric Theory, is highly technical (and challenging) and is the most important of the book, aiming to provide a deeper understanding of the geometric structure of the Kepler Problem. Here especially the sixth chapter, on conformal regularization, is fundamental for the subsequent development. In the other chapters of this part, topics such as, e.g., spinorial regularization and geometric quantization are clearly developed.In the third part, the Perturbation Theory of the Kepler Problem is being faced and, firstly, a preliminary, useful presentation of the methods of general perturbation theory is given. Soon after, the specific perturbations of the Kepler Problem are studied and a method is presented which avoids the drawbacks of coordinates which are not global, by the adoption of the Fock parameters. Such parameters are well suited also for the numerical integration, and a method involving them is used and implemented in the nice KEPLER program, which is provided on a CD ROM attached to the book.Surely this is a challenging book and, maybe, the availability of a number of exercises - presently missing - would contribute to confirm the non-specialist on the correct understanding of the concepts. In any case, the clear expository style, together with the support of the useful Appendices, considerably help the reader, who can appreciate the fascinating character of this matter.

The proof given by Newton of the fact that the motion ofplanets satisfies the three Kepler's laws is one of the mostimpressive and beautiful intellectual conquests of mankind.(I was little more than a child when one of my teachers toldme about this fact, and, after 30 years, I still rememberthat day.) After more than three centuries, the Keplerproblem still plays an important role in mathematics andphysics, having relations with group theory, spinor theory,separation of variables, perturbation theory, and quantummechanics.This book is pedagogically very effective, since itgradually leads the reader from the basic definitions ofconics (i.e., ellipse, parabola, and hyperbola) to theregularization of the Kepler problem, its quantization, andits perturbation theory. There are a lot of appendices wherethe mathematical tools are briefly (but precisely) recalled,and a huge number of beautiful figures. The author gaveimportant contributions to understand the mathematicalaspects of the Kepler problem and has a long teachingexperience. I think that this book can be enjoyed by a verywide range of readers.