This is a book dealing with the subject in an original way, presenting the many aspects of the modern KeplerProblem, 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 of planets satisfies the three Kepler's laws is one of the most impressive and beautiful intellectual conquests of mankind. (I was little more than a child when one of my teachers told me about this fact, and, after 30 years, I still remember that day.) After more than three centuries, the Kepler problem still plays an important role in mathematics and physics, having relations with group theory, spinor theory, separation of variables, perturbation theory, and quantum mechanics. This book is pedagogically very effective, since it gradually leads the reader from the basic definitions of conics (i.e., ellipse, parabola, and hyperbola) to the regularization of the Kepler problem, its quantization, and its perturbation theory. There are a lot of appendices where the mathematical tools are briefly (but precisely) recalled, and a huge number of beautiful figures. The author gave important contributions to understand the mathematical aspects of the Kepler problem and has a long teaching experience. I think that this book can be enjoyed by a very wide range of readers.
This book contains a comprehensive treatment of the Keplerproblem, i.e., the two body problem.
It is divided into four parts. In the first part the arguments are exposed elementarily, and the properties of the problem are recovered in a purely computational way. This part is written at an undergraduate student level.
In the second part a unifying point of view, originally due to the author, is presented which centers the exposition on the intrinsic group-geometrical aspects. This part requires more mathematical background which is presented 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 form the basis for a one-semester course.
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