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Lie pseudogroups and mechanics. (English) Zbl 0677.58003

Mathematics and its Applications, 16. New York etc.: Gordon & Breach Science Publishers. viii, 591 p. $ 215.00 (1988).
The main purpose of the book (as it is determined by the author) is to prove that the approach to mathematical physics using the group theory, Lie groups, connections on principal bundles etc., following the works by E. Cartan, is based on a confusion between two differential sequences, namely the Janet sequence and the Spencer sequence. These sequences are considered in the formal theory of systems of P.D.E. and Lie pseudogroups. The author would like to show that correcting this mathematical confusion leads the science to revisit the foundations of many physical theories (e.g. continuum mechanics, thermodynamics, gauge theories, relativity). Also the history of the question is described, the role of the book by brothers Cosserat (1909) on elasticity theory is underlined. The matter of the material included in the book can be found in the table of contents below. To make the author’s point of view more clear we should present some of the author’s remarks: continuum mechanics only depends on group theory through the nonlinear Spencer sequence; the classical variational calculus has been misunderstood and one must substitute a new kind of variational calculus; in the works of Spencer and co-workers the impossibility to use connections in (well-posed) gauge theories is shown; Einstein equations are not compatible with Spencer sequence, i.e. the geometry of gravitation must not come from sections of the second Spencer bundle (curvature, torsion) as it is believed.
The Contents: Preface. Introduction. Chapter I. Historical survey: A. Geometry. B. Physics.
Chapter II. Systems of partial differential equations: A. Jet theory. B. Nonliner systems of PDE. C. Linear systems.
Chapter III. Lie pseudogroups: A. Lie groups. B. Lie equations. C. Differential invariants. D. Differential sequences (1. Nonlinear Janet sequence. 2. Nonlinear Spencer sequence).
Chapter IV. Continuum mechanics: A. Variational calculus. B. Dynamics on Lie pseudogroups. C. Dynamics on Lie groups (1. Cosserat theory. 2. Birkhoff-Arnold theory).
Chapter V. Gauge theory: A. Classical gauge theory. B. Conformal approach. C. Generalized thermodynamics.
Chapter VI. Relativity theory: A. Special relativity. B. General relativity.
Chapter VII. Differential algebraic geometry: A. Differential Galois theory. B. Applications (1. Dynamical systems. 2. Control theory).
Chapter VIII. Deformation theory: A. Deformation of Lie algebras. B. Deformation of Lie equations.
Bibliography. Appendix. Index of notations. Index of definitions.
Reviewer: Yu.E.Gliklikh

MSC:

58-02 Research exposition (monographs, survey articles) pertaining to global analysis
58A20 Jets in global analysis
55S99 Operations and obstructions in algebraic topology
58H05 Pseudogroups and differentiable groupoids
58J90 Applications of PDEs on manifolds
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