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Transformation groups applied to mathematical physics

Author: N Kh Ibragimov
Publisher: Dordrecht ; Boston : D. Reidel ; Hingham, MA, U.S.A. : Sold and distributed in the U.S.A. and Canada by Kluwer Academic Publishers, ©1985.
Series: Mathematics and its applications (D. Reidel Publishing Company)., Soviet series.
Edition/Format:   Print book : EnglishView all editions and formats
Summary:

Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non-trivially) in - gional and theoretical  Read more...

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Material Type: Internet resource
Document Type: Book, Internet Resource
All Authors / Contributors: N Kh Ibragimov
ISBN: 9027718474 9789027718471 1402003390 9781402003394
OCLC Number: 11067160
Notes: Translation of: Gruppy preobrazovaniǐ v matematicheskoǐ fizike.
Description: xv, 394 pages ; 24 cm.
Contents: I: Point Transformations.- Introductory Chapter: Group and Differential Equations.- 1. Continuous groups.- 1.1 Topological groups.- 1.2 Lie groups.- 1.3 Local groups.- 1.4 Local Lie groups.- 2. Lie algebras.- 2.1 Definitions.- 2.2 Lie algebras and local Lie groups.- 2.3 Inner automorphisms.- 2.4 The Levi-Mal'cev theorem.- 3. Transformation groups.- 3.1 Local transformation groups.- 3.2 Lie's equation.- 3.3 Invariants.- 3.4 Invariant manifolds.- 4. Invariant differential equations.- 4.1 Prolongation of point transformations.- 4.2 The defining equation.- 4.3 Invariant and partially invariant solutions.- 4.4 The method of invariant majorants.- 5. Examples.- 5.1 Let x ? ?n, a ? ?.- 5.2 Let us illustrate the algorithm for computing the group admitted by a differential equation by means of the example of a second-order equation.- 5.3 The Korteweg-de Vries equation.- 5.4 Consider the equation of motion of a polytropic gas.- 1: Motions in Riemannian Spaces.- 6. The general group of motions.- 6.1 Local Riemannian manifolds.- 6.2 Arbitrary motions in Vn.- 6.3 The defect of a group of motions in Vn.- 6.4 Invariant family of spaces.- 7. Examples of motions.- 7.1 Isometries.- 7.2 Conformal motions.- 7.3 Motions with ? = 2.- 7.4 Nonconformal motions with ? = 1.- 7.5 Motions with given invariants.- 8. Riemannian spaces with nontrivial conformal group.- 8.1 Conformally related spaces.- 8.2 Spaces of constant curvature.- 8.3 Conformally-flat spaces.- 8.4 Spaces with definite metric.- 8.5 Lorentzian spaces.- 9. Group analysis of Einstein's equations.- 9.1 Harmonic coordinates.- 9.2 The group admitted by Einstein's equations.- 9.3 The Lie-Vessiot decomposition.- 9.4 Exact solutions.- 10. Conformally-invariant equations of second order.- 10.1 Preliminaries.- 10.2 Linear equations in Sn.- 10.3 Semilinear equations in Sn.- 10.4 Equations admitting an isometry group of maximal order.- 10.5 The wave equation in Lorentzian spaces.- 2: A Group-Theoretical Approach to the Huygens Principle.- 11. General considerations and some history of the problem.- 11.1 Hadamard's problem.- 11.2 Hadamard's criterion.- 11.3 The Mathisson-Asgeirsson Theorem.- 11.4 The necessary conditions of Gunther and McLenaghan.- 11.5 The Lagnese-Stellmacher transformation.- 11.6 The present state of the art and generalizations of Hadamard's problem.- 12. The wave equation in V4.- 12.1 Computation of the geodesic distance in a plane-wave metric.- 12.2 Conformal invariance and the Huygens principle.- 12.3 The solution of the Cauchy problem.- 12.4 The case of a trivial conformal group.- 13. The Huygens principle in Vn+1.- 13.1 Preliminary analysis of the solution.- 13.2 The Fourier transform of the Bessel function J0(a|?|).- 13.3 The descent method. Representation of solution for arbitrary n.- 13.4 Summary of the Huygens principle.- 13.5 Failure of the connection between Huygens' principle and conformal invariance.- II: Tangent Transformations.- 3: Introduction to the Theory of Lie-Backlund Groups.- 14. Heuristic considerations.- 14.1 Contact transformations.- 14.2 Finite-order tangent transformations.- 14.3 Bianchi-Lie transformation.- 14.4 Backlund transformations. Examples.- 14.5 The concept of infinite-order tangent transformation.- 15. Formal groups.- 15.1 Lie's equation for formal one-parameter groups.- 15.2 Invariants and invariant manifolds.- 16. One-parameter groups of Lie-Backlund transformations.- 16.1 Definition and the infinitesimal criterion.- 16.2 Lie-Backlund operators. Canonical operators.- 16.3 Examples.- 17. Invariant differential manifolds.- 17.1 A criterion of invariance.- 17.2 Examples of solutions of the defining equation.- 17.3 Ordinary differential equations.- 17.4 The isomorphism theorem.- 17.5 Linearization by means of Lie-Backlund transformations.- 4: Equations with Infinite Lie-Backlund Groups.- 18. Typical examples.- 18.1 The heat equation.- 18.2 The Korteweg-de Vries equation.- 18.3 A fifth-order equation.- 18.4 The wave equation.- 19. Evolution equations.- 19.1 The algebra AF.- 19.2 The Faa de Bruno formula.- 19.3 The algebra LF.- 19.4 Differential substitutions.- 19.5 Equivalence transformations defined by ordinary differential equations.- 20. Analysis of second- and third-order evolution equations.- 20.1 m = 2.- 20.2 m = 3.- 20.3 Two systems of nonlinear equations.- 21. The equation F(x,y,z,p,q,r,s,t) = 0.- 21.1 Analysis of the general case.- 21.2 Classification of the equations s = F(z).- 21.3 A system of two nonlinear equations.- 5: Conservation Laws.- 22. Fundamental theorems.- 22.1 The Noether identity.- 22.2 The Noether theorem.- 22.3 Invariance on the extremals.- 22.4 The action of the adjoint algebra.- 22.5 First integrals of evolution equations.- 23. Examples.- 23.1 Motion in de Sitter space.- 23.2 The equation utt + ?2u = 0.- 23.3 The non-steady-state transonic gas flow.- 23.4 Short waves.- 24. The Lorentz group.- 24.1 Conservation laws in relativistic mechanics.- 24.2 A nonlinear wave equation.- 24.3 Dirac equation.- 25. The Galilean group.- 25.1 Motion of a particle.- 25.2 Perfect gas.- 25.3 Incompressible fluid.- 25.4 Shallow-water flow.- 25.5 A basis of conservation laws for the K-dV equation.- References.
Series Title: Mathematics and its applications (D. Reidel Publishing Company)., Soviet series.
Other Titles: Gruppy preobrazovaniǐ v matematicheskoǐ fizike.
Responsibility: Nail H. Ibragimov.

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