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Applications of Lie groups to differential equations. (English) Zbl 0588.22001
Graduate Texts in Mathematics, 107. New York etc.: Springer-Verlag. XXVI, 497 p. DM 158.00 (1986).
The book is a very fine gift of a mathematician to the physicists. It will be reviewed from this point of view. (The physicists are of course well aware of the importance of the symmetry induced by groups. The Nobelist E. Wigner took as the three basic foundations of physics: natural laws, initial conditions and symmetry.)
Chapter 1 “Introduction to Lie groups” (76 pp.) covers the introduction to manifolds, Lie groups and algebras, vector fields and differential forms and ends with Stokes’ theorem. Solved examples illuminate the broad theme, at the end of the chapter exercises are given.
Chapter 2 “Symmetry groups of differential equations” (90 pp.) gives a method to compute the symmetry groups both to partial and ordinary differential equations. The strength of the method is demonstrated e.a. on the one spatial dimension heat equation, nonlinear Burgers’ equation, two spatial wave equation, Euler ideal fluid equation in three space dimensions. To handle the algebraic manipulation for even more complicated equations and the systems of them, the MACSYMA computer software system can be of help. Again the exercises both expand the text and suggest new problems.
Chapter 3 “Group-invariant solutions” (106 pp.) is written with respect to the physicist who wishes to analyse his particular differential equation as soon as possible, nevertheless the rigorous justification is supplemented. The solved examples continue the analysis of the equations of Chapter 2 and moreover of the Korteweg-de Vries equation. With the apparatus as developed before, it is possible to cover the dimensional analysis rigorously on several pages. As in the other chapters the author illuminates the historical development of the continuous group approach to symmetries of differential equations appraising mainly Lie, Poincaré, Birkhoff and Ovsyannikov.
Chapter 4 “Symmetry groups and conservation laws” (46 pp.) briefly reviews the calculus of variations with conservation of physical quantities as given by the benchmark paper of E. Noether. Thoroughfull examples from electrostatics and \(n\) body systems are analysed – moreover the analysis of the wave equation in two spatial dimensions is continued.
Chapter 5 “Generalized symmetries” treats what had been given – according to the view of the author – “the misleading misnomer” of Lie-Bäcklund transformations. The author presents as examples the analysis of the spatial dimensional nonlinear wave equation, Burgers’ equation, Kepler problem, sine-Gordon equation, two dimensional wave equation. The variational complex is introduced to solve the inverse variational problems.
Chapter 6 “Finite-dimensional Hamiltonian systems” (45 pp.) is motivated by the importance of Hamiltonian methods in fluids, plasmas and elastic media. The author presents Poisson and symplectic structures. In the examples the \(n\) body problem and rigid body motion are analysed.
Chapter 7 “Hamiltonian methods for evolution equations” (34 pp.) is concerned with infinite-dimensional generalization of evolution equations. In the examples the Korteweg-de Vries and Euler ideal fluid equations are analysed.
References (18 pp.) cover the theme of the book from Lie’s 1874 seminal paper to Champagne-Winternitz’ 1985 preprint of a MACSYMA program used to find the differential equations’ symmetries.
The author’s book was published by Springer-Verlag, keeping the high standard not only of the text but even of the production – the typesetting was provided by Arrowsmith, printing and bounding by Donneley. A single flaw: the cataloging data contains the new term lie (sic!) groups – not the right (Sophus) Lie groups.

58-02 Research exposition (monographs, survey articles) pertaining to global analysis
35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
58J70 Invariance and symmetry properties for PDEs on manifolds
35A30 Geometric theory, characteristics, transformations in context of PDEs
35Q31 Euler equations
35Q53 KdV equations (Korteweg-de Vries equations)
35K05 Heat equation
37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)
37K35 Lie-Bäcklund and other transformations for infinite-dimensional Hamiltonian and Lagrangian systems
37L05 General theory of infinite-dimensional dissipative dynamical systems, nonlinear semigroups, evolution equations
22E70 Applications of Lie groups to the sciences; explicit representations