Geometry from dynamics, classical and quantum. (English) Zbl 1364.81001
Dordrecht: Springer (ISBN 978-94-017-9219-6/hbk; 978-94-017-9220-2/ebook). xxv, 719 p. (2015).
This impressive book takes the reader onto a long journey through the topological and geometrical settings behind the main topics of classical and quantum mechanics. As it is pointed out in the introduction, the possible motto of this work could be: “All geometric structures used in the description of the dynamics of a given physical system should be dynamically determined”. The book is completely self-contained, besides from references to the original literature (every chapter finishes with a reference list), for all of the presented results also the proofs are given.
The first chapter is an engrossing presentation of several examples, in both finite and infinite dimensions, where some of the main ideas in dealing with dynamical systems like constants of motion, symmetries, Lagrangian and Hamiltonian frameworks are recalled. Here, the authors give an excellent picture of their philosophy that “geometrically thinking will be always used as a guide, almost as a metalanguage, in analysis of the problems” (p. xi). Chapter 2 takes up the basic geometrical structures needed to continue the discussion started here: manifolds, bundles, vector fields, Lie groups, etc. Chapter 3 contains a detailed description of the geometrical setting of linear structures based on Euler or dilatation vector fields. Chapter 4 deals with the Poisson structures of the Hamiltonian world and a number of examples from group theory and harmonic analysis are discussed. The classical formulation of Lagrangian and Hamiltonian dynamics are the subject of Chapter 5. The next chapter turns to the Hermitian geometry of quantum mechanics. The rich technique of reduction is discussed into Chapter 7 while to the fundamental problem of integrability is devoted the whole Chapter 8. This long excursion is closed with a special class of dynamical systems called Lie-Scheffers. An enormous amount of information is packed in a hundred pages of the appendices.
Reading this book provides an excellent opportunity to practice and understand the geometric mechanics.
The first chapter is an engrossing presentation of several examples, in both finite and infinite dimensions, where some of the main ideas in dealing with dynamical systems like constants of motion, symmetries, Lagrangian and Hamiltonian frameworks are recalled. Here, the authors give an excellent picture of their philosophy that “geometrically thinking will be always used as a guide, almost as a metalanguage, in analysis of the problems” (p. xi). Chapter 2 takes up the basic geometrical structures needed to continue the discussion started here: manifolds, bundles, vector fields, Lie groups, etc. Chapter 3 contains a detailed description of the geometrical setting of linear structures based on Euler or dilatation vector fields. Chapter 4 deals with the Poisson structures of the Hamiltonian world and a number of examples from group theory and harmonic analysis are discussed. The classical formulation of Lagrangian and Hamiltonian dynamics are the subject of Chapter 5. The next chapter turns to the Hermitian geometry of quantum mechanics. The rich technique of reduction is discussed into Chapter 7 while to the fundamental problem of integrability is devoted the whole Chapter 8. This long excursion is closed with a special class of dynamical systems called Lie-Scheffers. An enormous amount of information is packed in a hundred pages of the appendices.
Reading this book provides an excellent opportunity to practice and understand the geometric mechanics.
Reviewer: Mircea Crâşmăreanu (Iaşi)
MSC:
| 81-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to quantum theory |
| 70-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to mechanics of particles and systems |
| 70H03 | Lagrange’s equations |
| 70H05 | Hamilton’s equations |
| 70H33 | Symmetries and conservation laws, reverse symmetries, invariant manifolds and their bifurcations, reduction for problems in Hamiltonian and Lagrangian mechanics |
| 70G60 | Dynamical systems methods for problems in mechanics |
| 70G65 | Symmetries, Lie group and Lie algebra methods for problems in mechanics |
| 81Q05 | Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics |
| 81Q10 | Selfadjoint operator theory in quantum theory, including spectral analysis |
