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Book review

Book review Acta Applicandae Mathematicae 11 (1988). 285 Paulette Libermann and Charles-Michel Marie: Symplectic Geometry and Analy- tical Mechanics, D. Reidel, Dordrecht, 1987, xvi + 526 pp. The language of symplectic geometry has become, in many domains of modern calculus, as well as in mathematical physics, irreplaceable to the same extent as the language of the usual linear algebra. Hamiltonians, Poisson brackets, and related notions that were originally connected in principle only with analytical mechanics, are nowadays standard objects for such domains as noncommutative harmonic analysis, microlocal analysis, the theory of dynamical systems, the theory of management, etc., not to mention modern quantum field theory. Crucial in this system of geometrical patterns that has turned out to be so successful, is the notion of a Lagrange manifold. The point is that, if we view a differential of a smooth function as a section in the cotangent fibre bundle of a manifold, it proves to be a Lagrange submanifold of the cotangent fibre bundle. Also, differentials of a wide class of many-valued functions with singularities arising as solutions of differential equations, turn out to be Lagrange sub- manifolds. This connection recalls an analogous relationship between Riemann surfaces and analytical functions in complex http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Applicandae Mathematicae Springer Journals

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References (6)

Publisher
Springer Journals
Copyright
Copyright
Subject
Mathematics; Computational Mathematics and Numerical Analysis; Applications of Mathematics; Partial Differential Equations; Probability Theory and Stochastic Processes; Calculus of Variations and Optimal Control; Optimization
ISSN
0167-8019
eISSN
1572-9036
DOI
10.1007/BF00140122
Publisher site
See Article on Publisher Site

Abstract

Acta Applicandae Mathematicae 11 (1988). 285 Paulette Libermann and Charles-Michel Marie: Symplectic Geometry and Analy- tical Mechanics, D. Reidel, Dordrecht, 1987, xvi + 526 pp. The language of symplectic geometry has become, in many domains of modern calculus, as well as in mathematical physics, irreplaceable to the same extent as the language of the usual linear algebra. Hamiltonians, Poisson brackets, and related notions that were originally connected in principle only with analytical mechanics, are nowadays standard objects for such domains as noncommutative harmonic analysis, microlocal analysis, the theory of dynamical systems, the theory of management, etc., not to mention modern quantum field theory. Crucial in this system of geometrical patterns that has turned out to be so successful, is the notion of a Lagrange manifold. The point is that, if we view a differential of a smooth function as a section in the cotangent fibre bundle of a manifold, it proves to be a Lagrange submanifold of the cotangent fibre bundle. Also, differentials of a wide class of many-valued functions with singularities arising as solutions of differential equations, turn out to be Lagrange sub- manifolds. This connection recalls an analogous relationship between Riemann surfaces and analytical functions in complex

Journal

Acta Applicandae Mathematicae – Springer Journals

Published: Jun 9, 2004

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