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Geometric mechanics and symmetry. From finite to infinite dimensions. (English) Zbl 1175.70001

Oxford Texts in Applied and Engineering Mathematics 12. Oxford: Oxford University Press (ISBN 978-0-19-921291-0/pbk; 978-0-19-921290-3/hbk). xvi, 515 p. (2009).
According to authors’ statements, this “is a textbook for graduate students that introduces geometric mechanics in finite and infinite dimensions”. The book has two rather diverse parts. Again in the authors’ words, “Part I is meant to be used as a textbook in an upper-level course on geometric mechanics. It contains many detailed explanations and exercises. Although a wide range of topics is treated, the introduction to each of them is meant to be gentle. Part II addresses a more advanced reader and focuses on recent applications of geometric mechanics in soliton theory, image analysis and fluid mechanics”.
This description gives a fair account of the book. In fact, the first part (of about 350 pages) is suitable for use as a textbook, with a blend of standard topics – both foundational and dealing, as promised in the title, with the interplay of mechanics and symmetry. The special attention is paid to the Poincaré formulation, as first laid down in a short note [H. Poincaré, “Sur une forme nouvelle des équations de la mécanique”, C. R. Acad. Sci. 132, 369–371 (1901; JFM 32.0715.01)], and to Euler-Poincaré reduction by symmetry, which is the leading theme of the book.
The first chapter gives a short account of Lagrangian and Hamiltonian mechanics, which includes a discussion of constraints. The second chapter is devoted to introducing manifolds and vectors and differentials; the connection between Lie groups, specifically matrix groups, and manifolds is emphasized. Chapter three presents the geometry of manifolds, vector fields and differential forms; this allows to introduce Riemannian symplectic geometry. In chapter four the material introduced so far is used to discuss mechanics – both Lagrangian and Hamiltonian – on manifolds; this is also the occasion for a first look at symmetry, reduction and conserved quantities.
Chapter five provides an introduction to Lie groups and algebras; the subsequent chapter six deals then with group actions, symmetries, quotient spaces and Poisson symmetry reduction. Chapter seven is devoted to Euler-Poincaré reduction, discussing rigid body dynamics and heavy tops, as well as Euler-Poincaré systems with advected parameters. In chapter eight the authors discuss the momentum map, while chapter nine provides a careful discussion of Lie-Poisson reduction. Finally, chapter ten ends Part I with a treatment of pseudo-rigid bodies.
As promised in the introduction, computations are given in full detail when appropriate, and the style of the book is constantly aware of the needs by students first exposed to the topics dealt with.
All chapters in Part I are complemented by exercises, and include solutions of selected ones (these are written by D. C. P. Ellis), which will prove specially helpful and are in many ways a relevant part of the book.
As for Part II (of about 150 pages), this is written in a quite different style and roughly speaking deals with infinite-dimensional mechanics as related to Arnold’s view of fluid mechanics as a geodesic flow on the group of diffeomorphisms [V. I. Arnold, Ann. Inst. Fourier 16, No. 1, 319–361 (1966; Zbl 0148.45301); V. I. Arnold and B. A. Khesin, Topological methods in hydrodynamics. New York, NY: Springer (1998; Zbl 0902.76001); D. G. Ebin and J. E. Marsden, Ann. Math. (2) 92, 102–163 (1970; Zbl 0211.57401)]. This part is not intended to be used as a textbook, and accordingly much less (but still enough) details are given; this is an unavoidable choice in view of the amount of material (and of its advanced level) which is presented in Part II. In fact, this part is reminiscent of the spirit behind the appendices in books [V. I. Arnold, Mathematical methods of classical mechanics. Translated by K. Vogtman and A. Weinstein. New York-Heidelberg-Berlin: Springer-Verlag (1978; Zbl 0386.70001); V. I. Arnold and A. Avez, Problèmes ergodiques de la mécanique classique. Paris: Gauthier-Villars (1967; Zbl 0149.21704)], and discusses applications of the Euler-Poincaré equation to continuum theories [D. D. Holm, J. E. Marsden and T. S. Ratiu, Adv. Math. 137, 1–81 (1988; Zbl 0951.37020)], providing the necessary foundational material as well as discussion of selected applications.
