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Global aspects of classical integrable systems. (English) Zbl 0882.58023

Basel: Birkhäuser. xvi, 435 p. (1997).
This is in some sense a heterodox book. Indeed, it starts with examples and includes the theory in five appendices at the end. So the reader can choose to go to the theory and after that work out the examples, or vice versa. Both ways are possible and fruitful. A third possibility is to browse along the book as the authors ask in the introduction.
The book is devoted to the study of global geometric features of five classical integrable systems, one per chapter. They are the harmonic oscillator (Chapter I), the Kepler problem (Chapter II), the Euler top (Chapter III), the spherical pendulum (Chapter IV), and the Lagrange top (Chapter V). Actually, the authors also consider the simple pendulum in the introduction.
More or less, the treatment is similar in each chapter. The qualitative behavior of the reduced system is discussed, and the energy momentum mapping is carefully analyzed. Also, the reconstruction of the dynamics and the integration of the equations of motion are studied. It should be noticed that the reduction process is performed using invariant theory. The main advantage is that the authors construct the reduced space together with an embedding in Euclidean space, and thus the Poisson structure is easily obtained. This approach also permits to treat the singular case. As a matter of curiosity, the authors give a beautiful explanation of a twisting phenomenon which a tennis racket-like rigid body performs (this is just the subject of the book cover).
The appendices provide a good introduction to Hamiltonian mechanics. Appendix A is devoted to symplectic linear algebra, symplectic manifolds, Hamilton equations and Poisson brackets. Maybe the name of cosymplectic manifold is not the most appropriate since it is usually reserved for a kind of almost contact structure on odd-dimensional manifolds. Appendix B presents Hamiltonian systems with symmetry and the reduction theorem, even for the singular case. Ehresmann connections are discussed in Appendix C, and the existence of local action angle coordinates for an integrable Hamiltonian system is discussed in Appendix D, the obstruction to be global is just given by the monodromy (monodromy is indeed a leitmotiv in this book). Finally, a review of basic Morse theory is given in Appendix E. The book ends with a very interesting list of historical notes.
The book is well written, and the reader will find detailed proofs as well as a lot of visual explanations. The reviewer’s final advice is: please, buy the book and enjoy it!

MSC:

37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
37-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to dynamical systems and ergodic theory
70-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to mechanics of particles and systems
70H05 Hamilton’s equations
70H33 Symmetries and conservation laws, reverse symmetries, invariant manifolds and their bifurcations, reduction for problems in Hamiltonian and Lagrangian mechanics
37C75 Stability theory for smooth dynamical systems
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
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