Since a comprehensive review of the first English edition of this book is available [see (1988; Zbl 0716.53005); for the Russian original see (1988; Zbl 0751.53002)], we shall mention new features of this edition only. The original text is essentially unchanged, but a whole new chapter devoted to recent results on the topological classification of integrable non-degenerate Hamiltonian equations with two degrees of freedom is appended. The only visible changes to the first 5 chapters consist in the omission of sections 3.3.3, 5.8, and the slight extension of the original preface, explaining briefly the aims of the new chapter.
Already the first edition could be described as an unusual mathematical monograph, combining quite classical exposition of mathematical definitions, theorems and proofs, sketchy surveys with many intuitive links and remarks, and even bibliographic surveys. Now, the 6th chapter provides the reader with a mathematical essay starting with a nearly philosophical dispute on what the “physically relevant” integrable systems are. The actual mathematical problem is the study of integrable Hamiltonian systems on four dimensional symplectic phase manifolds, i.e., having two degrees of freedom, subject to several strong regularity conditions. Such a system can be reduced to a three-dimensional surface \(Q^3\) of constant energy, which is always assumed to be a smooth compact manifold. Then the restriction of the second integral \(f\) to \(Q^3\) must have critical points which cannot be isolated in general. The non-degenerate case is when the critical points of \(Q^3\) form smooth submanifolds. Thus \(f\) has to be a Bott function on each such \(Q^3\), the case found as most typical in physical systems [cf. A. T. Fomenko, ‘Integrability and nonintegrability in geometry and mechanics’, Kluwer Academic Publishers (1988; Zbl 0675.58018)].
Now, there is the obvious topological equivalence of two such Hamiltonian systems, requiring the existence of a diffeomorphism which preserves the Liouville’s foliations. In order to distinguish more easily the non-equivalent systems, the author introduces the so-called rough topological equivalence, a much coarser equivalence allowing surgeries along regular Liouville tori. The ultimate classification with respect to these equivalence relations is described in sections 6.5-6.10, summarizing a vast number of recent publications by the author and many of his collaborators, let us mention at least A. T. Fomenko and H. Zieschang [Math. USSR , Izv. 36, No. 3, 567–596 (1991); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 54, No. 3, 546-575 (1990; Zbl 0723.58024)], A. V. Bolsinov, S. V. Matveev and A. T. Fomenko, [Russ. Math. Surv. 45, No. 2, 59–94 (1990); translation from Usp. Mat. Nauk 45, No. 2(272), 49–77 (1990; Zbl 0705.58025)], A. T. Fomenko and T. Z. Nguyen [Adv. Sov. Math. 6, 267–296 (1991; Zbl 0744.58033)], A. A. Oshemkov [Adv. Sov. Math. 6, 67–146 (1991; Zbl 0745.58028)]. The algorithms computing the necessary topological invariants for both types of equivalences are followed by a historical survey and the last two sections present deep applications of the theory.
Reviewer’s remark: As mentioned in the review of the first edition, it is a pity that such an interesting topic is presented somewhat carelessly. In particular, most assumptions are made only implicitly and often even the results are presented in a quite fuzzy way. All this can be excused by the general features of the book, presenting an essay on very nice mathematics (accessible for wider audience), rather than a strict monographic exposition (aimed at a small group of experts). However, it is even harder to accept the poor quality of the English translation, which was already heavily attacked in the original reviews. At any places, mere following of the sense of sentences is quite difficult. It is hard to believe that this remained unnoticed by the publisher.