Chapter eleven introduces the Euler-Poincaré equation on diffeomorphisms, called EPDiff for short [V. I. Arnold, Ann. Inst. Fourier, loc. cit.; D. G. Ebin and J. E. Marsden, loc. cit.; D. D. Holm, J. E. Marsden and T. S. Ratiu, loc. cit.]. In chapter twelve the behaviour of solutions to EPDiff is studied, with special attention to peakons [R. Camassa and D. D. Holm, Phys. Rev. Lett. 71, No. 11, 1661–1664 (1993; Zbl 0972.35521)] and pulsons [O. B. Fringer and D. D. Holm, Physica D 150, No. 3–4, 237–263 (2001; Zbl 0980.35134)]. Chapter thirteen analyzes the integrability of EPDiff in one spatial dimension; here attention is focused on the Camassa-Holm equation [loc. cit.]. Chapter fourteen tackles EPDiff in higher dimensions, dealing in particular with singular momentum solutions and considering in more detail the geometry of the singular momentum map (the momentum map on the lower-dimensional support of singular solutions). With this background, the authors pass to discuss image recognition: in chapter fifteen “computational anatomy” and contour matching are discussed, and in chapter sixteen they deal with image matching. The two final chapters are devoted to fluid dynamics: chapter seventeen deals with continuum equations with advection, discussing in particular the Euler-Poincaré equation for continua and Kelvin-Noether theorem; chapter eighteen applies this discussion to geophysical fluid dynamics.
The volume is completed by a bibliography of 111 entries.
In the introduction, the authors state that “one of the intentions in writing the first part was to bridge the gap between the standard mechanics books such as “Classical mechanics” by Goldstein and “Mechanics” by Landau and Lifshitz, and the more advanced books such as “Foundations of mechanics” by Abraham and Marsden and “Introduction to mechanics and symmetry” by Marsden and Ratiu”. As hinted by the authors’ choice of these examples, the notation is chosen to be coherent with the one used in the two standard books just quoted. Also the choice of arguments included in (or excluded from) the book is coherent with the goal mentioned in the phrase quoted above. For example, we mention three topics (whose missing is pointed out by the authors) which bear some relation to the theme of symmetry: no discussion is provided of Hamilton-Jacobi theory and separation of variables, nor of integrable systems, nor of bifurcations (including symmetry-breaking ones).
Actually the book makes much more than merely allowing students to pass from one book to the other, and turns out to be a very well written and self-contained treatment of the interplay between mechanics and symmetry, with a view – which will be most welcome by several readers and students – to physical and engineering applications beside and beyond the construction of an appealing mathematical formalism. More than that, the applications discussed in Part II on the one side could provide a strong motivation to application-oriented students (and researchers not already in the field) to master the material presented in Part I and the more advanced one given in the initial chapters of Part II; on the other side these applications are of great interest, and discussing these in book format will help student and researchers to adopt a unified view on them.

MSC:

70-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to mechanics of particles and systems
37-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to dynamical systems and ergodic theory
70G45 Differential geometric methods (tensors, connections, symplectic, Poisson, contact, Riemannian, nonholonomic, etc.) for problems in mechanics
37J15 Symmetries, invariants, invariant manifolds, momentum maps, reduction (MSC2010)
37Jxx Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
37K05 Hamiltonian structures, symmetries, variational principles, conservation laws (MSC2010)
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
37N10 Dynamical systems in fluid mechanics, oceanography and meteorology
53D05 Symplectic manifolds (general theory)
70G40 Topological and differential topological methods for problems in mechanics
70G65 Symmetries, Lie group and Lie algebra methods for problems in mechanics
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