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**Hamiltonian systems: chaos and quantization.**
*(English)*
Zbl 0734.58002

Cambridge Monographs on Mathematical Physics. Cambridge etc.: Cambridge University Press (ISBN 0-521-38670-5/pbk; 0-521-34531-6/hbk; 978-0-511-56416-1/ebook). ix, 238 p. (1988).

The discovery of chaotic behaviour in deterministic dynamical systems has had a profound effect in many areas of physics. There is now a large literature on this subject, however, a physicist just entering this beautiful field, or an advanced graduate student, still finds the need for a concise introduction to the basic concepts in an unsophisticated mathematical language. This is in fact the declarating goal of this well written book.

The material is organized in two parts. Part I - Classical dynamics - with six chapters namely: Linear dynamical systems, Nonlinear systems, Chaotic motion, Normal forms, Maps on the circle and Integrable and quasi-integrable systems and Part II - Quantum dynamics - with three chapters: Torus quantization, Quantization of ergodic systems and Periodic orbits in quantum mechanics.

Part I begins with linearization around a fixed point and proceeds to consider its structural stability, that is, the qualitative effect of a nonlinear perturbation. Chaotic motion appears as the natural consequence of homoclinic intersections of the separatrices emanating from unstable fixed points. The author then applies some of the basic concepts of ergodic theory to the study of the proliferation of periodic orbits in chaotic systems. Infinite families of fixed points are calculated analytically as an application of the theory of normal forms. In the last chapter the focus turns to the invariant tori of integrable systems and the delicate problem of their preservation under perturbations - the so called KAM theory.

Part II is dedicated to the semiclassical limit of quantum mechanics. It presents mainly the traditional semiclassical theory of quantization of Lagrangian surfaces, which goes back to the work of Dirac, Einstein, Brillouin, Keller and Maslov. The stationary states correspond to the invariant tori of classically integrable systems. This identification becomes even tighter in the Wigner-Weyl phase space representation, which is also nicely discussed. The time evolution of an open Lagrangian surface corresponds to a nonstationary state with which he starts his study. In the last chapter the author returns to the classical periodic orbits and explains their remarkable effect on the eigenstates and spectrum of the quantum Hamiltonians.

Written with the hand of a specialist the book will be very useful to all mathematicians and physicists who work in these modern and exciting topics.

The material is organized in two parts. Part I - Classical dynamics - with six chapters namely: Linear dynamical systems, Nonlinear systems, Chaotic motion, Normal forms, Maps on the circle and Integrable and quasi-integrable systems and Part II - Quantum dynamics - with three chapters: Torus quantization, Quantization of ergodic systems and Periodic orbits in quantum mechanics.

Part I begins with linearization around a fixed point and proceeds to consider its structural stability, that is, the qualitative effect of a nonlinear perturbation. Chaotic motion appears as the natural consequence of homoclinic intersections of the separatrices emanating from unstable fixed points. The author then applies some of the basic concepts of ergodic theory to the study of the proliferation of periodic orbits in chaotic systems. Infinite families of fixed points are calculated analytically as an application of the theory of normal forms. In the last chapter the focus turns to the invariant tori of integrable systems and the delicate problem of their preservation under perturbations - the so called KAM theory.

Part II is dedicated to the semiclassical limit of quantum mechanics. It presents mainly the traditional semiclassical theory of quantization of Lagrangian surfaces, which goes back to the work of Dirac, Einstein, Brillouin, Keller and Maslov. The stationary states correspond to the invariant tori of classically integrable systems. This identification becomes even tighter in the Wigner-Weyl phase space representation, which is also nicely discussed. The time evolution of an open Lagrangian surface corresponds to a nonstationary state with which he starts his study. In the last chapter the author returns to the classical periodic orbits and explains their remarkable effect on the eigenstates and spectrum of the quantum Hamiltonians.

Written with the hand of a specialist the book will be very useful to all mathematicians and physicists who work in these modern and exciting topics.

Reviewer: M.Puta (Timişoara)

### MSC:

58-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to global analysis |

37J99 | Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems |

53D50 | Geometric quantization |

37D45 | Strange attractors, chaotic dynamics of systems with hyperbolic behavior |

37G05 | Normal forms for dynamical systems |

81-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to quantum theory |