This book is a remarkable introduction from various points of view (differential, topological, analytical and algebro-geometric) of a historical, but nowadays in top of research, theory, namely that of completely integrable Hamiltonian systems. Although, the most recent framework of these systems is provided by Poisson geometry, the author restricts to the case of symplectic manifolds in order to handle a lot of examples with a mechanical or geometrical flavor such as Hénon-Heiles, the simple and spherical pendulum, the anharmonic oscillator, the rigid body with a fixed point and so on.
The organization of the book is excellent. Its first chapter contains all the background material in symplectic geometry which is needed. The technical part of the book covers the next three chapters and is presented in a highly professional way. More precisely, the second chapter provides an introduction to the main tool of the theory, namely action-angle variables, through the celebrated Liouville-Arnold theorem. The next chapter is devoted to the algebraic approach based on differential Galois theory. More specifically, the author emphasizes the role of a theorem by J. J. Moralez-Ruiz and J. P. Ramis [Methods Appl. Anal. 8, No. 1, 33–95 (2001; Zbl 1140.37352); ibid., 97–111 (2001; Zbl 1140.37354)] in order to show that certain Hamiltonian systems are not completely integrable. The last chapter turns to algebraic geometry which in some cases is able to give a more tractable version of the Liouville-Arnold theorem. Finally, two appendices summarize useful definitions and properties from differential Galois theory and the theory of algebraic curves, respectively.
As a reference work, this monograph is invaluable while as a work of pedagogy, through its several examples and exercises, could be used in a graduate course being an extremely helpful textbook. Complemented sometimes with the author’s previous work [Spinning tops. A course on integrable systems, Cambridge Studies in Advanced Mathematics, 51, Cambridge: Cambridge Univ. Press (1996; Zbl 0867.58034)] this book will become a standard reference in this field.