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


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2 of 2 people found the following review helpful
5.0 out of 5 stars Original, modern book on the Kepler Problem, January 9, 2004
By 
"vdirgeod" (Florence, Italy) - 
This review is from: The Kepler Problem: Group Theoretical Aspects, Regularization and Quantization, with Application to the Study of Perturbations (Progress in Mathematical Physics) (Hardcover)
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.

3 of 3 people found the following review helpful
5.0 out of 5 stars Kepler, what's your problem?, November 24, 2003
By 
Marco Pedroni (Milan, Italy) - 
This review is from: The Kepler Problem: Group Theoretical Aspects, Regularization and Quantization, with Application to the Study of Perturbations (Progress in Mathematical Physics) (Hardcover)
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.